Lie Optimal Solutions of Heat Transfer in a Liquid Film over an Unsteady Stretching Surface with Viscous Dissipation and External Magnetic Field

Abstract

In this thesis, we explore the fluid flow and heat transfer in thin liquid films over an unsteady stretching surface with viscous dissipation under the influence of an external magnetic field. To investigate the behavior of this system, we use similarity transformations with the analytical and numerical solution techniques. Specifically, we derive the Lie point symmetries of the system of partial differential equations that describes the flow and heat transfer in a thin liquid layer. A 5-dimensional Lie point symmetry algebra is derived. We develop new similarity transformations for the given model using a pair of the admitted Lie point symmetry generators. Arbitrary coefficients are used in the Lie optimal system and these arbitrary coefficients may be employe to the control the flow and heat transfer. By constructing the optimum system of Lie sub-algebras, related invariants, and similarity transformations, we reduce the number of independent variables in this flow model and convert the partial differential equations into ordinary differential equations for simplifying the solution procedure. This conversion requires double reduction, and the resulting transformations allow us to reduce the model to nonlinear ordinary differential equations. To examine the proposed magnetohydrodynamic (MHD) flow and heat transfer, we build analytic solutions for the obtained system of ordinary differential equations using the Homotopy analysis approach. The results are presented in the form of tables and figures, which demonstrate how the magnetic parameter, Prandtl number, Eckert number, and unsteadiness parameter affect fluid velocity, film thickness, and heat transfer. We compare these changes in velocity and temperature profiles with those previously reported for flow and heat transfer inside a thin film under the influence of viscous dissipation and an external magnetic field. Overall, our findings provide valuable insights into the behavior of fluid flow and heat transfer in thin films over unsteady stretching surfaces with viscous dissipation and external magnetic fields. These insights have important practical applications in fields such as chemical engineering, material sciences, and energy transfer.