The story about the crown of Archimedes is, according to tradition, the following. More than 2000 years ago, King Hieron II of Syracuse reigned. He commissioned a goldsmith to make his crown out of pure gold. After Hieron received the crown, however, he could not shake the feeling that the crown was not made of pure gold at all. To make sure that the goldsmith had not cheated him, he asked the scientist Archimedes to look into the matter and check the crown.

Archimedes did not know at first how to prove or disprove that the crown was made of pure gold. To think better, Archimedes took a warm bath in a wooden barrel filled to the brim with water. As the water overflowed when he got into the barrel, an idea came to him. He then examined various bodies, each of which he placed in a vessel filled to the brim with water. Some objects floated on the water surface and some sank to the ground. For the objects that floated on the water, the buoyant force was therefore just as great as the weight force.

For the objects that sank to the bottom, the buoyant force was less than the weight. He was able to determine the buoyant force in this case by determining the apparent weight of the object when it is completely immersed in water. The buoyant force makes the object appear lighter. The amount by which the body appears to be lighter is just equal to the buoyant force.

In each case, he also examined how much water overflows from the brim-full vessel when each of the various objects was placed in it. He found that the weight of the displaced fluid (overflowing water) is just equal to the buoyancy. This is also known as Archimedes’ principle.

Archimedes’ principle states that the buoyant force of a body immersed in a fluid is equal to the weight of the displaced fluid!

With the help of this principle, Archimedes had now found a way to check the purity of the crown. According to tradition, Archimedes ran through the streets shouting “Eureka! Eureka!” (the word “eureka” comes from the ancient Greek and means something like “I have found [the solution]“). If the crown should not be made out of pure gold, but mixed with cheaper silver, then the crown would have a greater volume due to the lower density of silver (gold counts to the 10 heaviest elements, silver is about half the weight).

Due to its larger volume, the fake crown would thus displace a greater mass of water when immersed. According to Archimedes’ principle, the buoyant force would therefore be greater than for a crown made of pure gold. Archimedes therefore took a beam balance and placed the crown on the disk. On the opposite disk he then placed an ingot of pure gold, which was as heavy as the crown.

If the crown should indeed be made of pure gold, then the volume of the crown and the ingot should also be the same. Note that metals cannot be compressed, i.e. the volume of a metal object cannot be changed, but the volume can only be changed into a different form. Archimedes now placed both disks together with the crown or ingot in two vessels and filled them with water. The beam balance now tilted in the direction of the gold ingot.

Evidently, the crown experienced a greater buoyancy than the gold ingot. Archimedes concluded that the crown displaced a greater amount of water and thus the volume of the crown must be greater than the volume of the gold ingot. The crown could therefore not be made of pure gold, since the silver contained in it increased the volume of the crown.

If the goldsmith had used tungsten instead of the lighter silver, the cheating might not have been noticed, since tungsten (19.35 g/cm³) has almost the same density as gold (19.32 g/cm³), and thus the volumes are also almost identical for the same mass.

Cooking on Mount Everest

With increasing altitude above sea level, the air pressure decreases more and more (see also the article on barometric formula). This shows the phenomenon that water begins to boil at significantly lower temperatures than one is used to at lower altitudes. At sea level at a pressure of 1.013 bar, water begins to boil at a temperature of 100 °C.

However, on Mount Everest at an altitude of 8849 m, the air pressure is only around 0.325 bar. Due to this significantly reduced pressure, the water already begins to boil at a temperature of around 71°C. However, since the temperature does not rise any further during boiling, the cooking of foods such as potatoes or pasta thus takes significantly longer (see also the article Why does the temperature remain constant during a change of state?).

Explanation with the particle model

The fact that the boiling point depends on the ambient pressure applies not only to water, but ultimately to all liquids. In particular, it is true that the boiling point decreases with decreasing pressure. This phenomenon can be explained qualitatively with the particle model of matter.

During boiling, the liquid vaporizes and becomes gaseous. In this vaporization process, energy is absorbed by the liquid and added to the molecules, allowing them to break free from the molecular binding forces of the liquid and enter the gas phase. At an ambient air pressure of 1 bar, vaporization of water takes place at a temperature of 100 °C.

However, if the ambient air pressure is increased, the air molecules collide more strongly with the surface of the liquid. In the process, the air molecules push the liquid molecules back into the liquid, so to speak. It thus becomes more difficult for the molecules in the liquid to pass into the gas phase. The water molecules consequently require greater energy and thus a higher temperature in order to escape the liquid phase. For this reason, with increased ambient air pressure, a higher boiling temperature is required to vaporize a liquid or bring it to a boil.

The boiling temperature of a liquid increases with increasing ambient pressure!

Increase of boiling temperature at elevated ambient pressure (pressure cooker)

Under high ambient pressure, water consequently also boils at higher temperatures. This is used, for example, in so-called pressure cookers to heat the water to over 100 °C. A pressure cooker seals the pot of water gas-tight. During vaporization, water normally expands 1700 times. However, since this is not possible with a sealed pot, the pressure consequently increases. A pressure relief valve usually limits the pressure to a maximum of 2 bar. The boiling temperature rises to around 120 °C at this increased pressure. As a result, food prepared in the pot is no longer cooked at just 100 °C, but at 120 °C!

Decrease of boiling temperature at reduced ambient pressure

If an increase in the ambient pressure leads to an increase in the boiling temperature, then in the opposite case this means that a decrease in the ambient pressure results in a decrease in the boiling temperature. And this is exactly what explains why water on Mount Everest boils at already 71 °C due to the lower pressure of only 0.325 bar. Preparing food that normally requires a temperature of 100 °C in water is therefore not so easy at high altitudes. At this point, one would have to use the pressure cooker already explained to obtain increased pressure and raise the boiling temperature.

The following experiment provides an impressive demonstration of the decrease in boiling temperature with decreasing pressure. For this purpose, a glass with water is placed under a vacuum chamber. A thermometer is placed in the glass to observe the temperature. The thermometer indicates a temperature of 20 °C. Now the vacuum pump is switched on and thus the pressure is reduced step by step. Below a pressure of about 0.023 bar, one then observes small bubbles rising in the water. This is the typical phenomenon when water boils, with the temperature still at 20 °C. And indeed, at a pressure of 0.023 bar, the water already begins to boil at 20 °C.

Euler equation

In the article Derivation of the Euler equation the following equation was derived to describe the motion of frictionless flows:

\begin{align}
&\boxed{\frac{\partial \vec v}{\partial t} + \left(\vec v \cdot \vec \nabla \right) \vec v + \frac{1}{\rho} \vec \nabla p = \vec g}~~~\text{Euler equation} \\[5px]
\end{align}

The assumption of a frictionless flow means in particular that the viscosity of fluids is neglected (inviscid fluids). In practice, however, every fluid has a viscosity (even ideal gases!). The viscosity leads to frictional forces within the fluid. Taking the viscosity into account in the Euler equation finally leads to the Navier-Stokes equation.

Normal force acting on a fluid element

For the derivation of the Navier-Stokes equations we consider a fluid element and the forces acting on it. At first we will consider only the motion of the fluid in x-direction. The motion of the fluid element is influenced by the pressure forces acting on the front and back surface of the cubic volume element. These pressure forces basically represent normal stresses and are therefore denoted in the following by the Greek symbol σ instead of p. We will see later that in addition to the pressure forces, viscosity-related forces also act perpendicular to the surfaces and contribute to the normal stress.

However, the normal stresses acting on the front and back of the fluid element are different, since, for example, the pressure in the direction of flow generally decreases. Thus, a pressure gradient and thus a normal stress gradient ∂σ_x/∂x is present the considered point x. If the normal stress σ_x acts at the position x, then it changes along the distance dx by ∂σ_x/∂x⋅dx. The following pressure-related normal forces thus act at the position x and x+dx on the volume element:

\begin{align}
& \underline{F_\text{x} = \sigma_\text{x} \cdot \text{d}A_\text{yz}} ~~~~~\text{normal force at point }x\\[5px]
& \underline{F_\text{x+dx} = \left(\sigma_\text{x} + \frac{\partial \sigma_\text{x}}{\partial x}\text{d}x\right) \text{d}A_\text{yz}} ~~~~~\text{normal force at point }x+\text{d}x \\[5px]
\end{align}

Since these forces obviously act in different directions, the resulting normal force F_σx on the fluid element is the difference between the two forces:

\begin{align}
\require{cancel}
& F_{\sigma_\text{x}} = F_\text{x} – F_\text{x+dx} \\[5px]
& F_{\sigma_\text{x}} = \sigma_\text{x} \cdot \text{d}A_\text{yz} – \left(\sigma_\text{x}+ \frac{\partial \sigma_\text{x}}{\partial x}\cdot \text{d}x\right) \cdot \text{d}A_\text{yz}\\[5px]
& F_{\sigma_\text{x}} = \cancel{\sigma_\text{x} \cdot \text{d}A_\text{yz}} – \cancel{\sigma_\text{x} \cdot \text{d}A_\text{yz}} – \frac{\partial \sigma_\text{x}}{\partial x} \cdot \underbrace{\text{d}x \cdot \text{d}A_\text{yz}}_{\text{d}V} \\[5px]
& \boxed{F_{\sigma_\text{x}} = ~ – \frac{\partial \sigma_\text{x}}{\partial x} \cdot \text{d}V} ~~\text{resultant normal force in x-direction} \\[5px]
\end{align}

Shear force acting on a fluid element

Due to the viscosity η, shear forces or shear stresses in x-direction act on the lateral surfaces of the fluid element and on the top and bottom surfaces. In contrast to the normal stresses considered above, which are directed perpendicular to the surface, the shear stresses act parallel to the surface.

The shear stresses τ_ij are determined according to Newton’s law of fluid friction using the velocity gradient∂v_j/∂i (also called shear rate); more on the validity of this law for a continuum later:

\begin{align}
\label{newton}
& \boxed{\tau_\text{ij} = \eta \cdot \frac{\partial v_\text{j}}{\partial i}} ~~~\text{Newton’s law of fluid friction}\\[5px]
\end{align}

In this equation i (=y,z) denotes the direction of the surface normals for which the shear stress is determined and j (=x) denotes the direction in which the shear stress τ_ij acts. Thus, the designation τ_yx results for the lateral surface at the position y:

\begin{align}
&\underline{\tau_\text{yx}}~~~~~\text{shear stress at point }y \\[5px]
\end{align}

Along the width dy of the fluid element, the velocity gradient ∂v_x/∂y will change in general. Thus, a different shear stress will act at the position y+dy. A corresponding shear stress gradient ∂τ_yx/∂y can be defined at this point so that the following shear stress results at point y+dy:

\begin{align}
&\underline{\tau_\text{yx}+\frac{\partial \tau_\text{yx}}{\partial y}\text{d}y}~~~~~\text{shear stress at point }y+\text{d}y \\[5px]
\end{align}

In concrete terms, a positive shear stress gradient means that according to Newton’s law of fluid friction the velocity gradient increases in the y direction. The flow velocity at the location y+dy is thus greater than at the location y. Looking in the direction of flow (positive x-direction), the fluid flows slower to the right of the fluid element and faster to the left.

The fluid element is slowed down on the right side, so to speak, and on the left side it is carried along by the flow, i.e. accelerated. The shear stress at the point y is thus directed in negative x direction and at the point y+dy in positive direction. The resulting shear stress of the two lateral surfaces is therefore the difference between the two shear stresses:

\begin{align}
&\tau_\text{yx,res} = \underbrace{\tau_\text{yx}+\frac{\partial \tau_\text{yx}}{\partial y}\cdot \text{d}y}_{\text{shear stress at }y+\text{d}y} – \underbrace{\tau_\text{yx}}_{\text{shear stress at }y} \\[5px]
&\underline{\tau_\text{yx,res}=\frac{\partial \tau_\text{yx}}{\partial y}\cdot \text{d}y}~~~~~\text{resultant shear stress on the fluid element} \\[5px]
\end{align}

This shear stress (force per unit area) multiplied by the area dA_xz finally results in the resulting shear force F_τyx acting on the lateral surfaces of the fluid element:

\begin{align}
&F_{\tau_\text{yx}}= \tau_\text{yx,res} \cdot \text{d}A_\text{xz}\\[5px]
&F_{\tau_\text{yx}}=\frac{\partial \tau_\text{yx}}{\partial y}\cdot \underbrace{\text{d}y \cdot \text{d}A_\text{xz}}_{\text{d}V}\\[5px]
\label{for}
&\boxed{F_{\tau_\text{yx}} =\frac{\partial \tau_\text{yx}}{\partial y}\cdot \text{d}V} \\[5px]
\end{align}

For the resulting shear stress or the resulting shear force F_τzx on the top and bottom surface of the fluid element applies quite analogously:

\begin{align}
&\boxed{F_{\tau_\text{zx}} =\frac{\partial \tau_\text{zx}}{\partial z}\cdot \text{d}V} \\[5px]
\end{align}

Weight force acting on a fluid element

In addition to normal and shear forces, gravity generally acts on a fluid element. The considered flow in x-direction does not necessarily have to be horizontal, but can run at any angle. In these cases, only that component of the weight force F_gx is relevant which points in x direction g_x denotes the component of the gravitational acceleration in x direction):

\begin{align}
&F_{\text{g}_\text{x}} = \text{d}m \cdot g_\text{x} \\[5px]
&\boxed{F_{\text{g}_\text{x}} = \rho g_\text{x} \cdot \text{d}V } ~~~\text{weight force on the fluid element in x direction} \\[5px]
\end{align}

In contrast to normal or shear forces (so-called surface forces), the weight force does not act on surfaces, but on the entire “body” of the fluid element. The weight force is therefore a so-called body force. Other body forces acting on the fluid element may be, for example, electrical or magnetic forces. These forces must therefore also be taken into account. At this point, the weight force is therefore only an example of other other forces.

Substantial, local and convective acceleration

The sum of normal force F_σx, shear force F_τyx and F_τzx and body force F_gx finally gives the overall resulting force F_xres acting on the fluid element in x direction:

\begin{align}
&F_{\text{x}_\text{res}} = F_{\sigma_\text{x}} + F_{\tau_\text{yx}} + F_{\tau_\text{zx}} + F_{\text{g}_\text{x}} \\[5px]
&F_{\text{x}_\text{res}} =
– \frac{\partial \sigma_\text{x}}{\partial x} \cdot \text{d}V
+ \frac{\partial \tau_\text{yx}}{\partial y} \cdot \text{d}V
+ \frac{\partial \tau_\text{zx}}{\partial z} \cdot \text{d}V
+ \rho g_\text{x} \cdot \text{d}V
\\[5px]
\end{align}

According to the Newton’s second law, this resultant force leads to the following acceleration a_x in x direction, where dm=ϱ⋅dV denotes the mass of the fluid element:

\begin{align}
\require{cancel}
&a_\text{x} = \frac{F_{\text{x}_\text{res}}}{\text{d}m} = \frac{F_{\text{x}_\text{res}}}{\rho~ \text{d}V} \\[5px]
&a_\text{x} =
\frac{- \frac{\partial \sigma_\text{x}}{\partial x} \cdot \cancel{\text{d}V}
+ \frac{\partial \tau_\text{yx}}{\partial y} \cancel{\text{d}V}
+ \frac{\partial \tau_\text{zx}}{\partial z} \cancel{\text{d}V}
+ \rho g_\text{x} \cdot \cancel{\text{d}V}}{\rho~ \cancel{\text{d}V}}
\\[5px]
&a_\text{x} =
\frac{- \frac{\partial \sigma_\text{x}}{\partial x}
+ \frac{\partial \tau_\text{yx}}{\partial y}
+ \frac{\partial \tau_\text{zx}}{\partial z}
+ \rho g_\text{x}}{\rho}
\\[5px]
\end{align}

Therefore the following equation applies:

\begin{align}
\label{a}
&\underline{a_\text{x}~ \rho=
– \frac{\partial \sigma_\text{x}}{\partial x}
+ \frac{\partial \tau_\text{yx}}{\partial y}
+ \frac{\partial \tau_\text{zx}}{\partial z}
+ \rho g_\text{x} }
\\[5px]
\end{align}

The change in velocity of the fluid element in x direction, also called substantial acceleration a_x, can be expressed by the change in velocity over time at a fixed location (local acceleration: ∂v_x/∂t) and on the other hand by the change in velocity due to the change in location of the fluid element (convective acceleration: ∂v_x/∂x⋅v_x + ∂v_x/∂y⋅v_y + ∂v_x/∂z⋅v_z):

\begin{align}
\label{sub}
&\underbrace{~~a_\text{x}~~}_{\text{substantial}\\\text{acceleration}} = \underbrace{\frac{\partial v_\text{x}}{\partial t}}_{\text{local}\\\text{acceleration}} + \underbrace{\frac{\partial v_\text{x}}{\partial x}v_\text{x} + \frac{\partial v_\text{x}}{\partial y}v_\text{y} + \frac{\partial v_\text{x}}{\partial z}v_\text{z}}_\text{convective acceleration} \\[5px]
\end{align}

This relationship between substantial, local and convective acceleration is described in detail in the article Derivation of the Euler equation and will not be explained further here. If the equation for the substantial acceleration (\ref{sub}) is put into equation (\ref{a}), then the following relationship applies:

\begin{align}
\label{nav}
&\boxed{\left(\frac{\partial v_\text{x}}{\partial t} + \frac{\partial v_\text{x}}{\partial x}v_\text{x} + \frac{\partial v_\text{x}}{\partial y}v_\text{y} + \frac{\partial v_\text{x}}{\partial z}v_\text{z}\right) ~ \rho=- \frac{\partial \sigma_\text{x}}{\partial x} + \frac{\partial \tau_\text{yx}}{\partial y} + \frac{\partial \tau_\text{zx}}{\partial z} + \rho g_\text{x}} \\[5px]
\end{align}

Viscous stress tensor

In three-dimensional flow, the velocity or velocity gradient changes in all three directions. Due to the viscosity, the resulting shear stress is no longer only due to the velocity gradient in a certain direction, but also to a velocity gradient perpendicular to it.

