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. Fluid mechanics or fluid dynamics is the study of the macroscopic physical behaviour of fluids . ...

Partial differential equations for incompressible fluids are as follows: In mathematics, and in particular analysis, a partial differential equation (PDE) is an equation involving partial derivatives of an unknown function. ...

The last three equations imply that the gradient of the density of an incompressible fluid is zero: In vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of change of the scalar field, and whose magnitude is the greatest rate of change. ...

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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. Note that all the examples given below express the same idea (i. ... In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ...

Proof

The continuity equation is

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An identity of vector calculus states that Vector calculus is a field of mathematics concerned with multivariate real analysis of vectors in 2 or more dimensions. ...

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

Q.E.D. Q. E. D. is an abbreviation of the Latin phrase quod erat demonstrandum (literally, that which was to be demonstrated). This is a translation of the Greek oper edei deixai which was used by many early mathematicians including Euclid and Archimedes. ...

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). This article is in need of attention. ... The title given to this article is incorrect due to technical limitations. ...

Otherwise, if an incompressible fluid also has a curl of zero, so that it is also irrotational, then the velocity field is actually Laplacian. In fluid mechanics, an irrotational vector field is a vector field whose curl is zero. ... In vector calculus, a Laplacian vector field is a vector field which is both irrotational and incompressible. ...