3.2.4.13. Brinkman Momentum

The Brinkman (Brinkman-Forchheimer) momentum equation is a porous media specialization of the fluid momentum equation (3.21) given by

(3.67)\pt{} \left( \frac{\density}{\varphi} \v \right) + \frac{\density}{\varphi^{2}} \v\bcdot\grad\v -\div\T + \left[ \frac{\density \hat{c}}{\sqrt k} \| \v \| + \frac{\viscosity}{k} \right] \v - g = \vector{0}

where \varphi is the matrix porosity, \T is the fluid stress tensor \g is a body force and \mu is termed the flowing liquid viscosity. The bracketed terms are often referred to as the Forchheimer or inertial flow contribution and the Darcy term respectively. In the limit of low velocities the equation reduces to a balance of pressure and a scaled velocity and Darcy’s law for a single component fluid is recovered [12]. This equation is most useful in cases where a clear fluid domain is bounded by a porous media. In most applications of this equation the hydrostatic contribution will be negligible but is retained here for completeness. We construct the G/FEM residual form of (3.67) 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

(3.68)\symRes_{m,k}^i & = \int\limits_\Vol \left[ \pt{} \left( (\frac{\density}{\varphi}\v ) + \frac{\density}{\varphi^{2}}\v\bcdot\grad\v + \left[ \frac{\density \hat{c}}{\sqrt k} \| \v \| + \frac{\viscosity}{k} \right] - 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

Here we note that the viscosity associated with the stress tensor \T is not the viscosity of the fluid but instead a so-called Brinkman viscosity \mu_{B}. In Aria, each term in (3.68) is specified separately as identified in equation (3.18).

(3.69)\symRes_{m,k}^i & = \underbrace{\int\limits_\Vol \pt{}(\frac{\density}{\varphi}\v)\bcdot\e_k\phi^i\dV}_\mathrm{MASS} + \underbrace{\int\limits_\Vol \frac{\density}{\varphi^{2}}\v\bcdot\grad\v\bcdot\e_k\phi^i\dV}_\mathrm{ADV} - \underbrace{\int\limits_\Vol \g - \left[ \frac{\density \hat{c}}{\sqrt k} \| \v \| + \frac{\viscosity}{k} \right] \v \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