# Non-Newtonian fluids: From concepts to modeling

Most of us, engineers and non-engineers alike, tend to think we can tell a "solid" from a "fluid". But how do we define the characteristic difference?

Most physics textbooks use the concepts of "elastic/plastic deformation" to define solids and "viscous deformation" to define fluids. In simple terms, a solid undergoes deformation under stress, while a fluid exhibits continual deformation (i.e., it flows). And when we think of fluids, we assume the concept of viscosity (the ratio of applied stress to the rate of deformation) to be a material property, independent of flow conditions or applied stresses.

As always, real-world materials do not always conform to these simple constructs. To be sure, for a large class of fluids of engineering relevance, viscosity can be modeled as a material property—these Newtonian fluids include water, air, hydrocarbons, and oil. We know how to model the fluid flow and heat transfer for these fluids. The simple relationship between shear stress and the rate of deformation (or the strain rate) allows us to solve mass, momentum, and energy transport equations and leads to the well-known formulations used in hand-calculations, software tools in Xist®, and CFD.

Other fluids do not conform to this simple construct. These non-Newtonian fluids include polymers, suspensions (like toothpaste and ceramics), foams, and slurries. Models have been developed to describe many non-Newtonian behaviors.

One such behavior is shear thickening, where application of shear stress increases the resistance to flow (or apparent viscosity). The material oobleck (a mixture of cornstarch and water, not just a made-up word by Dr. Seuss) shows classic shear-thickening behavior: the mixture is runny when stirred with a spoon but hardens when slapped, punched, or even walked on! Fluids like ketchup and paint act the opposite way, that is, they are shear thinning.

Both shear-thinning (pseudoplastic) and shear-thickening (dilatant) fluids have a non-linear relationship between shear stress and strain rate, where a power law can describe the apparent viscosity. Note that the apparent viscosity is no longer a physical property but just a convenient (albeit inaccurate) method of trying to model non-Newtonian fluids within the Newtonian framework.

Another non-Newtonian behavior is time dependences of apparent viscosity: rheopectic fluids show an increase in apparent viscosity with time, while thixotropic fluids show a reduction with time. Such behaviors point to the complex molecular structure of such fluids, as well as how molecular interactions can degrade or modify over time or under applied stress.

Hysteresis in the shear stress and strain rate curve is often a manifestation of such behavior. For example, for a fluid that is both thixotropic and pseudoplastic, ramping up the strain rate from low to high leads to a drop in viscosity. If this test is followed by a ramp down back to the original low strain rate, the apparent viscosity does not build back up to the original value.

The topic of non-Newtonian behaviors is like quicksand (another non-Newtonian fluid). It is easy to get drawn into and hard not to drown in the complexities.

Along with the aforementioned behaviors, some materials exhibit viscoelasticity, a combination of solid-like and fluid-like behaviors. The concept of yield stress further blurs the line between solids and fluids. There is fierce debate on the very concept of a solid: some have claimed, like Heraclitus, that "panta rei"—everything flows. Professor Eugene C. Bingham performed experiments on stones and determined their viscosities in the order of 1016 Pa-s: extending time to geological scales, even mountains flow.

As practical engineers, we can set aside such philosophical considerations. To characterize a non-Newtonian fluid, we must first understand the behaviors exhibited within reasonable time scales. To do so, we can perform standard rheological tests followed by controlled fluid flow and heat transfer experiments.

HTRI selected ketchup as an entry point to understanding and modeling non-Newtonian fluids, based on a member survey that indicated interest in pseudoplastic fluids. Figure 1 shows some pressure drop results obtained by running ketchup through the Tubeside Single-Phase Unit (TSPU), driven by a progressive cavity pump. Ketchup exhibited well-defined power-law behavior, with the consistency index K showing a linear trend with temperature. We also observed thixotropy both over the short and long term, with the K value reducing over time, and the power n remaining virtually unchanged. Although rheological tests indicated the presence of yield stress, accounting for it was not important once the ketchup started flowing.

Our ketchup tests in the TSPU have concluded. With steam heating and water cooling the ketchup, we were able to identify methods to calculate the thermal performance and pressure drop wall correction factor.

And while ketchup is just one of many non-Newtonian fluids, this project provided critical data that allow us to validate and modify power law fluid heat transfer and pressure drop models. This step enables us to incorporate improved models of non-Newtonian fluids in HTRI software.