Free surface problem of stationary non-Newtonian fluids
The authors study a stationary free-surface problem for certain class of non-Newtonian fluids, called sometimes power-law fluids. They consider the viscous part of the stress tensor in the form \((\mu_0 + \mu_1|e(u)|^{p-2})e_{ij}(u)\), where \(e(u)\) is the symmetric part of the gradient of velocity \(u\), and \(1<p<\infty\). It is proved that in dimensions \(2\leq m\leq 5\) there exists a unique solution to the stationary free-surface problem provided that either \(\mu_0\) or \(\mu_1\) is sufficiently large or the forcing term is sufficiently small. The proof is based on a fixed point argument, and the upper bound for the space dimension comes from the requirement of compact imbedding of \(W^{1,2}\) into \(L^3\).
scientific article; zbMATH DE number 1472256