This situation is similar to the deformation of a fluid element. Imagine shear forces that deform the fluid element. In this way, inclined surfaces are formed so that the shear forces acting on the surface now affect another spatial dimension. In fact, in fluids, the stresses acting in additional spatial directions are not due to an actual deformation (magnitude of shearing), but to the shear rate.

This insight leads to the introduction of a so-called viscous stress tensor, which describes the stresses (normal and shear stresses) acting on a fluid element:

\begin{align}
\label{matr1}
& \boxed{\tau =
\begin{pmatrix}
\color{red}{\tau_\text{xx}} & \tau_\text{xy} & \tau_\text{xz}
\\\
\tau_\text{yx} & \color{red}{\tau_\text{yy}} & \tau_\text{yz}
\\\
\tau_\text{zx} & \tau_\text{zy} & \color{red}{\tau_\text{zz}}
\end{pmatrix}} ~~~\text{viscous stress tensor}\\[5px]
\end{align}

For an isotropic Newtonian fluid, the components of the viscous stress tensor τ can be determined from the viscosity η as follows:

\begin{align}
\label{matr2}
& \boxed{\large\tau = \eta \cdot
\begin{pmatrix}
&\color{red}{\left(\frac{\partial v_\text{x}}{\partial x} + \frac{\partial v_\text{x}}{\partial x}\right)}
& \left(\frac{\partial v_\text{x}}{\partial y} + \frac{\partial v_\text{y}}{\partial x}\right)
& \left(\frac{\partial v_\text{x}}{\partial z} + \frac{\partial v_\text{z}}{\partial x}\right)
\\\
& \left(\frac{\partial v_\text{y}}{\partial x} + \frac{\partial v_\text{x}}{\partial y}\right)
&\color{red}{\left(\frac{\partial v_\text{y}}{\partial y} + \frac{\partial v_\text{y}}{\partial y}\right)}
& \left(\frac{\partial v_\text{y}}{\partial z} + \frac{\partial v_\text{z}}{\partial y}\right)
\\\
& \left(\frac{\partial v_\text{z}}{\partial x} + \frac{\partial v_\text{x}}{\partial z}\right)
& \left(\frac{\partial v_\text{z}}{\partial y} + \frac{\partial v_\text{y}}{\partial z}\right)
&\color{red}{\left(\frac{\partial v_\text{z}}{\partial z} + \frac{\partial v_\text{z}}{\partial z}\right)}
\end{pmatrix}} \\[5px]
\end{align}

The individual entries of the viscous stress tensor can also be determined with the following formula:

\begin{align}
\label{t}
&\boxed{\tau_\text{ij} = \eta \left(\frac{\partial v_\text{i}}{\partial j} + \frac{\partial v_\text{j}}{\partial i}\right)} ~~~\text{viscous stress}\\[5px]
\end{align}

At this point the difference to Newton’s approach according to equation (\ref{newton}) becomes obvious: In three-dimensional flows of a continuum (continuum mechanics) the shear stress depends on a further velocity gradient. This finding and the introduction of the viscous stress tensor is essentially the achievement of Stokes.

For the shear stresses τ_yx and τ_zx in equation (\ref{nav}) applies according to equation (\ref{t}):

\begin{align}
\label{tyx}
&\underline{\tau_\text{yx} = \eta \left(\frac{\partial v_\text{y}}{\partial x} + \frac{\partial v_\text{x}}{\partial y}\right)}\\[5px]
\label{tzx}
&\underline{\tau_\text{zx} = \eta \left(\frac{\partial v_\text{z}}{\partial x} + \frac{\partial v_\text{x}}{\partial z}\right)}\\[5px]
\end{align}

Taking into account the viscous normal stress

For the stresses indicated with identical indices in equation (\ref{matr1}) or (\ref{matr2}) (marked red), both the surface normal and the effective direction of the force obviously point in the same direction. These are therefore not shear stresses but normal stresses, which are often denoted by σ for better differentiation:

\begin{align}
& \tau_\text{xx} = \sigma_{\eta~\text{x}} \\[5px]
& \tau_\text{yy} = \sigma_{\eta~\text{y}} \\[5px]
& \tau_\text{zz} = \sigma_{\eta~\text{z}} \\[5px]
\end{align}

However, this obviously also means that the viscosity generates not only shear stresses but also normal stresses. These normal stresses resulting from the viscosity are therefore additionally designated with the symbol for viscosity η. For the viscosity-related normal stress σ_ηx acting in the x-direction, the following formula (\ref{t}) applies:

\begin{align}
& \tau_\text{xx} =\underline{\sigma_{\eta~\text{x}}}= \eta \left(\frac{\partial v_\text{x}}{\partial x} + \frac{\partial v_\text{x}}{\partial x}\right) = \underline{2\eta \frac{\partial v_\text{x}}{\partial x}} \\[5px]
\end{align}

This viscous normal stress acts as a normal force on the fluid element in addition to the (static) pressure. In contrast to the static pressure, which acts against the surface normal of the fluid element (i.e. is directed towards the fluid element), the viscous normal stress acts in the direction of the surface normal (i.e. away from the fluid element). The resulting normal stress σ_x acting on the fluid element in x-direction thus results from the static pressure p, minus the viscous normal stress σ_ηx:

\begin{align}
& \sigma_\text{x} = p~ – \sigma_{\eta~\text{x}} \\[5px]
\label{sx}
& \underline{\sigma_\text{x} = p~ – 2\eta \frac{\partial v_\text{x}}{\partial x}} \\[5px]
\end{align}

The Navier-Stokes equation for incompressible fluids

If the equations (\ref{tyx}), (\ref{tzx}) and (\ref{sx}) are put into the right side of equation (\ref{nav}), the following equation results (for clarity only the right side of the equation is shown):

\begin{align}
&…=- \frac{\partial \sigma_\text{x}}{\partial x} + \frac{\partial \tau_\text{yx}}{\partial y} + \frac{\partial \tau_\text{zx}}{\partial z} + \rho g_\text{x} \\[5px]

&…=- \frac{\partial}{\partial x}\left(p~ – 2\eta \frac{\partial v_\text{x}}{\partial x}\right) +\eta ~\frac{\partial}{\partial y} \left(\frac{\partial v_\text{y}}{\partial x} + \frac{\partial v_\text{x}}{\partial y}\right) + \eta ~\frac{\partial}{\partial z} \left(\frac{\partial v_\text{z}}{\partial x} + \frac{\partial v_\text{x}}{\partial z}\right)+ \rho g_\text{x} \\[5px]

&…=- \frac{\partial p}{\partial x} + \color{red}{ 2\eta \frac{\partial^2 v_\text{x}}{\partial x^2}} +\eta \frac{\partial}{\partial y} \left(\frac{\partial v_\text{y}}{\partial x}\right) + \eta \frac{\partial^2 v_\text{x}}{\partial y^2}+ \eta \frac{\partial}{\partial z} \left(\frac{\partial v_\text{z}}{\partial x} \right)+ \eta \frac{\partial^2 v_\text{x}}{\partial z^2}+ \rho g_\text{x} \\[5px]
\end{align}

We can also write the term marked in red in a slightly different way:

\begin{align}
…=&- \frac{\partial p}{\partial x} + \color{red}{\eta \frac{\partial^2 v_\text{x}}{\partial x^2} +\eta \frac{\partial}{\partial x} \left(\frac{\partial v_\text{x}}{\partial x}\right)} +\eta \frac{\partial}{\partial y} \left(\frac{\partial v_\text{y}}{\partial x}\right) + \eta \frac{\partial^2 v_\text{x}}{\partial y^2} \\[5px]&
+ \eta \frac{\partial}{\partial z} \left(\frac{\partial v_\text{z}}{\partial x} \right) + \eta \frac{\partial^2 v_\text{x}}{\partial z^2}+ \rho g_\text{x} \\[5px]
\end{align}

Now we rearrange this equation:

\begin{align}
…=&- \frac{\partial p}{\partial x}
+ \eta \left(\frac{\partial^2 v_\text{x}}{\partial x^2}
+ \frac{\partial^2 v_\text{x}}{\partial y^2}
+ \frac{\partial^2 v_\text{x}}{\partial z^2}\right)\\[5px]&
+ \eta \frac{\partial}{\color{red}{\partial x}} \left(\frac{\partial v_\text{x}}{\color{blue}{\partial x}}\right)
+ \eta \frac{\partial}{\color{red}{\partial y}} \left(\frac{\partial v_\text{y}}{\color{blue}{\partial x}}\right)
+ \eta \frac{\partial}{\color{red}{\partial z}} \left(\frac{\partial v_\text{z}}{\color{blue}{\partial x}} \right)
+ \rho g_\text{x} \\[5px]
\end{align}

We can also swap the coordinates of the partial derivatives, because it makes no difference, if we first derive with respect to y and then x or first derive with respect to x and then y. Thus we swap the red marked operators outside the brackets with the blue marked operators inside the brackets:

\begin{align}
…=&- \frac{\partial p}{\partial x}
+ \eta \left(\frac{\partial^2 v_\text{x}}{\partial x^2}
+ \frac{\partial^2 v_\text{x}}{\partial y^2}
+ \frac{\partial^2 v_\text{x}}{\partial z^2}\right)\\[5px]&
+ \eta \frac{\partial}{\color{red}{\partial x}} \left(\frac{\partial v_\text{x}}{\color{blue}{\partial x}}\right)
+ \eta \frac{\partial}{\color{red}{\partial x}} \left(\frac{\partial v_\text{y}}{\color{blue}{\partial y}}\right)
+ \eta \frac{\partial}{\color{red}{\partial x}} \left(\frac{\partial v_\text{z}}{\color{blue}{\partial z}} \right)
+ \rho g_\text{x} \\[5px]
\end{align}

Now we factor out the operator ∂/∂x and the viscosity η:

\begin{align}
…=&- \frac{\partial p}{\partial x}
+ \eta \left(\frac{\partial^2 v_\text{x}}{\partial x^2}
+ \frac{\partial^2 v_\text{x}}{\partial y^2}
+ \frac{\partial^2 v_\text{x}}{\partial z^2}\right)
+ \eta \frac{\partial}{\partial x}
\color{red}{\underbrace{\left(
\frac{\partial v_\text{x}}{\partial x}
+\frac{\partial v_\text{y}}{\partial y}
+\frac{\partial v_\text{z}}{\partial z}
\right)}_{=0}}
+ \rho g_\text{x} \\[5px]
\end{align}

The term marked in red is zero for an incompressible fluid due to the continuity equation (this applies in very good approximation to gases at not too high flow velocities as well)! This simplification finally leads to the following equation, which describes the motion in x direction of an incompressible fluid as a continuum:

\begin{align}
&\left(\frac{\partial v_\text{x}}{\partial t}+ \frac{\partial v_\text{x}}{\partial x}v_\text{x} + \frac{\partial v_\text{x}}{\partial y}v_\text{y} + \frac{\partial v_\text{x}}{\partial z}v_\text{z}\right) \rho
=
– \frac{\partial p}{\partial x}
+ \eta \left(\frac{\partial^2 v_\text{x}}{\partial x^2}
+ \frac{\partial^2 v_\text{x}}{\partial y^2}
+ \frac{\partial^2 v_\text{x}}{\partial z^2}\right)
+ \rho g_\text{x}
\\[5px]
\end{align}

For the motion in y- and z-direction the analogous equations apply. These equations are finally called Navier-Stokes equations:

\begin{align}
&\boxed{\left(\frac{\partial \color{red}{v_\text{x}}}{\partial t}
+ \frac{\partial \color{red}{v_\text{x}}}{\partial x}v_\text{x}
+ \frac{\partial \color{red}{v_\text{x}}}{\partial y}v_\text{y}
+ \frac{\partial \color{red}{v_\text{x}}}{\partial z}v_\text{z}
\right) \rho
=
– \frac{\partial p}{\partial \color{red}{x}}
+ \eta \left(\frac{\partial^2 \color{red}{v_\text{x}}}{\partial x^2}
+ \frac{\partial^2 \color{red}{v_\text{x}}}{\partial y^2}
+ \frac{\partial^2 \color{red}{v_\text{x}}}{\partial z^2}\right)
+ \rho \color{red}{g_\text{x}}}
\\[5px]

&\boxed{\left(\frac{\partial \color{green}{v_\text{y}}}{\partial t}
+ \frac{\partial \color{green}{v_\text{y}}}{\partial x}v_\text{x}
+ \frac{\partial \color{green}{v_\text{y}}}{\partial y}v_\text{y}
+ \frac{\partial \color{green}{v_\text{y}}}{\partial z}v_\text{z}
\right) \rho
=
– \frac{\partial p}{\partial \color{green}{y}}
+ \eta \left(\frac{\partial^2 \color{green}{v_\text{y}}}{\partial x^2}
+ \frac{\partial^2 \color{green}{v_\text{y}}}{\partial y^2}
+ \frac{\partial^2 \color{green}{v_\text{y}}}{\partial z^2}\right)
+ \rho \color{green}{g_\text{y}}}
\\[5px]

&\boxed{\left(\frac{\partial \color{blue}{v_\text{z}}}{\partial t}
+ \frac{\partial \color{blue}{v_\text{z}}}{\partial x}v_\text{x}
+ \frac{\partial \color{blue}{v_\text{z}}}{\partial y}v_\text{y}
+ \text{ }\frac{\partial \color{blue}{v_\text{z}}}{\partial z}v_\text{z}
\right) \rho
=
– \frac{\partial p}{\partial \color{blue}{z}}
+ \eta \left(\frac{\partial^2 \color{blue}{v_\text{z}}}{\partial x^2}
+ \text{ }\frac{\partial^2 \color{blue}{v_\text{z}}}{\partial y^2}
+ \text{ }\frac{\partial^2 \color{blue}{v_\text{z}}}{\partial z^2}\right)
+ \rho \color{blue}{g_\text{z}}}
\\[5px]
\end{align}

In vektorieller Schreibweise stellt sich die Navier-Stokes-Gleichung wie folgt dar:

\begin{align}
&\boxed{ \left[\frac{\partial \vec v}{\partial t} + \left(\vec v \cdot \vec \nabla \right) \vec v \right] \rho = – \vec \nabla p+ \eta \left( \vec \nabla^2 \vec v \right) + \rho \vec g}~~~\text{Navier-Stokes equation} \\[5px]
&\underbrace{\left[\underbrace{\frac{\partial \vec v}{\partial t}}_\text{local acceleration} + \underbrace{\left(\vec v \cdot \vec \nabla \right) \vec v}_\text{convective acceleration} \right]}_\text{(substantial) acceleration} \rho = \underbrace{- \vec \nabla p}_{\text{“pressure force”}\\\text{(normal force)}}+ \underbrace{\eta \left( \vec \nabla^2 \vec v \right)}_{\text{“frictional force”}\\\text{(shear force)}} + \underbrace{\rho \vec g}_{\text{“weight force”}\\\text{(body force)}} \\[5px]
\end{align}

Note that the Navier-Stokes equation shown here only applies to incompressible fluids and approximately also to relatively slow flowing gases!

In general, the Navier-Stokes equations cannot be solved analytically. Only in special cases can a solution or approximate solution be found under simplified assumptions. For example, the special case of an incompressible flow in a pipe results in the law of Hagen-Poiseuille. For fluids for which the viscosity can be neglected (η=0), the Euler equation for inviscid flows is obtained:

\begin{align}
& \left[\frac{\partial \vec v}{\partial t} + \rho \left(\vec v \cdot \vec \nabla \right) \vec v \right] \rho = – \vec \nabla p+ \rho \vec g \\[5px]
&\boxed{\frac{\partial \vec v}{\partial t} + \left(\vec v \cdot \vec \nabla \right) \vec v + \frac{1}{\rho} \vec \nabla p = \vec g}~~~\text{Euler equation} \\[5px]
\end{align}

Pressure forces on a fluid element

The Euler equation is based on Newton’s second law, which relates the change in velocity of a fluid particle to the presence of a force. Associated with this is the conservation of momentum, so that the Euler equation can also be regarded as a consequence of the conservation of momentum.

For the derivation of the Euler equation we consider an infinitesimal fluid volume dV with mass dm. We describe the motion of the fluid element from a fixed coordinate system (so-called Eulerian approach). By definition, the considered fluid element moves along an arbitrarily oriented streamline.

