Exact Solutions for MHD Flow of a Viscoelastic Fluid with the Fractional Burgers' Model in an Annular Pipe
I. Introduction Fluid non-Newtonian does not show a linear relationship between stress and strain rate and received a lot of attention due to the increasing applications of industrial and technological field, such as polymer solutions, paints, blood and heavy oils. No model alone that can describe the behavior of all fluids non-Newtonian because of complex behavior. Thus, it was suggested many of the foundational equations of non-Newtonian fluid models. The Burgers' fluid is one of them which
... one of them which cannot be described a typical relation between shear stress and the rate of strain, for this reason many models of constitutive equations have been proposed for these fluids [7, 8, 11] . Many applications of this type of fluid can be found in [1, 2, 15, 18] . Many of the developments in the theory of viscoelastic flows have been mainly restricted to the formulations of the basics equation and constitutive models [12, 16] , and many applications of fractional calculus can be found in turbulence and fluid dynamics, stochastic dynamical system and nonlinear control theory [3, 14, 19] . Recently, Tong and Liu  studied the unsteady rotating flows of a non-Newtonian fluid in an annular pipe with Oldroyd-B fluid model. Tong ... etc  discussed the flow of Oldroyd-B fluid with fractional derivative in an annular pipe. Hyder ... etc  discussed the flow of a viscoelastic fluid with fractional Burgers' model in an annular pipe. Tong ... etc  discussed the flow of generalized Burgers' fluid in an annular pipe. Khan  investigated the (MHD) flow of generalized Oldroyd-B fluid in a circular pipe. Later on  investigated the slip effects on MHD flow of a generalized Oldroyd-B fluid with fractional derivative. In this paper, our aim is to steady the effects of MHD on the unsteady flow of a viscoelastic fluid with fractional generalized Burgers' fluid model in an annular pipe. The exact solution for velocity distribution is established by using the finite Hankel transform and discrete Laplace transform of the sequential fractional derivatives. D are the fractional differentiation operators of order and based on the Riemann-Liouville definition, defined as Abstract: This paper presents an analytical study for the magnetohydrodynamic (MHD) flow of a generalized Burgers' fluid in an annular pipe. Closed from solutions for velocity is obtained by using finite Hankel transform and discrete Laplace transform of the sequential fractional derivatives. Finally, the figures are plotted to show the effects of different parameters on the velocity profile.