Exact Solutions for a Rotational Flow of Generalized Second Grade Fluids Through a Circular Cylinder

TL;DR

This paper derives exact analytical solutions for the velocity and shear stress in a generalized second grade fluid flowing rotationally inside an infinite circular cylinder, using integral transforms and special functions.

Contribution

It provides explicit solutions for generalized second grade fluids under rotational flow, including special cases for Newtonian and ordinary second grade fluids.

Findings

Solutions expressed in integral and series forms using generalized G-functions.

The velocity and shear stress are decomposed into Newtonian and non-Newtonian parts.

Special cases recover classical Newtonian and second grade fluid solutions.

Abstract

In this note the velocity field and the associated tangential stress corresponding to the rotational flows 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 $\beta = 1$.