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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 \( S=\alpha _{31}(1+|E|^2)(1+|\nabla v|^2)^{\frac{p-2}{2}}\nabla v\).

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