Similarity Analysis of a Class of Non Newtonian Boundary Layer Flows

Abstract

In the field of fluid mechanics, despite its well-established foundations, numerous physical phenomena still need to be more adequately understood. One of these is the boundary layer flow of non-Newtonian fluids. Almost all the fluids used in the industries are non-Newtonian, and most of the flows can be numerically modelled by a flow past a stretching/shrinking sheet, flow near a stagnation point, and magnetohydrodynamics flow over a porous shrinking sheet. The present research aims to understand better the behaviour of different non-Newtonian fluids in the laminar boundary layer flow on the aforementioned geometries. The non-Newtonian fluid models are not only important because of their technological significance but also in view of the interesting mathematical features presented by the partial differential equations (PDEs) governing the flows. Lie group analysis is utilized to transform the governing partial differential equations into ordinary differential equations (ODEs), enabling a more simplified mathematical approach and facilitating deeper insights into the physical behaviour of the system. The primary goal of this thesis is to derive suitable similarity variables of the system of PDEs in order to obtain certain classes of group invariant solutions. The self-similar equations are solved numerically. It is interesting to observe that some of these self-similar equations admit dual solutions while others have a unique solution. Our secondary goal is to perform linear temporal stability analysis to determine whether the numerical solutions are physically acceptable and reliable. For dual solutions, linear temporal stability analysis revealed that the upper (first) branch solutions are stable and practically reliable, while the lower (second) branch solutions are unstable. Moreover, in the case of a unique solution, the stability analysis helps us to validate the obtained numerical solution. The stability analysis of these obtained solutions is determined by a sign of the smallest eigenvalue, where the positive or negative sign of the smallest eigenvalue leads to a stable or unstable solution, respectively. Effective numerical schemes have been used to determine the smallest eigenvalue. The effects of various material and flow parameters on the skin friction coefficient, Nusselt number, Sherwood number, shear stresses, velocity, temperature profiles, and boundary and thermal layer thicknesses are shown through graphs and tabular forms.