Stability analysis for multiple solutions of boundary layer flow towards a shrinking sheet: Analytical solution by using least square method

Abstract

The Purpose:In some cases, most of the boundary value problems contain multiple solutions in which one of the solution is stable or physical reliable and other solutions are unstable. In this study, the problem of hydromagnetic flow due to shrinking sheet is considered and calculated dual solutions against velocity ratio parameter in case of shrinking sheet. A stability analysis is performed for the checking, which solution is stable. The main materials and methods used:Initially, due to more than one independent variable, the flow is governed by partial differential equations (PDEs). For simplicity the conversion of these PDEs were made in form of ODEs (ordinary differential equations) and solved analytically by using weighted residual method named as least square method. The resulting nonlinear system of equations is linearized by using Newton method (Burden and Faires, 1991). For the accuracy of this method, a comparison is made with the previous published work (Wang, 2008). The algorithm of least square method is constructed in Mathematica software and is very easy to apply as compare to analytical method namely: homotopy analysis method. In case of shrinking sheet there exists more than one solution. To check which solution is physical reliable or stable and unstable, a present problem is converted into time dependent problem and then stability analysis is performed and BVP4c method is used for calculating the eigenvalues. The main results obtained:In case of shrinking sheet there exists dual solutions and dual solutions are calculated for some values of velocity ratio parameter λand magnetic parameterM by using least square method. The range of dual solutions is shown by plotting the graphs of skin friction coefficient and local Nusselt number. After performing the stability analysis, the smallest eigenvalues are calculated for various values of λ and M. The main conclusions of the work:It is observed that positive eigenvalue corresponds to stable solution or physical reliable i.e. the disturbance in the solution decay initially and negative eigenvalue corresponds to unstable solution i.e. it produces initial growth of disturbance. It is also observed that eigenvalues become higher in the presence of Lorentz force.