Incompressible Flow

The equation simplifies further when we consider an incompressible flow where density $\rho$ is a constant. Consequently we have,

$$~~\rho~~{\partial \over \partial t} \int_{CV} d ~ \forall~~+~~\rho~~ \int_{CS}~ \overrightarrow{V}.d\overrightarrow{A}~=~0$$ (3.33)

Dividing by density, $\rho$,

$$~~{\partial \over \partial t} \int_{CV} d ~ \forall~~+~~ \int_{CS}~ \overrightarrow{V}.d\overrightarrow{A}~=~0$$ (3.34)

The first term is the rate of change of volume within a control volume, which for a fixed control volume is zero by definition. This gives a simple form of the equation for the conservation of mass for the control volume as

$$\int_{CS}~ \overrightarrow{V}.d\overrightarrow{A}~=~0$$ (3.35)

Thus for an incompressible flow the continuity equation is the same irrespective of whether the flow is steady or unsteady.

(c) Aerospace, Mechanical & Mechatronic Engg. 2005
University of Sydney