Equations of Incompressible Fluid Flow

In this case, the continuity equation (1.40) reduces to

(1.77) |

We conclude that, as a consequence of mass conservation, an incompressible fluid must have a divergence-free, or solenoidal, velocity field. This immediately implies, from Equation (1.42), that the volume of a co-moving fluid element is a constant of the motion. In most practical situations, the initial density distribution in an incompressible fluid is uniform in space. Hence, it follows from Equation (1.76) that the density distribution remains uniform in space and constant in time. In other words, we can generally treat the density, , as a uniform constant in incompressible fluid flow problems.

Suppose that the volume force acting on the fluid is conservative in nature (see Section A.18): that is,

(1.78) |

where is the potential energy per unit mass, and the potential energy per unit volume. Assuming that the fluid viscosity is a spatially uniform quantity, which is generally the case (unless there are strong temperature variations within the fluid), the Navier-Stokes equation for an incompressible fluid reduces to

where

(1.80) |

is termed the

The complete set of equations governing incompressible flow is

Here, and are regarded as known constants, and as a known function. Thus, we have four equations--namely, Equation (1.81), plus the three components of Equation (1.82)--for four unknowns--namely, the pressure, , plus the three components of the velocity, . Note that an energy conservation equation is redundant in the case of incompressible fluid flow.