Stokes Flow in a Lid-Driven Cavity#
Author: Ahmed Ratnani
Mathematical Model#
Consider the steady-state Stokes equations for flow in a 2D lid-driven cavity:
\[\begin{split}
\begin{align}
- \nabla p + \mu \nabla^2 \mathbf{v} &= 0 \\
\nabla \cdot \mathbf{v} &= 0
\end{align}
\end{split}\]
Boundary Conditions#
Lid: \(\mathbf{v} = (1, 0)\) (lid-driven at constant velocity).
Other boundaries: \(\mathbf{v} = (0, 0)\) (no-slip condition).
Outlet: \(\frac{\partial p}{\partial n} = 0\) (zero-pressure gradient).
Initial Conditions#
Start with an initial guess for velocity and pressure.
Weak Formulation#
Find \(\mathbf{v} \in H_0^1(\Omega)\) and \(p \in L^2(\Omega)\) such that:
\[\begin{split}
\begin{align}
&\int_{\Omega} \mu \nabla \mathbf{v} : \nabla \mathbf{w} \, d\Omega - \int_{\Omega} p \nabla \cdot \mathbf{w} \, d\Omega \\
&= 0
\end{align}
\end{split}\]
for all test functions \(\mathbf{w} \in H_0^1(\Omega)\).