Axial　magnetohydrodynamic　(MHD)　flows　for　Oldroyd-B　fluid　are　investigated　between　two　cylinders.　The　motion　of　the　fluid　is　produced　by　the　two　oscillating　cylinders.　The　fractional　calculus　approach　is　introduced　to　establish　the　constitutive　relationship　of　a　viscolastic　fluid.　Velocity　field　and　shear　stress　of　the　motion　are　determined　in　terms　of　Bessel　function　and　generalized　Mittag-Leffler　function　by　using　Laplace　transform　and　Hankel　transform.　The　influence　of　pertinent　parameters　on　the　flows　is　delineated　and　appropriate　conclusions　are　drawn.