Analysis of unsteady flow of second grade fluid with power law spatially distributed viscosity

This paper investigates the influence of spatial inhomogeneity of fluid viscosity in an unsteady channel flow of a viscous second grade fluid. The viscosity model is assumed to follow a power-law spatial relation which may arise from concentration gradient, thermal gradient or boundary induced phase transition in the weakly elastic fluid. Two flow problems are considered (i) the Generalized Couette and (ii) the time-periodic plane Poiseuille flows. The associated partial differential equation governing each flow setup is decoupled into steady and transient state problems which are analyzed for possible closed form solutions. The analytical technique explored in this study is premised on theoretical analysis of the resulting ordinary differential equations. Corresponding results to the deduced ordinary differential equations with variable coefficient are presented as trigonometric and hyperbolic functions. In cases where analytical results are not attainable, numerical solutions are obtained via finite volume techniques. On comparing instances with the same parameter values, both the numerical and analytical solutions show good agreement. The spatial variation in the viscosity results in either a plug flow for cases where the viscosity index is negative or a fast flow throw the axis in instances where the index is positive. With graphical and tabular illustrations the influence of associated material parameters on the flow are presented and discussed.