Condition on Velocity for Incompressible Flow
Consulting our chapter road map in Figure 3.4, we have completed the left branch dealing with Bernoulli’s equation. We now begin a more general consideration of incompressible flow, given by the center branch in Figure 3.4. However, before introducing Laplace’s equation, it is important to establish a basic condition on velocity in an incompressible flow, as follows.
First, consider the physical definition of incompressible flow, namely, p = constant. Since p is the mass per unit volume and p is constant, then a fluid element of fixed mass moving through an incompressible flow field must also have a fixed, constant volume. Recall Equation (2.32), which shows that V • V is physically the time rate of change of the volume of a moving fluid element per unit volume. However, for an incompressible flow, we have just stated that the volume of a fluid element is constant [e. g., in Equation (2.32), D(SV)/Dt = 0]. Therefore, for an incompressible flow,
The fact that the divergence of velocity is zero for an incompressible flow can also be shown directly from the continuity equation, Equation (2.52):
For incompressible flow, p = constant. Hence, dp/dt = 0 and V • (pV) = pV ■ V.
Equation (2.52) then becomes
O + pV • V = 0
or V • V = 0
which is precisely Equation (3.39).