Heat transfer of fourth-grade fluid flow in the plane duct under an externally applied magnetic field with convection on walls

Abstract

In this paper, a fully developed steady flow of a fourth-grade fluid in the plane duct under an externally applied magnetic field with convection on walls was analyzed numerically by FlexPDE software package. The governing equations, continuity, momentum and energy for this problem are reduced to nonlinear ordinary forms. The momentum nonlinear equation and the resulting energy equation with Robin mixed boundary condition are solved with finite element method (FEM). For validity, the results compare with 4th order Runge–Kutta method. The effect of different physical parameters such as the non-Newtonian parameter, the Biot number, the Hartmann number, the Prandtl number and the Eckert number on the dimensionless velocity profiles, dimensionless temperature profiles, and dimensionless gradient temperature profiles have been discussed. It was concluded that by increasing of non-Newtonian parameter and Biot number the dimensionless velocity, temperature and temperature gradient profiles reduce and thus the heat transfer of fluid flow on the walls decreases. Also, by decreasing the Biot number, increases the dimensionless temperature and the dimensionless temperature profile becomes more uniform. Increasing of Prandtl number decreases the dimensionless temperature and more uniform dimensionless temperature profile within the duct. It was also found that, the FlexPDE software has successfully solved the equations and has offered reasonable results.