The work presented in this thesis focuses on the mathematical modeling of pressure-driven flows in thin domains by analyzing the asymptotic behavior of solutions as a small parameter tends to zero.
The problem of describing asymptotic flows in thin domains arises in many scientific fields, where various physical phenomena are modeled, such as lubrication, liquid molding of fiber-reinforced polymer composites, fluid conduction in thin tubes, and blood circulation in capillaries. In such cases, the flow exhibits different characteristic lengths in different directions, particularly when the domain takes the form of a thin film or a slender tube. Mathematically, the flow is described by a set of partial differential equations defined in a thin domain, depending on a small parameter ε related to the geometry, such as the ratio of two characteristic lengths. Lower-dimensional models, which retain the essential features of the original problem, are derived by letting ε approach zero. In this limiting process, all variables (e.g., velocity and pressure) depend on ε, and the resulting limit problem yields a simplified model of the flow. To address these problems, several mathematical approaches have been developed, including asymptotic expansions and two-scale convergence for thin domains.
The thesis summarizes the work presented in five papers, referred to as papers I through V, with complementary appendices. The results are discussed in a broader context in an introduction that also provides an overview of the subject. In all papers, the flow is assumed to be governed by the Stokes system posed in a three-dimensional thin domain, subject to a mixed boundary condition. The no-slip and no-penetration conditions require that the velocity vanishes on the solid surfaces of the domain. This is complemented by a normal stress condition on the remaining boundary, defined by an external pressure. Physically, this means that the fluid motion is driven by an external pressure gradient, acting parallel to the surfaces.
The thesis aims to provide a clearer explanation of the novel pressure-driven flow introduced in the papers, while also offering a deeper understanding of the properties of the solutions to such equations formulated in thin domains.
Two types of fluid configurations are considered in the thesis: In papers I and II, the fluid is confined within a generalized Hele-Shaw cell, a type of thin film domain, whereas in the remaining papers, the fluid flows through thin tubular domains. These tubular domains are further classified into two types: the thin straight tube, analyzed in papers III and IV, and the thin curved tube, examined in paper V.
In papers I, III, and V, a stationary incompressible Newtonian fluid is considered in thin domains, with results based on the formal asymptotic expansion method. The primary result is the construction of an approximate solution in an appropriate manner, which is rigorously justified by estimating the error, i.e., the difference between the exact solution and the approximation. In papers III and V, the approximate solution is refined by incorporating boundary layer corrections at the inlet and outlet boundaries.
In papers II and IV, the situation is similar that described in papers I and III, but the fluid follows a non-Newtonian law, modeled by a power-law. The results are obtained by developing functional analysis and calculus of variations techniques to justify theorems concerning the existence and uniqueness of weak solutions, along with a priori estimates for the corresponding Stokes problem. The limit problem is derived using compactness, the two-scale convergence procedure, and arguments such as monotonicity and variational inequality. Finally, the uniqueness and regularity of the solution to the limit problem are proved. In paper IV, strong two-scale convergence for the solution is also considered.