Unsteady Motions of a Maxwell Fluid Due to Longitudinal and Torsional Oscillations of an Infinite Circular Cylinder

Flows in the neighborhood of spinning or oscillating bodies are of interest to both academic workers and industry. Among them, the flows between oscillating cylinders are some of the most important and interesting problems of motion near oscillating bodies. As early as 1886, Stokes [1] established an exact solution for the rotational oscillations of an infinite rod immersed in a classical linearly viscous fluid. Casarella and Laura [2] obtained an exact solution for the problem of the rod undergoing both torsional and longitudinal oscillations in a Newtonian fluid. Later, Rajagopal [3] presents two simple but elegant solutions for the flow of a second grade fluid induced by the longitudinal and torsional oscillations of an infinite rod. Their solutions have been recently extended to Oldroyd-B fluids by Rajagopal and Bhatnagar [4]. However, we want to point out that all previous solutions are steady-state solutions, while in order to obtain a starting solution, describing the flow at small and large times after the start of the boundary wall, a transient solution has to be added to the steady-state solution. The aim of this paper is to study the motion of a Maxwell fluid due to the longitudinal and torsional oscillations of an infinite circular cylinder. Actually, we establish the starting solutions corresponding to such flows between infinite concentric circular cylinders and through a circular cylinder. Starting solutions for the motion of a non-Newtonian fluid due to an oscillating wall have been recently established in [5, 6]. These solutions, depending of the initial conditions, are presented as sum of the steady-state and transient solutions. For large times they tend to the steady-state solutions which are independent of initial conditions and periodic in time. Following Rajagopal [3], the steady-state solutions corresponding to the mentioned problems are also presented in simpler forms, in terms of the modified Bessel functions. In the special case, when the relaxation time , all solutions are going to those for a Newtonian fluid. 0 → λ