Multi-layer flows of immiscible fractional second grade fluids in a rectangular channel

Multi-layer laminar unsteady flows of immiscible fractional second grade fluids in a rectangular channel made by two parallel plates are studied. The fluid motion is produced by the motion of parallel walls in their plane and by the time-dependent pressure gradient in the presence of the linear fluid-fluid interface conditions. The mathematical model is based on the generalized constitutive equations for the shear stress described by the time-fractional Caputo derivative. Integral transforms (finite Fourier sine transform and Laplace transform) have been used to obtain analytical and semi-analytical solutions for velocity, shear stress and the temperature fields. In the case of semi-analytical solutions, the Talbot's algorithms are used for the inverse Laplace transform. The numerical calculations are carried out with the help of Mathcad software, and the results are illustrated graphically. It has been found that the memory effects have a significant influence on the motion of the fluids.