An unsteady flow of incompressible Newtonian fluids with linear dependence of viscosity on the pressure between two infinite horizontal parallel plates is analytically studied. The fluid motion is induced by the upper plate that applies an arbitrary time-dependent shear stress to the fluid. General expressions for the dimensionless velocity and shear stress fields are established using a suitable change of independent variable and the finie Hankel transform. These expressions, that satisfy all imposed initial and boundary conditions, 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 some important observations and graphical representations are provided. An interesting relationship is found between the solutions corresponding to motions induced by constant or ramptype shear stresses on the boundary. Furthermore, for validation of the results, the steady-state solutions corresponding to oscillatory motions are presented in different forms whose equivalence is graphically proved.
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For liquids such as water, for a range of pressures for which the density remains nearly constant, the viscosity could change by several orders of magnitude. Thus, such liquids could be approximated as incompressible liquids whose viscosity depends on the pressure. Stokes (1945) recognized this possibility and the fact that the viscosity could be considered to be independent of the pressure in only special flows. That this is indeed so has been verified in the case of numerous liquids (Bridgman, 1931). In this short study, we allow the viscosity of the fluid to be dependent on the pressure and investigate the consequence of the effect of gravity in simple flows such as Couette flows between parallel plates. We find that gravity can have a profound effect on the structure of the flow. Its presence leads to the concentration of vorticity adjacent to one of the plates.
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