Abstract
This paper presents exact expressions for dimensionless starting solutions corresponding to some oscillatory motions of fluids with exponential dependence of viscosity on pressure. These expressions are established in terms of standard Bessel functions of order zero and one. Fluid motion between two infinite horizontal parallel plates is generated by the lower plate, which applies oscillatory shear stresses to the fluid. The corresponding solutions, which are currently absent in the literature, are presented as sums of steady-state and transient components. These are useful for experimentalists who wish to eliminate transients from their experiments. For completeness, the dimensionless velocity field corresponding to the motion due to the lower plate applying a constant shear stress to the fluid is determined as a limiting case. Furthermore, to verify the results, it is shown that diagrams of the present steady-state solutions coincide with those of ordinary Newtonian fluids performing the same motions as the dimensionless pressure–viscosity coefficient tends to zero. The spatial distributions of starting solutions and some transversal sections are also presented and discussed.
Highlights
The concept of a fluid with pressure-dependent viscosity has been considered in the past;[1] the assumption of constant viscosity may induce errors when studying flows at high pressures
The fluid motion between two infinite horizontal parallel plates was assumed to be generated by the lower plate, which applies oscillatory or constant shear stresses to the fluid
Exact solutions were established for the velocity fields uc(y, t), us(y, t), and uC(y, t), and for the corresponding nontrivial shear stresses
Summary
The concept of a fluid with pressure-dependent viscosity has been considered in the past;[1] the assumption of constant viscosity may induce errors when studying flows at high pressures. Numerical solutions for the velocity field corresponding to a variant of Stokes’s problems for fluids in which viscosity is given by Eq (1) have been provided by Srinivasan and Rajagopal.[11] They presented a list of experimental works showing that many incompressible fluids exhibit the phenomenon of pressure-dependent viscosity. An interesting analytical study of the laminar flows between vertical parallel plates filled with two immiscible viscous and couple stress fluids in a composite porous medium modeled by the Brinkman equation was carried out by Umavathi et al.[18]. We provide the exact solutions for the motion of these fluids between two infinite horizontal parallel plates induced by the lower plate, which applies oscillatory shear stresses to the fluid Such solutions, which are lacking in the existing literature, are presented as sums of steady-state (permanent or long-time14) and transient components. The influence of the dimensionless pressure– viscosity coefficient on the fluid motion is graphically underlined and discussed
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