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In physics and chemistry, a non-Newtonian fluid is a fluid that does not follow Newton's law of viscosity, that is, it has variable viscosity dependent on stress. In particular, the viscosity of non-Newtonian fluids can change when subjected to force. Ketchup, for example, becomes runnier when shaken and is thus a non-Newtonian fluid. Many salt solutions and molten polymers are non-Newtonian fluids, as are many commonly found substances such as custard, toothpaste, starch suspensions, corn starch, paint, blood, melted butter and shampoo.

Most commonly, the viscosity (the gradual deformation by shear or tensile stresses) of non-Newtonian fluids is dependent on shear rate or shear rate history. Some non-Newtonian fluids with shear-independent viscosity, however, still exhibit normal stress-differences or other non-Newtonian behavior. In a Newtonian fluid, the relation between the shear stress and the shear rate is linear, passing through the origin, the constant of proportionality being the coefficient of viscosity. In a non-Newtonian fluid, the relation between the shear stress and the shear rate is different. The fluid can even exhibit time-dependent viscosity. Therefore, a constant coefficient of viscosity cannot be defined.

Although the concept of viscosity is commonly used in fluid mechanics to characterize the shear properties of a fluid, it can be inadequate to describe non-Newtonian fluids. They are best studied through several other rheological properties that relate stress and strain rate tensors under many different flow conditions—such as oscillatory shear or extensional flow—which are measured using different devices or rheometers. The properties are better studied using tensor-valued constitutive equations, which are common in the field of continuum mechanics.

For non-Newtonian fluid's viscosity, there are pseudoplastic, plastic, and dilatant flows that are time-independent, and there are thixotropic and rheopectic flows that are time-dependent. Three well-known time-dependent non-newtonian fluids which can be identified by the defining authors are the Oldroyd-B model, Walters’ Liquid B and Williamson fluids.

Time-dependent self-similar analysis of the Ladyzenskaya-type model with a non-linear velocity dependent stress tensor was performed unfortunately no analytical solutions could be derived, however a rigorous mathematical existence theorem was given for the solution.

For time-independent non-Newtonian fluids the known analytic solutions are much broader

Types of non-Newtonian behavior

Summary

Shear thickening fluid

The viscosity of a shear thickening – i.e. dilatant – fluid appears to increase when the shear rate increases. Corn starch suspended in water ("oobleck", see below) is a common example: when stirred slowly it looks milky, when stirred vigorously it feels like a very viscous liquid.

Shear thinning fluid

A familiar example of the opposite, a shear thinning fluid, or pseudoplastic fluid, is wall paint: The paint should flow readily off the brush when it is being applied to a surface but not drip excessively. Note that all thixotropic fluids are extremely shear thinning, but they are significantly time dependent, whereas the colloidal "shear thinning" fluids respond instantaneously to changes in shear rate. Thus, to avoid confusion, the latter classification is more clearly termed pseudoplastic.

Another example of a shear thinning fluid is blood. This application is highly favoured within the body, as it allows the viscosity of blood to decrease with increased shear strain rate.

Bingham plastic

Fluids that have a linear shear stress/shear strain relationship but require a finite yield stress before they begin to flow (the plot of shear stress against shear strain does not pass through the origin) are called Bingham plastics. Several examples are clay suspensions, drilling mud, toothpaste, mayonnaise, chocolate, and mustard. The surface of a Bingham plastic can hold peaks when it is still. By contrast Newtonian fluids have flat featureless surfaces when still.

Rheopectic or anti-thixotropic

There are also fluids whose strain rate is a function of time. Fluids that require a gradually increasing shear stress to maintain a constant strain rate are referred to as rheopectic. An opposite case of this is a fluid that thins out with time and requires a decreasing stress to maintain a constant strain rate (thixotropic).

Examples

Many common substances exhibit non-Newtonian flows. These include:

• Soap solutions, cosmetics, and toothpaste
• Food such as butter, cheese, jam, mayonnaise, soup, taffy, and yogurt
• Natural substances such as magma, lava, gums, honey, and extracts such as vanilla extract
• Biological fluids such as blood, saliva, semen, mucus, and synovial fluid
• Slurries such as cement slurry and paper pulp, emulsions such as mayonnaise, and some kinds of dispersions

Oobleck

An inexpensive, non-toxic example of a non-Newtonian fluid is a suspension of starch (e.g., cornstarch/cornflour) in water, sometimes called "oobleck", "ooze", or "magic mud" (1 part of water to 1.5–2 parts of corn starch). The name "oobleck" is derived from the Dr. Seuss book Bartholomew and the Oobleck.

