On pressure-driven Hele-Shaw flow

Abstract: The present licentiate thesis is devoted to the rigorous derivation of the equations governing thin-film flow of incompressible Newtonian and non-Newtonian fluids. More precisely, we consider flow in a generalized Hele-Shaw cell, which is a thin three-dimensional domain confined between two surfaces connected by cylindrical obstacles of various shapes.Thin-film flows arise naturally in several applications. For instance, it is commonly used when the domain itself has different characteristic lengths in different directions, i.e. when the domain is a thin layer or a slender tube. Mathematically, the flow is described by a set of partial differential equations in a thin domain which depends on a small parameterε, e.g. the ratio of two characteristic lengths. By letting ε tend to zero, one can obtain a better understanding of the properties of solutions of such equations. In this limit process, all variables involved (e.g. velocity and pressure) depend on ε and the resulting limit problem yields a simplified model of the flow. There exist several mathematical techniques that have been developed to deal with such problems, e.g. asymptotic expansions, two-scale convergence for thin domains, etc.The scientific results in this thesis are presented in two papers (I and II) and a complimentary appendix. The results are discussed in a more general context in an introduction which also gives an overview of the subject. In both papers, we assume that the flow is governed by the Stokes system posed in a generalized Hele-Shaw cell satisfying a mixed boundary condition. The so-called no-slip and no-penetration conditions require that the velocity vanishes on the solid surfaces of the cell. This condition is complemented by the normal stress condition on the lateral boundary which is defined by an external pressure. Physically this means that the motion of the fluid is caused by the external pressure gradient, which acts in a direction parallel to the surfaces. One of the main objectives of this thesis is to develop a rigorous mathematical description of pressure-driven flow in thin domains.In paper I, we consider Hele-Shaw flow of an incompressible Newtonian fluid. The results are based on the formal asymptotic expansion method, i.e. by introducing a small parameter ε representing the thickness of the domain, rescaling the problem to a fixed domain, and considering solutions in the form of power series of ε. It is shown that the leading term of the velocity satisfies the so-called Poiseuille-law, i.e. the velocity is a linear function of the pressure gradient, whereas the leading pressure term satisfies the generalized Hele-Shaw equation. The main result is the construction of an approximate solution, which is justified by estimating the L2-norm of the error, i.e. the difference between the exact solution and the approximation.In paper II, the situation is similar to that of paper I, but the fluid obeys a more general constitutive relationship between the stress and the shear rate. More precisely, the functional relationship between the viscosity and the symmetrical part of the velocity gradient is given by a power-law. We develop techniques of functional analysis and calculus of variations in order to justify theorems concerning the existence and uniqueness of weak solutions of the corresponding Stokes system. The nonlinear Poiseuille-law, i.e. the limit velocity and the limit pressure gradient follow a power-law, is derived by using a two-scale convergence procedure and monotonicity arguments. Finally, uniqueness and regularity results for the solution of the limit problem are proved.