Cross Power Law Fluid Flow in a Channel

Mathematical Model

Consider the cross power-law model for non-Newtonian fluids:

\[\begin{split} \begin{align} \sigma_{ij} &= -p\delta_{ij} + \lambda \varepsilon_{kk}\delta_{ij} + 2\mu\left(\varepsilon_{ij} + \alpha\varepsilon_{ij}^2\right) \\ \varepsilon_{ij} &= \frac{1}{2}\left(\frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i}\right) \\ \end{align} \end{split}\]

where \(\sigma_{ij}\) is the stress tensor, \(p\) is the pressure, \(\lambda\) and \(\mu\) are material constants, \(v_i\) is the velocity, \(\varepsilon_{ij}\) is the strain rate, and \(\alpha\) is a power-law index.

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

\[\begin{split} \begin{align} &\int_{\Omega} \sigma_{ij} \varepsilon_{ij} \, d\Omega + \int_{\Omega} \frac{\partial p}{\partial x_i}v_i \, d\Omega \\ &= 0 \end{align} \end{split}\]

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