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

Růžička, Michael

C. R. Acad. Sci., Paris, Sér. I, Math. 329, No. 5, 393-398 (1999).

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

Reviewer: Vladimir Shelukhin (Novosibirsk)

MSC:

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

Keywords:

coercive uniformly monotone operator; weak solution; electrorheological fluids; quasi-static Maxwell equations; incompressible non-Newtonian fluid equations; stress tensor; velocity gradient; existence