Transient magnetohydrodynamic flow and heat transfer of fractional Oldroyd-B fluids in a microchannel with slip boundary condition

The unsteady magnetohydrodynamic flow of viscoelastic fluids through a parallel plate microchannel under the combined influence of magnetic, electro-osmotic, and pressure gradient forcings is investigated. The fractional Oldroyd-B fluid is used for the constitutive equation to simulate the viscoelastic behavior of fluid in the microchannel. Considering the important role of slip boundary condition in microfluidics, the Navier slip model at wall is adopted. The Laplace and Fourier cosine transforms are performed to derive the analytical expression of velocity distribution. Then, by employing the finite difference method, the numerical solution of the velocity distribution is given. In order to verify the validity of our numerical approach, numerical solutions and analytical solutions of the velocity distribution are contrasted with the exact solutions of the Newtonian fluid in previous work, and the agreements are excellent. Furthermore, based on the values of the velocity distribution for the fully developed flow, the energy equation including volumetric Joule heating, electromagnetic couple effect, and energy dissipation is solved to give the temperature distribution in the microchannel by using the finite difference method. Finally, the influence of fractional parameters and pertinent system parameters on the fluid flow and heat transfer performance and the dependence of the dimensionless Nusselt number Nu on the Hartmann number Ha and Brinkman number Br are discussed graphically.