Mass Governing Equations

Derivation

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.

Compressible case

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 $

The mass-average velocity is typically denoted $ \boldsymbol{v} $,<ref name="Taylor">Taylor, R.; Krishna, R. (1993). Multicomponent Mass Transfer. Wiley and Sons. </ref> and this notation yields the more familiar continuity equation:

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

Incompressible case

In the case of incompressible flow, the density is constant, so the temporal derivative is zero. Further, the density term can be factored out of the divergence, which leaves the incompressible mass governing equation:

$ \nabla \cdot \boldsymbol{v} = 0 $

What velocity is $ \boldsymbol{v} $? Mass-average velocity

Could we define our velocity to be different? Would this equation still hold? (Why did we choose the mass-average velocity?

A material volume moves at a velocity that prevents the flux of the property (in this case, mass)

References

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