The temperature distribution and the heat transfer coefficient for forced convection in laminar channel flow with viscous dissipation are derived for the simplified Phan-Thien–Tanner fluid with a linear stress coefficient. Fully-developed thermal and hydrodynamic conditions are assumed with a constant wall heat flux imposed on both walls. As a simplifying assumption the effect of temperature variations on the material parameters is neglected.
The results show that, in all circumstances, ie for wall heating and cooling and regardless of the magnitude of viscous dissipation, an increase of fluid elasticity and/or an increase of ε results in enhanced heat transfer. As a beneficial consequence the range of temperatures inside the duct is reduced. There is also a coupling effect of viscous dissipation and fluid elasticity: heat transfer enhancement by fluid elasticity is stronger in the presence of a more intense viscous dissipation. For positive wall heat fluxes, ie wall cooling, whenever the Brinkman number exceeds a threshold value, the viscous dissipation overcomes the wall cooling effect and the fluid heats up longitudinally. Fluid elasticity delays this critical Brinkman number to higher values.