Darcy-Brinkman convection equations

Governing equations for convection coupled to Darcy-Brinkman flow, working in the Boussinesq approximation with constant viscosity and Newtonian deviatoric stress.

Dimensional equations

\[\begin{split} \begin{align*} &\text{Find} \\ &c(\textbf{x}, t): \Omega\times[0, \infty) \to \mathbb{R}, \\ &\textbf{u}(\textbf{x}, t): \Omega\times[0, \infty) \to \mathbb{R}^d, \\ &p(\textbf{x}, t): \Omega\times[0, \infty) \to \mathbb{R} \\ &\text{such that} \\ &\mathbb{IBVP}_{\textbf{u},p,c}\begin{cases} \phi\frac{\partial c}{\partial t}+\textbf{u}\cdot\nabla c=\nabla\cdot(\mathsf{D}(\phi, \textbf{u})\cdot\nabla c) & \\ \nabla\cdot\textbf{u} = 0 & \\ \rho_{\text{ref}}\left(\frac{\partial\textbf{u}}{\partial t}+\textbf{u}\cdot\nabla(\phi^{-1}\textbf{u})\right)=-\phi\nabla p + \nabla\cdot\tau + \phi\rho g\,\textbf{e}_g - \mu\phi\mathsf{K}^{-1}\cdot\textbf{u} \\ c=c_0 & \forall(\textbf{x}, t)\in\Omega\times\{0\} \\ \textbf{u}=\textbf{u}_0 & \forall(\textbf{x}, t)\in\Omega\times\{0\} \\ p=p_0 & \forall(\textbf{x}, t)\in\Omega\times\{0\} \\ c=c_{\text{D}} & \forall(\textbf{x}, t)\in\partial\Omega_{\text{D}} \times [0,\infty) \\ \textbf{n}\cdot(\mathsf{D}\cdot\nabla c) = c_{\text{N}} & \forall(\textbf{x}, t)\in\partial\Omega_{\text{N}} \times [0,\infty)~,~\partial\Omega_{\text{N}}=\partial\Omega/\partial\Omega_{\text{D}} \\ \textbf{u} = \textbf{u}_{\text{E}} & \forall(\textbf{x}, t)\in\partial\Omega_{\text{E}} \times [0,\infty) \\ (-p\mathsf{I}+\tau)\cdot\textbf{n} = \boldsymbol{\tau}_{\text{N}} & \forall(\textbf{x},t)\in\partial\Omega_{\text{N}}\times[0, \infty)~,~\partial\Omega_{\text{N}}=\partial\Omega/\partial\Omega_{\text{E}} \end{cases} \\ &\text{given} \\ &\mathbb{S}_{\textbf{u},p,c}= \begin{cases} \Omega\subset\mathbb{R}^d & \text{domain}\\ \mu_{\text{ref}} & \text{constant viscosity} \end{cases} \end{align*} \end{split}\]

Non-dimensionalization

Scalings

Quantity	\(\vert\textbf{x}\vert\)	\(\vert\textbf{u}\vert\)	\(t\)	\(c\)	\(\rho g\)	\(p\)	\(\psi\)
Scaling	\(\mathcal{L}\)	\(\mathcal{U}\)	\(\mathcal{T}\)	\(\Delta c\)	\(g \Delta\rho\)	\(\mu_{\text{ref}}\,\mathcal{U}\mathcal{L}/K_{\text{ref}}\)	\(\mathcal{U}\mathcal{L}\)

\(\mu\)	\(\phi\)	\(K\)	\(\vert\mathsf{D}\vert\)	\(\vert\mathsf{G}\vert\)	\(R\)	\(Q\)
\(\mu_{\text{ref}}\)	\(\phi_{\text{ref}}\)	\(K_{\text{ref}}\)	\(D_{\text{ref}}\)	\(G_{\text{ref}}\)	\(\Delta R\)	\(\Delta Q\)

Generic dimensionless numbers

\[ Ad=\frac{\mathcal{U}\mathcal{T}}{\mathcal{L}} ~,~ Di=\frac{D_{\text{ref}}\mathcal{T}}{\mathcal{L}^2} ~,~ Vi=\frac{\mu_{\text{ref}}\mathcal{T}}{\rho_{\text{ref}}\mathcal{L}^2} ~,~ Bu=\frac{\,g\Delta\rho\,\mathcal{T}}{\rho_{\text{ref}}\,\mathcal{U}} ~,~ Pm=\frac{\mu_{\text{ref}}\,\,\mathcal{T}}{K_{\text{ref}}\,\rho_{\text{ref}}\,\mathcal{U}} ~,~ X=\frac{\mathcal{L}_\Omega}{\mathcal{L}} \]

Physical dimensionless numbers

Definition	Name	Physical interpretation
\(Pr=\frac{\mu_{\text{ref}}}{\rho_{\text{ref}}D_{\text{ref}}}\)	Prandtl	Ratio of kinematic viscosity to diffusivity, defined with respect to solutal transport
\(Ra=\frac{\mathcal{L}_\Omega^3g\Delta\rho}{\mu_{\text{ref}}D_{\text{ref}}}\)	Rayleigh	Ratio of convective to diffusive speeds, defined with respect to solutal transport and domain length scale.
\(Dr = \frac{K_{\text{ref}}}{\mathcal{L}_\Omega^2}\)	Darcy	Ratio of permeability to squared length scale.
\(Le=\frac{G_{\text{ref}}}{D_{\text{ref}}}\)	Lewis	Ratio of thermal to solutal diffusivities.

Scaling choice

Name	\(\mathcal{L}\)	\(\mathcal{U}\)	\(\mathcal{T}\)	\(Ad\)	\(Di\)	\(Vi\)	\(Bu\)	\(Pm\)	\(X\)
advective	\(\mathcal{L}_\Omega\)	\(\sqrt{\mathcal{L}g\Delta\rho/\rho_{\text{ref}}}\)	\(\mathcal{L}/\mathcal{U}\)	\(1\)	\(1/\sqrt{Ra\,Pr}\)	\(\sqrt{Pr/Ra}\)	\(1\)	\(\sqrt{Pr/(Ra\,Dr^2)}\)	\(1\)

Non-dimensional time-discretized equations

Strong form

\[\begin{split} \begin{align*} &\text{Find}~c^{n+1},~\textbf{u}^{n+1},~p^{n+1}~\text{such that}~\forall n\geq0 \\ &\begin{cases} \frac{c^{n+1}-c^n}{\Delta t^n}+Ad\,\mathcal{D}_{\textbf{u},c}(\textbf{u}\cdot\nabla c)=Di\nabla\cdot\mathcal{D}_{\mathsf{D},c}(\mathsf{D}\cdot\nabla c) \\ \nabla\cdot\textbf{u}^{n+1}=0 \\ \frac{\textbf{u}^{n+1}-\textbf{u}^n}{\Delta t^n}+Ad\,\mathcal{D}_{\textbf{u}}(\textbf{u}\cdot\nabla(\phi^{-1}\textbf{u})) = -\phi\mathcal{D}_{p}(\nabla p) + Vi\,\nabla^2\mathcal{D}_{\textbf{u}}(\textbf{u}) + Bu\,\mu\phi\mathcal{D}_{\rho}(\rho\,\textbf{e}_g) - Pm\,\phi\mathsf{K}^{-1}\cdot\mathcal{D}_{u}(\textbf{u})\\ \vdots \\ \end{cases} \end{align*} \end{split}\]