Mathematical Analysis of Some Boundary Layer Flows

Abstract

The research presented in this thesis focuses on the analysis of some boundary layer flows. This work is divided into three parts, each corresponding to a different flow scenario. The first part regards the mathematical formulation of the boundary layer equations for two- or three-dimensional flow, which are the highly nonlinear partial differential equations (PDEs) of parabolic type. Moreover, the PDEs are converted into nonlinear ordinary differential equations (ODEs) using suitable similarity variables. It is worth recommending that the ODE boundary value problems (BVPs) governing these flow are intrinsic nonlinear, which becomes difficult to find an analytical solution than numerical results. The reduced boundary value problem is then analyzed in the subsequent parts of this thesis to understand the flow behaviours. Previous works were mainly focused on numerical simulations, and several conjectures regarding the existence and the behaviour of the solutions are discussed from the computational results. The ambition of this work is to review these conjectures mathematically. The second and major part contains the existence of a solution to the BVPs for the entire appropriate the physical parameters values. Uniqueness results are also presented for some (but not all) values of the parameters. It is to be mentioned here that the solutions are concave or convex for different parameter values, and in some cases, the differences in proofs and results may be significant. Both the cases are discussed thoroughly. Moreover, the last part contains the numerical results of the governing BVPs. The numerical results elucidate through table and graphs. In the limiting cases, the results are in great agreement with previously reported solutions in the literature. To demonstrate detail physical aspects of the flow domain, the velocity profiles and streamlines are presented graphically for governing parameters, and the results are discussed in detail.