Darcy ABC convection-reaction equations

Governing equations for triple solutal convection with an \(\text{A}+\text{B}\to\text{C}\) reaction coupled to Darcy flow.

Dimensional equations

\[\begin{split} \begin{align*} &\text{Find} \\ &a(\textbf{x}, t): \Omega\times[0, \infty) \to \mathbb{R}, \\ &b(\textbf{x}, t): \Omega\times[0, \infty) \to \mathbb{R}, \\ &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,a,b,c}\begin{cases} \phi\frac{\partial a}{\partial t} + \textbf{u}\cdot\nabla a = \nabla\cdot(\mathsf{D}_a(\phi, \textbf{u})\cdot \nabla a) - \Sigma \\ \phi\frac{\partial b}{\partial t} + \textbf{u}\cdot\nabla b = \nabla\cdot(\mathsf{D}_b(\phi, \textbf{u})\cdot \nabla b) - \Sigma \\ \phi\frac{\partial c}{\partial t} + \textbf{u}\cdot\nabla c = \nabla\cdot(\mathsf{D}_c(\phi, \textbf{u})\cdot \nabla c) + \Sigma \\ \nabla\cdot\textbf{u}=0 \\ \textbf{u}=-\frac{\mathsf{K}}{\mu}\cdot\left(\nabla p - \rho g\,\textbf{e}_g\right) & \forall(\textbf{x}, t)\in\Omega\times[0,\infty)\\ w=w_0 & \forall(\textbf{x}, t)\in\partial\Omega_{\text{D},w} \times [0,\infty)\quad\forall w\in\{a, b, c\}\\ w = w_{\text{D},w} & \forall(\textbf{x}, t)\in\partial\Omega_{\text{D},w} \times [0,\infty)\quad\forall w\in\{a, b, c\}\\ \textbf{n}\cdot(\mathsf{D}_w\nabla w) = w_{\text{N}, w} & \forall(\textbf{x}, t)\in\partial\Omega_{\text{N},w} \times [0,\infty)\quad\forall w\in\{a, b, c\}\\ \textbf{n}\cdot\textbf{u} = u_{\text{E}} & \forall(\textbf{x}, t)\in\partial\Omega_{\text{E}} \times [0,\infty) \\ p = p_{\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,a,b,c}= \begin{cases} \Omega\subset\mathbb{R}^d & \text{domain}\\ w_0(\textbf{x})~\forall w\in\{a,b,c\} & \text{solutal initial conditions}\\ \phi(\textbf{x}) \\ \mathsf{K}(\phi) \\ \mathsf{D}_w(\phi, \textbf{u}) ~\forall w\in\{a,b,c\} & \text{solutal dispersions}\\ \rho(a,b,c) \\ \mu(a,b,c) \\ \Sigma(a,b,c) \end{cases} \end{align*} \end{split}\]

Non-dimensionalization

Scalings

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

\(\mu\)	\(\phi\)	\(K\)	\(\vert\mathsf{D}_a\vert\)	\(\vert\mathsf{D}_b\vert\)	\(\vert\mathsf{D}_c\vert\)	\(\Sigma\)
\(\mu_{\text{ref}}\)	\(\phi_{\text{ref}}\)	\(K_{\text{ref}}\)	\(D_{a, \text{ref}}\)	\(D_{b, \text{ref}}\)	\(D_{c, \text{ref}}\)	\(\Delta\Sigma\)