First we consider the motion of the fluid element only in x-direction. At a position x a pressure p is present. At this location the force F_x acts on the surface dA_yz of the fluid element:

\begin{align}
&\underline{F_\text{x} = p \cdot \text{d}A_\text{yz}} \\[5px]
\end{align}

The pressure is not constant in a flow, but changes locally. This is because pressure differences are ultimately the reason why a flow comes about at all. Therefore, the pressure in the x-direction will change. For a given pressure gradient ∂p/∂x (which can be positive or negative!), the following pressure change dp_x results over the length dx of the fluid element:

\begin{align}
&\underline{\text{d}p_\text{x} =\frac{\partial p}{\partial x} \cdot \text{d}x} ~~~~~\text{pressure change along }\text{d}x \\[5px]
\end{align}

This pressure change leads to the following force F_x+dx at the position x+dx:

\begin{align}
&F_{\text{x+d}x}= \left(p + \text{d}p_\text{x} \right) \cdot \text{d}A_\text{yz} \\[5px]
&\underline{F_{\text{x+d}x} = \left(p + \frac{\partial p}{\partial x} \cdot \text{d}x \right) \cdot \text{d}A_\text{yz}} \\[5px]
\end{align}

The forces F_x and F_x+dx acting on the surfaces dA_yz are in opposite directions, so that the following resultant pressure force F_px acts in x-direction, which causes/influences the motion of the fluid element in x-direction:

\begin{align}
\require{cancel}
&F_\text{px} = F_\text{x} – F_{\text{x+d}x} \\[5px]
&F_\text{px} = p \cdot \text{d}A_\text{yz} – \left(p + \frac{\partial p}{\partial x} \cdot \text{d}x \right) \cdot \text{d}A_\text{yz} \\[5px]
&F_\text{px} = \cancel{p \cdot \text{d}A_\text{yz}} – \cancel{p \cdot \text{d}A_\text{yz}} – \frac{\partial p}{\partial x} \cdot \underbrace{\text{d}x \cdot \text{d}A_\text{yz}}_{\text{d}V} \\[5px]
&\boxed{F_\text{px} = – \frac{\partial p}{\partial x} \cdot \text{d}V} ~~~~~\text{pressure force in x direction} \\[5px]
\end{align}

The negative sign indicates that with a positive pressure gradient ∂p/∂x>0 the force on the fluid element is directed against the positive x-direction, since the pressure obviously increases along the fluid element. The fluid element would thus be slowed down in the x direction if it were to move in positive x direction. The same considerations we have made for the x direction can be made for the y and z directions. This leads to the following equations:

\begin{align}
&\boxed{F_\text{py} = – \frac{\partial p}{\partial y} \cdot \text{d}V}~~~~~\text{pressure force in y direction} \\[5px]
&\boxed{F_\text{pz} = – \frac{\partial p}{\partial z} \cdot \text{d}V}~~~~~\text{pressure force in z direction} \\[5px]
\end{align}

In these equations, ∂p/∂y and ∂p/∂z indicate the pressure gradients in y and z direction, respectively.

Vector notation of the pressure force

The pressure forces F_px, F_py and F_pz calculated on the basis of the respective gradients can be written line by line as a vector F_p for the pressure force:

\begin{align}
&\vec{F_\text{p}} = \begin{pmatrix}F_\text{px} \\\ F_\text{py} \\\ F_\text{pz} \end{pmatrix} = \begin{pmatrix}-\frac{\partial p}{\partial x}\cdot \text{d}V \\\ -\frac{\partial p}{\partial y}\cdot \text{d}V \\\ -\frac{\partial p}{\partial z} \cdot \text{d}V \end{pmatrix} = -\begin{pmatrix}\large\frac{\partial p}{\partial x} \\\ \large\frac{\partial p}{\partial y} \\\ \large\frac{\partial p}{\partial z} \end{pmatrix} \cdot \text{d}V\\[5px]
\end{align}

The term in brackets corresponds to the pressure gradient and can be expressed by the del operator, denoted by the nabla symbol ∇ (often the arrow above the symbol is not shown; formally, however, the del operator is a vector!)

\begin{align}
&\boxed{\vec F_\text{p} =- \vec \nabla p \cdot \text{d}V } ~~~\text{mit:}~~~\boxed{\vec \nabla =\begin{pmatrix}\large\frac{\partial}{\partial x}\\\ \large\frac{\partial }{\partial y} \\\ \large\frac{\partial}{\partial z} \end{pmatrix}} ~~~\text{del operator}\\[5px]
& \vec \nabla p = \text{pressure gradient} \\[5px]
\end{align}

Note that applying the del operator to a scalar field (here: spatial pressure distribution) results in a vector field. The vectors point in the direction of the greatest increase of the scalar quantity (here: in the direction of the greatest pressure increase).

It should again be noted that the negative sign in the formula above indicates that the pressure force F_p acting on a fluid volume dV is directed against the pressure gradient ∇p. A fluid element is thus accelerated in the direction of decreasing pressure, provided that no further forces act on the fluid element.

Shear forces and field forces

In a flow, the velocity is generally neither constant in time nor in space. Thus, the fluid flows either slower or faster past the lateral surfaces of the considered fluid element. Therefore internal frictional forces occur, which are the greater the more viscous the fluid is. However, since we assume a friction-free flow and thus a inviscid fluid, such shear forces do not occur in the flow.

Other forces that act on the fluid element but cannot be neglected in general are field forces such as those caused by gravity. For example, air flow on earth or flows in pipes are significantly influenced by gravity. For example, in flows containing ferromagnetic fluid particles, an external magnetic field also influences the flow.

Also electrically charged particles in a fluid are possible, so that the flow is then influenced by an external electric field. Water is already influenced by the dipole interaction of the H₂O molecules in an electric field without the need to contain electrically charged particles. This can be demonstrated with an electrically charged plastic rod. If the charged plastic rod is brought into the vicinity of a thin water jet, it is observed that the jet is deflected.

Thus, a fluid element is affected not only by the pressure force F_p, but also by field forces F_g (the weight force should only be representative of other possible field forces!) The sum of both forces then corresponds to the accelerating force F_a (resultant force), which influences the motion of the fluid element:

\begin{align}
& \vec{F}_\text{a} = \vec F_\text{g}+ \vec F_\text{p}\\[5px]
\label{besch}
&\boxed{\vec{F}_\text{a} = \vec F_\text{g}- \vec \nabla p \cdot \text{d}V } ~~~~~\text{accelerating force on a fluid element}\\[5px]
\end{align}

Newton’s second law (substantial acceleration)

According to Newton’s second law, the accelerating force F_a leads to a change in velocity (acceleration), which depends on the mass dm of the fluid element. This material acceleration is also called substantial acceleration a_sub:

