3.2.4.4. Conservation of Fluid Momentum

The Cauchy momentum equation is given by

(3.21) $\density\pt{\v} + \density\v\bcdot\grad\v - \g -\div\T = \vector{0}$

where $\T$ is the fluid stress tensor and $\g$ is a body force. We construct the G/FEM residual form of (3.21) by contracting with the unit coordinate vector in the $k$ -direction, $\e_k$ , multiplying by the weight function $\phi^i$ and integrating over the volume. Using the vector identity $(\div\T)\bcdot\e_k\phi^i = \div(\T\bcdot\e_k\phi^i) - \T^t:\grad(\e_k\phi^i)$ and integrating by parts gives

(3.22) $\symRes_{m,k}^i = \int\limits_\Vol \left[ \left(\density\pt{\v} + \density\v\bcdot\grad\v - \g\right)\bcdot\e_k\phi^i + \T^t:\grad\left(\e_k\phi^i\right)\right]\dV - \int\limits_\Surf \n\bcdot\T\bcdot\e_k\phi^i\dS = 0$

In Aria, each term in (3.22) is specified separately as identified in equation (3.23).

(3.23) $\symRes_{m,k}^i & = \underbrace{\int\limits_\Vol \density\pt{\v}\bcdot\e_k\phi^i\dV}_\mathrm{MASS} + \underbrace{\int\limits_\Vol \density\v\bcdot\grad\v\bcdot\e_k\phi^i\dV}_\mathrm{ADV} - \underbrace{\int\limits_\Vol \g\bcdot\e_k\phi^i\dV}_\mathrm{SRC} \\ & + \underbrace{\int\limits_\Vol \T^t:\grad\left(\e_k\phi^i\right)\dV}_\mathrm{DIFF} - \int\limits_\Surf \n\bcdot\T\bcdot\e_k\phi^i\dS = 0$