Stability Analysis of Boundary Layer Flows Admitting Multiple Solutions

Abstract

The theme of this thesis is to investigate some boundary layer flow geometries driven by a stretching (shrinking) surface. The governing partial differential equations are reduced to fully coupled, non-linear system of ordinary differential equations by suitable similarity variables. The self-similar equations are solved numerically. This work is divided into two parts: steady and unsteady flow. All the problems investigated in this thesis are incorporated with a simple energy equation without viscous dissipation effect. It is interesting to observe that these governing equations admit dual solutions in a certain range of flow parameters. Both the solutions satisfy the terminal boundary conditions asymptotically. Different characteristics of these dual solutions on the velocity and temperature profiles, skin friction coefficient, heat transfer rate, and shear stresses raise questions on the physically acceptable and reliable solution. Thus emphasis has been laid to carry a linear temporal stability analysis which reveals the upper (first) branch solution is the stable solution and practically reliable and the lower (second) branch solution is unstable. The stability analysis is performed by the 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. In addition, asymptotic solution behavior for large stretching and suction parameters are discussed for the steady flow problems. The effects of various material and flow parameters on the skin friction coefficient, Nusselt number, shear stresses, velocity and temperature profiles, and boundary and thermal layer thicknesses are shown through graphs and tabular forms. Emphasis has been given to the physical interpretation of the new findings of the considered problems for upper branch solution only as this is the only stable solution. It is observed that the momentum and thermal boundary thicknesses in upper branch solutions are thinner than the lower branch solutions in each flow problem.