Flow of a second grade fluid with fractional derivatives due to a quadratic time dependent shear stress

Abstract

The velocity field and shear stress for unsteady flow of a second grade fluid with fractional derivatives through an infinite long circular cylinder are evaluated. The fluid is initially at rest and at t=0+, the fluid inside the cylinder starts to move longitudinally due to tangential shear stress. Semi analytical solutions are obtained with the help of Laplace transformation and modified Bessel equation. This hybrid technique, which we have used has less computational efforts and time cost as compared to other schemes that are commonly used. The solutions in the transformed domain, are presented in terms of modified Bessel functions I0(·) and I1(·) and satisfy all given conditions. Inverse Laplace transformations have been numerically calculated by using MATLAB. The semi analytical solutions for the motion of second grade fluid with fractional derivatives are reduced to the similar solutions for ordinary second grade and Newtonian fluid. Finally, the effect of different parameters of the flow is graphically examined. Keywords: Second grade fluid, Velocity field, Quadratic shear stress, Fractional derivatives, Laplace transformation, Modified Bessel function