Analysis of unsteady flow of second grade fluid with power law spatially distributed viscosity
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Analysis of unsteady flow of second grade fluid with power law spatially distributed viscosity I. E. Ireka1
· S. S. Okoya 2
Received: 5 July 2019 / Accepted: 25 April 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020
Abstract 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. Keywords Unsteady second grade fluid · Spatial inhomogeneity of fluid viscosity · Generalized Couette flow · Pulsating Poiseuille device · Closed form solutions · Finite volume techniques Mathematics Subject Classification 35C05 · 35Q35 · 65N08 · 76E05
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S. S. Okoya [email protected] I. E. Ireka [email protected]
1
Fraunhofer Institute for Industrial Mathematics, Kaiserslautern, Germany
2
Department of Mathematics, Obafemi Awolowo University, Ile-Ife, Nigeria
123
I. E. Ireka , S. S. Okoya
1 Introduction Flow of inhomogeneous fluids between parallel plates continue to attract considerable research due to its immense theoretical and experimental relevance. Such spatial inhomogeneity may arise from gradients in the material concentration, system temperature or flow phase transition due to non-uniformity in thermal boundary conditions. The theoretical analysis of Anand and Rajagopal [1] show that approximating such flows as homogeneous under mild variation in fluid properties could lead to significant error when computing global or local flow attributes. In this article, the classical channel flow of an incompressible viscous second grade fluid presented by Rajagopal [2] is revisited. Assuming a constant effective viscosity and adopting the method of separation of variables to resolve
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