Equations of Incompressible Fluid Flow

In most situations of general interest, the flow of a conventional liquid, such as water, is incompressible to a high degree of accuracy. A fluid is said to be incompressible when the mass density of a co-moving volume element does not change appreciably as the element moves through regions of varying pressure. In other words, for an incompressible fluid, the rate of change of $\rho$ following the motion is zero: that is,

$$\frac{D\rho}{Dt} = 0.$$ (1.76)

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

$$\nabla\cdot{\bf v} = 0.$$ (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, $\rho$ , 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,

$${\bf F} = - \rho\,\nabla{\mit\Psi},$$ (1.78)

where ${\mit\Psi}({\bf r}, t)$ is the potential energy per unit mass, and $\rho\,{\mit\Psi}$ 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

$$\frac{D{\bf v}}{Dt}= - \frac{\nabla p}{\rho}-\nabla{\mit\Psi} + \nu\,\nabla^{\,2} {\bf v},$$ (1.79)

where

$$\nu = \frac{\mu}{\rho}$$ (1.80)

is termed the kinematic viscosity, and has units of meters squared per second. Roughly speaking, momentum diffuses a distance of order $\sqrt{\nu\,t}$ meters in $t$ seconds as a consequence of viscosity. The kinematic viscosity of water at $20^\circ\,{\rm C}$ is about $1.0\times 10^{-6}\,{\rm m^2/s}$ (Batchelor 2000). It follows that viscous momentum diffusion in water is a relatively slow process.

The complete set of equations governing incompressible flow is

$$\nabla\cdot{\bf v} = 0,$$ (1.81)

$$\frac{D{\bf v}}{Dt} = - \frac{\nabla p}{\rho} -\nabla{\mit\Psi} + \nu\,\nabla^{\,2} {\bf v}.$$ (1.82)

Here, $\rho$ and $\nu$ are regarded as known constants, and ${\mit\Psi}({\bf r}, t)$ 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, $p({\bf r},t)$ , plus the three components of the velocity, ${\bf v}({\bf r},t)$ . Note that an energy conservation equation is redundant in the case of incompressible fluid flow.