Caputo-Fabrizio Fractionalized Second Grade Fluid in a Circular Cylinder with Uniform Magnetic Field

In this communication, magnetohydrodynamics (MHD) fractional second grade fluid in a circular pipe is observed. Initially the whole system which consists on a circular cylinder filled with fluid having infinite length is at rest. Suddenly, the pipe set in translational motion along its axis due to stress i.e time dependent, on the boundary of pipe. The fluid inside the circular pipe which is electrically conducting, gradually starts moving by the effect of cylinder's motion. In the governing equations of second grade fluid, an innovative formulation of fractional derivative (FD) with specific kernel which is without singularity in a prescribed domain, given by Caputo and Fabrizio jointly is used, which is more suitable for viscoelastic and electromagnetic systems as compare to usual fractional derivative. To analyze above flow problem we use Laplace transform and modified Bessel functions. For the calculation of inverse Laplace transform, we apply Stehfest's algorithm with the help of MATHCAD software instead of lengthy and complicated calculations. For the validity of the results, a very good comparison between existing analytical solution and our approximate results is obtained, which shows that both the solutions are equivalent. Finally, the effect of different parameters and their comparison are explained graphically.