Analysis, control and simulation of non-Newtonian fluids

Overview & Synopsis

This thesis is devoted to the mathematical analysis of a class of non-Newtonian flows. In it, we first establish the existence of weak solutions for shear-thinning flows, namely such that the viscosity of the fluid decreases as the stresses applied to it increase. The solutions discussed are shown in the two-dimensional and three-dimensional cases, and are established in the form of variational inequalities. A remarkable property of shear-thinning flows is that they stop in finite time, and it is shown that solutions in the form of variational inequalities shed light on this property. A natural question then is whether it is possible, by applying spatially localized forces, to control the time at which the flow stops: this is a problem of internal controllability. Based on the Hilbert Uniqueness Method, we then show, by means of a fixed-point method carried out in a weak topology, that it is possible to establish such control for equations close to those describing shear-thinning or shear-thickening flows, which are given by a class of quasilinear parabolic equations. An important special case is then the existence of an approximate global control for the parabolic p-Laplacian equation when p > 3/2, the control being in fact exact for 3/2 < p < 2. Numerical simulations based on a mixed finite element method using Hsieh-Clough-Tocher finite elements complement these controllability results. In order to highlight the properties specific to the particular viscosity of these fluids, a sedimentation problem involving particles suspended in them is also studied. This is modeled by a Stokes-transport system, in which a degenerate viscosity is allowed as a function of the density of particles suspended in the mixture. By making use of renormalized solutions in a Friedrich approximation scheme, we then establish the existence of weak solutions for this system. Finally, the thesis concludes with a purely numerical study applied to medicine, consisting in the determination of diffusion coefficients of chemical species through a porous membrane from a data set. These coefficients, known as Beavers-Joseph coefficients, are then calculated using a combination of adaptive gradient descent and a Newton method.