Because of its dilatant properties, oobleck is often used in demonstrations that exhibit its unusual behavior. A person may walk on a large tub of oobleck without sinking due to its shear thickening properties, as long as the individual moves quickly enough to provide enough force with each step to cause the thickening. Also, if oobleck is placed on a large subwoofer driven at a sufficiently high volume, it will thicken and form standing waves in response to low frequency sound waves from the speaker. If a person were to punch or hit oobleck, it would thicken and act like a solid. After the blow, the oobleck will go back to its thin liquid-like state.

Flubber (slime)

Flubber, also commonly known as slime, is a non-Newtonian fluid, easily made from polyvinyl alcohol–based glues (such as white "school" glue) and borax. It flows under low stresses but breaks under higher stresses and pressures. This combination of fluid-like and solid-like properties makes it a Maxwell fluid. Its behaviour can also be described as being viscoplastic or gelatinous.

Chilled caramel topping

Another example of non-Newtonian fluid flow is chilled caramel ice cream topping (so long as it incorporates hydrocolloids such as carrageenan and gellan gum). The sudden application of force—by stabbing the surface with a finger, for example, or rapidly inverting the container holding it—causes the fluid to behave like a solid rather than a liquid. This is the "shear thickening" property of this non-Newtonian fluid. More gentle treatment, such as slowly inserting a spoon, will leave it in its liquid state. Trying to jerk the spoon back out again, however, will trigger the return of the temporary solid state.

Silly Putty

Silly Putty is a silicone polymer based suspension that will flow, bounce, or break, depending on strain rate.

Plant resin

Plant resin is a viscoelastic solid polymer. When left in a container, it will flow slowly as a liquid to conform to the contours of its container. If struck with greater force, however, it will shatter as a solid.

Quicksand

Quicksand is a shear thinning non-Newtonian colloid that gains viscosity at rest. Quicksand's non-Newtonian properties can be observed when it experiences a slight shock (for example, when someone walks on it or agitates it with a stick), shifting between its gel and sol phase and seemingly liquefying, causing objects on the surface of the quicksand to sink.

Ketchup

Ketchup is a shear thinning fluid. Shear thinning means that the fluid viscosity decreases with increasing shear stress. In other words, fluid motion is initially difficult at slow rates of deformation, but will flow more freely at high rates. Shaking an inverted bottle of ketchup can cause it to transition to a lower viscosity through shear thinning, making it easier to pour from the bottle.

Dry granular flows

Under certain circumstances, flows of granular materials can be modelled as a continuum, for example using the μ(I) rheology. Such continuum models tend to be non-Newtonian, since the apparent viscosity of granular flows increases with pressure and decreases with shear rate. The main difference is the shearing stress and rate of shear.

See also

• Complex fluid
• Dilatant
• Dissipative particle dynamics
• Generalized Newtonian fluid
• Herschel–Bulkley fluid
• Liquefaction
• Navier–Stokes equations
• Newtonian fluid
• Pseudoplastic
• Quicksand
• Quick clay
• Rheology
• Superfluids
• Thixotropy
• Weissenberg effect

