Papanastasiou Fluid Flow in a Channel

Mathematical Model

Consider the Papanastasiou model for non-Newtonian fluids:

\[\begin{split} \begin{align} \sigma_{ij} &= -p\delta_{ij} + 2\mu\left(\varepsilon_{ij} + \frac{\tau_0}{2\mu}\right)\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right) \\ \varepsilon_{ij} &= \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right) \\ \end{align} \end{split}\]

where \(\sigma_{ij}\) is the stress tensor, \(p\) is the pressure, \(\mu\) is the viscosity, \(u_i\) is the velocity, \(\varepsilon_{ij}\) is the strain rate, and \(\tau_0\) is the yield stress.

Boundary Conditions

• Inlet and outlet: Specify velocity or pressure conditions.

• Channel walls: No-slip conditions.

Initial Conditions

Start with an initial guess for velocity and pressure.

Weak Formulation

Find \(u_i \in H_0^1(\Omega)\) and \(p \in L^2(\Omega)\) such that:

\[\begin{split} \begin{align} &\int_{\Omega} \sigma_{ij} \left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right) \, d\Omega + \int_{\Omega} \frac{\partial p}{\partial x_i}u_i \, d\Omega \\ &= 0 \end{align} \end{split}\]

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