\begin{align}
& \boxed{\vec a_\text{sub} = \frac{\vec F_\text{a}}{\text{d}m}} ~~~~~\text{Newton’s second law}\\[5px]
& \vec a_\text{sub} = \frac{\vec F_\text{g}- \vec \nabla p \cdot \text{d}V}{\text{d}m} \\[5px]
& \vec a_\text{sub} = \frac{\vec F_\text{g}}{\text{d}m}- \frac{\vec \nabla p \cdot \text{d}V}{\text{d}m}
~~~\text{where} ~~~\text{d}m=\rho \cdot \text{d}V ~~~\text{:}\\[5px]
& \vec a_\text{sub} = \underbrace{\frac{\vec F_\text{g}}{\text{d}m}}_{\text{specific field force } g}- \frac{\vec \nabla p \cdot \text{d}V}{\rho \cdot \text{d}V} \\[5px]
\label{eu}
& \boxed{\vec a_\text{sub} = \vec g- \frac{1}{\rho} \vec \nabla p} ~~~~~\text{substantial acceleration}\\[5px]
\end{align}

The quotient of field force F_g and mass dm can generally be interpreted as field force per unit mass. In the case of gravity this quotient corresponds to the gravitational acceleration g. If other field forces such as magnetic or electrical forces are also involved, these must also be taken into account in the equation as mass-specific field forces.

Temporal and convective acceleration

The substantial acceleration, i.e. the actually observable acceleration of a fluid element, can be traced back to two causes. On the one hand, a flow and thus the velocity of a fluid element changes not only in time but also in place. Substantial acceleration is thus attributed to a temporal component (temporal acceleration) and a local component (convective acceleration).

We can imagine an air flow on a windy day. The wind blows with a constantly changing direction (unsteady flow). We observe the situation how the wind blows around the corner of a house. A fluid element viewed at a certain location changes its velocity from one second to the next. This is a consequence of the permanently changing flow over time. The resulting acceleration is therefore also called temporal acceleration or local acceleration a_loc, because it is a consequence of the changing velocity at a fixed location.

One could now conclude that in a steady flow, where the speed does not change in time, there is no acceleration on the fluid particles. However, this is not the case. Although the speed of a stationary flow does not change in time at a fixed location, a fluid particle generally has to change its velocity permanently while flowing. For example, the speed of a pipe flow is greater at a constricted section compared to a point at a larger cross-section. Thus, a fluid particle is generally still accelerated when it changes location, despite the lack of temporal change in the flow.

Even the steady flow of wind around a corner constantly requires the fluid particle to adapt to the new flow direction and thus to accelerate. The acceleration due to the change of location is therefore also called convective acceleration a_con.

In summary, it can be concluded that

• The temporal or local acceleration is due to the time-varying flow velocity of an unsteady flow at a fixed location.
• The convective acceleration is due to the flow velocity changing from place to place.
• Both acceleration components together result in the observable acceleration, which is also called substantial or material acceleration.

Note that there is no local acceleration in a steady flow, because the flow velocity does not change in time!

Relationship between local, convective and substantial acceleration

For the sake of simplicity, we first look at a fluid particle on a streamline s and we describe the motion on this streamline (one-dimensional motion). The substantial change in velocity dv, i.e. the actually observable change in velocity, is obtained in this case from the temporal change of velocity ∂v/∂t within the time dt (temporal component) and from a spatial change of velocity ∂v/∂s (gradient) within the distance ds (convective component):

\begin{align}
&\underbrace{\text{d}v}_{\text{substantial change}} = \underbrace{\frac{\partial v}{\partial t} \text{d}t}_{\text{temporal change}} + \underbrace{\frac{\partial v}{\partial s} \text{d} s}_{\text{convective change}}\\[5px]
\end{align}

If this equation is divided by the time duration dt, the following formula for the substantial acceleration a_sub in tangential direction of the streamline is obtained:

\begin{align}
\require{cancel}
&\boxed{a_\text{sub} = \frac{\text{d}v}{\text{d}t}} = \frac{\partial v}{\partial t} \frac{\cancel{\text{d}t}}{\cancel{\text{d}t}}+ \frac{\partial v}{\partial s} \underbrace{\frac{\text{d}s}{\text{d}t}}_{v} \\[5px]
&a_\text{sub} = \underbrace{~~~~~\frac{\partial v}{\partial t}~~~~~}_{\text{local acceleration}} + \underbrace{~~~~~\frac{\partial v}{\partial s}v~~~~~}_{\text{convective acceleration}} \\[5px]
\end{align}

\begin{align}
& \boxed{a_\text{sub} = \frac{\partial v}{\partial t} + \frac{\partial v}{\partial s}v} &&\text{substantial acceleration}\\[5px]
& \boxed{a_\text{loc}=\frac{\partial v}{\partial t}} &&\text{local acceleration} \\[5px]
& \boxed{a_\text{con}= \frac{\partial v}{\partial s}v} &&\text{convective acceleration} \\[5px]
\end{align}

The term of convective acceleration obviously depends on the flow velocity. This becomes clear, because if the fluid element flows very fast, it covers a relatively large distance within a certain time. For a given velocity gradient ∂c/∂s this means a large change in velocity and thus a large acceleration.

Vector notation of the substantial acceleration

For the sake of simplicity, we have only considered the one-dimensional motion of a fluid particle on a streamline. In the three-dimensional case, i.e. when describing the motion from a fixed coordinate system, the local acceleration can also still be written relatively easily in vector notation:

\begin{align}
& \boxed{\vec a_\text{loc}=\frac{\partial \vec v}{\partial t} = \begin{pmatrix}\Large\frac{\partial v_x}{\partial t} \\\ \Large\frac{\partial v_y}{\partial t} \\\ \Large\frac{\partial v_z}{\partial t} \end{pmatrix} } ~~~~~\text{local acceleration} \\[5px]
\end{align}

In contrast to the local acceleration, the vector notation of the convective acceleration is much more complicated, since it is a location-dependency in three dimensions. Each velocity component changes not only due to a local change in one dimension, but is the consequence of a change in all three dimensions!

The velocity component in x direction will generally not only change when a fluid particle moves in x direction. For example, the velocity in x direction will also change if the fluid particle moves in y direction, because the flow velocity in x direction is lower there. A change of location in z direction also generally results in a change of velocity in the x direction, because the flow velocity in the x direction is higher there, for example.

The component of the convective acceleration in x direction (a_con,x) is thus due to the change of location in all three dimensions:

\begin{align}
& \underline{a_\text{con,x}=\frac{\partial v_\text{x}}{\partial x}v_\text{x} + \frac{\partial v_\text{x}}{\partial y}v_\text{y} + \frac{\partial v_\text{x}}{\partial z}v_\text{z}} ~~~~~\text{convective acceleration in }x\text{ direction} \\[5px]
\end{align}

Just for clarification: The velocity of a fluid particle in x direction will generally not only change when the position of the particle changes in x direction (velocity gradient ∂v_x/∂x), but also in case of a change of location in y direction (velocity gradient ∂v_x/∂y) or z direction (velocity gradient ∂v_x/∂z).

For the components of the convective acceleration in y- and z-direction the analogous considerations apply:

\begin{align}
& \underline{a_\text{con,y}=\frac{\partial v_\text{y}}{\partial x}v_\text{x} + \frac{\partial v_\text{y}}{\partial y}v_\text{y} + \frac{\partial v_\text{y}}{\partial z}v_\text{z}} ~~~~~\text{convective acceleration in }y\text{ direction} \\[5px]
& \underline{a_\text{con,y}=\frac{\partial v_\text{z}}{\partial x}v_\text{x} + \frac{\partial v_\text{z}}{\partial y}v_\text{y} + \frac{\partial v_\text{z}}{\partial z}v_\text{z}} ~~~~~\text{convective acceleration in }z\text{ direction} \\[5px]
\end{align}

Thus, the vector of the convective acceleration a_con is represented by 9 terms:

\begin{align}
& \boxed{\vec a_\text{con}= \large \begin{pmatrix} a_\text{con,x}\\\ a_\text{con,y}\\\ a_\text{con,z} \end{pmatrix} 
= \Large \begin{pmatrix}
\frac{\partial v_\text{x}}{\partial x}v_\text{x} + \frac{\partial v_\text{x}}{\partial y}v_\text{y} + \frac{\partial v_\text{x}}{\partial z}v_\text{z}
\\\
\frac{\partial v_\text{y}}{\partial x}v_\text{x} + \frac{\partial v_\text{y}}{\partial y}v_\text{y} + \frac{\partial v_\text{y}}{\partial z}v_\text{z}
\\\
\frac{\partial v_\text{z}}{\partial x}v_\text{x} + \frac{\partial v_\text{z}}{\partial y}v_\text{y} + \frac{\partial v_\text{z}}{\partial z}v_\text{z}
\end{pmatrix} 
} \\[0px]
\end{align}

The convective acceleration can be represented much more clearly using the del operator ∇:

\begin{align}
& \boxed{\vec a_\text{con} = \left(\vec v \cdot \vec \nabla \right) \vec v}~~~\text{convective acceleration} \\[5px]
\end{align}

For the vector of the substantial acceleration a_sub, as the sum of local and convective acceleration, thus the following formula applies:

\begin{align}
\label{en}
&\boxed{\vec a_\text{sub} = \frac{\partial \vec v}{\partial t} + \left(\vec v \cdot \vec \nabla \right) \vec v} ~\text{substantial acceleration} \\[5px]
& \boxed{\vec a_\text{sub}
= \Large \begin{pmatrix}
\frac{\partial v_x}{\partial t} + \frac{\partial v_\text{x}}{\partial x}v_\text{x} + \frac{\partial v_\text{x}}{\partial y}v_\text{y} + \frac{\partial v_\text{x}}{\partial z}v_\text{z}
\\\
\frac{\partial v_y}{\partial t} + \frac{\partial v_\text{y}}{\partial x}v_\text{x} + \frac{\partial v_\text{y}}{\partial y}v_\text{y} + \frac{\partial v_\text{y}}{\partial z}v_\text{z}
\\\
\frac{\partial v_z}{\partial t} + \frac{\partial v_\text{z}}{\partial x}v_\text{x} + \frac{\partial v_\text{z}}{\partial y}v_\text{y} + \frac{\partial v_\text{z}}{\partial z}v_\text{z}
\end{pmatrix} \\[5px]
}
\end{align}

Using equation (\ref{en}) in equation (\ref{eu}), one finally obtains the equation of motion of a fluid particle in a frictionless, unsteady flow. This equation is also known as the Euler equation of motion and applies to both incompressible and compressible fluids (as long as they are inviscid):

\begin{align}
& \vec a_\text{sub} = \vec g- \frac{1}{\rho} \vec \nabla p \\[5px]
&\frac{\partial \vec v}{\partial t} + \left(\vec v \cdot \vec \nabla \right) \vec v = \vec g- \frac{1}{\rho} \vec \nabla p \\[5px]
&\boxed{\frac{\partial \vec v}{\partial t} + \left(\vec v \cdot \vec \nabla \right) \vec v + \frac{1}{\rho} \vec \nabla p = \vec g}~~~\text{Euler equation of motion} \\[5px]
\end{align}

The Euler equation can also be given in components:

\begin{align}
& \boxed{\frac{\partial v_\text{x}}{\partial t} + \frac{\partial v_\text{x}}{\partial x}v_\text{x} + \frac{\partial v_\text{x}}{\partial y}v_\text{y} + \frac{\partial v_\text{x}}{\partial z}v_\text{z} + \frac{1}{\rho} \frac{\partial p}{\partial x} = g_\text{x}}~~~\text{Euler eq. in x-direction} \\[5px]
& \boxed{\frac{\partial v_\text{y}}{\partial t} + \frac{\partial v_\text{y}}{\partial x}v_\text{x} + \frac{\partial v_\text{y}}{\partial y}v_\text{y} + \frac{\partial v_\text{y}}{\partial z}v_\text{z} + \frac{1}{\rho} \frac{\partial p}{\partial y} = g_\text{y}}~~~\text{Euler eq. in y-direction} \\[5px]
& \boxed{\frac{\partial v_\text{z}}{\partial t} + \frac{\partial v_\text{z}}{\partial x}v_\text{x} + \frac{\partial v_\text{z}}{\partial y}v_\text{y} + \frac{\partial v_\text{z}}{\partial z}v_\text{z} + \frac{1}{\rho} \frac{\partial p}{\partial z} = g_\text{z}}~~~\text{Euler eq. in z-direction} \\[5px]
\end{align}

Note the following procedure when calculating convective acceleration using the Nabla operator:

\begin{align}
\vec a_\text{con} &= \left(\vec v \cdot \vec \nabla \right) \vec v\\[5px]
&= \left[\begin{pmatrix} v_x \\\ v_y \\\ v_z \end{pmatrix}
\cdot \begin{pmatrix} \frac{\partial}{\partial x} \\\ \frac{\partial}{\partial y} \\\ \frac{\partial}{\partial z} \end{pmatrix} \right] \cdot \begin{pmatrix} v_x \\\ v_y \\\ v_z \end{pmatrix}\\[5px]
&=\left[ v_x \frac{\partial}{\partial x} + v_y \frac{\partial}{\partial y} + v_z \frac{\partial}{\partial z} \right] \cdot \begin{pmatrix} v_x \\\ v_y \\\ v_z \end{pmatrix}\\[5px]
&=\Large \begin{pmatrix}
v_\text{x} \frac{\partial v_\text{x}}{\partial x} + v_\text{y} \frac{\partial v_\text{x}}{\partial y} +v_\text{z} \frac{\partial v_\text{x}}{\partial z}
\\\
v_\text{x} \frac{\partial v_\text{y}}{\partial x} + v_\text{y} \frac{\partial v_\text{y}}{\partial y} +v_\text{z} \frac{\partial v_\text{y}}{\partial z}
\\\
v_\text{x} \frac{\partial v_\text{z}}{\partial x} +v_\text{y} \frac{\partial v_\text{z}}{\partial y} +v_\text{z} \frac{\partial v_\text{z}}{\partial z}
\end{pmatrix} 
\end{align}

Using the Euler equation along a streamline (Bernoulli equation)

We would like to apply the Euler equation to a streamline and describe the change of state of a flow along this streamline. In this case we are dealing with a flow in only one dimension: in direction of the streamline. The locally changing velocity and pressure are denoted by v and p respectively. With s as the coordinate along the streamline, the Euler equation is as follows:

\begin{align}
&\frac{\partial v}{\partial t} + \frac{\partial v}{\partial s}v + \frac{1}{\rho} \frac{\partial p}{\partial s} = – g \cdot \cos(\alpha) \\[5px]
\end{align}

The angle α is the angle between the vertical z direction and the tangent of the streamline s. The term -g⋅cos(α) thus describes the component of the gravitational acceleration acting against the streamline, which leads to a decelerating force of the fluid particle at a positive angle α (hence the negative sign). An infinitesimal change on the streamline ds is related by the angle α to the infinitesimal change dz in z direction as follows:

\begin{align}
&\cos(\alpha) = \frac{\text{d}z}{\text{d}s}\\[5px]
\end{align}

Therefore the Euler equation of motion along the streamline is:

\begin{align}
&\frac{\partial v}{\partial t} + \frac{\partial v}{\partial s}v + \frac{1}{\rho} \frac{\partial p}{\partial s} = – g \frac{\text{d}z}{\text{d}s}\\[5px]
\end{align}

For the sake of simplicity, we assume in the following that the flow is steady and incompressible. In this case the velocity is not a function of time and thus the partial derivative of the velocity with respect to time is zero (∂v/∂t=0). Due to the incompressibility, the density is not a function of the location either. Instead of the partial derivatives with respect to the location, we can now write the total derivatives. We then get the following relationship between the infinitesimal changes of pressure, velocity and height along the streamline:

\begin{align}
\require{cancel}
&\cancel{\frac{\partial v}{\partial t}} + \frac{\text{d}v}{\text{d}s}v + \frac{1}{\rho} \frac{\text{d}p}{\text{d}s} = -g \frac{\text{d}z}{\text{d}s}\\[5px]
&\frac{\text{d} v}{\text{d}s}v + \frac{1}{\rho} \frac{\text{d}p}{\text{d}s} = -g \frac{\text{d}z}{\text{d}s}&&|\cdot \text{d}s\\[5px]
&v~\text{d}v + \frac{1}{\rho} \text{d}p = -g \text{d}z&&|\cdot \rho\\[5px]
&\underline{\text{d}p + \rho v ~\text{d}v ~ + \rho g ~\text{d}z = 0}\\[5px]
\end{align}

This equation describes the relationship between the infinitesimal changes in pressure, velocity and height along a streamline. We can now integrate this equation to obtain the relationships between the quantities themselves rather than the relationships between the changes in the quantities. The density and acceleration due to gravity, which are considered constant, can be written before the integral sign.

\begin{align}
&\int \left(\text{d}p + \rho v ~\text{d}v ~ + \rho g ~ \text{d}z \right)= \text{konstant}\\[5px]
&\int \text{d}p + \rho \int v ~\text{d}v ~ + \rho g ~\int \text{d}z = \text{konstant}\\[5px]
&\boxed{p + \frac{\rho}{2} v^2 + \rho g ~z = \text{konstant}}~~~\text{Bernoulli’s equation}\\[5px]
\end{align}

We finally obtain the Bernoulli equation for incompressible and frictionless fluids! This equation states that the sum of pressure energy, kinetic energy and positional energy along the streamline is constant (conservation of energy). Note that this equation is only valid for frictionless flows, i.e. especially for inviscid fluids. Two points on a streamline are thus linked in this friction-free case as follows:

\begin{align}
&\boxed{p_1 + \frac{\rho}{2} v_1^2 + \rho g ~z_1 =p_2 + \frac{\rho}{2} v_2^2 + \rho g ~z_2}\\[5px]
\end{align}

Exercises with solutions based on the Bernoulli equation can be found in the linked article.

Continuity equation for one-dimensional flows

Experience shows that mass can neither be created out of nothing nor annihilated. Mathematically, this conservation of mass in flows is formulated in the so-called continuity equation.

Mass flux at an finite volume element

To derive the continuity equation we first consider a very small volume element (control volume). The length of the control volume is denoted by Δx and has the width Δy and the height Δz. A compressible fluid flows through this finite volume element. The fluid can enter the volume element through the lateral surfaces and leave it through other surfaces.

For the sake of simplicity, we will first consider only a flow in the x-direction. Within an infinitesimal time dt a certain mass dm_x,in flows into the volume element. On the opposite side a certain mass dm_x,out leaves the volume element at the same moment.

The inflowing or outflowing mass results from the speed with which the flow enters or leaves the volume element. If the flow enters the volume element with a velocity v_x,in, it travels the infinitesimal distance dx_x,in=v_x,in⋅dt within the time dt. Thus, the following volume dV_x,in flows into the control volume:

\begin{align}
&\text{d}V_\text{x,in}=\Delta A \cdot \text{d}x_\text{x,in} =\Delta A \cdot v_\text{x,in} \cdot \text{d}t\\[5px]
\end{align}

Since we consider an infinitesimal distance dx_x,ein, a constant density can be assumed within the inflowing volume. If the density of the fluid is ϱ_x,in at the point of entry into the volume element, the inflow mass dm_x,in is calculated as follows:

\begin{align}
&\text{d}m_\text{x,in}=\text{d}V_\text{x,in} \cdot \rho_\text{x,in} =\Delta A \cdot \rho_\text{x,in} \cdot v_\text{x,in} \cdot \text{d}t \\[5px]
\end{align}

The mass flowing into the control volume per unit area is therefore only dependent on density and velocity. This area-specific mass flow rate \(\dot m^\text{*}\) is also called mass flux and is marked with an asterisk (*) for better differentiation from the mass flow rate:

\begin{align}
&\dot m_\text{x,in}^\text{*} = \frac{\text{d}m_\text{x,in}}{\Delta A \cdot \text{d}t} = \rho_\text{x,in} \cdot v_\text{x,in} \\[5px]
\label{m_ein}
&\underline{\dot m_\text{x,in}^\text{*} =\rho_\text{x,in} \cdot v_\text{x,in}} \\[5px]
&\boxed{\dot m^\text{*} = \rho \cdot v}~~~\text{mass flux} \\[5px]
\end{align}

The product of density and flow velocity in a flow field of a fluid is called mass flux. It indicates the mass flowing in the flow direction per unit time and unit area!

For compressible fluids, the density in a flow field is generally not constant, but usually differs from one point to another. For example, if the fluid accumulates in the considered control volume, the density and the flow velocity at the outflow of the control volume will differ from the inflow of the control volume. The mass flux at the outflow is calculated in the same way as the mass flux at the inflow [see equation (\ref{m_ein})]:

\begin{align}
&\underline{\dot m_\text{x,out}^\text{*} =\rho_\text{x,out} \cdot v_\text{x,out}} \\[5px]
\end{align}

If more mass flows into the control volume than flows out, the mass inside increases (see animation below). The rate of change of mass \(\dot m_\text{CV}\) in the control volume (CV) results from the difference of the mass flowing in and out (the index x only indicates that the change of the mass in the control volume is due to a flow in x-direction – later on we will consider flow components in y- and z-direction as well):

\begin{align}
\label{m}
& \underbrace{~~\dot m_\text{CV}~~}_\text{change of mass inside the CV} = \underbrace{~~\dot m_\text{x,in}~~}_\text{inflowing mass into the CV} – \underbrace{~~\dot m_\text{x,out}~~}_\text{outflowing mass from the CV} \\[5px]
& \dot m_\text{CV} ~~=~~ \dot m_\text{x,in}^\text{*} \cdot \Delta A~~ -~~ \dot m_\text{x,out}^\text{*} \cdot \Delta A \\[5px]
& \boxed{\dot m_\text{CV} ~~=~~ \dot m_\text{x,in}^\text{*} \cdot \Delta y\cdot \Delta z~~ -~~ \dot m_\text{x,out}^\text{*} \cdot \Delta y \cdot \Delta z} \\[5px]
\end{align}

Mass flux at an infinitesimal volume element

Let us now consider an infinitesimal volume element (control volume) in a compressible flow, where the mass flux and thus the product of density and velocity changes along the x coordinate. Thus, along this x coordinate we can define a gradient of mass flux: ∂(ϱ⋅v_x)/∂x. Over the distance dx (length of the infinitesimal volume element) the following change in mass flux \(\text{d} \dot m_\text{x}^\text{*}\) results in x direction:

\begin{align}
& \underline{\text{d} \dot m_\text{x}^\text{*} =\frac{\partial (\rho v_\text{x})}{\partial x} \cdot \text{d}x} ~~~~~\text{change in mass flux along } \text{d}x \\[5px]
