Mass Governing Equations

The governing equation for total mass in a fluid flow can be obtained from Reynolds Transport Theorem, by setting the extensive property $ B $ equal to total mass, $ B = m $. The corresponding intensive property $ b = 1 $.

The material volume corresponding to the total mass is $ V_{mass}^{M}(t) $, whose boundary moves in such a way that the mass flux across the boundary is zero. The surface area of this material volume is $ S_{mass}^{M} $.

The material volume moves at the velocity $ \boldsymbol{v}_{mass} $, the average velocity of the mass of fluid.

The conservation of mass stipulates that

$ \frac{dB}{dt} = 0 $

and therefore Reynolds Transport Theorem becomes

$ \iiint_{V_{mass}^{M}} \frac{\partial \rho}{\partial t} dV + \iint_{S_{mass}^{M}} \rho \boldsymbol{v}_{mass} \cdot \boldsymbol{n} dS = 0 $

This can be rewritten using Gauss's Theorem:

$ \iiint_{V_{mass}^{M}} \frac{\partial \rho}{\partial t} + \nabla \cdot \left( \rho \boldsymbol{v}_{mass} \right) dV = 0 $

This equation can be used in the finite volume formulation. Alternatively, the integral volume can be dropped to yield the differential form, applicable for finite difference:

$ \frac{ \partial \rho }{ \partial t } + \nabla \cdot \left( \rho \boldsymbol{v}_{mass} \right) = 0 $