Incompressible flow

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In fluid mechanics, an incompressible flow is a fluid flow in which the divergence of velocity is zero. It is an idealization used to simplify analysis. In reality, all fluids are compressible to some extent. Note that this is a property of the flow and not the fluid. However, by making this assumption, the governing equations of fluid flow can be simplified to a good degree.

The Partial differential equation for incompressible flows is:

${\nabla \cdot \mathbf{u} = 0}$ .

From the equation of continuity

${\partial \rho \over \partial t} + \nabla \cdot (\rho \mathbf{u}) = 0$ ,

i.e.

${\frac{D\rho}{Dt}} + \rho\nabla\cdot \mathbf{u} = 0$ ,

and the fact that

ρ > 0

we see that a fluid is incompressible if and only if

${\frac{D\rho}{Dt}} = 0$ ,

that is, the mass density is constant following the fluid.

[edit] Relation to compressibility factor

In some fields, a measure of the incompressibility of a flow is the change in density as a result of the pressure variations. This is best expressed in terms of the Compressibility_factor Z.

$Z = {\frac{1}{\rho}} {\frac{d\rho}{dp}}$

If the compressibility factor is acceptably small, the flow is considered to be incompressible.

[edit] Relation to solenoidal field

An incompressible flow is described by a velocity field which is solenoidal. But a solenoidal field, besides having a zero divergence, also has the additional connotation of having non-zero curl (i.e., rotational component).

Otherwise, if an incompressible flow also has a curl of zero, so that it is also irrotational, then the velocity field is actually Laplacian.