Study On The Flow And Heat Transfer Of Viscoelastic Fluid Based On The Theory Of Fractional Calculus

This paper deals with the flow and heat transfer of generilized viscoelastic fluid with fractional derivative. Viscoelastic fluid, a subclass of Non-Newtonian fluids, does not obey the Newtonian postulate that the stress tensor is directly proportional to the rate of deformation tensor. The fractional derivative models of the viscoelastic fluids are modified by replacing the time derivative of an integer order by precisely non-integer order integrals or derivatives from classical equations. The solutions for the velocity field, shear stress and temperature field are established by the integral transforms and numerical methods.Two sections are studied in this paper. The first is to study the flow of fractional viscoelastic fluid by analyze method. In this part, paper research the unsteady rotating flows of a generalized Maxwell fluid with oscillating pressure gradient between coaxial cylinders, and presen an analysis for helical flows of a heated generalized Oldroyd-B fluid subject to a time-dependent shear stress in porous medium, where the motion is due to the longitudinal time-dependent shear stress and the oscillating velocity in boundary. The exact analytical solutions are established by using the sequential fractional derivatives Laplace transform coupled with finite Hankel transforms, presenting in terms of generalized G function and Mittag-Leffler functions. This paper presents research on fractional Magnetohydrodynamic (MHD) Maxwell fluids coupled flow and heat transfer over an infinite plate. The generalized fractional order theory of thermoelectric MHD with relaxation time is considered with Modified Fourier’s and Ohm’s Laws. The analytical solutions are obtained using a Laplace transform method coupled with state-space approach techniques.The other section is to study the flow and heat transfer of fractional viscoelastic fluid with variable viscosity and modified Fourier’s law. First part presents a numerical investigation for unsteady MHD flow of a generalized fractional Maxwell fluid with variable viscosity which the fluid viscosity is considered as power-law-dependent on the strain rate. This paper focuses on the coupling medels of flow and heat transfer of generalized Maxwell fluid in a pourous medium or with free convection. Unlike most classical works, the modified Fourier’s law is described by fractional derivative and the variable viscosity is first taken into consideration to describe the effects of temperature on the fractional viscoelastic flow fields. This paper propose computationally effective implicit numerical methods for this coupled governing equation, which the stability and convergence of the implicit numerical methods are proved by the energy method. And numerical examples are presented to show the effectiveness of approximation.Furthermore, the effects of various pertinent physical parameters (relaxation time, fractional parameter, permeability and porosity, viscosity, Pr, etc) on the flow and heat transfer are analyzed in detail through several graphical illustrations.