The wonderful realm of non-Newtonian fluids

The wonderful realm of non-Newtonian fluids

Newton's Viscosity Law and Non-Newtonian Fluids

In 1687, Newton published experimental results on one-dimensional shear flow using water as the working medium. The experiment was conducted with water filling the space between two parallel plates (Figure 1): the lower plate was fixed, while the upper plate moved to the right at a constant velocity U in its own plane. At this time, the velocities of fluid particles attached to the upper and lower plates were U and 0, respectively, and the velocity distribution between the two plates was linear. From this, the famous Newton's viscosity law was derived:

Figure 1: Fluid between two relatively moving plates

In the formula,  is the shear stress acting on the fluid plane of the upper plate, \(du/dy\) is the shear strain rate, and the slope is the viscosity coefficient.

In 1845, based on Newton's experimental law, Stokes made three assumptions: the stress tensor is a linear function of the strain rate tensor, the fluid is isotropic, and the strain rate is zero when the fluid is at rest. From these, he derived the widely used linear constitutive equation in fluid mechanics research, as well as the now widely applied Navier-Stokes equations.

Subsequent research revealed that Newton's viscosity law (and the Navier-Stokes equations established on its basis) is suitable for describing low-molecular-weight fluids such as water and air, but not for high-molecular-weight fluids. For the latter, the relationship between shear stress and shear strain rate no longer satisfies linearity. To distinguish them, fluids where shear stress and shear strain rate have a linear relationship are called Newtonian fluids, while those that do not are called non-Newtonian fluids.

A Variety of Non-Newtonian Fluids

Non-Newtonian fluids existed long before the emergence of humans, as most biological fluids fall into the category of non-Newtonian fluids as defined today [1]. Various body fluids in humans, such as blood, lymph, and cyst fluid, as well as "semi-fluids" like cytoplasm, are non-Newtonian fluids. Nowadays, one of the items in hospital blood tests is no longer referred to as "blood viscosity test" but "hemorheology test" (abbreviated as blood rheology). This is because for blood, the relationship between shear stress and shear strain rate is no longer linear, making it impossible to describe the mechanical properties of blood with a single slope (i.e., viscosity).

One of the main drivers for the rapid development of non-Newtonian fluid research in recent decades is the growth of the polymer industry. Polyethylene, polyacrylamide, polyvinyl chloride, nylon 6, PVS, celluloid, polyester, rubber solutions, various engineering plastics, and melts and solutions of chemical fibers are all non-Newtonian fluids.

Petroleum, mud, coal-water slurry, ceramic slurry, paper pulp, paint, ink, toothpaste, regenerated solution of silk fibroin, well-flushing fluid and completion fluid for drilling, magnetic slurry, coating fluids of some photosensitive materials, foam, liquid crystals, high-sediment-laden flow, debris flow, and the Earth's mantle are also non-Newtonian fluids.

Non-Newtonian fluids are also common in the food industry , such as tomato juice, starch solution, egg white, apple pulp, vegetable soup, thick sugar water, soy sauce, jam, condensed milk, agar, potato pulp, melted chocolate, dough, rice flour dough, and various minced food materials like fish paste and meat paste.

In summary, various complex fluids encountered in daily life and industrial production—such as polymer solutions, melts, pastes, gels, cross-linked systems, and suspension systems—are almost all non-Newtonian fluids. Sometimes, for industrial production purposes, adding polymers to a Newtonian fluid can improve its performance while converting it into a non-Newtonian fluid, such as fracturing fluids used to increase oil production and new lubricants.

Wonderful Properties and Applications of Non-Newtonian Fluids

1. Jet Swelling

When a non-Newtonian fluid is forced to flow from a large container into a capillary and then out of the capillary, it can be observed that the diameter of the jet is larger than that of the capillary (Figure 2). The ratio of the jet diameter to the capillary diameter is called the die swell ratio (also known as the extrudate swell ratio). For Newtonian fluids, this ratio depends on the Reynolds number and ranges approximately from 0.88 to 1.12. For polymer melts or concentrated solutions, the ratio is much larger, even exceeding 10. Generally, the die swell ratio is a function of the flow rate and capillary length.

