Alexis Aposporidis
A Primal-Dual Formulation for the Bingham Flow

Emory University
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aapospo@emory.edu
Alessandro Veneziani
Eldad Haber

The Bingham flow is an example of a Stokes-type equation with shear-dependent viscosity. If $\textbf{Du}=\frac{1}{2}(\nabla \textbf{u}+ \nabla \textbf{u}^T)$ and $\vert\textbf{Du}\vert=\sqrt{\textnormal{tr}(\textbf{Du}^2)}$, the equations read

$$\left\{ \begin{array}{rl} -\nabla \cdot \tau +\nabla p &= \textbf{f}, \\ -\nabla \cdot \textbf{u}&=0 \\ +B.C., \end{array} \right.$$

and

$$\left\{ \begin{array}{rl} \tau=2\mu \textbf{Du}+\tau_s \frac{\tex... ... \leq \tau_s, & \textnormal{ if } \vert\textbf{Du}\vert=0, \end{array} \right.$$

where the velocity $\textbf{u}\in \mathbb{R}^n$, $n=2,3$ and $p \in \mathbb{R}$ are the unknowns and $\mu$, $\tau_s$ are given constants. Due to its non-differentiability for $\textbf{Du}=0$, a regularization of the form $\textbf{Du}= \sqrt{\textnormal{tr}(\textbf{Du}^2)+\varepsilon}$ ( $\varepsilon >0$) is necessary. It is a well-known fact that applying a nonlinear solver such as Newton or Picard to these equations results in a high number of outer iterations, especially for small choices of $\varepsilon$ [1]. In this talk we suggest an alternative approach inspired by [6]: We introduce a dual variable $\textbf{W}= \frac{\textbf{Du}}{\vert\textbf{Du}\vert}$, the equations for the Bingham flow are then reformulated as

$$\left\{ \begin{array}{rl} -\nabla \cdot \left( 2\mu \textbf{Du}+ ... ... \\ \textbf{W}\vert\textbf{Du}\vert &=\textbf{Du}\\ +B.C. \end{array} \right.$$

We address a few properties of this formulation and its numerical solution. Moreover, we perform several numerical experiments for solving the Bingham equations in this formulation, including the lid-driven cavity test and an example where the analytical solution is known. These experiments indicate a significant reduction in the number of nonlinear iterations over the nonlinear solvers of the equations in primal variables.

Acknowledgement:
We thank M. Olshanskii for fruitful discussions.