Читать книгу Vibroacoustic Simulation - Alexander Peiffer - Страница 56

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

(2.7)

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

(2.8)

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