Analytical solutions of upper-convected Maxwell fluid flow with exponential dependence of viscosity on the pressure

Abstract

Exact and simple expressions for the permanent solutions corresponding to two oscillatory motions of incompressible upper-convected Maxwell fluids with exponential dependence of viscosity on the pressure between parallel plates have been established using suitable changes of the spatial variable and the unknown function and the Laplace transform technique. The solutions that have been obtained satisfy the boundary conditions and governing equations but are independent of the initial conditions. They are important for those who want to eliminate the transients from their experiments. The similar solutions for the simple Couette flow of the same fluids as well as known results for the Newtonian fluids performing the same motions were obtained as limiting cases. The convergence of starting solutions to the corresponding permanent components that has been graphically proved could constitute an asset on the correctness of obtained results. The influence of pertinent parameters on the fluid motion and the spatial profiles of starting solutions have been graphically depicted and discussed. The oscillations’ amplitude is an increasing function with respect to the dimensionless pressure–viscosity coefficient and the Weissenberg number. It is lower for the shear stress as compared to the fluid velocity. The three-dimensional distribution of the starting velocity fields has been numerically visualized by means of the two-dimensional contour graphs.