Numerical analysis of two-dimensional MHD flow and heat transfer of generalized Maxwell fluid through a rectangular pipe

The study of viscoelastic fluids is of great significance and has received considerable attention in recent decades. In this work, the numerical analysis for the two-dimensional MHD flow and heat transfer of generalized Maxwell fluid through the rectangular pipe is implemented and the nonlinear behaviors of the fluid are characterized. Temporal and spatial fractional derivatives are introduced to depict the transport mechanism of the fluid. And the coupled nonlinear fractional governing equations are proposed to character the flow and heat transfer processes of the fluid under the action of magnetic fields and pressure gradients. Then, finite difference algorithm is developed and the coupled alternating direction implicit algorithms are presented for the coupled fractional systems. The validity and accuracy of the numerical algorithm is tested by comparing numerical solutions with exact solutions of a numerical example. Moreover, the consistency between our numerical solutions and the results of previous work also indicates the feasibility of the numerical method. The influences of pressure gradient, fractional parameters and other related parameters on fluid flow velocity and temperature are displayed, graphically. The results indicate that compared to temperature, pressure gradient has a greater impact on fluid velocity and promotes fluid flow. The changes in other model parameters result in different trends in fluid velocity and temperature distribution, reflecting the significant role of each parameter in fluid flow and heat transfer behavior.