Newtonian fluid

Lore & Background

A fluid is Newtonian only if the tensors that describe the viscous stress and the strain rate are related by a constant viscosity tensor that does not depend on the stress state and velocity of the flow. If the fluid is also isotropic, the viscosity tensor reduces to two real coefficients, describing the fluid's resistance to continuous shear deformation and continuous compression or expansion. While no real fluid fits the definition perfectly, many common liquids and gases, such as water and air, can be assumed to be Newtonian for practical calculations under ordinary conditions.

Reader's Guide

Newtonian fluids are significant because they provide the simplest mathematical model for viscosity, enabling practical calculations for many common fluids like water and air under ordinary conditions. The constitutive equation for an incompressible isotropic Newtonian fluid in laminar flow relates shear stress to strain rate via a constant dynamic viscosity. This linear relationship allows engineers and scientists to predict flow behavior in pipes, channels, and around objects. However, non-Newtonian fluids are relatively common and include oobleck (which becomes stiffer when vigorously sheared) and non-drip paint (which becomes thinner when sheared). Other examples include many polymer solutions, molten polymers, many solid suspensions, blood, and most highly viscous fluids. The concept is named after Isaac Newton, who first used the differential equation to postulate the relation between the shear strain rate and shear stress for such fluids.

Did You Know?

• A Newtonian fluid's rate of flow cannot be altered by shaking, pumping, or stirring the fluid.
• If the fluid is also isotropic, the viscosity tensor reduces to two real coefficients.
• No real fluid fits the definition of a Newtonian fluid perfectly.
• Non-Newtonian fluids include oobleck, non-drip paint, polymer solutions, molten polymers, blood, and most highly viscous fluids.

Defining the Non-Newtonian Condition

In a Newtonian fluid, the relationship between shear stress and shear rate is a simple linear one passing through the origin, with the slope representing a constant viscosity coefficient. Non-Newtonian fluids break this rule entirely. Their viscosity is not a fixed number but shifts in response to applied stress, meaning the same substance can behave as a thick paste one moment and a thin liquid the next. This variability can be tied to the current shear rate, to the history of shear the fluid has experienced, or to both. Some non-Newtonian fluids even display time-dependent viscosity, making it impossible to assign a single constant coefficient of viscosity. While the standard viscosity concept from fluid mechanics still applies to shear properties, it proves insufficient on its own. Researchers therefore turn to broader rheological properties that connect stress and strain-rate tensors across various flow conditions, including oscillatory shear and extensional flow, measured with specialized instruments called rheometers. The mathematical framework for these fluids relies on tensor-valued constitutive equations, a staple of continuum mechanics.

The Spectrum of Flow Behaviors

Non-Newtonian behavior is not a single phenomenon but a family of distinct responses. Shear-thickening, or dilatant, fluids grow more viscous as the shear rate rises; cornstarch in water is the classic case, appearing milky and pourable when stirred gently but feeling almost solid under vigorous agitation. The opposite, shear-thinning or pseudoplastic behavior, is seen in wall paint, which flows easily off a brush yet resists dripping, and in blood, whose viscosity drops as shear strain rate increases—a trait the circulatory system exploits. Bingham plastics occupy another category: they require a finite yield stress before any flow begins at all, so their stress-strain plot does not cross the origin. Toothpaste, mayonnaise, mustard, and drilling mud all fall here, and their still surfaces can hold peaks rather than lying flat. Finally, time-dependent behaviors split into thixotropic fluids, which thin over time under constant shear, and rheopectic fluids, which thicken over time and demand ever-greater stress to sustain the same strain rate.

Oobleck and the Kitchen Cabinet

Few substances make non-Newtonian physics as tangible as a simple mixture of corn or potato starch and water, popularly called oobleck, ooze, or magic mud. The name traces back to the Dr. Seuss book Bartholomew and the Oobleck. Mixed at roughly one part water to one-and-a-half or two parts starch, the result is inexpensive and non-toxic, making it a favorite for classroom demonstrations. Because of its dilatant character, a person can sprint across a large tub of oobleck without sinking, provided each step delivers enough force to trigger the thickening response. Placed atop a powerful subwoofer, the mixture responds to low-frequency sound waves by stiffening and forming visible standing waves. A punch or slap drives it into a near-solid state, only for it to relax back into a pourable liquid once the impact passes. Beyond the lab, the same principles govern everyday substances: ketchup that loosens when shaken, toothpaste that holds its shape on a brush, shampoo, melted butter, custard, and even blood all exhibit non-Newtonian character.

Mathematical Models and Analytical Limits

Capturing non-Newtonian behavior in equations is far from straightforward. For time-independent fluids—pseudoplastic, plastic, and dilatant—the landscape of known analytic solutions is relatively broad, giving engineers and physicists a workable toolkit. Time-dependent cases, however, present a steeper challenge. Three well-known models named after their defining authors are the Oldroyd-B model, Walters' Liquid B, and Williamson fluids. When researchers applied time-dependent self-similar analysis to the Ladyzenskaya-type model, which features a non-linear velocity-dependent stress tensor, they could not extract closed-form analytic solutions. Instead, they established a rigorous mathematical existence theorem guaranteeing that a solution does in fact exist, even if it cannot be written down explicitly. This gap between existence and explicit form underscores why non-Newtonian rheology remains an active frontier: the physics is rich enough to defy simple algebra yet structured enough to yield to theorems in continuum mechanics.