References

1. Ouellette, Jennifer (2013). "An-Ti-Ci-Pa-Tion: The Physics of Dripping Honey". Scientific American.
2. Oldroyd, J. (1950). "On the Formulation of Rheological Equations of State". Proc. R. Soc. Lond. Ser. A Math. Phys. Sc. 200: 523–541. doi:10.1098/rspa.1950.0035.
3. Walters, K. (1963). "Non -Newtonian effects in some elastic-viscous liquids whose behavior at small rates of shear is characterized by a general linear equations of state". Quart. J. Mech. Appl. Math. 6: 63.
4. Williamson, R.V. "The flow of pseudoplastic materials". Ind. Eng. Chem. 21: 1108. doi:10.1021/ie50239a035.
5. I.F., Barna; Bognár, G.; Hriczó, K. (2016). "Self-Similar Analytic Solution of the Two-Dimensional Navier-Stokes Equation witha Non-Newtonian Type of Viscosity". Mathematical Modelling and Analysis. 21 (1): 83–94. arXiv:1410.1303. doi:10.3846/13926292.2016.1136901.
6. Wei, D.; Al-Ashhab, S. (2019). "Existence of self-similar solutions of the two-dimensional Navier–Stokes equation for non-Newtonian fluids". Arab Journal of Mathematical Sciences. 26 ((1/2)): 167. doi:10.1016/j.ajmsc.2019.04.001.
7. Guedda, M.; Hammouch, Z. (2008). "Similarity flow solutions of a non-Newtonian power-law fluid". International Journal of Nonlinear Science. 6 (3): 255–264. doi:10.48550/arXiv.0904.0315.
8. Guedda, M.; Kersner, R. (2011). "Non-Newtonian pseudoplastic fluids: Analytical results and exact solutions". International Journal of Non-Linear Mechanics. 46 (7): 949–957. doi:10.1016/j.ijnonlinmec.2011.04.009.
9. Wei, D.H.; Al-Ashhab, S. (2014). "Similarity solutions for non-newtonian power-law fluid flow". Applied Mathematics and Mechanics. 35 (9): 1155–1166. doi:10.1007/s10483-014-1854-6.
10. Bognár, G. (2009). "Similarity solution of boundary layer flow for non-Newtonian fluids". International Journal of Nonlinear Sciences and Numerical Simulation. 10: 555–1566. doi:10.1016/j.camwa.2010.09.039.
11. ^ Tropea, Cameron; Yarin, Alexander L.; Foss, John F. (2007). Springer handbook of experimental fluid mechanics. Springer. pp. 661, 676. ISBN 978-3-540-25141-5.
12. ^ Garay, Paul N. (1996). Pump Application Desk Book (3rd ed.). Prentice Hall. p. 358. ISBN 978-0-88173-231-3.
13. Rao, M. A. (2007). Rheology of Fluid and Semisolid Foods: Principles and Applications (2nd ed.). Springer. p. 8. ISBN 978-0-387-70929-1.
14. Schramm, Laurier L. (2005). Emulsions, Foams, and Suspensions: Fundamentals and Applications. Wiley VCH. p. 173. ISBN 978-3-527-30743-2.
15. Chhabra, R.P. (2006). Bubbles, Drops, and Particles in Non-Newtonian Fluids (2nd ed.). Hoboken: Taylor & Francis Ltd. pp. 9–10. ISBN 978-1-4200-1538-6.
16. Astarita, G.; Marucci, G. (1972). Principles of Non-Newtonian Fluid Mechanics. McGraw-Hill. ISBN 9780070840225.
17. Fridtjov, I. (2014). Rheology and Non-Newtonian Fluids. Springer. ISBN 9783319010526.{{cite book}}: CS1 maint: date and year (link)
18. Patel, M.; Timol, M. (2020). Non-Newtonian Fluid Models and Boundary Layer Flow. LAP Lambert Academic Publishing. ISBN 9786203198614.
19. Hori, Y. (2006). Hydrodynamic Lubrication. Springer. ISBN 9784431278986.{{cite book}}: CS1 maint: date and year (link)
20. Böhme, G. (1987). Non-Newtonian Fluid Mechanics. North-Holland. ISBN 9780444567826.{{cite book}}: CS1 maint: date and year (link)
21. This demonstration of oobleck is a popular subject for YouTube videos.
22. ^ "Oobleck: The Dr. Seuss Science Experiment". instructables.com.
23. "Outrageous Ooze". Exploratorium. 7 March 2023.
24. Rupp, Rebecca (1998). "Magic Mud and Other Great Experiments". The Complete Home Learning Source Book. Three Rivers Press. pp. 235–236. ISBN 978-0-609-80109-3.
25. Glurch Meets Oobleck Archived 6 July 2010 at the Wayback Machine. Iowa State University Extension.
26. Barra, Giuseppina (2004). The Rheology of Caramel (PhD). University of Nottingham.
27. Cartwright, Jon (2 September 2011). "Microscopy reveals why ketchup squirts". Chemistry World. Royal Society of Chemistry.

External links

• Classical experiments with Non-Newtonian fluids by the National Committee for Fluid Mechanics on YouTube

• Continuum mechanics
• Fluid dynamics
• Non-Newtonian fluids
• Viscosity
• Polymers
• Tribology