Fractional Analysis of Thin Film Flow of Non-Newtonian Fluid

Abstract

Modeling and analysis of thin film flow with respect to magneto hydro dynamical effect has been an important theme in the field of fluid dynamics, due to its vast industrial applications. The analysis involves studying the behavior and response of governing equations on the basis of various parameters such as thickness of the film, film surface profile, shear stress, liquid velocity, volumetric flux, vorticity, gravity, viscosity among others, along with different boundary conditions. In this article, we extend this analysis in fractional space using a homotopy based scheme, considering the case of a Non-Newtonian Pseudo-Plastic fluid for lifting and drainage on a vertical wall. After applying similarity transformations, the given problems are reduced to highly non-linear and inhomogeneous ordinary differential equations. Moreover, fractional differential equations are obtained using basic definitions of fractional calculus. The Homotopy Perturbation Method (HPM), along with fractional calculus is used for obtaining approximate solutions. Physical quantities such as the velocity profile, volume flux and average velocity respectively for lift and drainage cases have been calculated. To the best of our knowledge, the given problems have not been attempted before in fractional space. Validity and convergence of the obtained solutions are confirmed by finding residual errors. From a physical perspective, a comprehensive study of the effects of various parameters on the velocity profile is also performed. Study reveals that Stokes number St, non-Newtonian parameter β and magnetic parameter M have inverse relationship with fluid velocity in lifting case. In the drainage case, Stokes number St and non-Newtonian parameter β have direct relationship with fluid velocity, but magnetic parameter M shows inverse relationship with velocity. The investigation also shows that the fractional parameter α has direct relationship with the fluid velocity in lifting case, while it has inverse relationship with velocity in the drainage case.