Bingham Plastic Flow in a Pipe

Mathematical Model

Consider the Bingham plastic model for non-Newtonian fluids with yield stress:

\[\begin{split} \begin{align} \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} &= -\frac{1}{\rho}\frac{\partial p}{\partial x} + \frac{\tau_0}{\rho}\frac{\partial^2 u}{\partial y^2} \\ \frac{\partial^2 u}{\partial y^2} &\geq 0 \quad \text{(Yield stress condition)} \end{align} \end{split}\]

where \(u\) is the velocity, \(p\) is the pressure, \(\rho\) is the density, \(\tau_0\) is the yield stress, and \(y\) is the radial coordinate.

Boundary Conditions

• Inlet: Specify velocity or pressure conditions.

• Outlet: Specify outflow conditions.

• Pipe walls: No-slip conditions.

Initial Conditions

Specify initial conditions for velocity and pressure.

Weak Formulation

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

\[\begin{split} \begin{align} &\int_{\Omega} \frac{\partial u}{\partial t}v \, d\Omega + \int_{\Omega} u\frac{\partial u}{\partial x}v \, d\Omega + \int_{\Omega} \frac{1}{\rho}\frac{\partial p}{\partial x}v \, d\Omega \\ &+ \int_{\Omega} \frac{\tau_0}{\rho}\frac{\partial^2 u}{\partial y^2}\frac{\partial v}{\partial y} \, d\Omega \\ &= 0 \end{align} \end{split}\]

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