The Divergence of a Vector Field
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Continuity equation of a compressible fluid flow
Fluids comprise liquids and gases. Liquids are almost incompressible, and may be treated as incompressible; gases are compressible, which means that the density of a gas, v, is a function of position , [Equation goes here - download the original pdf to see it.] , and may also be a function of time, t.. Consider a compressible fluid (gas) confined to a region, R, where there are no sources or sinks - meaning that fluid is neither introduced into this region nor escapes from it The velocity of the fluid at a point v may be described by a vector field [Equation goes here - download the original pdf to see it.] Let [Equation goes here - download the original pdf to see it.] be a rectangular subspace of R of dimensions [Equation goes here - download the original pdf to see it.] where the orientation of [Equation goes here - download the original pdf to see it.] is i, j, k respectively. Let u be the mass flow per second at the point [Equation goes here - download the original pdf to see it.] ; that is [Equation goes here - download the original pdf to see it.] We will assume that v and u are continuously differentiable vector functions. The flux across a boundary is the total mass of fluid flowing across it per second. The flux across the rectangular face of [Equation goes here - download the original pdf to see it.] given by [symbol] with the point [Equation goes here - download the original pdf to see it.] at one corner during the time interval [symbol] is approximated by [Equation goes here - download the original pdf to see it.] Here the subscript y serves to distinguish this face of the rectangle from the one opposite it, containing the point [Equation goes here - download the original pdf to see it.] The flux across the opposite rectangle is [Equation goes here - download the original pdf to see it.] The difference is [Equation goes here - download the original pdf to see it.] This is the approximate loss or gain of mass across these two sides. Similarly, the flux across the other two parallel faces is [Equation goes here - download the original pdf to see it.] So the total flux loss or gain is [Equation goes here - download the original pdf to see it.] This is equal to the change in density per unit time; i.e. [Equation goes here - download the original pdf to see it.] Then [Equation goes here - download the original pdf to see it.] If we take the limit [equation] then this becomes [Equation goes here - download the original pdf to see it.] or [Equation goes here - download the original pdf to see it.] or [Equation goes here - download the original pdf to see it.] This is called the continuity equation of a compressible fluid flow and provides the condition for the conservation of mass in a fluid. If the fluid is steady, then it does not depend on time; whence [Equation goes here - download the original pdf to see it.] and [Equation goes here - download the original pdf to see it.] If the fluid is incompressible (a liquid) then p is constant, and [Equation goes here - download the original pdf to see it.] This condition is known as the condition of incompressibility. Thus, the physical meaning of divergence is expressed by this condition of incompressibility.
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Contents of
The Divergence of a Vector Field
1 The divergence of a vector field
2 Invariance of divergence
3 The divergence and the Laplacian
4 Continuity equation of a compressible fluid flow
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