Incompressible fluid

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In fluid mechanics, an incompressible fluid is a fluid whose density (often represented by the Greek letter ρ) is constant: it is the same throughout the field and it does not change through time. It is an idealization used to simplify analysis. In reality, all fluids are compressible to some extent.

Partial differential equations for incompressible fluids are as follows:

<math> {\partial \rho \over \partial t} = 0, <math>

<math> {\partial \rho \over \partial x} = 0, <math>

<math> {\partial \rho \over \partial y} = 0, <math>

<math> {\partial \rho \over \partial z} = 0. <math>

The last three equations imply that the gradient of the density of an incompressible fluid is zero:

<math> \nabla \rho = 0 <math>.

The continuity equation can be applied to obtain another criterion for an incompressible fluid: the divergence of the velocity field v of an incompressible fluid is zero.

Proof

The continuity equation is

<math> {\partial \rho \over \partial t} = - \nabla \cdot ( \rho \mathbf{v} ) \qquad \qquad (1) <math>.

An identity of vector calculus states that

<math> \nabla \cdot ( \rho \mathbf{v} ) = \rho \nabla \cdot \mathbf{v} + \mathbf{v} \cdot \nabla \rho. \qquad \qquad (2) <math>

But the gradient of the density of an incompressible fluid is zero, therefore (combining equations (1) and (2)):

<math> {\partial \rho \over \partial t} = - \rho \nabla \cdot \mathbf{v}, <math>

which is equivalent to

<math> {1 \over \rho} {\partial \rho \over \partial t} = - \nabla \cdot \mathbf{v} . \qquad \qquad (3) <math>

Then, since the partial derivative of density with respect to time is zero (for an incompressible fluid), equation (3) becomes

<math> \nabla \cdot \mathbf{v} = 0. <math>

Q.E.D.

Relation to Solenoidal Field

An incompressible fluid 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. a rotational component).

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