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In this research, a fully developed steady flow of a third-grade fluid in a pipe under an externally applied magnetic field with convection on wall is investigated. The governing equations including momentum and energy in the form of partial differential equations are reduced to ordinary differential equations which are solved numerically by using a finite element method (FEM) as part of the FlexPDE software package. For validity, the results are compared with the 4 th  order Runge-Kutta method. The effect of different physical parameters such as the non-Newtonian parameter, the Biot number, the Hartmann number, the Eckert number on the dimensionless velocity profiles, the dimensionless velocity gradient profiles, the dimensionless temperature profiles, and the dimensionless gradient temperature profiles have been discussed. It is concluded that by increasing the non-Newtonian parameter and Hartman number the dimensionless velocity, the velocity gradient, the temperature and temperature gradient profiles reduce and thus the heat transfer of fluid flow, the shear stress and the skin friction on the pipe wall decrease. Increasing the Biot number caused a decrease of the temperature and a more uniform dimensionless temperature profile of the fluid within the pipe. Besides, with a decrease of the Prandtl number, the dimensionless temperature decreases inside the pipe. In fact, the dimensionless temperature profile becomes flat. For this reason, the dimensionless temperature gradient decreases on the pipe wall which causes the reduction of the heat transfer rate on the pipe wall. Further, by increasing the Eckert number, the dimensionless temperature of the fluid within the pipe and the heat transfer from the fluid to the pipe wall increases. Applying the FlexPDE software for solving governing equations numerically seems to lead to appropriate and reasonable results.

Heat Transfer of Third-grade Fluid Flow in a Pipe under an Externally Applied Magnetic Field with Convection on Wall