Navier-Stokes convection equations

Governing equations for thermosolutal convection coupled to Navier-Stokes flow, working in the Boussinesq approximation.

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} \\ &\mathbb{IBVP}\begin{cases} \frac{\partial c}{\partial t}+\textbf{u}\cdot\nabla c=\nabla\cdot(\mathsf{D}(\textbf{u})\cdot\nabla c) & \\ \frac{\partial\theta}{\partial t}+\textbf{u}\cdot\nabla\theta=\nabla\cdot(\mathsf{G}(\textbf{u})\cdot\nabla\theta) & \\ \nabla\cdot\textbf{u} = 0 & \\ \rho_{\text{ref}}\left(\frac{\partial\textbf{u}}{\partial t}+\textbf{u}\cdot\nabla\textbf{u}\right)=-\nabla p + \nabla\cdot\tau + \rho g\,\textbf{e}_g & \forall(\textbf{x}, t)\in\Omega\times[0,\infty) \\ c=c_0 & \forall(\textbf{x}, t)\in\Omega\times\{0\} \\ \theta=\theta_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}, 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}\\ \textbf{u}_0(\textbf{x}) & \text{velocity initial condition}\\ p_0(\textbf{x}) & \text{pressure 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

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

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 the transport of \(c\)
\(Ra=\frac{\mathcal{L}_\Omega^3g\Delta\rho}{\mu_{\text{ref}}D_{\text{ref}}}\)	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	\(\mathcal{L}\)	\(\mathcal{U}\)	\( \mathcal{T}\)	\(\{Ad, Di, Vi, Bu, X\}\)	Examples
advective	\(\mathcal{L}_\Omega\)	\(g\Delta\rho\mathcal{L}_\Omega^2/\mu_{\text{ref}}\)	\(\mathcal{L}/\mathcal{U}\)	\(\{1, 1/Ra, Pr/Ra, Pr/Ra, 1\}\)	…
diffusive	\(\mathcal{L}_\Omega\)	\(D_{\text{ref}}/\mathcal{L}\)	\(\mathcal{L}/\mathcal{U}\)	\(\{1, 1, Pr, PrRa, 1\}\)	…

Non-dimensional time-discretized equations

Strong form

\[\begin{split} \begin{align*} &\text{Find}~c^{n+1}, \theta^{n+1},~\textbf{u}^{n+1},~p^{n+1}~\text{such that}~\forall n\geq0 \\ &\begin{cases} \frac{c^{n+1}-c^n}{\Delta t^n}+Ad\,\mathcal{D}_{\textbf{u},c}(\textbf{u}\cdot\nabla c)=Di\nabla\cdot\mathcal{D}_{\mathsf{D},c}(\mathsf{D}\cdot\nabla c) \\ \frac{\theta^{n+1}-\theta^n}{\Delta t^n}+Ad\,\mathcal{D}_{\textbf{u},\theta}(\textbf{u}\cdot\nabla\theta)=\frac{Di}{Le}\nabla\cdot\mathcal{D}_{\mathsf{G},\theta}(\mathsf{G}\cdot\nabla\theta) \\ \nabla\cdot\textbf{u}^{n+1}=0 \\ \frac{\textbf{u}^{n+1}-\textbf{u}^n}{\Delta t^n}+Ad\,\mathcal{D}_{\textbf{u}}(\textbf{u}\cdot\nabla\textbf{u})= Vi\,\nabla\cdot\mathcal{D}_{\tau}(-p\mathsf{I}+\tau) + Bu\,\mathcal{D}_{\rho}(\rho)\,\textbf{e}_g \\ c^0=c_0 \\ \theta^0=\theta_0 \\ c^n\vert_{\partial\Omega_{\text{D}, c}}=c^n_{\text{D}} \\ \left(\textbf{n}\cdot(\mathsf{D}^n\cdot\nabla c^n)\right)\vert_{\partial\Omega_{\text{N}, c}} = c_{\text{N}}^n \\ \theta^n\vert_{\partial\Omega_{\text{D}, \theta}}=\theta^n_{\text{D}} \\ \left(\textbf{n}\cdot(\mathsf{G}^n\cdot\nabla\theta^n)\right)\vert_{\partial\Omega_{\text{N}, \theta}} = \theta_{\text{N}}^n \\ \textbf{u}^n\vert_{\partial\Omega_{\text{E}}} = \textbf{u}^n_{\text{E}}\\ p^n\vert_{\partial\Omega_{\text{N}}} = p^n_{\text{N}} \\ \left(\textbf{n}\cdot(-p^n\mathsf{I}+\tau^n)\right)\vert_{\partial\Omega_{\text{N}, \theta}} = \boldsymbol{\tau}_{\text{N}}^n \\ \end{cases} \end{align*} \end{split}\]

Weak forms

…