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Quasi-static Bingham fluid

Let $G$ be a bounded domain in $\R^n$. The flow of a Bingham fluid through $G$ is described by the following system of equations: $$\begin{eqnarray*}&&div\; u=0 \quad \mbox{(imcompressibility)}\\ [2ex]&&Div\; T+ f=0\quad \mbox{(balance law of the momentum)}\end{eqnarray*}$$ Here $f$ is the density of the applied outer forces. The tensor $T$ satisfied the following constituent law: $$\begin{eqnarray*}T=p\vert{\cal E}u\vert^{p-2}{\cal E} u+\beta\frac{{\cal E} u}{\vert{\cal E}u\vert} \quad\mbox{for}\quad {\cal E} u\neq 0\end{eqnarray*}$$ If ${\cal E} u=0$ then we claim $\vert T\vert<\beta.$ Furthermore let $p\ge 2$ and $\beta >0$. ${\cal E} u$ denotes the strain velocity and $T$ the corresponding stress. Strain velocity can be expressed in terms of the velocity field $u=\partial _t w$ where $w=w(x,t)$ is the vector field of displacements. We have $$\begin{eqnarray*}{\cal E}u=\partial _t{\cal E}w:=\frac12\partial _t\left(\nabla w+\nabla w^T\right).\end{eqnarray*}$$ Note that $Div$ is the so-called tensor divergence, which iimage is a vector. $\nabla$ is the gradient for spatial variables. Boundary conditions are imposed for $T$ and $u$.
 

Bingham fluids have a quasi-static viscoelastic material behaviour. A good reference for this model is the book written by G. Astarita and G. Marrucci (Principles of Non-Newtonian Fluid Mechanics, McGraw-Hill, London 1974). If $\beta=0$ then we have the case of the so-called Norton fluid. In the last time energy minimising concepts become important. Note that the energy can be given by the functional

$$\begin{eqnarray*}I(u)=\int\limits _G(\vert{\cal E}u\vert^p+\beta\vert{\cal E} u\vert)dx.\end{eqnarray*}$$ A late reference: M. Fuchs, J.F. Grotowski and Reuling J. (1996), On variational models for Quasi-static Bingham Fluids MMAS, 19, 991-1015.