Stokes convection equations#
Governing equations for thermosolutal convection coupled to Stokes flow.
Dimensional equations#
\[\begin{split}
\begin{align*}
&\text{Find} \\
&c(\textbf{x}, t): \Omega\times[0, \infty) \to \mathbb{R}, \\
&\theta(\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} \\
&\begin{cases}
\frac{\partial c}{\partial t}+\textbf{u}\cdot\nabla c=\nabla\cdot(\mathsf{D}\cdot\nabla c) & \\
\frac{\partial\theta}{\partial t}+\textbf{u}\cdot\nabla\theta=\nabla\cdot(\mathsf{G}\cdot\nabla\theta) & \\
\nabla\cdot\textbf{u} = 0 & \\
\textbf{0}=-\nabla p + \nabla\cdot\tau + \rho g\,\textbf{e}_g & \forall(\textbf{x}, t)\in\Omega\times[0,\infty) \\
c(\textbf{x},t=0)=c_0 & \forall\textbf{x}\in\Omega \\
\theta(\textbf{x},t=0)=\theta_0 & \forall\textbf{x}\in\Omega \\
c=c_{\text{D}} & \forall(\textbf{x}, t)\in\partial\Omega_{\text{D}, c} \times [0,\infty] \\
\textbf{n}\cdot(\mathsf{D}\cdot\nabla c) = c_{\text{N}} & \forall(\textbf{x}, t)\in\partial\Omega_{\text{N}, c}
\times [0,\infty]~,~\partial\Omega_{\text{N}, c}=\partial\Omega/\partial\Omega_{\text{D}, c} \\
\theta=\theta_{\text{D}} & \forall (\textbf{x}, t)\in\partial\Omega_{\text{D}, \theta} \times [0,\infty] \\
\textbf{n}\cdot(\mathsf{G}\cdot\nabla \theta) = \theta_{\text{N}} & \forall(\textbf{x}, t)\in\partial\Omega_{\text{N}, \theta}
\times [0,\infty]~,~\partial\Omega_{\text{N}, \theta}=\partial\Omega/\partial\Omega_{\text{D}, \theta} \\
\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}\begin{cases}
\Omega\subset\mathbb{R}^d & \text{domain}\\
c_0(\textbf{x}) & \text{concentration initial condition}\\
\theta_0(\textbf{x}) & \text{temperature initial condition}\\
c_{\text{D}}(\textbf{x}, t)~,~\partial\Omega_{\text{D},c} & \text{concentration Dirichlet boundary condition} \\
\theta_{\text{D}}(\textbf{x}, t)~,~\partial\Omega_{\text{D},\theta} & \text{temperature Dirichlet boundary condition} \\
c_{\text{N}}(\textbf{x}, t)~,~\partial\Omega_{\text{N},c} & \text{concentration Neumann boundary condition} \\
\theta_{\text{N}}(\textbf{x}, t)~,~\partial\Omega_{\text{N}, \theta} & \text{concentration Neumann boundary condition} \\
\textbf{u}_{\text{E}}(\textbf{x}, t)~,~\partial\Omega_{\text{E}} & \text{velocity essential boundary condition} \\
\boldsymbol{\tau}_{\text{N}}(\textbf{x}, t)~,~\partial\Omega_{\text{N}} & \text{traction natural boundary condition} \\
\tau(\mu,\textbf{u}) & \text{deviatoric stress} \\
\mathsf{D}(\textbf{u}) & \text{solutal dispersion}\\
\mathsf{G}(\textbf{u}) & \text{thermal dispersion}\\
\rho(c, \theta) & \text{density}\\
\mu(c, \theta) & \text{viscosity}\\
\end{cases}
\end{align*}
\end{split}\]
Non-dimensionalization#
Scalings#
Quantity |
\(\vert\textbf{x}\vert\) |
\(\vert\textbf{u}\vert\) |
\(t\) |
\(c\) |
\(\theta\) |
\(\rho g\) |
\(p\) |
|---|---|---|---|---|---|---|---|
