oscillating motion of an oldroyd-b fluid with fractional derivatives in a circular cylinder

The velocity field and tangential shear stress for unsteady flow of an oldroyd-b fluid with caputo fractional derivatives through an infinite long cylinder are evaluated. the fluid in the infinitely long cylinder is initially at rest and at t = 0+, due to shear, the fluid starts to oscillate longitudinally. we have solved the fractional model with the tool of laplace and finite hankel transformations. the solutions are in series form and are written in generalized g-function to avoid the entanglement. in limiting cases, the solutions of ordinary oldroyd-b fluid, maxwell fluid with fractional as well as ordinary and newtonian fluid are derived. finally, behavior of different physical parameters on fluid is illustrated by graphs.