Incompressible Flow Past a Cylinder

Mathematical Model:

Consider the steady, incompressible Navier-Stokes equations for fluid flow around a cylinder:

\[\begin{split} \begin{align} \rho (\mathbf{v} \cdot \nabla) \mathbf{v} &= -\nabla p + \mu \nabla^2 \mathbf{v} + \rho \mathbf{g} \\ \nabla \cdot \mathbf{v} &= 0 \end{align} \end{split}\]

Boundary Conditions

• Inlet: \(\mathbf{v} = (U, 0)\) (uniform velocity).

• Outlet: \(\frac{\partial \mathbf{v}}{\partial x} = 0\) (zero gradient).

• Cylinder surface: \(\mathbf{v} \cdot \mathbf{n} = 0\) (no-slip condition).

Initial Conditions

Assume a quiescent fluid, i.e., \(\mathbf{v} = (0, 0)\) at \(t = 0\).

Weak Formulation

Find \(\mathbf{v} \in H_0^1(\Omega)\) and \(p \in L^2(\Omega)\) such that:

\[\begin{split} \begin{align} \int_{\Omega} \rho (\mathbf{v} \cdot \nabla) \mathbf{v} \cdot \mathbf{w} \, d\Omega & = -\int_{\Omega} p \nabla \cdot \mathbf{w} \, d\Omega + \mu \int_{\Omega} \nabla \mathbf{v} : \nabla \mathbf{w} \, d\Omega + \int_{\Omega} \rho \mathbf{g} \cdot \mathbf{w} \, d\Omega \\ \int_{\Omega} \nabla \cdot \mathbf{v} q \, d\Omega & = 0 \end{align} \end{split}\]

where \(\mathbf{w} \in H_0^1(\Omega)\) and \(q \in L^2(\Omega)\).