Simultaneous Flow of Two Immiscible Fractional Maxwell Fluids with the Clear Region and Homogeneous Porous Medium

One-dimensional transient flows of two layers immiscible fractional Maxwell fluids in a rectangular channel is investigated. The studied problem is based on a mathematical model focused on the fluids with memory described by a constitutive equation with time-fractional Caputo derivative. The flow domain is considered two regions namely one clear region and another filled with a homogeneous porous medium saturated by a generalized Maxwell fluid. Semianalytical and analytical solutions to the problem with initial-boundary conditions and interface fluid-fluid conditions are determined by employing the integral transform method (the Laplace transform and the finite sine-Fourier transform). Talbot’s algorithm for the numerical inversion of the Laplace transforms is employed. The memory effects and the influence of the porosity coefficient on the fluid motion are studied. Numerical results and graphical illustrations obtained using the Mathcad software are utilised to analyze the fluid behavior. The influence of the memory on the fluid motion is significant at the beginning of motion and it is attenuated as time passes by.