Unsteady helical flows of a generalized Oldroyd-B fluid

Abstract

Helical flows for Oldroyd-B fluid are studied in concentric cylinders and a circular cylinder. The fractional calculus approach in the constitutive relationship model Oldroyd-B fluid is introduced and a generalized Jeffreys model with the fractional calculus has been built. Exact solutions of some unsteady helical flows of Oldroyd-B fluid in an annular pipe are obtained by using Hankel transform and Laplace transform for fractional calculus. For α = β = 1 , λ 1 = 0 , λ 2 = 0 or λ 1 = λ 2 = 0 they reduce to the similar solutions for a Maxwell fluid, a second grade or Newtonian fluid, respectively. Numerical results for helical flow in pipe are presented.