Generic dimensionless numbers

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

Physical dimensionless numbers

Definition	Name	Physical interpretation
\(Ra=\frac{\mathcal{L}_\Omega K_{\text{ref}}g\Delta\rho}{\mu_{\text{ref}}D_{\text{ref}}}=\underbrace{\frac{K_{\text{ref}}\,g\Delta\rho}{\mu_{\text{ref}}}}_{\text{convective speed}} \big/ \underbrace{\frac{D_{a,\text{ref}}}{\mathcal{L}_\Omega}}_{\text{diffusive speed}}\)	Rayleigh	Ratio of convective to diffusive speeds, defined with respect to the transport of \(a\) and domain length scale.
\(Da=\frac{\mathcal{L}_\Omega \mu_{\text{ref}}\,\Delta\Sigma}{K_{\text{ref}}\,g\Delta\rho\Delta a} = \underbrace{\frac{\Delta\Sigma}{\Delta a}}_{\text{reaction rate}} \big/ \underbrace{\frac{K_{\text{ref}}\,g\Delta\rho}{\mathcal{L}_\Omega \mu_{\text{ref}}}}_{\text{convection rate}}\)	Damköhler	Ratio of reaction to convection rates, defined with respect to the transport of \(a\) and domain length scale.
\(Le_w=\frac{D_{w,\text{ref}}}{D_{a,\text{ref}}}\quad\forall w\in\{b,c\}\)	Lewis	Ratio of diffusivities.
\(Lr_w=\frac{\Delta w}{\Delta a}\quad\forall w\in\{b,c\}\)		Effective stoichiometric coefficient.

Scaling choice

Name	\(\mathcal{L}\)	\(\mathcal{U}\)	\(\mathcal{T}\)	\(Ad\)	\(Di\)	\(Ki\)	\(Bu\)	\(X\)
advective	\(\mathcal{L}_\Omega\)	\(K_{\text{ref}}\,g\Delta\rho/\mu_{\text{ref}}\)	\(\phi_{\text{ref}}\mathcal{L}/\mathcal{U}\)	\(1\)	\(1/Ra\)	\(Da\)	\(1\)	\(1\)
diffusive	\(\mathcal{L}_\Omega\)	\(D_{\text{ref}}/\mathcal{L}\)	\(\phi_{\text{ref}}\mathcal{L}/\mathcal{U}\)	\(1\)	\(1\)	\(RaDa\)	\(Ra\)	\(1\)
advective-diffusive	\(D_{\text{ref}}/\mathcal{U}\)	\(K_{\text{ref}}\,g\Delta\rho/\mu_{\text{ref}}\)	\(\phi_{\text{ref}}\mathcal{L}/\mathcal{U}\)	\(1\)	\(1\)	\(Da/Ra\)	\(1\)	\(Ra\)
reactive	\(\sqrt{D_{\text{ref}}\mathcal{T}/\phi_{\text{ref}}}\)	\(\phi_{\text{ref}}\mathcal{L}/\mathcal{T}\)	\(\phi_{\text{ref}}\Delta a/\Delta\Sigma\)	\(1\)	\(1\)	\(1\)	\(\sqrt{Ra/Da}\)	\(\sqrt{RaDa}\)

Non-dimensional time-discretized equations

Strong form

\[\begin{split} \begin{align*} &\text{Find}~a^{n+1}, b^{n+1}, c^{n+1},~\textbf{u}^n,~p^n~\text{such that}~\forall n\geq0 \\ &\begin{cases} \phi\frac{a^{n+1}-a^n}{\Delta t^n} + Ad\,\mathcal{D}_{\textbf{u}, a}(\textbf{u}\cdot\nabla a) = Di\,\nabla\cdot\mathcal{D}_{\mathsf{D}_a,a}(\mathsf{D}_{a}(\phi, \textbf{u})\cdot \nabla a) - Ki\,\Sigma \\ \phi\frac{b^{n+1}-b^n}{\Delta t^n} + Ad\,\mathcal{D}_{\textbf{u}, b}(\textbf{u}\cdot\nabla b) = Le_bDi\,\nabla\cdot\mathcal{D}_{\mathsf{D}_b,b}(\mathsf{D}_b(\phi, \textbf{u})\cdot \nabla b) - Lr_b Ki\,\Sigma \\ \phi\frac{c^{n+1}-c^n}{\Delta t^n} + Ad\,\mathcal{D}_{\textbf{u}, c}(\textbf{u}\cdot\nabla c) = Le_cDi\,\nabla\cdot\mathcal{D}_{\mathsf{D}_c,c}(\mathsf{D}_c(\phi, \textbf{u})\cdot \nabla c) + Lr_c Ki\,\Sigma \\ \nabla\cdot\textbf{u}^n=0 \\ \textbf{u}^n=-\frac{\mathsf{K}}{\mu^n}\cdot\left(\nabla p^n + Bu\,\rho^ng\,\textbf{e}_g\right) \\ \end{cases} \end{align*} \end{split}\]