Unsteady flows of Maxwell fluids with shear rate memory and pressure-dependent viscosity in a rectangular channel

Unsteady flows of the incompressible upper-convected Maxwell fluids with the shear rate memory and pressure-dependent viscosity of exponential form, within a rectangular channel are studied. To consider memory effects, the generalized constitutive equations described by fractional differential equations with time-fractional Caputo-Fabrizio derivative have been considered in the mathematical model. The comparison between the two models, fractional and ordinary is made by means of the weight function from the expression of the shear stress. Semi-analytical solutions for velocity, shear stress, and normal stress are determined using the Laplace transform, the Stehfest's numerical algorithm for the inversion of the Laplace transforms and an adequate numerical scheme for the Caputo-Fabrizio derivative. A particular case characterized by the Neumann condition on the bottom plate and Dirichlet condition on the upper plate is investigated. It is found that the memory effect is significant only for small values of the time t. For large values of the time t, the differences between fractional and ordinary models are negligible.