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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:
The last three equations imply that the gradient of the density of an incompressible fluid is zero:
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.
The continuity equation is
An identity of vector calculus states that
But the gradient of the density of an incompressible fluid is zero, therefore (combining equations (1) and (2)):
which is equivalent to
Then, since the partial derivative of density with respect to time is zero (for an incompressible fluid), equation (3) becomes
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).
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