Problem on stationary flow of electrorheological fluids at the generalized conditions of slip on the boundary (Q2469408)

The stationary flow of electrorheological fluid is considered in the paper. The stress tensor for such fluids has a form \[ \sigma_{ij}(p,u,E)=-p\,\delta_{ij}+2\varphi(I(u),| E| ,\mu(u,E))\varepsilon_{ij},\quad i,j=1,\dots,n,\quad n=2,3 \] Here \(u=(u_1,\dots,u_n)\) is the velocity, \(p\) is the pressure of the fluid, \(E=(E_1,\dots,E_n)\) is the electric field strength, \(\varepsilon_{ij}(u)\) are the components of the rate of strain tensor, \(I(u)\) is the second invariant of the rate of strain tensor \[ I(u)=\sum\limits_{i,j=1}^n(\varepsilon_{ij}(u))^2, \] \(\varphi\) is the viscosity function. Function \(\mu(u,E)\) describes the anisotropy of the electrorheological fluid. The author considers the stationary flow with mixed boundary conditions. It means that the surface forces are given on the part of the boundary (the inflow and outflow of the fluid), and on the other part, relevant to hard immovable boundary, the regularized conditions of slip are prescribed. The existence of a generalized solution to the problem is proved for different forms of the function \(\mu\). It is shown that the Galerkin approximations can be used in solving the problem. The existence and uniqueness of classical solution are proved if the slip conditions are given on the whole boundary. Some unsolved problems are formulated at the end of the paper. Derivatives are denoted in the paper by nonstandard symbols.