Scaling |
\(\mathcal{L}\) |
\(\mathcal{U}\) |
\(\mathcal{T}\) |
\(\Delta c\) |
\(\Delta\theta\) |
\(g \Delta\rho\) |
\(\mu_{\text{ref}}\,\mathcal{U}/\mathcal{L}\) |
\(\mu\) |
\(\vert\tau\vert\) |
\(\vert\mathsf{D}\vert\) |
\(\vert\mathsf{G}\vert\) |
|---|---|---|---|
\(\mu_{\text{ref}}\) |
\(\mu_{\text{ref}}\,\mathcal{U}/\mathcal{L}\) |
\(D_{\text{ref}}\) |
\(G_{\text{ref}}\) |
Abstract dimensionless numbers#
\[
Ad=\frac{\mathcal{U}\mathcal{T}}{\mathcal{L}}~,~
Di=\frac{D_{\text{ref}}\mathcal{T}}{\mathcal{L}^2}~,~
Bu=\frac{\mathcal{L}^2g\Delta\rho}{\mu_{\text{ref}}\,\mathcal{U}}~,~
X=\frac{\mathcal{L}_\Omega}{\mathcal{L}}
\]
Physical dimensionless numbers#
Definition |
Name |
Physical interpretation |
|---|---|---|
\(Ra=\frac{\mathcal{L}_\Omega^3g\Delta\rho}{\mu_{\text{ref}}D_{\text{ref}}}\) |
Rayleigh |
Ratio of convective to diffusive speeds, defined with respect to the transport of \(c\) and domain length scale. |
\(Le=\frac{G_{\text{ref}}}{D_{\text{ref}}}\) |
Lewis |
Ratio of thermal to solutal diffusivities. |
Scaling choice#
Name |
\(\mathcal{L}\) |
\(\mathcal{U}\) |
\( \mathcal{T}\) |
\(\{Ad, Di, Bu, X\}\) |
Examples |
|---|---|---|---|---|---|
advective |
\(\mathcal{L}_\Omega\) |
\(g\Delta\rho\mathcal{L}_\Omega^2/\mu_{\text{ref}}\) |
\(\mathcal{L}/\mathcal{U}\) |
\(\{1, 1/Ra, 1, 1\}\) |
… |
diffusive |
\(\mathcal{L}_\Omega\) |
\(D_{\text{ref}}/\mathcal{L}\) |
\(\mathcal{L}/\mathcal{U}\) |
\(\{1, 1, Ra, 1\}\) |
… |
Non-dimensional time-discretized equations#
Strong form#
\[\begin{split}
\begin{align*}
&\text{Find}~c^{n+1}, \theta^{n+1},~\textbf{u}^{n},~p^{}~\text{such that}~\forall n\geq0 \\
&\begin{cases}
\frac{c^{n+1}-c^n}{\Delta t^n}+Ad\,\mathcal{D}_{\textbf{u},c}(\textbf{u}\cdot\nabla c)=Di\nabla\cdot\mathcal{D}_{\mathsf{D},c}(\mathsf{D}\cdot\nabla c) \\
\frac{\theta^{n+1}-\theta^n}{\Delta t^n}+Ad\,\mathcal{D}_{\textbf{u},\theta}(\textbf{u}\cdot\nabla\theta)=\frac{Di}{Le}\nabla\cdot\mathcal{D}_{\mathsf{G},\theta}(\mathsf{G}\cdot\nabla\theta) \\
\nabla\cdot\textbf{u}^{n}=0 \\
\textbf{0}= \nabla\cdot(-p^n\mathsf{I}+\tau^n) + Bu\,\rho^n\,\textbf{e}_g \\
c^0=c_0 \\
\theta^0=\theta_0 \\
c^n\vert_{\partial\Omega_{\text{D}, c}}=c^n_{\text{D}} \\
\left(\textbf{n}\cdot(\mathsf{D}^n\cdot\nabla c^n)\right)\vert_{\partial\Omega_{\text{N}, c}} = c_{\text{N}}^n \\
\theta^n\vert_{\partial\Omega_{\text{D}, \theta}}=\theta^n_{\text{D}} \\
\left(\textbf{n}\cdot(\mathsf{G}^n\cdot\nabla\theta^n)\right)\vert_{\partial\Omega_{\text{N}, \theta}} = \theta_{\text{N}}^n \\
\textbf{u}^n\vert_{\partial\Omega_{\text{E}}} = \textbf{u}^n_{\text{E}}\\
p^n\vert_{\partial\Omega_{\text{N}}} = p^n_{\text{N}} \\
\left(\textbf{n}\cdot(-p^n\mathsf{I}+\tau^n)\right)\vert_{\partial\Omega_{\text{N}, \theta}} = \boldsymbol{\tau}_{\text{N}}^n \\
\end{cases}
\end{align*}
\end{split}\]
Weak forms#
…