A flow is said to be incompressible if the density of a fluid element does not change during its motion. It is a property of the flow and not of the fluid. The rate of change of density of a material fluid element is given by the material derivative
From the continuity equation we have
Hence the flow is incompressible if the divergence of the velocity field is identically zero. Note that the density field need not be uniform in an incompressible flow. All that is required is that the density of a fluid element should not change in time as it moves through space. For example, flow in the ocean can be considered to be incompressible even though the density of water is not uniform due to stratification.
The Navier-Stokes equations for incompressible flow are
- Continuity equation
- Momentum equation
- Energy equation
- is the Laplacian operator
- E is the internal energy per unit mass
- is the rate of dissipation of mechanical energy per unit mass
- is the kinematic viscosity
- k is the coefficient of thermal conductivity
- T is the temperature
A consequence of incompressible flow is that there is no equation of state for pressure unlike in compressible flow. Since there is no separate equation for pressure it must be obtained from the continuity and momentum equations. The main role of pressure is to satisfy the zero divergence condition of the velocity field. Note that pressure is only determined upto a constant.
If the viscosity is assumed to be constant then the energy equation is decoupled from the continuity and momentum equations.