\end{align}

At the outflow of the control volume, the mass flux is thus determined as follows:

\begin{align}
& \dot m_\text{x,out}^\text{*} =\dot m_\text{x,in}^\text{*} +\text{d} \dot m_\text{x} \\[5px]
& \underline{\dot m_\text{x,out}^\text{*} =\dot m_\text{x,in}^\text{*} + \frac{\partial (\rho v_\text{x})}{\partial x} \cdot \text{d}x} ~~~~~\text{outflowing mass flux}\\[5px]
\end{align}

The resulting temporal change of mass \(\dot m_\text{CV}\) inside the control volume can be determined with equation (\ref{m}), whereby the infinitesimal dimensions dy and dz are used at this point:

\begin{align}
\require{cancel}
& \dot m_\text{CV} = \dot m_\text{x,in}^\text{*} \cdot \text{d}y \cdot \text{d}z ~-~ \dot m_\text{x,out}^\text{*} \cdot \text{d}y \cdot \text{d}z \\[5px]
& \dot m_\text{CV} = \dot m_\text{x,in}^\text{*}\cdot \text{d}y \cdot \text{d}z ~-~ \left(\dot m_\text{x,in}^\text{*}+ \frac{\partial (\rho v_\text{x})}{\partial x} \cdot \text{d}x \right) \cdot \text{d}y \cdot \text{d}z \\[5px]
& \dot m_\text{CV} = \cancel{\dot m_\text{x,in}^\text{*}\cdot \text{d}y \cdot \text{d}z} ~-~ \cancel{\dot m_\text{x,in}^\text{*} \cdot \text{d}y \cdot \text{d}z}~-~ \frac{\partial (\rho v_\text{x})}{\partial x} \cdot \text{d}x \cdot \text{d}y \cdot \text{d}z \\[5px]
&\dot m_\text{CV} = ~- \frac{\partial (\rho v_\text{x})}{\partial x} \cdot \underbrace{\text{d}x \cdot \text{d}y \cdot \text{d}z}_{\text{d}V} \\[5px]
\label{a}
&\underline{\dot m_\text{CV} = ~- \frac{\partial (\rho v_\text{x})}{\partial x} \cdot \text{d}V } \\[5px]
\end{align}

The negative sign indicates that in case of a positive gradient the mass in the volume element decreases over time, because obviously the outgoing mass flow is larger than the incoming mass flow. However, if the mass in the volume element generally changes over time, then its density ϱ also changes over time:

\begin{align}
\label{b}
&\underline{\dot m_\text{CV} = \frac{\partial \rho}{\partial t} \cdot \text{d}V} \\[5px]
\end{align}

Note that we are looking at an infinitesimal volume element, to which a single density, changing over time, can be assigned. In the next step, we will see that the size of the volume element does not matter anyway and thus can indeed be chosen infinitesimally small. Furthermore, note that the density in flows generally varies not only over time (e.g. in unsteady flows), but also changes from one point to another (e.g. in a pipe reducer). The temporal change of density is therefore a partial derivative of the density function with respect to time.

Using equation (\ref{b}) in equation (\ref{a}), the following relationship is finally obtained between the gradient of the mass flux and the resulting temporal change of the density in one point of the flow:

\begin{align}
\require{cancel}
&\frac{\partial \rho}{\partial t} \cdot \cancel{\text{d}V} = ~- \frac{\partial (\rho v_\text{x})}{\partial x} \cdot \cancel{\text{d}V} \\[5px]
&\boxed{\frac{\partial \rho}{\partial t} = ~- \frac{\partial (\rho v_\text{x})}{\partial x}}~~~\text{continuity equation for one-dimensional flows} \\[5px]
\end{align}

This equation is finally called continuity equation and is valid in this form for one-dimensional flows. As seen, this equation results from the conservation of mass. The continuity equation serves to determine the temporal change of density in arbitrary points of a flow by means of the existing flow field (represented by the vectors of the mass flux). Thus statements about the development of unsteady flows of compressible fluids are possible. According to the continuity equation, the following statement applies:

The gradient of the mass flux in a point of a flow corresponds to the temporal change of the density in this point!

Note that the mass flux is a vector and points in the same direction as flow velocity, since it is ultimately a product of a vector (velocity) and a scalar (density). The mass flux is basically a flow velocity weighted by the density. To illustrate flow fields, see also the article Streamlines, pathlines, streaklines and timelines.

Continuity equation for three-dimensional flows

The previous consideration was limited to a one-dimensional flow in x-direction. In general, however, a flow is three-dimensional, i.e. the flow velocity has components in all three directions. Therefore, not only the mass flow through a volume element in x direction may be considered, but also the flow components in y and z direction must be taken into account. Mass does not only flow into and out of the volume element through the front and rear surface, but also through the lateral surfaces (y direction) and the bottom and top surface (z-direction).

For the y and z direction, the changes of mass inside the control volume can be determined in the same way as in equation (\ref{a}):

\begin{align}
\label{aa}
&\dot m_\text{x} = ~- \frac{\partial (\rho v_\text{x})}{\partial x} \cdot \text{d}V \\[5px]
\label{c}
&\dot m_\text{y} = ~- \frac{\partial (\rho v_\text{y})}{\partial y} \cdot \text{d}V \\[5px]
\label{d}
&\dot m_\text{z} = ~- \frac{\partial (\rho v_\text{z})}{\partial z} \cdot \text{d}V \\[5px]
\end{align}

In this equation the velocities v_x, v_y and v_z denote the components of the flow vector v. The resulting change in mass in the infinitesimal control volume is now no longer only due to a flow in x direction, but also to the flow components in y and z direction. As with equation (\ref{b}) for a one-dimensional flow, the following formula now applies to a three-dimensional flow:

\begin{align}
&\underbrace{\frac{\partial \rho}{\partial t} \cdot \text{d}V}_\text{change in mass inside the control volume} = \underbrace{\dot m_\text{x}+\dot m_\text{y} +\dot m_\text{z}}_{\text{mass flowing over the boundaries of the control volume}} \\[5px]
\end{align}

Equations (\ref{aa}) to (\ref{d}) used in this equation finally provides the continuity equation for three-dimensional flows:

\begin{align}
\require{cancel}
&\frac{\partial \rho}{\partial t} \cdot \cancel{\text{d}V}= ~- \frac{\partial (\rho v_\text{x})}{\partial x}\cdot \cancel{\text{d}V} – \frac{\partial (\rho v_\text{y})}{\partial y}\cdot \cancel{\text{d}V} – \frac{\partial (\rho v_\text{z})}{\partial z}\cdot \cancel{\text{d}V} \\[5px]
&\boxed{\frac{\partial \rho}{\partial t} = ~- \left[\frac{\partial (\rho v_\text{x})}{\partial x}+ \frac{\partial (\rho v_\text{y})}{\partial y}+ \frac{\partial (\rho v_\text{z})}{\partial z}\right]}~~~\text{continuity equation} \\[5px]
\end{align}

The sum of the partial derivatives with respect to the different directions (sum of the gradients of mass flux) is also called divergence (div) in mathematics. The divergence is a mathematical operator, which balances flows through a volume element in case of a flow field. The divergence can also be written as scalar product of del operator ∇ and vector field of mass flux ϱv:

\begin{align}
&\boxed{\frac{\partial \rho}{\partial t} = ~- \text{div}\left(\rho \vec v \right) }~~~\text{where}~~~\text{div}\left(\rho \vec v \right) = \vec \nabla \cdot \rho \vec v =\frac{\partial (\rho v_\text{x})}{\partial x}+ \frac{\partial (\rho v_\text{y})}{\partial y}+ \frac{\partial (\rho v_\text{z})}{\partial z} \\[5px]
\end{align}

For incompressible fluids the density does not change and is therefore constant over time. The partial derivative of the density with respect to time is therefore zero (∂ϱ/∂t=0). Furthermore, the density is also spatially constant and can therefore be written before the divergence operator. For incompressible fluids, the continuity equation has therefore the following form:

\begin{align}
&\boxed{\text{div}\left(\vec v \right) = 0} ~~~\text{continuity equation for incompressible fluids} \\[5px]
\end{align}

Interpretation of the divergence

Vector fields can be visualized with arrows. The length of the arrows indicates the magnitude of the quantity and the arrowheads show the direction. Vector fields can also be visualized by field lines. The density of the field lines indicate the magnitude of the field strength and the tangent to the field lines show the direction. In a similar way the flow field of fluids can be visualized by streamlines (“field lines”). The density of the streamlines is a measure for the flow velocity and the tangent to the streamlines show the flow direction.

The figure below schematically shows a two-dimensional flow field. You can imagine pouring water into a sink and leaving the drain open at the same time. The resulting velocity field would then correspond in a simplified way to the flow field shown.

At the point where the water meets the sink, there is a water source. At this point the water is created in the flow plane, so to speak (note that we are looking at the two-dimensional case here and not the three-dimensional case, where obviously water cannot be created out of nothing). At the point where the drain is, the water disappears from a two-dimensional perspective. At this point, the water in the flow plane is annihilated, so to speak.

If you would apply the divergence operator to this vector field, you would get a positive value at the point where the water is created. At the point where the water is annihilated, you would get a negative value. By applying the divergence operator to a vector field, you can visualize the sources and sinks of vector fields in a vivid way. Field lines or flow lines start at points of positive divergence (sources) and end at points of negative divergence (sinks). The fact, that field lines or streamlines diverge strongly at sources or sinks, is finally the reason for the name of the operator.

The divergence operator converts a vector field into a scalar field, and the scalar quantities are a measure for the strength of a field source! Positive values indicate sources and negative values indicate sinks of field lines!

Instead of the flow field shown above you can also imagine two electric charges. The resulting field lines would look very similar. In fact, the divergence as a measure for the strength of a source (measure of the electric charge) plays a major role in this field, too. The divergence operator is therefore of great importance for many vector fields.

But there are also vector fields that have no sources or sinks and are therefore source-free. These include, for example, magnetic fields whose field lines have no beginning and no end, but are always closed. Even incompressible flows are obviously source-free due to the fact that mass cannot be created or annihilated (note: the above example of the 2D flow field only served as an illustrative example).

Definition of viscosity

In the article Viscosity, the cause of viscosity was mainly attributed to attractive forces between the layers of a fluid. These forces act similar to frictional forces, so that the individual fluid layers try to slow each other down. For the definition of viscosity one can imagine a fluid between two plates. The lower plate is at rest and the upper plate is moved at constant speed.

Due to the no-slip condition, both the upper and the lower fluid layer adhere to these plates. The lower fluid layer thus remains at rest and the upper fluid layer moves at the same speed as the upper plate. A linear velocity profile is formed between the plates. A considered fluid layer thus always moves slower than the fluid layer above it. Due to the molecular forces between the molecules, a lower fluid layer always tries to slow down the fluid layer above.

These frictional forces between the layers must be compensated if the uppermost layer is to be moved at a constant speed. The more viscous a fluid is, the stronger the internal friction forces and the greater the force required to move the top plate. The viscosity η shows the relationship between the area-related force F/A (called shear stress τ), which is required to move the fluid layers, and the slope of the velocity profile dv/dy (called velocity gradient):

\begin{align}
\label{t}
&\frac{F}{A}=\boxed{\tau= \eta \cdot \frac{\text{d} v}{\text{d} y}} ~~~~~\text{Newton’s law of fluid friction}\\[5px]
\end{align}

Viscosity of gases (momentum transfer)

The cause of viscosity due to frictional forces acting between the fluid layers can be clearly understood for liquids. In gases, however, the molecules exert almost no molecular forces on each other. Frictional forces between the fluid layers due to intermolecular cohesion are therefore almost non-existent. However, practice shows that even gases have a considerable viscosity and that fluid layers are slowed down in flows. How can this behavior be explained without the presence of inter-molecular forces?

The slowing down effect of fluid layers in gases is mainly due to the momentum transfer of the gas molecules when they diffuse from a slower layer into a faster layer. This process also takes place in liquids, but it is negligible compared to the decelerating effect due to the intermolecular forces.

Let us again consider the already mentioned laminar flow, which this time consists of an ideal gas as fluid. If one gas molecule collides with another, an momentum exchange takes place, i.e. a slower molecule takes up part of the momentum of the faster molecule. However, such a momentum transfer does not only take place within a fluid layer. Due to the random molecular motion (Brownian motion), molecules also diffuse into adjacent fluid layers. What happens if a slower molecule diffuses into a layer of faster particles? The fast molecules are slowed down by this diffused gas molecule and the layer slows down.

One can illustrate the situation with a cart and a ball. The cart is to be moved at a constant speed when suddenly the heavy ball is put in while passing by. The ball represents the slower gas molecule (in this case even stands still), which diffuses into the faster fluid layer (illustrated by the cart). Since the ball has a lower speed than the cart when put in, the ball must be accelerated to the speed of the cart if the speed is to remain constant. This requires a force corresponding to the mass of the ball (force = mass x acceleration). This means that if this force would not be applied, the cart would be slowed down, similar to a frictional force.

The diffusion of gas molecules between the layers of a laminar flow leads to a momentum transfer on which the viscosity of gases is mainly based!

Faster layers thus transfer part of their momentum by diffusion into slower layers. All in all there is a transport of momentum from the moving plate at the top of the fluid to the resting plate at the bottom of the fluid. This momentum transport (which ultimately corresponds to a force between the layers) is directed in the direction of decreasing velocity, so to speak, i.e. against the velocity gradient. The transport of momentum becomes particularly clear when the upper plate is set in motion from rest. First, only the layer directly adhering to the upper plate is set in motion. By the transport of momentum onto the layer underneath, this layer starts to move, etc. The momentum thus gradually spreads through the layers, so to speak.

The resistance that one experiences when moving the upper plate is caused by the fact that the movement of the gas layers is hindered because the lower plate is fixed. Holding the bottom plate in place ultimately requires the same force as maintaining the movement of the upper plate. The momentum is, so to speak, introduced at the upper plate and is transferred from layer to layer and exits at the lower plate. Finally, in a state of equilibrium, there is no net momentum flow, which is transferred to the fluid layers, so that these finally move at constant but different speeds according to the linear velocity profile.

Derivation of the viscosity of ideal gases

Shear stress as momentum flux

A force F can generally be determined from the change in momentum per unit time:

\begin{align}
&F= \frac{\text{d} p}{\text{d} t} = \dot p ~~~~~\Rightarrow~~~~~\boxed{\text{force = momentum flow rate}}\\[5px]
\end{align}

The force acting on the individual fluid layers is thus caused by the change in momentum and can therefore also be understood as momentum flow rate p*. If one relates the force and thus the momentum flow rate to the surface area, this corresponds to a shear stress, which in turn can be interpreted as momentum flux p*_A (change in momentum per unit time and unit area):

\begin{align}
&\tau = \frac{F}{A}= \frac{\dot p}{A} = \dot p_\text{A} ~~~~~\Rightarrow~~~~~\boxed{\text{shear stress = momentum flux}}\\[5px]
\end{align}

Newton’s law of fluid friction (\ref{t}) can thus also be represented as follows:

\begin{align}
\label{tt}
&\boxed{\dot p_\text{A} = – \eta \cdot \frac{\text{d} v}{\text{d} y}} \\[5px]
\end{align}

The negative sign was introduced to account for the fact that the momentum flux is transferred away from faster layers to slower layers, i.e. in the direction of decreasing velocity gradient. At this point an interesting analogy can be drawn to other transport mechanisms like heat transport and mass transport, which are ultimately described in a similar way:

Momentum transfer between the layers

With the help of the kinetic theory of gases, the viscosity of ideal gases can be calculated. To derive a formula, we consider a laminar flow, where an ideal gas is placed between two plates. The lower plate is fixed and the upper plate moves with a constant speed.

We observe the gas flow on a microscopic level and move along with a fluid layer. The mean distance a gas molecule travels between two collisions is called mean free path λ. We therefore consider gas layers that have a distance λ from each other, so that diffusion processes result in a collision inside those layers and thus in a momentum transfer. We now look at a layer at any height y. The mean velocity of the gas molecules in x-direction with respect to a coordinate system fixed at the resting plate is denoted by v_x(y).

The mean velocityv_x(y+λ) of the gas molecules in the layer above at the distance λ can be determined using the velocity gradient dv/dy:

\begin{align}
&v_{x}(y+\lambda)= v_{x}(y) + \lambda \cdot \frac{\text{d}v}{\text{d}y} \\[5px]
\end{align}

In the same way, the mean velocity v_x(y-λ) of the gas particles in the layer below at the distance λ can be determined:

\begin{align}
&v_{x}(y-\lambda)= v_{x}(y) – \lambda \cdot \frac{\text{d}v}{\text{d}y} \\[5px]
\end{align}

If n*_A denotes the particle flux, i.e. the number of particles per unit time and unit area that diffuse from the upper layer or lower layer into the middle layer, then the respective momentum fluxes can be determined with the following formulas. Note that the particle flux is identical for both layers if we assume an incompressible gas flow where the particle density is the same at every point in the flow.

\begin{align}
&\dot p_{A}(y+\lambda) = \dot n_\text{A} \cdot \overbrace{m \cdot v_{x}(y+\lambda)}^{\text{momentum of one molecule}} = \dot n_\text{A} \cdot m \cdot \left(v_{x}(y) + \lambda \cdot \frac{\text{d}v}{\text{d}y} \right) \\[5px]
&\dot p_{A}(y-\lambda) = \dot n_\text{A} \cdot m \cdot v_{x}(y-\lambda) = \dot n_\text{A} \cdot m \cdot \left(v_{x}(y) – \lambda \cdot \frac{\text{d}v}{\text{d}y} \right) \\[5px]
\end{align}

The net momentum flux p*_A(y) in the layer at the height y is finally the sum of both momentum fluxes. According to the chosen coordinate system the bottom-up momentum flux (pointing in positive y direction) corresponds to a positive value and the downward facing momentum flux to a negative value.

\begin{align}
\dot p_\text{A}(y) &= \dot p_{A}(y-\lambda) ~-~ \dot p_{A}(y+\lambda) \\[5px]
&= \dot n_\text{A} \cdot m \cdot \left(v_{x}(y) – \lambda \cdot \frac{\text{d}v}{\text{d}y} \right)- \dot n_\text{A} \cdot m \cdot \left(v_{x}(y) + \lambda \cdot \frac{\text{d}v}{\text{d}y} \right) \\[5px]
&= \dot n_\text{A} \cdot m \cdot \left(v_{x}(y) ~- \lambda \cdot \frac{\text{d}v}{\text{d}y} ~-~ v_{x}(y) ~- \lambda \cdot \frac{\text{d}v}{\text{d}y}\right) \\[5px]
\end{align}

\begin{align}
&\boxed{\dot p_\text{A}= – 2~ \dot n_\text{A} \cdot m \cdot \lambda \cdot \frac{\text{d}v}{\text{d}y}}~~~\text{net momentum flux} \\[5px]
\end{align}

Viscosity of ideal gases as a function of particle flux

According to the above equation, the net momentum flux is obviously no longer a function of the variable y and thus identical in every point of the flow! If one compares this formula with Newton’s law of fluid friction (\ref{tt}), it is apparent that the expression 2⋅n*_A⋅m⋅λ obviously corresponds to the viscosity η:

\begin{align}
&\dot p_\text{A} = – \eta \cdot \frac{\text{d} v}{\text{d} y} \\[5px]
&\dot p_\text{A}=- \underbrace{2~ \dot n_\text{A} \cdot m \cdot \lambda}_{\eta} \cdot \frac{\text{d}v}{\text{d}y} \\[5px]
\label{eta}
&\boxed{\eta= 2~ \dot n_\text{A} \cdot m \cdot \lambda} ~~~\text{viscosity of ideal gases}\\[5px]
\end{align}

The viscosity of an (ideal) gas is therefore only dependent on the mass of a gas particle, the mean free path and the particle flux. The area-related particle flow n*_A, which diffuses in from a layer above or below, depends in turn on how strongly the gas molecules move due to the random diffusion motion (Brownian motion). This in turn is determined by the temperature.

