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PDE conversion

We want to convert the integral mass conservation equation for a fixed control volume,

or

into a PDE that is pointwise valid. To do so, we need to get rid of the surface integral.

The divergence theorem takes surface integrals to volume integrals and vice-versa.

Apply this to mass conservation:

Conservation form of the continuity equation:
(3)
(Pure derivatives, simply put the integrals back in to go to the conservation equations.)

Index notation (with Einstein summation convention):
(4)

Nonconservative form:
(5)

Material fluxion of density:
(6)

For incompressible fluids:
(7)
The density does not have to be constant from one fluid element to the next (eg, oil and water, varying salinity, ...)


Next: Example Up: Continuity Previous: FCV conversion
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