# Navier-Stokes convection equations

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

## Dimensional equations

$$
\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*}
$$

## Non-dimensionalization

### Scalings

| Quantity | $\vert\textbf{x}\vert$ | $\vert\textbf{u}\vert$ | $t$ | $c$ | $\theta$ | $\rho g$ | $p$ |
| -------- | ------- | ------- | ------- | ------- | ------- |  ------- |  ------- | 
| **Scaling** | $\mathcal{L}$ | $\mathcal{U}$ |$\mathcal{T}$ | $\Delta c$ | $\Delta\theta$ | $g \Delta\rho$ | $\mu_{\text{ref}}\,\mathcal{U}/\mathcal{L}$ |

| $\mu$ | $\vert\tau\vert$  | $\vert\mathsf{D}\vert$ | $\vert\mathsf{G}\vert$ |
| ------- | ------- | ------- | ------- |
| $\mu_{\text{ref}}$ |  $\mu_{\text{ref}}\,\mathcal{U}/\mathcal{L}$ | $D_{\text{ref}}$ | $G_{\text{ref}}$ |


### Generic dimensionless numbers

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

### 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 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. |
| $Le=\frac{G_{\text{ref}}}{D_{\text{ref}}}$  |  Lewis  | Ratio of thermal to solutal diffusivities. |


### Scaling choice

| Name | $\mathcal{L}$ | $\mathcal{U}$ | $\mathcal{T}$ | $Ad$ | $Di$ | $Vi$ | $Bu$ | $X$ |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| 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

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

### Weak forms

...