Analytic solutions for the MHD flow and heat transfer in a thin liquid film over an unsteady stretching surface with Lie symmetry and homotopy analysis method
Muhammad Safdar, M. Ijaz Khan, Rahmat Ali Khan, S. Taj, Fakhar Abbas, Samia Elattar, Ahmed M. Galal
Abstract
In thin films on unsteady stretching surfaces, flow and heat transfer under different physical conditions are studied to refine countless industrial products. In this work we consider such a flow with an external magnetic field and apply Lie symmetry method on it to analyze the flow and heat transfer. We construct Lie point symmetry generators for this magnetohydrodynamic flow and heat transfer in a thin liquid film over an unsteady stretching surface. We obtain a six dimensional Lie point symmetry algebra and by employing a few of these symmetry generators we deduce invariants and similarity transformations for the flow model. We employ these transformations on the flow equations that are partial differential equations to reduce their independent variables from three to one. This procedure reveals corresponding systems of ordinary differential equations. The similarity transformations obtained here are different from those that are derived earlier for the considered flow problem. Hence by employing these transformations on the flow model leads to new classes of ordinary differential equations. It implies existence of additional classes of Lie similarity transformations to study the MHD flow and heat transfer in a thin film on an unsteady stretching sheet. We obtain analytic solutions for these reduced systems of equations with the homotopy analysis method to present effects of the Prandtl number, magnetic and unsteadiness parameters on the velocity and temperature profiles. These are new solutions for the considered flow model as Lie symmetries have not been employed earlier on it. All results are presented with the help of graphs and tables.