Exact solution of the Graetz–Brinkman problem extended to non-Newtonian nanofluids flow in elliptical microchannels

The article delves into the Graetz–Brinkman problem, extending it to non-Newtonian nanofluids flow in elliptical microchannels. It begins by discussing the phenomenon of viscous heating, which arises from the internal friction of fluid layers moving at differential velocities. The study builds on earlier works by Hersey and Brinkman, who mathematically confirmed that viscous dissipation is caused by high shear rates and can be quantified by the Brinkman number. The authors present an exact analytical solution using the Generalized Integral Transform Technique (GITT), a method not previously applied to this specific problem. The solution considers the effects of viscous dissipation on the temperature distribution in non-Newtonian fluids, which is crucial for understanding heat transfer in microchannels. The article also includes a numerical approach using the Dynamic Alternating Direction Implicit Method (DADI) for validation purposes. The practical application of the methodology is demonstrated with a non-Newtonian nanofluid flowing in a micro-elliptical channel. This research is significant for industries requiring high-performance thermal management devices, as it provides insights into controlling viscous dissipation to prevent undesirable side effects such as material degradation and heat exchange deficiency.