Exact solutions for unsteady motion between parallel plates of some fluids with power-law dependence of viscosity on the pressure

Abstract

Unsteady motion between two infinite horizontal parallel plates of incompressible viscous fluids with a power-law dependence of viscosity on the pressure is studied analytically. The fluid motion is generated by the lower plate that is moving in its plane with a time dependent velocity. General solutions for dimensionless velocity and shear stress fields are established using suitable changes of independent variable and unknown function. They satisfy all imposed initial and boundary conditions and can generate exact solutions for any motion of this kind of respective fluids. Consequently, the problem in discussion is completely solved. For illustration, three particular cases with engineering applications are considered and graphical representations are presented and discussed. The solutions corresponding to some motions due to an accelerated plate are connected to those of the simple Couette flow by means of Riemann-Liouville fractional integral operator. The solutions corresponding to oscillating motions are presented as sums of steady-state and transient components and, for the validation of results that have been obtained, the steady-state solutions are presented in different forms whose equivalence is graphically proved. The required time to reach the steady-state is graphically determined both for oscillating motions and the simple Couette flow.