If the temperature is high, diffusion take place more strongly and more particles diffuse between the layers. Thus, the particle flux is high and so is the momentum transfer. This results in an increasing force, which is necessary to maintain the macroscopic flow (movement of the plate)! The viscosity of gases therefore generally increases with temperature and not decreases as with liquids!

At this point it is also evident that with ideal gases, pressure has no influence on viscosity. Although the particle density and thus the diffusing particle flow increases proportionally with increasing pressure, the mean free path decreases to the same extent. Both effects cancel each other out.

With ideal gases the viscosity is independent of pressure and increases with increasing temperature!

Viscosity of ideal gases as a function of temperature

At this point we would like to explicitly derive the dependence of the viscosity of ideal gases on temperature. For this purpose, a correlation between the particle flux diffusing perpendicular to the flow and the temperature must be found.

For this we move in thoughts with a layer, so that this layer rests relative to us. The gas molecules themselves, however, are by no means at rest on the microscopic level. Due to Brownian motion, they move in all directions in a completely random manner. According to the Maxwell-Boltzmann distribution, the mean speed of a gas particle v_T is linked to the temperature T of the gas as follows

\begin{align}
\label{a}
&\boxed{ \overline{v_\text{T}} = \sqrt{\frac{8 k_B T}{\pi m}}} ~~~\text{arithmetic mean speed} \\[5px]
\end{align}

In this equation m denotes the mass of a gas molecule and k_B is the Boltzmann constant. Note that the speed v_T represents the mean speed relative to the moving fluid layers and does not include the superposition of the macroscopic flow motion. The latter has no influence on the temperature anyway; after all, the temperature of a gas does not depend on whether the gas is at rest or moving.

Let us now consider a directed flow in which all particles move in the same direction with the (mean) velocity v. The number of particles flowing per unit time and area (particle flux) can be determined as follows. We consider an area element dA through which the particles flow with the (mean) velocity v within a time dt. The particles cover the distance dl=v⋅dt. Thus the particles obviously flow through the volume element dV:

\begin{align}
&\text{d}V =\text{d}A \cdot \text{d}l = \text{d}A \cdot \overline{v} \cdot \text{d}t \\[5px]
\end{align}

For a given particle density n (number of particles per unit volume), the following number of particles dN will be found this volume element:

\begin{align}
&\text{d}N = n \cdot \text{d}V =n \cdot \text{d}A \cdot \overline{v} \cdot \text{d}t \\[5px]
\end{align}

The number of particles per unit time and area (particle flux n*_A) passing through an area perpendicular to the flow can thus be calculated with the following formula:

\begin{align}
&\dot n_\text{A} = \frac{\text{d}N}{\text{d}A \cdot \text{d}t} =n \cdot \overline{v} \\[5px]
&\boxed{\dot n_\text{A} =n \cdot \overline{v} } ~~~\text{particle flux of a directed motion}\\[5px]
\end{align}

Let us now look again at our laminar flow and we move along in our thoughts with a fluid layer. From this point of view, the flow is no longer directed, but completely random with a mean molecular velocity denoted by v_T. The molecules do not move in any preferred direction. This means that only a sixth of the particles move downwards and diffuse into a gas layer below. The particle flux directed perpendicular to the main flow (bulk motion) is thus only one-sixth as large:

\begin{align}
\label{na}
&\boxed{\dot n_\text{A} = \frac{1}{6} n \cdot \overline{v_\text{T}} } ~~~\text{particle flux of a random motion}\\[5px]
\end{align}

Using (\ref{a}) in equation (\ref{na}) results in the following diffusing particle flux as a function of temperature:

\begin{align}
&\dot n_\text{A} = \frac{1}{6} n \cdot \underbrace{\sqrt{\frac{8 k_B T}{\pi m}}}_{\overline{v_\text{T}}} \\[5px]
\end{align}

This equation used in formula (\ref{eta}) finally shows the following relationship between viscosity and temperature:

\begin{align}
&\eta= 2~ \dot n_\text{A} \cdot m \cdot \lambda \\[5px]
&\eta= 2~ \frac{1}{6} n \cdot \sqrt{\frac{8 k_B T}{\pi m}} \cdot m \cdot \lambda \\[5px]
\label{ac}
&\boxed{\eta= \frac{1}{3} n \cdot \sqrt{\frac{8 k_B m T}{\pi}} \cdot \lambda} \\[5px]
\end{align}

Last but not least, the mean free path λ can be expressed by the particle density n and the diameter of the gas molecules d (for the derivation of this formula see article Mean free path & collision frequency):

\begin{align}
& \boxed{\lambda = \frac{1}{\sqrt{2}~n ~\pi d^2}} \\[5px]
\end{align}

Using this formula in equation (\ref{ac}), the following formula for calculating the viscosity of ideal gases is finally obtained:

\begin{align}
&\eta= \frac{1}{3} n \cdot \sqrt{\frac{8 k_B m T}{\pi}} \cdot \lambda \\[5px]
&\eta= \frac{1}{3} n \cdot \sqrt{\frac{8 k_B m T}{\pi}} \cdot \frac{1}{\sqrt{2}~n ~\pi d^2} \\[5px]
&\boxed{\eta= \sqrt{\frac{4 k_B m~T}{9\pi^3~d^4}}} \\[5px]
&\boxed{\eta \sim \sqrt{T}} \\[5px]
\end{align}

This formula shows that the particle density and thus the pressure has no influence on the viscosity of ideal gases. Only the temperature as a variable quantity influences the viscosity. The viscosity increases proportionally with the square root of the temperature!

Note that this formula only applies to laminar flows where the gas layers do not mix macroscopically and diffusion between the layers only occur at the microscopic level. In turbulent flows, the momentum exchange through the turbulence is greater and the viscosity is higher.

Comparing viscosity and thermal conductivity

Equation (\ref{na}) can also be put directly into the formula (\ref{eta}) for the viscosity, resulting in the following relationship:

\begin{align}
&\eta= 2~ \dot n_\text{A} \cdot m \cdot \lambda \\[5px]
&\eta= 2~ \frac{1}{6} n \cdot \overline{v_\text{T}} \cdot m \cdot \lambda \\[5px]
&\eta= \frac{1}{3} \underbrace{n \cdot m}_{\rho} \cdot \lambda \cdot \overline{v_\text{T}} \\[5px]
&\boxed{\eta= \frac{1}{3} \cdot \rho \cdot \lambda \cdot \overline{v_\text{T}}} \\[5px]
\end{align}

This derivation exploited the fact that the product of particle density and mass of a single particle equals the density ϱ of the gas. At this point an interesting analogy to thermal conductivity k of ideal gases can be seen (to avoid confusion with the mean free path, the thermal conductivity was not denoted by λ but k):

\begin{align}
& \boxed{k= \frac{1}{3} \cdot \rho \cdot \lambda \cdot c_v \cdot \overline{v_\text{T}} } \\[5px]
\end{align}

The thermal conductivity thus obeys in principle the same laws as the viscosity, i.e. it increases with increasing mean particle speed (increasing temperature). This is not surprising, since the diffusion of particles is not only associated with an momentum transfer, but also with a energy transfer in terms of heat.

Radial, axial and mixed flow pumps

Centrifugal pumps are turbomachines that pump liquids by means of centrifugal force generated by rotating impellers. Axially to the impeller, the liquid to be pumped enters the suction port of the pump on the suction side. The impeller rotating in the pump accelerates the liquid under the effect of centrifugal force.

The increase in speed raises the dynamic pressure in the fluid. When leaving the impeller, the liquid is slowed down again at the discharge port due to the accumulated liquid. Thus, the dynamic pressure of the liquid is converted into static pressure (more information about the relationship between dynamic and static pressure can be found in the article Venturi Effect). As a result, the static pressure at the discharge port increases very much. This high pressure enables the liquid to overcome a certain geodetic head.

Depending on the design of the pump or impeller, the liquid leaves the impeller in radial or axial direction. Pumps that generate a radial flow are also called radial flow pumps. The impellers of radial pumps are designed either as a closed impeller with a cover plate or as an open impeller (the latter is shown in the figures).

If the liquid leaves the impeller in axial direction, this design is also called axial flow pump. In this case, the impeller acts like a propeller of a ship, except that in this case the propeller is stationary and thus creates a flow inside the pipe. Therefore axial pumps are also called propeller pumps. They are often used as circulation pumps in chemical plants. There are also centrifugal pumps where the liquid does not leave the impeller exactly axially or radially. These pumps are then called diagonal flow pumps or mixed flow pumps.

Centrifugal pumps are turbomachines that deliver liquids by means of centrifugal forces! Depending on the flow generated in relation to the impeller, they are called radial flow pumps, axial flow pumps or mixed flow pumps!

In contrast to axial flow pumps, radial flow pumps can provide higher pressure heads, but they usually produce a relatively low flow rate. To achieve high flow rates, several radial flow pumps may have to be connected in parallel. Axial flow pumps offer high volumetric flow rates, but only relatively low pressure heads (approx. 15 m). In order to achieve large pressure heads with axial flow pumps, several pumps may have to be connected in series. A compromise between the advantages and disadvantages of both pump types is offered by mixed flow pumps.

Radial flow pumps provide high pressure heads at low flow rates, while axial flow pumps provide high flow rates at low pressure heads. Mixed flow pumps offer a compromise of both designs.

Radial impellers with one to a maximum of three vanes are used for pumping heavily contaminated liquids or liquids containing solids. The small number of vanes increases the flow cross-section and thus improves the flow through the impeller. Such impellers are also called channel impellers. In the case of propeller pumps, a screw design of the impeller is used when it comes to pumping liquids containing solids.

Cavitation

The pump sucks in the fluid due to the negative pressure generated at the suction port. More precisely: the higher ambient pressure outside the suction pipe pushes the fluid into the pump in the direction of the negative pressure (principle of drinking with a straw). Since the ambient pressure is 1 bar and a pump can create a vacuum at most, the pressure with which the fluid can be pressed into the pump is limited to a maximum of 1 bar. This allows only a limited geodetic suction head to be achieved. In the case of pumping water, the maximum suction lift when a perfect vacuum is created is theoretically 10 meters.

However, the suction head of a centrifugal pump is not only limited by the ambient pressure, but also by cavitation. Cavitation is the formation of vapor bubbles when the static pressure in a liquid falls below the vapor pressure. The vapor pressure of water at a temperature of 20 °C is 23 mbar. If the static pressure in water falls below this value, the water starts to evaporate locally even at this low temperature and small gas bubbles are formed (analogous to the rising vapor bubbles when water is boiling).

If the pressure in the pump subsequently rises again, the gas bubbles become unstable and implode. Since this is a gas-filled bubble, the particle density is relatively low. Thus, the bubble collapses with almost no resistance. The surrounding liquid is accelerated so strongly in the collapsing bubble that the resulting micro jets generate local pressures of several thousand atmospheres! If such micro jets hit the vanes of the impeller, this leads to damage over time. The occurrence of cavitation is often noticed by loud noises or vibrations of the pump.

Cavitation is the formation of vapor bubbles and their subsequent implosion, whereby the resulting micro-jets destroy the surfaces of components!

Areas in which the static pressure is relatively low are particularly susceptible to the formation of vapor bubbles. This is the case at the pump inlet for two reasons. On the one hand, the pump must have a negative pressure there anyway, so that the liquid can be delivered into the pump at all. In addition, according to the Venturi effect, the static pressure decreases when the flow velocity increases. Especially at the inlet to the impeller, the flow velocity is particularly high due to the reduced inlet cross section, and the static pressure is thus lowest. If the pressure drops below the vapor pressure there, gas bubbles are formed which finally implode due to the pressure increase in the pump.

NPSH value

The risk of cavitation is obviously particularly high when the pump has to generate a strong negative pressure to deliver the fluid. From the perspective of the piping system, this is the case with large geodetic suction heads and with large head losses of the suction pipe (caused by friction). From the perspective of the pump, the risk of cavitation is high with high flow rates, since this in turn means high flow velocities and thus leads to a strong drop in static pressure.

NPSH value of the piping system (NPSHA)

For cavitation free operation of centrifugal pumps, it must therefore be ensured that the total pressure at the inlet of the pump (center of the suction port as reference level) does not fall below the vapor pressure of the liquid to be pumped. The existing difference Δp between the total pressure at the inlet p_in,tot and the vapor pressure of the liquid p_vap is determined with the following equation, where the total pressure can be written as the sum of static pressure p_in and dynamic pressure ϱ/2⋅v_in² (v_in denotes the mean flow velocity at the inlet to the impeller):

\begin{align}
&\Delta p = p_\text{in,tot} – p_\text{vap} \\[5px]
\label{dp}
&\Delta p = \left(p_\text{in}+\frac{\rho}{2}v_\text{in}^2 \right) – p_\text{vap} \\[5px]
\end{align}

What is the meaning of this equation? Without taking losses into account, the total pressure at the inlet (term in the round brackets) corresponds to the total pressure with which the medium is pressed into the open end of the suction pipe due to the conservation of energy. This pressure is counteracted at the other end of the suction pipe (inlet to the impeller) by the vapor pressure of the liquid as minimum available pressure. Thus, the pressure difference Δp can be interpreted as net positive suction pressure with which the liquid could be sucked in at maximum without falling below the vapor pressure.

This net positive suction pressure can also be converted into an equivalent height (head) of a liquid column that could be raised with this suction pressure. This is called the Net Positive Suction Head (NPSH). To determine the available NPSH value of the piping system (index A for available), equation (\ref{dp}) is divided by the term ϱ⋅g:

\begin{align}
&\underbrace{\frac{\Delta p}{\rho g}}_{\text{NPSH}_\text{A}} = \left(\frac{p_\text{in}}{\rho g}+\frac{v_\text{in}^2}{2g}\right) – \frac{p_\text{vap}}{\rho g} \\[5px]
\label{a}
&\underline{\text{NPSH}_\text{A} = \left(\frac{p_\text{in}}{\rho g}+\frac{v_\text{in}^2}{2g} \right) – \frac{p_\text{vap}}{\rho g}} \\[5px]
\end{align}

Using the extended Bernoulli equation, a relationship can be established between the state of the liquid at the tank (0), where the liquid is sucked in, and the state at the inlet to the impeller (in). Pressure losses Δp_loss in the suction pipe are also taken into account, whereby kinetic energies are neglected due to the usually low rate of descent of the liquid level in the tank (v₀≈0).

\begin{align}
\require{cancel}
& p_0 + \frac{1}{2} \rho \cancel{v_0^2} +\rho g H_0= p_\text{in} + \frac{1}{2} \rho v_\text{in}^2 + \rho g H_\text{in} + \Delta p_\text{loss}\\[5px]
& p_0 + \rho g H_0= p_\text{in} + \frac{1}{2} \rho v_\text{in}^2 + \rho g H_\text{in} + \Delta p_\text{loss} \\[5px]
& \frac{p_0}{\rho g} + H_0= \frac{p_\text{in}}{\rho g} + \frac{v_\text{in}^2}{2g} + H_\text{in} + \underbrace{\frac{\Delta p_\text{loss}}{\rho g}}_{\text{head loss }H_\text{loss}} \\[5px]
& \frac{p_\text{in}}{\rho g} + \frac{v_\text{in}^2}{2g} = \frac{p_0}{\rho g} – \underbrace{(H_\text{in}~ -~ H_0)}_{\text{geodetic suction head }H_\text{s}} – H_\text{loss} \\[5px]
\label{pe}
& \underline{\frac{p_\text{in}}{\rho g} + \frac{v_\text{in}^2}{2g} = \frac{p_0}{\rho g} – H_\text{s} – H_\text{loss}} \\[5px]
\end{align}

The term in the round bracket in equation (\ref{a}) can now be replaced by equation (\ref{pe}):

\begin{align}
&\text{NPSH}_\text{A} = \left(\frac{p_0}{\rho g} – H_\text{s} – H_\text{loss} \right) – \frac{p_\text{vap}}{\rho g} \\[5px]
&\boxed{\text{NPSH}_\text{A} = \frac{p_0-p_\text{vap}}{\rho g} – H_\text{s} – H_\text{loss}} \\[5px]
\end{align}

In this equation, H_loss denotes the head loss due to friction and flow losses inside the suction pipe. The geodetic suction head H_s corresponds to the difference in height between the liquid level in the tank and the inlet to the impeller. For a so-called suction operation, i.e. when the tank is lower than the pump inlet, a positive value must be used for the suction head. If the tank is higher than the pump inlet, this is called pressure operation and the value for the “suction head” has a negative sign.

NPSH value of the pump (NPSHR)

The NPSH_A value corresponds to the available pressure head due to the characteristics of the piping system. For cavitation free operation, the pump with its provided pressure head must not exceed the NPSH_A value of the system. It is therefore absolutely necessary that the pump generates a lower NPSH value than is available from the perspective of the system. The NPSH value of the pump is therefore referred to as the NPSH_R value (index R for Required).

The NPSH_R values of centrifugal pumps are specified by the manufacturers in their data sheets. The NPSH_R value is strongly dependent on the rotational speed of the impeller and the delivered flow rate. For safety reasons the NPSH_A value of the system should be about 0.5 m higher than the NPSH_R value of the pump:

\begin{align}
&\boxed{\text{NPSH}_\text{A} \geq \text{NPSH}_\text{R} + 0,5 \text{ m}} ~~~\text{cavitation free operation} \\[5px]
\end{align}

Measures to prevent cavitation

If cavitation occurs during operation of a centrifugal pump, appropriate countermeasures must be taken. Reducing the flow velocity is crucial to prevent the pressure from dropping too much.

From the perspective of the pump, this can be achieved, for example, by reducing the rotational speed of the impeller. However, the flow rate decreases with this measure. If such a reduction of the flow rate is not wanted, a larger pump must be used, which delivers the same flow rate at lower speeds.

An excessive drop in pressure can also be prevented by the use of a so-called inducer. An inducer is an axial impeller which is connected upstream of the actual pump impeller and is mounted on the same shaft. Inducers increase the static pressure of the fluid before it enters the actual impeller (or before the first stage of several impellers) and thus reduce the risk of the pressure falling below the vapor pressure (reduction of the NPSHR value of the pump).

Centrifugal pumps with inducers are being operated in the partial load range, since cavitation would occur in the inducers under full load. Therefore, pumps with inducers cannot cover the entire load range compared to pumps without inducers. Due to the additional impellers, the energy conversion efficiencies are also lower.

Measures to reduce/avoid cavitation can also be taken from the perspective of the piping system. An increase in the cross-section of the suction pipe (at constant flow rate) also causes a reduction in the flow velocity. In this context, the suction pipe should also be checked for blockages.

Another possibility to avoid cavitation is to keep the geodetic suction head as low as possible. For this purpose either the suction tank must be placed higher or the pump lower. It may also be necessary to consider a changeover from suction operation to pressure operation.

Start-up and shut-down procedure of centrifugal pumps

Centrifugal pumps should not be just switched on and off, as this can cause damage to the pump. Therefore a certain procedure must be followed. First of all, the pump should not be switched on when no fluid is inside the pump and the pump must not run dry during operation. On the one hand, centrifugal pumps usually cannot generate a suction effect when running dry and thus cannot pump any fluid (exception: self priming pumps like side-channel pumps). On the other hand, the flowing liquid also serves to cool the pump. If this cooling is missing, the pump overheats very quickly. In addition, the liquid pressure exerts a certain centering effect on the rotating components within the pump and thus prevents direct contact with adjacent housings. For the reasons mentioned above, centrifugal pumps often have a dry running protection which immediately switches off the pump if there is no liquid.

Before starting the pump, it must therefore be filled with liquid (priming). When priming the pump, it must not be emptied again through the suction pipe. For this reason, a check valve or a foot valve (non-return valve) is usually installed in the suction pipe, which automatically prevents a backflow. A simple block valve or gate valve can also be installed, which must then first be closed manually.

