A note on the unsteady torsional sinusoidal flow of fractional viscoelastic fluid in an annular cylinder

In this note, the velocity field and the associated shear stress corresponding to the torsional oscillatory flow of a generalized second grade fluid, between two infinite coaxial circular cylinders, are determined by means of Laplace and Hankel transforms. Initially both cylinders and fluid are at rest and then the two cylinders suddenly start torsional oscillations around their common axis with simple harmonic motions having angular frequencies ω1 and ω2. The solutions that have been obtained are presented under integral and series forms in terms of the generalized G and R functions and satisfy the governing differential equation and all imposed initial and boundary conditions. The respective solutions for the motion between the cylinders, when one of them is at rest, can be obtained from our general solutions. Furthermore, the corresponding solutions for the similar flow of ordinary second grade fluid and Newtonian fluid are also obtained as limiting cases of our general solutions. At the end, flows corresponding to the Newtonian, second grade and generalized second grade fluids are shown graphically by plotting velocity profiles.