Numerical study of heat transfer of a third grade viscoelastic fluid in non-Darcy porous media with thermophysical effects

A numerical solution is presented for the natural convective dissipative heat transfer of an incompressible, third grade, non-Newtonian fluid flowing past an infinite porous plate embedded in a Darcy–Forchheimer porous medium. The mathematical model is developed in an (x,y) coordinate system. Using a set of transformations, the momentum equation is rendered one-dimensional and a partly linearized heat conservation equation is derived. The viscoelastic formulation presented by Akyildiz (2001 Int. J. Non-Linear Mechanics 36 349–52) is adopted, which generates lateral mass and viscoelastic terms in the heat conservation equation, as well as in the momentum equation. A number of special cases of the general transformed model are discussed. A finite element method is implemented to solve, with appropriate boundary conditions, the coupled third-order, second degree ordinary differential equation for momentum and the second-order, fourth degree heat conservation equation. We study the influence of the third grade viscoelastic parameter (β3), Darcian parameter (inversely proportional to permeability (kp)), the Forchheimer inertial parameter (b), transpiration velocity (Vo), the transpiration parameter in the heat equation (R) and the thermal conductivity parameter (S) on momentum and heat transfer. Additionally, we study the influence of the Forchheimer inertial parameter (b) on second-order viscoelastic non-Darcy free convection flow and also the effects of the third grade parameter (β3) on Darcian free convection. Velocities increase with rising permeability (Darcian parameter) for both second and third grade viscoelastic free convection regimes and decrease with rising Forchheimer parameter. The effects of the other parameters are described at length. The flow scenario is important in chemical engineering processes.