One-dimensional optimal system of Lie sub-algebra and analytic solutions for a liquid film fluid flow

Lie symmetry procedure enables reduction of the dependent and/or independent variables of the differential equations through similarity transformations, if they admit a Lie point symmetry algebra. A 7-dimensional Lie point symmetry algebra for the fluid flow and heat transfer in a thin liquid film due to an unsteady stretching sheet has been obtained earlier. Here we construct the 1-dimensional optimal system of Lie sub-algebras, corresponding invariants and similarity transformations. We use these transformations in reduction of the independent variables of the considered flow model. We achieve double reductions of the model that convert the governing partial differential equations into ordinary differential equations. We present all classes of ordinary differential equations that are obtainable through the invariants associated with each member of the deduced optimal system. In some cases, we construct analytic solutions for these reduced systems of differential equations using Homotopy analysis method. The selection of these cases is based on the form of stretching sheet velocity, temperature and film thickness, i.e., both the former remain functions of space and time variables while the latter is a function of time only.