First exact solutions for mixed boundary value problems concerning the motions of fluids with exponential dependence of viscosity on pressure

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.