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rho v Subscript normal x Baseline upper A right-parenthesis Subscript x plus d x Baseline plus ModifyingAbove m With dot"/> (2.1)

for mass conservation. Expanding the second term on the right hand side in a Taylor series gives

StartFraction partial-differential left-parenthesis rho upper A d x right-parenthesis Over partial-differential t EndFraction equals left-bracket left-parenthesis rho v Subscript normal x Baseline upper A right-parenthesis Subscript x Baseline minus left-parenthesis rho v Subscript normal x Baseline upper A right-parenthesis Subscript x Baseline minus StartFraction partial-differential left-parenthesis rho v Subscript normal x Baseline upper A right-parenthesis Subscript x Baseline Over partial-differential x EndFraction d x right-bracket plus ModifyingAbove m With dot

and finally

StartFraction partial-differential rho Over partial-differential t EndFraction plus StartFraction partial-differential left-parenthesis rho v Subscript normal x Baseline right-parenthesis Over partial-differential x EndFraction equals ModifyingAbove rho With dot Subscript normal s (2.2)

This one dimensional equation of mass conservation in x-direction can be extended to three dimensions:

StartFraction partial-differential rho Over partial-differential t EndFraction plus nabla left-parenthesis rho bold v right-parenthesis equals ModifyingAbove rho With dot Subscript normal s (2.3)

The second term of (2.3) may be confusing, but it says that the change of density is not only determined by a gradient in the velocity field but also by a gradient of the density.

2.2.2 Newton’s law – Conservation of Momentum

The same procedure is applied to the momentum of the fluid. As shown in Figure 2.2 we get for flow in x-direction:

1 The momentum of the control volume is ρvxdV=ρvxAdx.

2 The momentum flow into the volume (ρvx2A)x.

3 mass flow out of the volume (ρvx2A)x+dx.

4 The force at position x is (PA)x.

5 The force at position x+dx is (PA)x+dx.

6 External volume force density fx.

Figure 2.2 Momentum flow in x-direction through control volume. Source: Alexander Peiffer.

Thus, the conservation of momentum in x reads

(2.4)

Using Taylor expansions for (ρvx2A)x+dx and (PA)x+dx gives

(2.5)

Here, fx=Fx/(Adx) is the volume force density (force per volume). Using the chain law the partial derivatives of the first and second term lead to

(2.6)

The term in brackets is the homogeneous continuity Equation (2.1), and Equation (2.6) simplifies to

rho StartFraction partial-differential v Subscript normal x Baseline Over partial-differential t EndFraction plus rho v Subscript normal x Baseline StartFraction partial-differential v Subscript normal x Baseline Over partial-differential x EndFraction plus StartFraction partial-differential upper P Over partial-differential x EndFraction equals f Subscript x (2.7)

As with the conservation of mass, this can be extended to three dimensions:

rho left-brace StartFraction partial-differential bold v Over partial-differential t EndFraction plus left-parenthesis bold v nabla right-parenthesis bold v right-brace plus nabla upper P equals bold f (2.8)

This equation is the non-linear, inviscid momentum equation called the Euler equation.

2.2.3 Equation of State

The above equations relate pressure, velocity and density. For further reducing this set we need a third equation. The easiest way would be to introduce the . Here we start with the first law of thermodynamics in order to show the difference between isotropic (or adiabatic) equation of state and other relationships.

d q equals d u plus upper P d v minus d r (2.9)

With the following specific quantities per unit mass

(2.10)

With the specific entropy ds=dq+drT we get:

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