If the pump were switched on directly after filling and with the valve open, the pump would immediately deliver a high flow rate. To set the shaft of the electric motor in rotation and at the same time deliver a high flow rate would require a very high electrical power of the motor. This can lead to overheating. To prevent this from happening, the flow rate must be throttled when the pump is started.

However, the flow rate is not throttled by a block valve in the suction pipe, since a constriction in the suction pipe would increase the flow velocity and thus pose the risk of cavitation. For this reason there is another block valve in the discharge pipe, which is initially closed when the pump is switched on and is only gradually opened so that the flow rate slowly increases to its maximum.

In summary, centrifugal pumps are started up by the following procedure:

1. Close the valve in the suction pipe (prevents running dry of the pump)
2. Fill the pump with fluid (achieving a suction effect and protection against overheating)
3. Close the valve in the discharge pipe
4. Open the valve in the suction pipe completely (avoiding cavitation)
5. Switch on the pump motor
6. Slowly open the valve in the discharge pipe (preventing the motor from overheating)

When the pump is to be shut down, a certain procedure must also be followed. First the valve in the discharge pipe is slowly closed again. A too fast closing should be avoided, because otherwise the liquid behind the valve is tempted to continue flowing due to its inertia and thus a very high negative pressure is created. The negative pressure then pulls the liquid back again and hits the valve with full force. This can destroy both the valve and the pipe.

After closing the valve in the discharge pipe, the pump can now be switched off. The block valve in the suction pipe should then be closed again to prevent the pump from running empty. With non-return valves this happens automatically.

In summary, centrifugal pumps are shut down by the following procedure:

1. Slowly close the valve in the discharge pipe (preventing pressure surges)
2. Switch off the pump motor
3. Close the block valve in the suction pipe (prevents running dry of the pump)

Static head of a pump

Pumps are used to deliver gases or liquids. These fluids are usually pumped from a lower level to a higher level. The pump is located between these levels. The pump creates a negative pressure on the so-called suction side so that the fluid is sucked in. The pump then puts the fluid under high pressure. At the outlet, the fluid is now pushed through the pipe to the higher level. In the following we will only consider incompressible fluids such as liquids.

The difference in height the liquid can overcome depends on the power of the pump, the density of the fluid to be pumped and the volume flow rate. To show this, we consider a vertical pipeline. Through this pipe, a fluid of density ϱ is pumped up from a tank. The water is discharged at the open end of the pipe. The difference in height to be overcome is denoted by H. Friction losses are neglected in the following.

The time period until a liquid parcel has been delivered from the lower tank (liquid level) to the top of the pipe is denoted by t. Within this time t obviously once the total liquid of mass m=V⋅ϱ in the pipe must be lifted by the height H (the volume of the fluid inside the pipe is denoted by V). The amount of energy W_H required for this is given by the following formula:

\begin{align}
&W_\text{H} = m \cdot g \cdot H ~~~\text{where}~~~m =V \cdot \rho \\[5px]
&W_\text{H} = V \cdot \rho \cdot g \cdot H ~~~~~\text{energy of the pump} \\[5px]
\end{align}

This energy is obviously converted within the time t, resulting in the following power P_H that the pump delivers:

\begin{align}
&P_\text{H} = \frac{W_\text{H}}{t} \\[5px]
&P_\text{H} = \frac{V \cdot \rho \cdot g \cdot H}{t} \\[5px]
&P_\text{H} = \underbrace{\frac{ V}{t}}_{\dot V} \cdot \rho \cdot g \cdot H\\[5px]
&\boxed{P_\text{H} = \dot V \cdot \rho \cdot g \cdot H} ~~~~~\text{power of the pump} \\[5px]
\end{align}

When deriving this formula, it was exploited that the quotient of fluid volume V and time t corresponds to the delivered volume flow rate V*.

Conversely, this means: For a given power of the pump P_H and volume flow rate V* to be delivered, the fluid can only overcame a certain height H, depending on the density of the fluid ϱ (at a greater height, the hydrostatic pressure of the fluid column would be greater than the pressure generated by the pump and the fluid could not be pumped any higher):

\begin{align}
&P_\text{H} = \frac{W_\text{H}}{t} \\[5px]
&P_\text{H} = \frac{V \cdot \rho \cdot g \cdot H}{t} \\[5px]
&P_\text{H} = \underbrace{\frac{ V}{t}}_{\dot V} \cdot \rho \cdot g \cdot H\\[5px]
\label{p}
&\boxed{H = \frac{P_\text{H}}{\dot V \cdot \rho \cdot g}} ~~~~~\text{static head of the pump} \\[5px]
\end{align}

This maximum height a pump can deliver is also called static pressure head, static head or pressure head of a pump.

The static head of a pump is the maximum height a pump can deliver in a loss-free case due to the mechanical power transferred to the fluid at a given volume flow rate and density!

Note that the static head of the pump does not take into account friction losses inside the pipe or pressure losses due to valves, bends, fittings, etc. This is because these losses cannot be directly attributed to the pump, but depend on the the piping system for which the pump is to be used. Manufacturers of pumps can therefore not take such losses into account anyway, as they do not know the operating conditions of the pump. For this reason, such pressure losses are taken into account by a so-called head loss of the piping system (more about this later).

Pump efficiency

The power P_H in the above equations refers only to the power that the pump effectively delivers to the fluid, i.e. the power that is actually required to continuously lift the fluid to the height H! This power output is not identical with the power input P_in of the pump (power consumption), i.e. the power that an electric pump, for example, has to draw from the mains.

It is therefore necessary to take into account conversion losses that occur when converting the supplied electrical power into mechanical power for transporting the fluid. Also flow losses in the pump due to turbulence must be taken into account, especially at high volume flow rates. All this is summarized in a pump efficiency η:

\begin{align}
&\boxed{P_\text{H} = P_\text{in} \cdot \eta} \\[5px]
\end{align}

The pump efficiency is not a constant value, but depends on the volume flow rate! The efficiency first increases with increasing flow rate and then decreases again from a maximum point due to turbulences and the associated flow losses. Typical maximum efficiencies for pumps are between 70 % and 90 %.

The figure below shows for a given rotational speed of a centrifugal pump the typical curves for pressure head, pump efficiency and power consumption of the pump as a function of volume flow rate. Note that such curves are only valid for certain rotational speed of the pump.

Static head of a piping system

Elevation head (geodetic head)

The physical difference in height between a lower placed and a higher placed basin is called elevation head or geodetic head H_e. The respective liquid levels serve as reference points for the elevation head, provided that the higher-lying tank is filled from below. If it is filled from above, the reference point is the point at which the liquid flows out of the pipe. For the lower basin, it does not matter how deep the suction pipe is immersed in the liquid. The liquid surface must always be taken as a basis, since the liquid in the suction pipe rises to the outer liquid level by itself anyway without the help of the pump.

The elevation head of a piping system can be further divided into a suction head on the suction side of the pumpe and a discharge head on the discharge side of the pump. Both heads together comprise the elevation head of the piping system.

The elevation head of a piping system is the physical difference in height between a lower and a higher liquid level! It results from the sum of the suction head and discharge head.

In order for liquids to be pumped, the static head of the pump must always be greater than the elevation head of the system. However, this only applies if there are no friction losses or flow losses in the pipe due to installed components such as valves, bends, fittings or measuring instruments. Furthermore, the maximum suction head is limited by physical conditions.

Maximum geodetic suction head

On the suction side, the pump works like a drinking straw. This means that the pump does not suck in liquid at all. Rather, the (ambient) pressure on the surface of the liquid pushes the fluid into the pump. Since the ambient pressure is finite, even when a perfect vacuum is created, it is not possible to overcome any suction head. By neglecting friction and flow losses, the maximum suction lift is determined according to the following formula (see article How does a drinking straw work? for the derivation of this formula).

\begin{align}
\label{hmax}
&\boxed{h_\text{s,max} = \frac{p_0}{\rho \cdot g} } ~~~~~\text{maximum suction head} \\[5px]
\end{align}

In this formula, p₀ denotes the (ambient) pressure on the liquid surface and ϱ denotes the density of the liquid. For pumping water with a density of ϱ = 1000 kg/m³ the maximum suction head at an ambient pressure of 1 bar is thus 10 meters.

However, due to the fact that no pump can create a perfect vacuum and due to the viscosity of the pumped liquid, friction losses are inevitable, the maximum geodetic suction head for water is in practice only 8 metres. Note that when the tank is closed, the ambient pressure can be artificially increased, so that higher suction heads are possible.

Unlike the suction head, the discharge head on the discharge side of the pump is in principle not limited to a maximum value. Depending on the pressure generated on the discharge side, (almost) any discharge head can be achieved.

Friction head (head loss)

As already mentioned, in reality friction and flow losses in the piping system must be taken into account. So in practice, the pump must transfer a greater power to the fluid than in the friction-free case. Taking friction into account, the real system behaves as if a fictitious frictionless system had a higher head. This additional (fictitious) head, which includes friction and flow losses, is called friction head or head loss H_f.

If P_f denotes the power loss that occurs with a certain volumetric flow rate V*, then the friction head H_f can be determined according to equation (\ref{p}):

\begin{align}
&\boxed{H_\text{f} = \frac{P_\text{f}}{\dot V \cdot \rho \cdot g} } ~~~~~\text{friction head (head loss)} \\[5px]
\end{align}

The (fictitious) total static head of the piping system H_tot is therefore greater than the elevation head H_e by the amount of the friction head H_f:

\begin{align}
&\boxed{H_\text{tot} = H_\text{e} + H_\text{f}} ~~~~~\text{total static head of the piping system} \\[5px]
\end{align}

So one has to compare the static head of the pump with the (fictitious) static head of the piping system when it comes to selecting a suitable pump. The static head of the pump must be greater than the total static head of the system in order for the fluid to be conveyed. However, this does not yet take into account the fact that different pressures can exist on the liquid surface in the upper and lower tanks. For this reason, a pressure head must generally also be taken into account, which will be discussed in more detail in the next section.

Even in a horizontal pipe, friction inevitably occur due to the viscosity of the fluid. In this case, the associated head loss can indeed be shown very clearly. One can imagine small vertical tubes attached to the pipe. Due to the static pressure in the flowing liquid, the fluid in the vertical tubes is pressed upwards by a certain amount. Due to the friction losses in the pipe, however, the static pressure downstream decreases (assuming a constant pipe cross-section). The fluid in a downstream tube only reaches a lower height. The difference in the liquid levels corresponds to the head loss of the horizontal pipe.

Pressure head

Imagine the following situation. Water is to be pumped from a closed tank to an open tank 6 meters higher up. In the closed tank there is a positive pressure (overpressure) created by compressor. Even without the presence of a pump, this positive pressure will push the water upwards. At an overpressure of 0,1 bar the water theoretically rises to a height of 1 meter above the liquid level [see formula (\ref{hmax})]. So the pump only has to overcome the last 5 meters of height. From an energy point of view, the system has a total static head of only 5 meters from the pump’s point of view (without taking into account head loss).

Conversely, the total static head of the system increases when the lower tank is open and a positive pressure is generated in the higher tank. In this case the higher pressure in the upper tank pushes the water in the pipe downwards. This means that the pump now has to overcome a much greater difference in height. The effects only balance each other out if the pressures in both tanks are the same (for example ambient pressure in both tanks). Even without external compressors, pressure differences occur in closed tanks, since the air volume in the tank also changes when the liquid level changes, without air being able to escape or flow in.

The increased head or the reduced head due to the pressure difference Δp between the two tanks is called pressure head H_p. The pressure difference is determined by the difference between the pressure in the upper reservoir p₂ and the pressure in the lower reservoir p₁ In this way, the sign is also reproduced correctly, so that in the event of a negative pressure at the upper reservoir (or a positive pressure at the lower reservoir) a negative pressure head results, which reduces the total static head of the system.

\begin{align}
&H_\text{p} = \frac{\Delta p}{\rho \cdot g} \\[5px]
&\boxed{H_\text{p} = \frac{p_\text{2}-p_\text{1}}{\rho \cdot g} } ~~~~~\text{pressure head} \\[5px]
\end{align}

The total head of the system H_tot is thus generally determined from the sum of the elevation head H_e, friction head H_f and pressure head H_p:

\begin{align}
&\boxed{H_\text{tot} = H_\text{e} + H_\text{V} + H_\text{p} } ~~~~~\text{total static head of the piping system} \\[5px]
\end{align}

Head as an energy per unit weight

At this point the head according to equation (\ref{p}) shall be examined more closely and interpreted somewhat differently. For this purpose it will be exploited that power is defined as energy per unit time and volume flow flow ist defined as liquid volume per unit time.

\begin{align}
\require{cancel}
&H = \frac{P_\text{H}}{\dot V \cdot \rho \cdot g} ~~~~~\text{where}~~~P_\text{H}=\frac{W_\text{H}}{t}~~~~~\text{and}~~~\dot V = \frac{V}{t}~~~\text{:}\\[5px]
&H = \frac{\frac{W_\text{H}}{\bcancel{t}}}{\frac{V}{\bcancel{t}} \cdot \rho \cdot g} \\[5px]
&H = \frac{W_\text{H}}{\underbrace{V \cdot \rho}_{m} \cdot g} \\[5px]
&H = \frac{W_\text{H}}{m \cdot g} \\[5px]
&\boxed{H = \frac{W_\text{H}}{F_\text{g}}} \\[5px]
\end{align}

This formula shows that head can be interpreted as an energy per unit weight. Therefore, the following statements apply:

• static head of the pump = Energy of the pump transferred to a fluid element (related to the weight of the fluid element).
• static head of the piping = Energy required to deliver a fluid element (related to the weight of the fluid element).

System characteristic curve

While elevation head and pressure head are constant values of a piping system, the friction head or head loss depends on the volume flow rate. The pressure losses increase with increasing flow rate. In the article Pressure Loss in pipe systems (Darcy friction factor) the following equation has been derived, which describes the pressure loss Δp_f of a pipe with the friction factor f, the inner diameter d and the length L:

\begin{align}
& \boxed{\Delta p_\text{f} = f \cdot \frac{8\rho~L}{\pi^2} \cdot \frac{\dot{V}^2}{d^5}} ~~~\text{pressure loss in a straight pipe section} \\[5px]
\end{align}

This pressure loss is inevitably associated with a loss of mechanical energy and a corresponding head loss. The head loss thus increases (approximately) quadratically with the volume flow rate. “Approximately” because the friction factor is in turn influenced by the volume flow rate. The figure below shows qualitatively the characteristic curve of the total static head of a piping system as a function of the volume flow rate.

While the total head of the system increases with increasing volume flow rate, the static head of the pump decreases due to the increasing flow losses within the pump. During operation of the pump, depending on the volume flow rate, a common operating point (working point) is established which corresponds to the point of intersection between the pump characteristic curve and the system characteristic curve.

The system characteristic curve can be influenced by a throttle valve to control the flow rate. However, it must be noted that a centrifugal pump has a maximum efficiency at a certain volume flow rate. For energy-efficient operation, the working point should be as close as possible to this point of maximum efficiency. However, a change in the system characteristic curve caused by a throttle valve usually has a negative effect on the operating point – lower efficiencies result from the greater flow losses due to the throttling. A change in the rotational speed of the pump to control the volume flow rate can therefore be more sensible at this point.

Pressure loss

In the article about the Darcy friction factor the pressure loss in pipes and the associated formulas have already been discussed in detail. Therefore, in this article the formulas are only briefly summarized. The pressure loss Δp_l along a pipe with the inside diameter d and the length L depends on the fluid density ϱ and the (mean) flow velocity v:

\begin{align}
\label{def}
& \boxed{\Delta p_\text{l} = f\cdot \frac{\rho}{2} \bar v^2 \cdot \frac{L}{ d}} ~~~\text{pressure loss in a straight pipe section} \\[5px]
\end{align}

The influence of the flow type (laminar or turbulent) and roughness of the pipe wall on the pressure loss are taken into account by the so-called Darcy friction factor f (also referred to as resistance coefficient or just friction factor).

The pressure loss can also be expressed as a function of the volume flow rate V*:

\begin{align}
\label{volu}
& \boxed{\Delta p_\text{l} = f \cdot \frac{8\rho~}{\pi^2} \dot{V}^2 \cdot \frac{L}{d^5}} ~~~\text{pressure loss in a straight pipe section} \\[5px]
\end{align}

Friction factor for laminar pipe flows

For laminar flows, the friction factor is not dependent on the roughness of the pipe wall, but is only determined by the Reynolds number Re:

\begin{align}
\label{a}
&\boxed{f_\text{lam}= \dfrac{64}{Re}} ~~~\text{friction factor for laminar flows}\\[5px]
&Re= \frac{v \cdot d \cdot \rho}{\eta}
\end{align}

Friction factor for turbulent pipe flows

In turbulent flows, however, the influence of the roughness of the pipe wall is relatively large. The friction factor increases with increasing roughness and decreases with increasing flow velocity. If the Reynolds numbers are sufficiently high, the friction factor becomes independent of the flow velocity and is thus only dependent on the roughness of the pipe. The friction factor for turbulent flow is given by the Colebrook-White equation (\ref{cw}):

\begin{align}
\label{cw}
&\boxed{\color{red}{\frac{1}{\sqrt{f_\text{tur}}}}=-2\cdot \log_\text{10}\left(\frac{2.51}{Re} \cdot \color{red}{\frac{1}{\sqrt{f_\text{tur}}}} +\frac{\varepsilon}{3.71}\right)} ~~~\text{Colebrook-White equation} \\[5px]
\end{align}

In this equation ε denotes the relative roughness of the pipe wall, i.e. the ratio of surface roughness to pipe diameter.

As a simple approximation for the calculation of the friction factor, the Haaland equation given below can be used. It is also used to determine the starting value for the iterative solution of the Colebrook-White equation. For more information about the iterative solution of the Colebrook-White equation, see article Pressure loss in pipe systems.

\begin{align}
&\boxed{\color{red}{\frac{1}{\sqrt{f_\text{tur,0}}}}=-1.8\cdot \log_\text{10}\left(\frac{6.9}{Re} +\left(\frac{\varepsilon}{3.7}\right)^{1.11}\right)} ~~~\text{Haaland equation} \\[5px]
\end{align}

For hydraulically smooth pipes, where the viscous sublayer (laminar sublayer) completely covers the surface roughness, the relative roughness is zero (ε=0). The Colebrook-White equation for this special case is:

\begin{align}
&\boxed{\color{red}{\frac{1}{\sqrt{f_\text{tur}}}}=-2\cdot \log_\text{10}\left(\frac{2.51}{Re} \cdot \color{red}{\frac{1}{\sqrt{f_\text{tur}}}} \right)} ~~~\text{for hydraulically smooth pipes} \\[5px]
\end{align}

