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4 Material Balance Difference Equations in Two and Three Dimensions

The derivation of material balance difference equations in more than one space dimension very closely parallels the derivation for the one space variable case of the previous section. For the case of two space variables, we assume that some material M has a spatial density given by . Again, for simplicity, we begin by assuming that density is related to a field variable by a linear constitutive equation of the form

We assume the flow of material M to be down-gradient with respect to u. That is, the flux is given by

or