Thermal entry flow problem for Giesekus fluid inside an axis-symmetric tube through isothermal wall condition: a comparative numerical study between exact and approximate solution

The thermal entry flow problem also known as the Graetz problem is investigated for a Giesekus fluid model. Both analytical (exact) and approximate solutions for velocity are obtained. The nondimensional pressure gradient is numerically obtained via the mean flow rate relation. The energy equation along with the Giesekus fluid velocity is analytically solved for the constant wall temperature case by using the classical separation of variable method. This method transforms the energy equation into a Sturm-Liouville (SL) boundary value problem. The MATLAB solver bvp5c is employed to compute the eigenvalues and the related eigenfunctions numerically. The impact of mobility parameter and Weissenberg number on local Nusselt number, mean temperature, and average Nusselt number is discussed and displayed graphically. It is also found that the presence of the Weissenberg number elevates the Nusselt numbers. Further, the presence of the mobility parameter of the Giesekus fluid model delays the prevalence fully developed conditions in both entrance and fully developed regions. The comparison between approximate and exact solution is also presented. It reveals that both solutions have an exact match with each other for smaller values of mobility parameter and Weissenberg number. However, there is a deviation for larger values of both parameters.