Existence of free boundary for two non-Newtonian fluids (Q2717403)

The authors consider a slow flow of two immiscible fluids of Norton-Hoff type (i.e. the viscous part of stress tensor has the form \(T_{ij} = K |\varepsilon(u)|^{p-2} \varepsilon_{ij}(u)\) with \(\varepsilon(u)\) being the symmetric part of velocity gradient, \(K\) being a positive constant depending on fluid properties, and \(1<p\leq 2\)) with the same exponent \(p\), but with different values of \(K\). During the motion the fluids are separated by an interface. Both the time derivative of velocity and the convective term are neglected in the model; nevertheless, the interface between the fluids may change in time. Using a quasi-steady approximation, the authors prove existence of weak solution describing the evolution of two fluids. NEWLINENEWLINENEWLINEThe proof is based on fixed point argument. First, using the saddle point technique for a fixed configuration, the authors show the existence of solution to the equations describing steady slow flow of two immiscible fluids. Then the equation describing the evolution of interface is studied, and, using fine arguments from shape optimization, the authors prove the existence of a fixed point giving the solution to the original problem.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00036].