Incompressible flow
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In fluid mechanics, an incompressible flow is a fluid flow in which changes in the fluid density (often represented by the Greek letter ρ) have little effect on the variables of interest, such as the lift on a wing. Therefore, it is modeled as constant: the same throughout the field and unchanging with 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 flow: the divergence of the velocity field v of an incompressible flow is zero.
Proof
The continuity (conservation of mass) 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 flow 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 flow), equation (3) becomes
- <math> \nabla \cdot \mathbf{v} = 0. </math>
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.
it:flusso incomprimibile