Analytical solutions for some unsteady flows of fluids with linear dependence of viscosity on the pressure

Abstract

New exact solutions for unidirectional unsteady flows of incompressible viscous fluids with linear dependence of viscosity on the pressure between two infinite horizontal parallel plates are established when the lower plate is moving in its plane with an arbitrary time-dependent velocity. In addition to being useful solutions to idealizations of technologically relevant problems such exact solutions also serve to test the validity and the efficacy of numerical schemes that have been developed for much more complicated three-dimensional flows. The flows considered herein correspond to important solutions in the study of the classical Navier-Stokes fluid model. General results which are obtained can generate exact solutions for any motion of this type of the respective fluids. For illustration, three special cases with technical relevance are considered, and the variations of the fluid velocity and the non-trivial shear stress are graphically presented and discussed in some situations. The exact solutions corresponding to some motions generated by an accelerated plate are connected to the adequate solutions of the simple Couette flow.

KEYWORDS:

• Semi-inverse solutions
• pressure-dependent viscosity
• incompressible fluid
• parallel plates
• general solutions

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 76A05

Acknowledgements

The authors would like to express their sincere gratitude to the Editor and reviewers for their careful assessment, fruitful remarks and valuable suggestions and comments regarding the first two versions of the paper. The authors also would like to thank Prof. K.R. Rajagopal for bringing the present problem to their attention and for some fruitful and valuable suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).