Numerical investigation of thin film flow over a porous, non-flat moving sheet with nonlinear kinematics

Flow of viscous thin film has previously been studied over a flat, porous and moving plate, whereas, these investigations are carried out for particular cases of uniform and linearly variable injection, suction, stretching and shrinking velocities. Moreover, the mechanisms of injection/suction and stretching/shrining have been analyzed individually and jointly for such flows. The present simulation has generalized the steady flow models of viscous thin film over porous, stretching and shrinking sheet of nonuniform thickness. We explored new and multiple dimensions of classical problems of thin film flow and analyzed it. Here, we categorically emphasized on the steady nature and kinds of injection (suction) and moving velocities of the sheet, whereas, steady form and variable size of the sheet are also taken into account. We formed different variables and investigated nonlinear cases of steady nature and checked different options for the two components of velocity, defined at the surface of sheet, variable sizes of the thin film and that of sheet. By analyzing all possible cases, we identified exact similarities that allowed the system of partial differential equations and boundary conditions to be precisely converted into ordinary differential equations based on these new variables. The transformed equation of continuity has two dimensionless parameters, which show that mass can be added/removed to/from the flow regime through two sources i.e. mass can be added/removed to/from the system by (i) injection/suction through porous sheet (ii) condensation/evaporation into/from the free surface. The system of exact ordinary differential equations is solved by bvp4c technique. Whereas, exact solutions of the system are also found under some restrictions on the parameters value. Besides that the two solutions are exactly matched to each other. Moreover, the present simulations are completely matched with the previously published work of this type for appropriate choices of functions and parameters. Note that the strict behavioral changes in the velocity profiles and skin friction coefficient are observed by changing the sign of parameters.