3.2.4.1. Conservation of Mass

For a material with density $\density$ , letting $b = \density$ results in the conservation of mass. Since there is no net flow relative to the mass average velocity $\vector{f} = \vector{0}$ . Although there are no sources of mass, having such a source can be convenient in modeling and simulation; so, we let the mass source be $B_\Vol = q_m$ . Thus, equation (3.1) becomes

(3.6) $\pt{\density} + \density\div\v + \v\bcdot\grad\density = q_m.$

For the special but common case of constant density, this reduces to

(3.7) $\div \v = 0.$

Using equation (3.3), the G/FEM residual form is

(3.8) $\symRes_P^i = \int\limits_\Vol \left( -\pt{\density} - \density\div\v - \v\bcdot\grad\density + q_m \right) \phi^i\dV = 0.$

Attention

Equation (3.8) has been multiplied by -1 because this form results in a better linear system for the special case of incompressible flow. This is important to remember when defining mass source terms.

In Aria, each term in (3.8) is specified separately as identified in equation (3.9).

(3.9) $\symRes_P^i = \underbrace{\int\limits_\Vol -\pt{\density}\phi^i\dV}_\mathrm{MASS} + \underbrace{\int\limits_\Vol -\left(\v\bcdot\grad\density + \density\div\v\right)\phi^i\dV}_\mathrm{ADV} + \underbrace{\int\limits_\Vol q_m\phi^i\dV}_\mathrm{SRC} = 0$

For a purely incompressible form, Aria offers the alternative form given in (3.10);

(3.10) $\symRes_P^i = \underbrace{\int\limits_\Vol -\div\v\phi^i\dV}_\mathrm{DIV} + \underbrace{\int\limits_\Vol q_m\phi^i\dV}_\mathrm{SRC} = 0$