Stokes equations

Velocity-pressure formulation

Strong form

\[\begin{split} \begin{align*} &\text{Find}~\textbf{u}(\textbf{x}): \Omega \to \mathbb{R}^d~\text{and}~p(\textbf{x}): \Omega \to \mathbb{R}~\text{such that} \\ &\mathbb{BVP}_{\textbf{u}, p}\begin{cases} \nabla\cdot\textbf{u} = 0 & \\ \textbf{0}=-\nabla p + \nabla\cdot\tau + \textbf{f} & \forall\textbf{x}\in\Omega \\ \textbf{u} = \textbf{u}_{\text{E}} & \forall \textbf{x}\in\partial\Omega_{\text{E}} \\ (-p\mathsf{I}+\tau)\cdot\textbf{n} = \boldsymbol{\tau}_{\text{N}} & \forall\textbf{x}\in\partial\Omega_{\text{N}}=\partial\Omega/\partial\Omega_{\text{E}} \end{cases} \\ &\text{given} \\ &\mathbb{S}_{\textbf{u}, p} \begin{cases} \Omega & \text{domain}\\ \textbf{u}_{\text{E}}(\textbf{x})~,~\partial\Omega_{\text{E}} & \text{velocity essential boundary condition} \\ \boldsymbol{\tau}_{\text{N}}(\textbf{x})~,~\partial\Omega_{\text{N}} & \text{traction natural boundary condition} \\ \tau(\mu,\textbf{u}) & \text{deviatoric stress constitutive relation} \\ \textbf{f}(\textbf{x}) & \text{body force} \\ \end{cases} \end{align*} \end{split}\]

Weak form

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

Linear algebra

Monolithic structure

\[\begin{split} \begin{align*} &\begin{pmatrix}\textbf{v}\\q\end{pmatrix}=\boldsymbol{\xi}_i~,~\begin{pmatrix}\textbf{u}\\p\end{pmatrix}=\sum_jX_j\boldsymbol{\xi}_j \\ &\implies A_{ij}X_j=b_i \iff \mathsf{A}\cdot\textbf{X}=\textbf{b} \end{align*} \end{split}\]

Block structure

\[\begin{split} \begin{align*} F_{\textbf{u}\textbf{u}}(\textbf{u}, \textbf{v}) + F_{\textbf{u}p}(p, \textbf{v}) &= 0 \quad\forall \textbf{v}\in V_{\textbf{u}} \\ F_{p\textbf{u}}(\textbf{u}, q) + F_{pp}(p, q) &= 0 \quad\forall q\in V_{p} \\ \end{align*} \end{split}\]

\[\begin{split} \begin{align*} &\textbf{v}=\boldsymbol{\xi}^u_i~,~q=\xi^p_i~,~\textbf{u}=\sum_jU_j\boldsymbol{\xi}^u_j~,~u=\sum_jP_j\xi^p_j \\ &\implies \begin{pmatrix} \mathsf{A}_{\textbf{u}\textbf{u}} & \mathsf{A}_{\textbf{u}p} \\ \mathsf{A}_{p\textbf{u}} & \mathsf{A}_{pp} \end{pmatrix} \begin{pmatrix} \textbf{U} \\ \textbf{P} \end{pmatrix} \begin{pmatrix} \textbf{b}_{\textbf{u}} \\ \textbf{b}_{p} \\ \end{pmatrix} \end{align*} \end{split}\]

Streamfunction formulation

Strong form

\[\begin{split} \begin{align*} &\text{Find}~\psi(\textbf{x}): \Omega \to \mathbb{R}~\text{such that} \\ &\mathbb{BVP}_\psi\begin{cases} \mu\nabla^2(\nabla^2\psi) = \frac{\partial f_y}{\partial x}- \frac{\partial f_x}{\partial y} & \forall\textbf{x}\in\Omega \\ \psi=\psi_{\text{E}} & \forall \textbf{x}\in\partial\Omega \\ \mu\nabla^2\psi=\psi_{\text{W}} & \forall \textbf{x}\in\partial\Omega \end{cases}\\ &\text{given} \\ &\mathbb{S}_{\psi}\begin{cases} \Omega\subset\mathbb{R}^2 & \text{domain}\\ \psi_{\text{E}}(\textbf{x})~,~\partial\Omega_{\text{E}} & \text{essential boundary condition} \\ \psi_{\text{W}}(\textbf{x})~,~\partial\Omega_{\text{W}} & \text{weak boundary condition} \\ f_x(\textbf{x}), f_y(\textbf{x}) & \text{body force} \\ \end{cases}\\ &\text{where}\\ &\textbf{u}=\nabla\times\boldsymbol{\psi}=\textbf{u}=\nabla\times\psi\textbf{e}_z=\frac{\partial\psi}{\partial y}\textbf{e}_x - \frac{\partial\psi}{\partial x}\textbf{e}_y \iff \nabla\cdot\textbf{u}=0\\ &\textbf{f}=f_x\textbf{e}_x + f_y\textbf{e}_y \\ &\tau(\mu,\textbf{u}) = \tfrac{\mu}{2}\left(\nabla\textbf{u} + \nabla\textbf{u}^{\mathsf{T}}\right) \\ &\nabla\mu=\textbf{0} \\ \end{align*} \end{split}\]

Weak form

\[\begin{split} \begin{aligned} &\text{Find}~\psi\in V~\text{such that} \\ &\begin{align*} F(\psi, v) &= \int_\Omega\text{d}\Omega~\mu\nabla^2v \nabla^2u - v\frac{\partial f_y}{\partial x} + v\frac{\partial f_x}{\partial y} \\ &\quad + \int_{\mathcal{F}}\text{d}\Gamma~\frac{\alpha\mu}{h}\left[\!\left[\nabla v\right]\!\right]\left[\!\left[\nabla u\right]\!\right] - \mu\left[\!\left[\nabla v\right]\!\right]\{\nabla^2u\} - \mu\{\nabla^2v\}\left[\!\left[\nabla u\right]\!\right] \\ &=0\quad\forall v\in V \end{align*} \\ &\text{where}~\alpha\in\mathbb{R}~\text{is a penalty parameter}\\ &\text{and}~~\text{is the local mesh cell size.} \end{aligned} \end{split}\]