### Exact analytic solutions for the flow of second grade fluid between two longitudinally oscillating cylinders

**Amir Mahmood, Najeeb Alam Khan, Corina Fetecau, Muhammad Jamil, Qammar Rubbab**

**Abstract:**The velocity field and associated shear stress corresponding to the longitudinal oscillatory flow of a second grade fluid, between two infinite coaxial circular cylinders, are determined by means of Laplace and Hankel transforms. The flow is due to both of the cylinders that at \(t = 0^+\) suddenly begin to oscillate along their common axis with different angular frequencies of their respective velocities. The 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 Newtonian fluid are also obtained as limiting case. The flows of second grade and Newtonian fluids are compared graphically by plotting their velocity profiles.