Oldroyd-B Fluid Flow in a Channel

Mathematical Model

Consider the Oldroyd-B model for viscoelastic fluids:

\[\begin{split} \begin{align} \frac{\partial \mathbf{u}}{\partial t} + \mathbf{u}\cdot\nabla\mathbf{u} &= -\frac{1}{\rho}\nabla p + \nabla\cdot\boldsymbol{\tau} \\ \frac{\partial \boldsymbol{\tau}}{\partial t} + \mathbf{u}\cdot\nabla\boldsymbol{\tau} &= \boldsymbol{\tau}\cdot\nabla\mathbf{u} - \frac{\boldsymbol{\tau}}{\lambda} + 2\mu\left(\nabla\mathbf{u} - \frac{\boldsymbol{\tau}}{\lambda}\right) \\ \end{align} \end{split}\]

where \(\mathbf{u}\) is the velocity, \(p\) is the pressure, \(\rho\) is the density, \(\boldsymbol{\tau}\) is the extra stress tensor, \(\lambda\) is the relaxation time, and \(\mu\) is the viscosity.

Boundary Conditions

• Inlet and outlet: Specify velocity or pressure conditions.

• Channel walls: No-slip conditions.

Initial Conditions

Specify initial conditions for velocity and pressure.

Weak Formulation

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

\[\begin{split} \begin{align} &\int_{\Omega} \frac{\partial \mathbf{u}}{\partial t}\cdot\mathbf{w} \, d\Omega + \int_{\Omega} \mathbf{u}\cdot\nabla\mathbf{u}\cdot\mathbf{w} \, d\Omega + \int_{\Omega} \frac{1}{\rho}\nabla p\cdot\mathbf{w} \, d\Omega \\ &+ \int_{\Omega} \frac{\partial \boldsymbol{\tau}}{\partial t}:\mathbf{v} \, d\Omega + \int_{\Omega} \mathbf{u}\cdot\nabla\boldsymbol{\tau}:\mathbf{v} \, d\Omega \\ &- \int_{\Omega} \boldsymbol{\tau}\cdot\nabla\mathbf{u}:\mathbf{v} \, d\Omega + \int_{\Omega} \frac{\boldsymbol{\tau}}{\lambda}:\mathbf{v} \, d\Omega \\ &+ \int_{\Omega} 2\mu\left(\nabla\mathbf{u} - \frac{\boldsymbol{\tau}}{\lambda}\right):\nabla\mathbf{w} \, d\Omega \\ &= 0 \end{align} \end{split}\]

for all test functions \(\mathbf{w} \in H_0^1(\Omega)\).