## Flow of shear dependent electrorheological fluids.(English. Abridged French version)Zbl 0954.76097

The model of electrorheological fluids under consideration consists of quasi-static Maxwell equations for electric field $$E$$, and incompressible non-Newtonian fluid equations with forces $$f+$$ $$\chi \text{div}(E\otimes E)$$, where $$f$$ is the mechanical force and $$\chi$$ is the constant of dielectric susceptibility. The stress tensor $$S$$ depends on $$E$$ and $$D$$, and on the symmetrical part of the velocity gradient $$\nabla v$$, according to the law $\begin{split} S=\alpha _{21}((1+|D|^2)^{\frac{p-1}{2}}-1)E\otimes E+\\ (\alpha _{31}+\alpha _{33}|E|^2)(1+|D|^2)^{\frac{p-2}{2}}D+ \alpha _{51}(1+|D|^2)^{\frac{p-2}{2}}(DE\otimes E+E\otimes DE), \end{split}$ where $$p=p(|E|^2)$$ is a $$C^1$$-function such that $$1<p_{\infty}\leq p\leq p_0 <\infty$$. The equations are studied in a bounded domain $$\Omega \subset R^3$$ with boundary conditions $$v=0$$ and $$(E-E_0)\cdot n =0$$ on $$\partial \Omega$$. The author formulates restrictions on the function $$p(|E|^2)$$ which ensure the existence of weak and strong steady solutions provided the operator $$S(E,D)$$ is coercive and uniformly monotone. The existence of weak unsteady solutions is proved in two cases. The first is given by $$\alpha _{21}=\alpha _{51}=0$$ and $$p=2$$. In the second case $$S=\alpha _{31}(1+|E|^2)(1+|\nabla v|^2)^{\frac{p-2}{2}}\nabla v$$.

### MSC:

 76W05 Magnetohydrodynamics and electrohydrodynamics 76A05 Non-Newtonian fluids 35Q35 PDEs in connection with fluid mechanics
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