Flows between two parallel plates of couple stress fluids with time-fractional Caputo and Caputo-Fabrizio derivatives

Abstract

The present work provides a comparative study of the unsteady flows between two parallel plates of a couple stress fluid with two different time-fractional derivatives, namely, Caputo time-fractional derivative (derivative with singular kernel) and Caputo-Fabrizio time-fractional derivative (derivative without singular kernel). The solutions to flows of the ordinary couple stress fluid are obtained as limiting cases, using the properties of the time-fractional derivatives. The analysis result shows that it is more advantageous to use the time-fractional derivatives without singular kernel. Advantages consist both in simpler calculations, and, especially, in the final expressions of solutions which are more appropriate for numerical computations. The solutions of the studied problems are obtained by means of the Laplace transform with respect to the time variable t and the finite Fourier transform with respect to the y-variable. It should be noted that by convenient manipulations of the inverse integral transforms, fluid velocity expressions are written as the sum between the steady-state solution (post-transient solution) and the transient solution. Some numerical calculations are carried out in order to study the influence of the time-fractional derivative order on the fluid velocity, shear stresses and couple stress. Also, the critical time at which the steady flow is obtained was numerically determined. Numerical results are illustrated graphically.