Figure 2: Jet Swelling

The die swell phenomenon is crucial in die design. When a polymer melt flows out of a nozzle with a rectangular cross-section, the swelling at the long side of the cross-section is more significant than at the short side, with the maximum swelling occurring at the center of the long side (Figure 3). Therefore, if the product is required to have a rectangular cross-section, the die shape cannot be rectangular but must be as shown in Figure 4.

Figure 3: Jet Swelling from a Rectangular Cross-Section Nozzle

Figure 4: Designed Shape of the Die

This jet swelling phenomenon is also called the Barus effect or Merrington effect.

2. Weissenberg Effect (Climbing Rod Effect)

In 1944, Weissenberg publicly demonstrated an interesting experiment at Imperial College London, UK. In a beaker containing a viscoelastic fluid (a type of non-Newtonian fluid), a rod was rotated. For Newtonian fluids, the liquid surface becomes concave due to centrifugal force (Figure 5(a)); for viscoelastic fluids, however, the fluid moves toward the center of the beaker and climbs up the rod, making the liquid surface convex (Figure 5(b)). This phenomenon can be observed even at very low rotation speeds of the rod.

The climbing rod effect, also known as the Weissenberg effect, must be considered in the design of mixers. Similarly, this effect should be taken into account and utilized in the design of pumps for transporting non-Newtonian fluids.

Figure 5: Climbing Rod Effect Experiment

3. Tube-Free Siphon

For Newtonian fluids, in a siphon experiment, if the siphon tube is lifted off the liquid surface, the siphon stops immediately. However, for polymer liquids—such as polyisobutylene in gasoline solution, 1% POX aqueous solution, or a slight gel system of polysaccharides in water—it is easy to demonstrate the tube-free siphon experiment. When the tube is slowly pulled out of the container, it can be seen that even though the tube is no longer inserted into the fluid, the fluid continues to be drawn from the beaker into the tube (Figure 6). Even more simply, without a siphon tube at all, tilting a beaker filled with such fluid to make the fluid flow down will start a process that does not stop until the beaker is empty (Figure 7). This tube-free siphon property is the basis for the spinnability of synthetic fibers.

Figure 6: Tube-Free Siphon

Figure 7: Tube-Free Siphon

4. Turbulent Drag Reduction

Another remarkable property of non-Newtonian fluids is turbulent drag reduction. It has been observed that adding a small amount of polymer to a Newtonian fluid can significantly reduce the pressure difference at a given flow rate. Figure 8 shows the relationship curves between the pipe friction coefficient f and the Reynolds number R for two polyethylene oxide solutions with different concentrations. Turbulence has long been an unsolved problem in fluid mechanics; however, adding a small amount of polymer additives to Newtonian fluids produces a drag reduction effect. Some reports indicate that after adding polymer additives, the burst period increases, which is believed to be due to the action of polymer chains.

Figure 8: Turbulent Drag Reduction

The drag reduction effect is also called the Toms effect. Although the principle is not fully understood, it has seen good applications. Adding a small amount of polyethylene oxide to fire-fighting water can more than double the head of water sprayed from fire truck nozzles. The use of polymer additives can also alter the occurrence and destructive effects of cavitation.

In addition to the above properties, non-Newtonian fluids have other interesting characteristics that have attracted attention, such as the "stringing effect" (small liquid rods connecting droplets formed by free jets), spinnability (ability to be stretched into extremely fine filaments, see [3]), shear thinning, and liquid rebound. Interested readers can learn more from relevant literature .

Since non-Newtonian fluids are involved in the processes, equipment, efficiency, and product quality of many industrial sectors, as well as human life and health, they have received increasing attention from scientists. At the 19th International Union of Theoretical and Applied Mechanics (IUTAM) Congress held at the Kyoto International Conference Center in Japan in August 1996, non-Newtonian fluid flow was one of the six key themes of the conference and the most actively participated theme in fluid mechanics [5]. Crochet's invited report pointed out that the properties of polymer solutions and melts are far different from those of Newtonian fluids, and that research on these abnormal properties is challenging.

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