For hydraulically rough pipes, where the surface roughness protrudes completely through the viscous sublayer, the following formula by Nikuradse applies. The Darcy friction factor is no longer dependent on the Reynolds number, but is mainly determined by the wall roughness.

\begin{align}
&\boxed{\color{red}{\frac{1}{\sqrt{f_\text{tur}}}}=-2\cdot \log_\text{10}\left(\frac{\varepsilon}{3.71} \right)} ~~~\text{for hydraulically rough pipes} \\[5px]
&f_\text{tur}=\frac{1}{\sqrt{2\cdot \log_\text{10}\left(\frac{3.71}{\varepsilon} \right)}} \\[5px]
\end{align}

Note that although the friction factor decreases with increasing Reynolds number, this does not mean that the pressure loss would be lower. The pressure loss increases according to equation (\ref{def}) with the square of the flow velocity, so that the pressure loss basically increases with increasing flow velocity.

Moody chart

The Moody diagram shows the friction factor as a function of the Reynolds number for selected roughnesses of the pipe wall. In the transition region between laminar and turbulent flow, however, a reliable indication of the friction factor is not possible.

The blue dashed curve in the diagram marks the area of hydraulically rough pipes for which the friction factor is independent of the Reynolds number. The following inequality applies to the hydraulically rough area:

\begin{align}
&\boxed{\sqrt{f}~Re~\varepsilon>200 } ~~~\text{for hydraulically rough pipes} \\[5px]
\end{align}

Introduction

When fluids flow through pipes, energy losses inevitably occur. On the one hand, this is due to friction that occurs between the pipe wall and the fluid (wall friction). On the other hand, frictional effects also occur within the fluid due to the viscosity of the fluid (internal friction). The faster the fluid flows, the greater the internal friction effect (see also the article on Poiseuille flow).

Further flow losses are caused by turbulences in the fluid, especially at fittings, which serve as obstacles for the flow. Although these turbulences contain kinetic energies, they do not transport them through the pipeline from a macroscopic point of view, but remain in place, so to speak.

In the article Venturi effect it was already shown in detail that pressure can also be understood as volume-specific energy. In this context, pressure indicates how much energy per unit volume is contained in a fluid. Thus, if pressure means energy, then a loss of energy inevitably means a loss of pressure. The friction and flow effects described above are thus accompanied by a corresponding pressure loss (pressure drop).

The pressure loss basically refers to the loss of static pressure (or loss of total pressure). The dynamic and hydrostatic pressure are not affected by the energy losses, as these are only the effect of the flow but not the cause. The hydrostatic pressures and dynamic pressures are predetermined by the geometry of the pipeline. Further pressure losses occur in individual components, e.g. valves, elbows or measuring equipment.

The (static) pressure loss in pipelines is associated with the loss of mechanical energy that inevitably occurs when a fluid flows through a pipe system.

In the following, we only consider incompressible flows such as liquids or slow flowing gases.

Pressure loss in pipes (Darcy friction factor)

Regardless of whether the flow is laminar or turbulent, the pressure loss or pressure drop through a pipeline is described by a dimensionless similarity parameter. This so-called Darcy friction factor f basically describes the relationship between pressure loss Δp_l (energy loss) and the kinetic energy contained in the flow in the form of the dynamic pressure Δp_dyn. In addition, the ratio of inner pipe diameter d and pipe length L must be taken into account.

\begin{align}
& f:= \frac{\Delta p_\text{l}}{p_\text{dyn}} \cdot \frac{d}{L}~~~\text{where}~~~ p_\text{dyn}=\tfrac{1}{2}\rho ~\bar v^2 ~~~\text{:} \\[5px]
\label{lambda}
& \boxed{f= \frac{\Delta p_\text{l}}{\tfrac{1}{2}\rho ~\bar v^2} \cdot \frac{d}{L}} ~~~\text{Darcy friction factor (resistance coefficient)} \\[5px]
\end{align}

In this equation d denotes the inside diameter of the pipe and L the length of the straight pipe section along which the pressure drop is Δp_l. The Darcy friction factor is also referred to as resistance coefficient or just friction factor.

The flow velocity refers to the mean flow velocity of the fluid in the pipe. Note that in both turbulent and laminar flow there is no uniform velocity distribution across the pipe cross section, but a typical velocity profile (see Poiseuille flow).

The Darcy friction factor (resistance coefficient) is a dimensionless similarity parameter to describe the pressure loss in straight pipe sections!

By defining the friction factor as a similarity parameter, the friction factor can be determined in a reduced model scale of the later pipe system. This can then be applied to the real scale and thus the pressure loss in the actual pipeline can be determined:

\begin{align}
\label{def}
& \boxed{\Delta p_\text{l} = f \cdot \frac{1}{2}\rho ~\bar v^2 \cdot \frac{L}{ d}} ~~~\text{pressure loss in a straight pipe section} \\[5px]
\end{align}

The friction factor can also be calculated mathematically based on the geometry of the pipe, as will be shown later.

Note that this formula only applies to straight pipe sections. In pipe elbows, further losses usually occur due to the redirection of the flow, which leads to pressure losses. These component-dependent pressure losses (individual resistances) are taken into account separately by a minor loss coefficient ζ. More about this later.

In practice, often not a certain flow velocity is required, but a certain volumetric flow rate. The mean flow velocity v is linked to the volume flow rate V* by the cross-sectional area of the pipe A:

\begin{align}
&\boxed{\dot V =\bar v \cdot A} ~~~\text{where}~A=\frac{\pi}{4}d^2~~~\text{:} \\[5px]
&\dot V = \bar v \cdot \frac{\pi}{4}~d^2 \\[5px]
\label{vol}
&\underline{\bar v = \frac{4\dot V}{\pi ~d^2}} \\[5px]
\end{align}

If equation (\ref{vol}) is used in equation (\ref{def}), then the following formula for the pressure loss at a given volume flow rate is finally obtained:

\begin{align}
& \Delta p_\text{l} = f \cdot \frac{1}{2}\rho ~\left( \frac{4\dot V}{\pi ~d^2}\right)^2 \cdot \frac{L}{ d}\\[5px]
\label{volu}
& \boxed{\Delta p_\text{l} = f \cdot \frac{8\rho~L}{\pi^2} \cdot \frac{\dot{V}^2}{d^5}} ~~~\text{pressure loss in a straight pipe section} \\[5px]
\end{align}

The diameter obviously influences the pressure loss to the fifth power and thus has a decisive influence. In general, the larger the diameter, the lower the pressure loss! Note that the friction factor f is, however, dependent on the flow velocity. The flow velocity in turn depends on the volumetric flow rate and thus on the pipe diameter! So, these variables generally influence each other.

Pressure loss for laminar flow

The pressure loss resulting from the viscosity due to the internal friction of the fluid (“toughness”) has already been derived in detail for a laminar flow in the article on the Hagen-Poiseuille equation:

\begin{align}
\label{lam}
& \boxed{\Delta p_\text{l,lam} = \frac{32~\eta ~ L}{d^2} \cdot \bar v} ~~~\text{pressure loss due to viscosity} \\[5px]
\label{lam2}
& \boxed{\Delta p_\text{l,lam} = \frac{128~\eta ~ L}{\pi d^4} \cdot \dot V} \\[5px]
\end{align}

In this equation Δp_l,lam denotes the pressure drop along a pipe section with the inside diameter d and the length L when a fluid with the dynamic viscosity η flows laminar through the pipe with the mean velocity v or the volume flow rate V*.

The pressure loss that inevitably occurs due to the ever-present viscosity of fluids must be compensated in any case if a fluid is to be pumped through a pipe. The pressure loss therefore corresponds to the pressure that a pump must generate in any case to keep the fluid flowing.

The Darcy friction factor f_lam for a laminar pipe flow can be determined by using formula (\ref{lam}) in equation (\ref{lambda}):

\begin{align}
\require{cancel}
f_\text{lam} &= \frac{\Delta p_\text{l,lam}}{\tfrac{1}{2}\rho ~\bar v^2} \cdot \frac{d}{L}\\[5px]
&= \frac{\frac{32~\eta ~ L}{d^2} \cdot \bar v}{\tfrac{1}{2}\rho ~\bar v^2} \cdot \frac{d}{L}\\[5px]
&= \frac{2 \cdot 32~\eta ~ \cancel{L}~\cancel{\bar v}}{d^{\cancel{2}}~\rho ~\bar v^{\cancel{2}}} \cdot \frac{\cancel{d}}{\cancel{L}}\\[5px]
&= \frac{64~\eta}{d~\rho ~\bar v}\\[5px]
&= \dfrac{64}{\color{red}{\dfrac{d \rho ~\bar v}{\eta}}}~~~\text{mit}~~~\color{red}{Re=\frac{d~\rho~\bar v}{\eta}}\\[5px]
\end{align}

\begin{align}
\label{a}
&\boxed{f_\text{lam}= \dfrac{64}{Re}} ~~~\text{Darcy friction factor for laminar flow}\\[5px]
\end{align}

With laminar flow, the friction factor depends only on the Reynolds number. The higher the Reynolds number, the lower the friction factor!

Note that although the friction factor decreases with increasing flow velocity (increasing Reynolds number), this does not mean that the pressure loss is reduced. According to equation (\ref{def}), the pressure loss increases with the second power of the flow velocity. The pressure loss thus increases proportionally with the flow velocity – see equation (\ref{lam})!

It has already been said that friction is not only present within the fluid itself, but that friction effects also occur in general between the fluid and the pipe wall. However, since the fluid adheres to the wall anyway (no-slip condition) and the laminar layers cover the roughness of the wall, this has no additional influence on the pressure loss. The total pressure loss is therefore given only by equation (\ref{a}) for a laminar flow. The situation is different for turbulent flows, which will be discussed in more detail in the next section.

Pressure loss for turbulent flow

Turbulence in a flow means a lot of vortices. These contain kinetic energy, but this energy is not really transported. Downstream, this energy does not find its way, so to speak, and is therefore lost in a technically sense. In addition to the pressure loss due to internal friction caused by the viscosity of the fluid, there is therefore an additional pressure loss due to the turbulence. The pressure loss is therefore greater in turbulent flow than in laminar flow.

Viscous sublayer

In turbulent flows, the roughness of the pipe wall has a great influence on the friction factor. For this purpose we take a closer look at the situation of the fluid on the rough pipe wall. First of all, even in turbulent flows, the fluid particles located directly on the wall adhere to it due to the no-slip condition. However, no turbulence can form in the immediate vicinity of the wall, as cross-flows are prevented by the wall (the fluid cannot flow through the wall). For this reason, a so-called laminar sublayer, also referred to as viscous sublayer, forms directly at the wall.

Depending on how thick this viscous sublayer is and how large the roughness is, the sublayer covers the roughness of the wall more or less. If the roughnesses are too large, then they influence the flow very strongly and lead to increased turbulence. These in turn cause a relatively large pressure loss. If, on the other hand, the surface roughness protruding from the viscous sublayer is relatively small, then the turbulence and thus the pressure loss is lower. If, on the other hand, the surface roughnesses are completely covered by the viscous sublayer, then the pressure loss due to the turbulence in the flow is lowest. In this case one also speaks of a hydraulically smooth pipe.

A pipe is considered hydraulically smooth when the viscous sublayer completely covers the surface roughness. The pressure loss is lowest in this case!

Relative roughness

The roughness of a surface is indicated by a roughness parameter k (also denoted by R_z). This roughness parameter describes the height between the lowest and highest points of a rough surface, averaged over several sections.

However, this roughness parameter as an absolute measure of the roughness of the pipe wall is not suitable for characterizing the influence on the turbulent flow. The roughness must always be considered in relation to the entire flow cross-section, i.e. the inner diameter of the pipe. The ratio of absolute roughness k and pipe diameter d is also called relative roughness ε:

\begin{align}
\label{e}
&\boxed{\varepsilon= \frac{k}{d}} ~~~\text{relative roughness}\\[5px]
\end{align}

The relative roughness indicates the percentage of the total pipe diameter taken up by the roughness.

Implicit Colebrook-White equation

The scientists Colebrook and White derived the following implicit function for determining the Darcy friction factor f_tur for turbulent pipe flows using empirical results:

\begin{align}
\label{cw}
&\boxed{\color{red}{\frac{1}{\sqrt{f_\text{tur}}}}=-2\cdot \log_\text{10}\left(\frac{2.51}{Re} \cdot \color{red}{\frac{1}{\sqrt{f_\text{tur}}}} +\frac{\varepsilon}{3.71}\right)} ~~~\text{Colebrook-White equation} \\[5px]
\end{align}

The term “implicit” means that this equation cannot be solved directly for the friction factor. Rather, with a given Reynolds number Re of the flow and a given relative roughness ε of the pipe wall, it is necessary to find a friction factor which then satisfies this equation. If this is the case, the friction factor found corresponds to the value searched for. In the next section Explicit Haaland equation, an iterative solution of this equation is described in more detail. With a so-called Moody chart, the friction factors can also be determined graphically.

For hydraulically smooth pipes, the viscous sublayer covers the roughness of the wall. In this case, the relative roughness ε in the Colebrook-White equation must be set to zero, regardless of the value actually obtained:

\begin{align}
&\boxed{\color{red}{\frac{1}{\sqrt{f_\text{tur}}}}=-2\cdot \log_\text{10}\left(\frac{2.51}{Re} \cdot \color{red}{\frac{1}{\sqrt{f_\text{tur}}}} \right)} ~~~\text{for hydraulically smooth pipes} \\[5px]
\end{align}

If the surface roughness of the pipe wall protrude completely from the viscous sublayer, the friction factor is determined almost exclusively by the wall roughness and is independent of the Reynolds number. The scientist Nikuradze derived the following explicit equation for such hydraulically rough pipes to determine the pipe friction coefficient:

\begin{align}
&\boxed{\color{red}{\frac{1}{\sqrt{f_\text{tur}}}}=-2\cdot \log_\text{10}\left(\frac{\varepsilon}{3.71} \right)} ~~~\text{for hydraulically rough pipes} \\[5px]
&f_\text{tur}=\frac{1}{\left[2\cdot \log_\text{10}\left(\frac{3.71}{\varepsilon} \right)\right]^2} \\[5px]
\end{align}

Explicit Haaland equation

There is a reason why the Colebrook-White equation (\ref{cw}) is given in this somewhat strange form. This allows an iterative procedure, so that the friction factor can be determined starting from a starting value 1/√f_tur,0. The starting value can be determined by the explicit equation proposed by Haaland:

\begin{align}
&\boxed{\color{red}{\frac{1}{\sqrt{f_\text{tur,0}}}}=-1.8\cdot \log_\text{10}\left(\frac{6.9}{Re} +\left(\frac{\varepsilon}{3,7}\right)^{1.11}\right)} ~~~\text{Haaland equation} \\[5px]
\end{align}

The value 1/√f_tur,0 explicitly determined with the Haaland equation can now be used in the right side of the Colebrook-White equation. This results in a new value 1/√f_tur,1 according to the left side of the equation. This value can then again be used in the right side of the equation. The value 1/√f_tur,2 obtained after two passes usually corresponds with sufficient accuracy to the value 1/√f_tur searched for. Finally, the pipe friction coefficient f_tur can be determined.

Alternatively to the Haaland equation, a value of 1/√f_tur,0=7.5} can be used as a starting value. This corresponds to the result of the Haaland equation for a hydraulically smooth pipe (ε=0) and a Reynolds number of 10⁵.

Power loss

Any pressure loss in a pipeline must be compensated by the power of an appropriate pump. The power loss P_l due to the pressure loss Δp_l depends on the volume flow rate V*:

\begin{align}
\label{v}
&P_\text{l} = \Delta p_\text{l} \cdot \dot V \\[5px]
\end{align}

If the pressure drop Δp_l according to equation (\ref{volu}) is put in equation (\ref{v}), then the following formula results:

\begin{align}
\label{dr}
&\boxed{P_\text{l} = f \cdot \frac{8\rho~L}{\pi^2} \cdot \frac{\dot{V}^3}{d^5}} ~~~\text{applies in general}\\[5px]
\end{align}

At this point it must again be noted that the friction factor f is dependent on the Reynolds number and thus on the flow velocity. The flow velocity, in turn, depends on the volume flow rate and the pipe diameter!

Only for laminar flows, there is an explicit relationship between the Reynolds number and the friction factor according to equation (\ref{a}). In this case, the pressure loss for laminar flows can be directly put into the formula for the power loss (\ref{v}) according to equation (\ref{lam2}). In the case of laminar pipe flows, the following relationship then applies:

\begin{align}
&P_\text{l,lam} = \Delta p_\text{l,lam} \cdot \dot V \\[5px]
&\boxed{P_\text{l,lam} = \frac{128~\eta ~ L}{\pi} \cdot \frac{\dot{V}^2}{d^4}} ~~~\text{only applies to laminar flows}\\[5px]
\end{align}

Pressure loss through individual components (minor loss coefficient)

A pipe system usually does not consist of a single straight pipe. A pipe system usually consists of several elbows, branches, reducers, valves, etc. These individual components also cause energy losses and thus pressure losses.

These pressure losses are each described by a minor loss coefficient ζ. The minor loss coefficient is defined analogously to the Darcy friction factor, i.e. as the ratio between pressure loss Δp_l in the component and the dynamic pressure of the flow p_dyn:

\begin{align}
& \zeta:= \frac{\Delta p_\text{l}}{p_\text{dyn}} ~~~\text{where}~~~ p_\text{dyn}=\tfrac{1}{2}\rho ~\bar v^2 ~~~\text{:} \\[5px]
\label{zeta}
& \boxed{\zeta= \frac{\Delta p_\text{l}}{\tfrac{1}{2}\rho ~\bar v^2} } ~~~\text{minor loss coefficient} \\[5px]
\end{align}

The meaning of the minor loss coefficient for objects through which a fluid flows is ultimately identical to the drag coefficient for bodies around which flow passes.

The minor loss coefficient is a dimensionless similarity parameter to describe the pressure loss in individual components (elbows, valves, reducers, etc.)!

The minor loss coefficients for the various components are usually determined experimentally and are given in table books. If the minor loss coefficient is known, the pressure loss through the component can then be determined as follows:

\begin{align}
\label{dez}
& \boxed{\Delta p_\text{l} = \zeta \cdot \frac{1}{2}\rho ~\bar v^2 } ~~~\text{pressure loss in individual components} \\[5px]
\end{align}

The flow velocity basically refers to the velocity of the fluid prior to the actual component and not to the flow velocity inside the component! A valve, for example, reduces the flow cross-section and thus increases the flow speed in the component. However, the flow velocity to be taken as a basis for the pressure loss refers to the flow velocity in the pipeline!

Eventually a minor loss coefficient can even be defined for a straight pipe section. In this way the straight pipe sections can also be considered as single component. In this case the minor loss coefficient is related to the Darcy friction factor f and the length of the pipe section L and the internal pipe diameter d as follows:

\begin{align}
&\boxed{\zeta_\text{p} = \frac{L}{d} f}~~~\text{minor loss coefficient of a straight pipe section} \\[5px]
\end{align}

Conversely, a so-called equivalent pipe length L_e can be given for individual components. These components can then be imagined as additional pipe sections. In the equation given below, the Darcy friction factor f corresponds to the friction factor of the actual pipes.

\begin{align}
&\boxed{L_\text{e}= \frac{d \cdot \zeta}{f}}~~~\text{equivalent pipe length of components} \\[5px]
\end{align}

With a pipe diameter of d = 1 cm, a minor loss coefficient of ζ=1 and a friction factor of f = 0.02, an equivalent pipe length of only 0.5 m is obtained. With very long piping systems and only few individual components (which is often the case), the pressure loss due to the installed components can therefore usually be neglected. For this reason, the resistance coefficient for individual components is called the minor loss coefficient. But note: In some cases the minor loss coefficient can have an enormous effect, especially for short piping systems!

In general, the following therefore applies: The sum of the pressure losses Δp_l.p through the individual straight pipe sections, plus the sum of the pressure losses Δp_l,c through the individual components, gives the total pressure loss Δp_l,total of the entire pipe system:

\begin{align}
&\boxed{\Delta p_\text{l,total} = \sum \Delta p_\text{l,p} +\sum \Delta p_\text{l,c} } \\[5px]
\end{align}

For only one pipeline of total length L with constant inside diameter d and thus constant mean flow velocity v the following formula applies:

\begin{align}
&\boxed{\Delta p_\text{l,total} = \left(f \frac{L}{d} + \sum \zeta \right) \frac{1}{2}\rho~\bar{v}^2} \\[5px]
\end{align}