Hydrothermal analysis of non-Newtonian fluid flow(blood) through the circular tube under prescrib e d non-uniform wall heat flux

The present article aims to investigate the Graetz-Nusselt problem for blood as a non-Newtonian fluid obeying the power-law constitutive equation and flowing inside the axisymmetric tube subjected to nonuniform surface heat flux. After the flow field is determined by solving the continuity and the momentum equations, the energy equation is handled by employing the separation of variables method. The resulting Eigen functions and Eigen values are numerically calculated using MATLAB built-in solver BVP4C. The analysis is first conducted for the situation of constant heat flux and subsequently generalized to apply to the case of sinusoidal variation of wall heat flux along the tube length, using Duhamel's Theorem. Furthermore, an approximate analytic solution is determined, employing an integral approach to solve the boundary layer equations. With respect to the comparison, the results of approximate solution display acceptable congruence with those of exact solution with an average error of 7.4%. Interestingly, with decreasing the power-law index, the discrepancy between the two presented methods significantly reduces. Eventually, the influences of the controlling parameters such as surface heat flux and power-law index on the non-Newtonian fluid flow's thermal characteristics and structure are elaborately discussed. It is found that switching from constant wall heat flux to non-uniform wall heat flux that sinusoidally varies along the tube length significantly improves the simulation's accuracy due to the better characterization of the heat transport phenomenon in non-Newtonian fluid flow through the tube. In the presence of sinusoidally varying wall heat flux with an amplitude of 200 W/m 2 and when the power-law index is 0.25, the maximum arterial wall temperature is found to be about 311.56 K.