Abstract
Some unsteady motions of incompressible upper-convected Maxwell (UCM) fluids with exponential dependence of viscosity on the pressure are analytically studied. The fluid motion between two infinite horizontal parallel plates is generated by the lower plate, which applies time-dependent shear stresses to the fluid. Exact expressions, in terms of standard Bessel functions, are established both for the dimensionless velocity fields and the corresponding non-trivial shear stresses using the Laplace transform technique and suitable changes of the unknown function and the spatial variable in the transform domain. They represent the first exact solutions for unsteady motions of non-Newtonian fluids with pressure-dependent viscosity. The similar solutions corresponding to the flow of the same fluids due to an exponential shear stress on the boundary as well as the solutions of ordinary UCM fluids performing the same motions are obtained as limiting cases of present results. Furthermore, known solutions for unsteady motions of the incompressible Newtonian fluids with/without pressure-dependent viscosity induced by oscillatory or constant shear stresses on the boundary are also obtained as limiting cases. Finally, the influence of physical parameters on the fluid motion is graphically illustrated and discussed. It is found that fluids with pressure-dependent viscosity flow are slower when compared to ordinary fluids.
Highlights
The concept of fluid with pressure-dependent viscosity is due to Stokes [1] who, in his celebrated paper on the fluid response, remarked that the liquid viscosity can depend on the pressure
Equation (1) of the incompressible upper-convected Maxwell (UCM) fluids are used to describe the viscoelastic response of such a fluid, which is moving in a horizontal rectangular channel
Similar solutions for the ordinary UCM fluids performing the same motions, as well as the solutions corresponding to motions of the incompressible Newtonian fluids with/without pressure-dependent viscosity induced by oscillatory S cos(ωt), S sin(ωt) or constant S shear stresses on the boundary, have been obtained as limiting cases of general results
Summary
The concept of fluid with pressure-dependent viscosity is due to Stokes [1] who, in his celebrated paper on the fluid response, remarked that the liquid viscosity can depend on the pressure. The first explicit exact solutions for the flow of the incompressible Newtonian fluids with pressure-dependent viscosity between infinite horizontal parallel plates were presented by Rajagopal [29] He established steady solutions for the motion of the same fluids over an inclined plane due to gravity [30]. Exact expressions are established for the velocity and shear stress fields corresponding to unsteady motions of incompressible UCM fluids with exponential dependence of viscosity on the pressure between two infinite horizontal parallel plates. They are the first exact solutions corresponding to unsteady motions of the non-Newtonian fluids with pressure-dependent viscosity. The influence of the physical parameters on the fluid motion is graphically underlined and discussed
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