Darcy equations

Mixed formulation

Strong form

\[\begin{split} \begin{align*} &\text{Find}~\textbf{u}(\textbf{x}): \Omega \to \mathbb{R}^d~\text{and}~p(\textbf{x}): \Omega \to \mathbb{R}~\text{such that} \\ &\mathbb{BVP}_{\textbf{u},p}\begin{cases} \nabla\cdot\textbf{u} = 0 & \\ \textbf{u} = -\frac{\mathsf{K}}{\mu}\cdot(\nabla p - \textbf{f}\,) & \forall\textbf{x}\in\Omega \\ \textbf{n}\cdot\textbf{u} = u_{\text{E}} & \forall \textbf{x}\in\partial\Omega_{\text{E}} \\ p = p_{\text{N}} & \forall\textbf{x}\in\partial\Omega_{\text{N}}=\partial\Omega/\partial\Omega_{\text{E}} \\ \int_{\partial\Omega}\text{d}\Gamma~u_{\text{E}}=0 & \text{if}~\partial\Omega_{\text{N}}=\varnothing\iff\partial\Omega_{\text{E}}=\partial\Omega \\ \end{cases} \\ &\text{given} \\ &\mathbb{S}_{\textbf{u},p}\begin{cases} \Omega\subset\mathbb{R}^d & \text{domain}\\ \mathsf{K}(\textbf{x}) & \text{permeability}\\ \mu(\textbf{x}) & \text{viscosity} \\ u_{\text{E}}(\textbf{x})~,~\partial\Omega_{\text{E}} & \text{normal velocity essential boundary condition} \\ p_{\text{N}}(\textbf{x})~,~\partial\Omega_{\text{N}} & \text{pressure natural boundary condition} \\ \end{cases} \end{align*} \end{split}\]

Weak form

\[\begin{split} \begin{aligned} &\text{Find}~(\textbf{u}, p)\in V_{\textbf{u}} \times V_p~\text{such that} \\ &\begin{align*} F(\textbf{u}, p, \textbf{v}, q) &= \int_\Omega\text{d}\Omega~q(\nabla\cdot\textbf{u}) + \textbf{v}\cdot(\mu\mathsf{K}^{-1}\cdot\textbf{u}) - p(\nabla\cdot\textbf{v}) - \textbf{v}\cdot\textbf{f} \\ &\quad +\int_{\partial\Omega_{\text{N}}}\text{d}\Gamma~p_{\text{N}}\,\textbf{v}\cdot\textbf{n} \\ &=0 \quad\forall(\textbf{v}, q)\in V_{\textbf{u}} \times V_p~. \end{align*} \end{aligned} \end{split}\]

Streamfunction formulation

General definition

\[\begin{split} \begin{align*} &\textbf{u}=\nabla\times\boldsymbol{\psi} \iff \nabla\cdot\textbf{u}=0 \\ &\implies\nabla\times\left(\mu\mathsf{K}^{-1}\cdot\nabla\times\boldsymbol{\psi}\right) = \nabla\times\textbf{f} \end{align*} \end{split}\]

Two-dimensional Cartesian definition

\[\begin{split} \begin{align*} &\boldsymbol{\psi}=\psi\textbf{e}_z\implies\textbf{u}=\frac{\partial\psi}{\partial y}\textbf{e}_x - \frac{\partial\psi}{\partial x}\textbf{e}_y \\ &\textbf{f}=f_x\textbf{e}_x + f_y\textbf{e}_y \end{align*} \end{split}\]

Strong form

\[\begin{split} \begin{align*} &\text{Find}~\psi(\textbf{x}): \Omega \to \mathbb{R}~\text{such that} \\ &\mathbb{BVP}_\psi\begin{cases} \nabla\cdot\left(\frac{\mu\mathsf{K}^{\mathsf{T}}\cdot\nabla\psi}{\text{det}(\mathsf{K})}\right)=-\frac{\partial(f_y)}{\partial x} + \frac{\partial(f_x)}{\partial y} & \forall\textbf{x}\in\Omega \\ \psi=\psi_{\text{D}} & \forall \textbf{x}\in\partial\Omega_{\text{D}} \\ \textbf{n}\cdot\left(\frac{\mu\mathsf{K}^{\mathsf{T}}\cdot\nabla\psi}{\text{det}(\mathsf{K})}\right) = \psi_{\text{N}} & \forall\textbf{x}\in\partial\Omega_{\text{N}}=\partial\Omega/\partial\Omega_{\text{D}} \end{cases} \\ &\text{given} \\ &\mathbb{S}_{\psi}\begin{cases} \Omega\subset\mathbb{R}^2 & \text{domain}\\ \mathsf{K}(\textbf{x}) & \text{permeability}\\ \mu(\textbf{x}) & \text{viscosity} \\ f_x(\textbf{x}), f_y(\textbf{x}) & \text{body force components} \\ \psi_{\text{D}}(\textbf{x})~,~\partial\Omega_{\text{D}} & \text{Dirichlet boundary condition} \\ \psi_{\text{N}}(\textbf{x})~,~\partial\Omega_{\text{N}} & \text{Neumann boundary condition} \\ \end{cases} \end{align*} \end{split}\]

Weak form

See Poisson equation.

Pressure formulation

Strong form

\[\begin{split} \begin{align*} &\text{Find}~p(\textbf{x}): \Omega \to \mathbb{R}~\text{such that} \\ &\mathbb{BVP}_p\begin{cases} \nabla\cdot\left(\frac{\mathsf{K}}{\mu}\cdot\nabla p\right)=\nabla\cdot\left(\frac{\mathsf{K}}{\mu}\cdot\textbf{f}\right) & \forall\textbf{x}\in\Omega \\ p=p_{\text{D}} & \forall \textbf{x}\in\partial\Omega_{\text{D}} \\ \textbf{n}\cdot\left(\frac{\mathsf{K}}{\mu}\cdot\nabla p\right) = p_{\text{N}} & \forall\textbf{x}\in\partial\Omega_{\text{N}}=\partial\Omega/\partial\Omega_{\text{D}} \end{cases} \\ &\text{given} \\ &\mathbb{S}_{p}\begin{cases} \Omega\subset\mathbb{R}^d & \text{domain}\\ \mathsf{K}(\textbf{x}) & \text{permeability}\\ \mu(\textbf{x}) & \text{viscosity} \\ \textbf{f}(\textbf{x}) & \text{body force} \\ p_{\text{D}}(\textbf{x})~,~\partial\Omega_{\text{D}} & \text{Dirichlet boundary condition} \\ p_{\text{N}}(\textbf{x})~,~\partial\Omega_{\text{N}} & \text{Neumann boundary condition} \\ \end{cases} \end{align*} \end{split}\]

Weak form

See Poisson equation.