Exact solutions for a rotational flow of generalized second grade fluids through a circular cylinder

In this note the velocity field and the associated tangential stress corresponding to the rotational flow of a generalized second grade fluid within an infinite circular cylinder are determined by means of the Laplace and Hankel transforms. At time $t=0$ the fluid is at rest and the motion is produced by the rotation of the cylinder, around its axis, with the angular velocity $\Omega t$. The velocity field and the adequate shear stress are presented under integral and series forms in terms of the generalized $G$-functions. Furthermore, they are presented as a sum between the Newtonian solutions and the adequate non-Newtonian contributions. The corresponding solutions for the ordinary second grade fluid and Newtonian fluid are obtained as particular cases of our solutions for $\beta=1$, respectively $\alpha=0$ and $\beta=1$.