**Significance **

Different types of fluids are important both in engineering applications and in nature. Among them, pseudoplastic and dilatant fluids have drawn significant research attention in recent years. Pseudoplastic fluids are purely viscous, time-independent, non-Newtonian fluids with no yield stress whose apparent viscosity decreases with increasing shear rate. Dilatant fluids are similar except that they exhibit increasing apparent viscosity. As such, their flow curves are characterized by five continuous regions, I-V. Beginning with Newtonian behavior (I) at near-zero shear rates, they transition smoothly (II) to power-law behavior (III) at intermediate shear rates, and then revert again smoothly (IV) to Newtonian behavior (V) at high shear rates. Thus, their apparent viscosity depends both on material properties and on the shear rates in the flow field, which can generally span multiple regions. This phenomenon may have a significant impact on hydrodynamic and heat transfer solutions of flow problems. For instance, using a model that extends Region III behavior throughout the entire shear rate regime produces results that are necessarily limited to cases where shear rates within the flow field are predominantly in the power-law region. Despite this being a necessary condition for the solution to apply, it is insufficient, as was shown in this study. The fluid must also satisfy another criterion based on fluid properties alone, namely, that the zero and infinite shear rate Newtonian viscosities differ by four orders of magnitude or more, here termed a “strong” non-Newtonian fluid (otherwise called “weak”). Other more complete models, for example, the modified power-law (MPL) model, describe the behavior across a broader span of the flow curve, in this case Regions I, II, and III, with Region III extending indefinitely with increasing shear rate. This applies to fluids that either exhibit no return to Newtonian behavior, or strong non-Newtonian fluids whose shear rates are mainly in those three regions.

Dr. Massimo Capobianchi and Dr. Richard Cangelosi from Gonzaga University, in collaboration with Dr. Patrick McGah from the University of Washington, investigated the heat transfer problem in laminar flows of dissipative pseudoplastic and dilatant fluids within the hydrodynamically and thermally fully-developed region (H&TFDR) of circular conduits. This problem has received much attention owing to its practical importance. But despite solutions existing using power law and MPL models, this problem is only partly understood for the reasons described above.

In particular, they solved the heat transfer problem using a model for the apparent viscosity, called the extended modified power law (EMPL) model, that is valid throughout all five regions of the flow curve. They determined the Nusselt number for two boundary conditions, constant temperature and constant heat flux, for various values of the Brinkman number, including 0 (non-dissipative fluids). The solutions are thus valid for whatever shear rates may exist in the flow field and for both strong and weak non-Newtonian fluids. The algorithm used to solve the governing equations numerically was described and validated, and the obtained results were discussed in detail. Their research work is currently published in the *Journal of Heat Transfer*.

Nusselt numbers were presented as a function of a dimensionless shear rate parameter whose value characterizes the shear rate region where the system is operating. Plotted on a log scale, the left endpoint of this parameter indicates operation in Region I, Region V operation is at the opposing endpoint, and Region III is at the midpoint. The results were u-shaped curves, concave down for pseudoplastic fluids and up for dilatants, with endpoints at the Newtonian values. However, solutions at the midpoints only approached power-law values as the limiting Newtonian viscosity ratio increased, ultimately reaching them within a few percent when it approached 10^{|4|}. Therefore, the Nusselt number was observed to always be between the Newtonian and power-law solutions, and the parameter values where these asymptotic behaviors reside were quantified.

In summary, the Nusselt numbers for laminar flows of dissipative pseudoplastic and dilatant fluids flowing within the H&TFDR of circular tubes were computed for the constant temperature and the constant heat flux boundary conditions. The presented approach allows the determination of the Nusselt number for any flow situation and thus supplements the current literature and quantifies the conditions where power law and Newtonian solutions apply.

**Reference**

Capobianchi, M., Cangelosi, R., & McGah, P. (2021). **Heat Transfer in Fully Developed, Laminar Flows of Dissipative Pseudoplastic and Dilatant Fluids in Circular Conduits**. Journal of Heat Transfer, 143(3), 031801-031813.