Convection-Diffusion Equation in a Channel

Mathematical Model

Consider the 2D convection-diffusion equation in a channel:

\[ \begin{align} \rho u \frac{\partial c}{\partial x} + \rho v \frac{\partial c}{\partial y} - \nabla \cdot (D \nabla c) = f \end{align} \]

Boundary Conditions

• Inlet: \(c = c_{\text{inlet}}\) with $\mathbf{v} \cdot \mathbf{n} < 0$ (inward flow).

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

• Lateral faces: \(-D \nabla c \cdot \mathbf{n} = 0\) (insulated).

Initial Conditions

Assume an initial concentration distribution \(c_0(x, y)\) at \(t = 0\).

Weak Formulation

Find \(c \in H_0^1(\Omega)\) such that:

\[\begin{split} \begin{align} &\int_{\Omega} \left(\rho u \frac{\partial c}{\partial x} + \rho v \frac{\partial c}{\partial y}\right) \phi \, d\Omega - \int_{\Omega} D \nabla c \cdot \nabla \phi \, d\Omega \\ &= \int_{\Omega} f \phi \, d\Omega \end{align} \end{split}\]

for all test functions \(\phi \in H_0^1(\Omega)\).