Darcy-Brinkman equations

Nonlinear strong form

\[\begin{split} \begin{align*} &\text{Find} \\ &\textbf{u}(\textbf{x}, t): \Omega\times[0, \infty) \to \mathbb{R}^d, \\ &p(\textbf{x}, t): \Omega\times[0, \infty) \to \mathbb{R} \\ &\text{such that} \\ &\mathbb{IBVP}_{\textbf{u},p}\begin{cases} \nabla\cdot\textbf{u} = 0 & \\ \rho\left(\frac{\partial\textbf{u}}{\partial t}+\textbf{u}\cdot\nabla(\phi^{-1}\textbf{u})\right)=-\phi\nabla p + \nabla\cdot\tau + \phi\,\textbf{f} - \mu\phi\mathsf{K}^{-1}\cdot\textbf{u} \\ \textbf{u}=\textbf{u}_0 & \forall(\textbf{x}, t)\in\Omega\times\{0\} \\ p=p_0 & \forall(\textbf{x}, t)\in\Omega\times\{0\} \\ \textbf{u} = \textbf{u}_{\text{E}} & \forall(\textbf{x}, t)\in\partial\Omega_{\text{E}} \times [0,\infty) \\ (-p\mathsf{I}+\tau)\cdot\textbf{n} = \boldsymbol{\tau}_{\text{N}} & \forall(\textbf{x},t)\in\partial\Omega_{\text{N}}\times[0, \infty)~,~\partial\Omega_{\text{N}}=\partial\Omega/\partial\Omega_{\text{E}} \end{cases} \\ &\text{given} \\ &\mathbb{S}_{\textbf{u},p}= \begin{cases} \Omega\subset\mathbb{R}^d & \text{domain} \\ \mu_{\text{ref}} & \text{constant viscosity} \\ \tau(\mu,\textbf{u}) & \text{deviatoric stress} \\ \end{cases} \end{align*} \end{split}\]