Multiple Lie symmetry solutions for effects of viscous on magnetohydrodynamic flow and heat transfer in non-Newtonian thin film

Numerous flow and heat transfer studies have relied on the construction of similarity transformations which map the nonlinear partial differential equations (PDEs) describing the flow and heat transfer, to ordinary differential equations (ODEs). For these reduced equations, one finds multiple analytic and approximate solution procedures as compared to the flow PDEs. Here, we aim at constructing multiple classes of similarity transformations that are different from those already existing in the literature. We adopt the Lie symmetry method to derive these new similarity transformations which reveal new classes of ODEs corresponding to flow equations when applied to them. With these multiple classes of similarity transformations, one finds multiple reductions in the flow PDEs to ODEs. On solving these ODEs analytically or numerically, we obtain different kinds of flow and heat transfer patterns that help in determining optimized solutions in accordance with the physical requirements of a problem. For the said purpose, we derive Lie point symmetries for the magnetohydrodynamic Casson fluid flow and heat transfer in a thin film on an unsteady stretching sheet with viscous dissipation. Linear combinations of these Lie symmetries that are again the Lie symmetries of the flow model are employed here to construct new similarity transformations. We derive multiple Lie similarity transformations through the proposed procedure which lead us to more than one class of reduced ODEs obtained by applying the deduced transformations. We analyze the flow and heat transfer by deriving analytic solutions for the obtained classes of systems of ODEs using the homotopy analysis method. Magnetic parameters and viscous dissipation influences on the flow and heat transports are investigated and presented in graphical and tabulated formats.