PDE conversion

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PDE conversion

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

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.

Scalar variable:
Vector variable:

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