Non-Newtonian Fluids

Core Idea

Non-Newtonian fluids are those whose shear stress is not linearly proportional to the strain rate — their apparent viscosity varies with shear rate, time, or flow history. The most common models are: shear-thinning (pseudoplastic) fluids like blood, paint, and polymer solutions, where viscosity decreases with increasing shear rate (modeled by the power-law τ = Kγ̇ⁿ with n < 1); shear-thickening (dilatant) fluids like cornstarch suspensions, where viscosity increases with shear rate (n > 1); and Bingham plastics like toothpaste and drilling mud, which behave as solids below a yield stress τ_y and flow as viscous fluids above it (τ = τ_y + μ_p·γ̇). Viscoelastic fluids (like polymer melts) exhibit both viscous and elastic behavior — they can store energy and exhibit phenomena like die swell, rod climbing (Weissenberg effect), and elastic recoil. Rheometry — controlled shear and extensional testing — is used to characterize these complex behaviors.

How It's Best Learned

Derive the velocity profile for a power-law fluid in a pipe (it changes from parabolic to blunted for n < 1 and pointed for n > 1) and compare it to the Newtonian parabolic profile. Sketch the shear stress vs. strain rate curves for Newtonian, shear-thinning, shear-thickening, and Bingham plastic fluids on the same axes. Study real examples: why does ketchup flow easily when shaken (thixotropy), why does paint stay on a wall after brushing (shear-thinning with recovery), and why does a polymer solution climb a rotating rod (normal stress differences).

Common Misconceptions

• Shear-thinning and thixotropy are not the same. Shear-thinning is an instantaneous decrease of viscosity with shear rate; thixotropy is a time-dependent decrease at constant shear rate (the fluid's structure gradually breaks down). Many real fluids exhibit both.
• The power-law model τ = Kγ̇ⁿ fails at very low and very high shear rates, where real shear-thinning fluids approach constant Newtonian plateaus (zero-shear viscosity η₀ and infinite-shear viscosity η∞). The Carreau or Cross models capture these limits.
• Yield stress is a practical concept, not always a sharp physical threshold. Whether a material has a "true" yield stress or simply a very high viscosity at low shear rates is debated (the "yield stress myth" controversy), but the Bingham model remains a useful engineering approximation.

Explainer

Your prerequisite on viscosity established the Newtonian model: shear stress τ is proportional to strain rate γ̇, with the constant of proportionality being the dynamic viscosity μ. Water, air, and most simple solvents obey this linear relationship — double the strain rate, double the stress. But this is a special case, not a universal rule. Non-Newtonian fluids are materials where the relationship between stress and strain rate is nonlinear, time-dependent, or both. Once you look, they are everywhere: blood, concrete, toothpaste, ketchup, polymer melts, paint, and drilling mud all behave in ways the Newtonian model cannot capture.

The simplest departure is the power-law (Ostwald-de Waele) model: τ = Kγ̇ⁿ, where K is a consistency index and n is the flow behavior index. When n = 1, you recover the Newtonian case. When n < 1, the apparent viscosity η_app = Kγ̇^(n-1) decreases as the fluid is sheared harder — this is shear-thinning (or pseudoplastic) behavior. Paint is a classic example: it flows easily under the high shear of a brush stroke, then quickly becomes more viscous and stays on the wall without dripping. Blood vessels exploit shear-thinning to allow red blood cells to flow efficiently through narrow capillaries at high shear while behaving more viscously in large vessels at low shear. When n > 1, viscosity increases with shear rate — this is shear-thickening (or dilatant) behavior. A dense cornstarch suspension is the prototype: at low agitation it flows freely, but a sudden impact solidifies it momentarily. This is why running on a cornstarch pool works until you slow down.

A physically distinct class is the Bingham plastic: fluids that do not flow at all below a threshold yield stress τ_y, then flow as viscous fluids above it. The constitutive relation is τ = τ_y + μ_p·γ̇ for τ > τ_y, and γ̇ = 0 otherwise. Toothpaste sits on your brush rather than flowing off — it has yielded in the tube (where pressure exceeds τ_y) but sits immobile once extruded. Concrete, mayonnaise, and drilling muds are all practical Bingham plastics. Engineering pipe flow calculations for these materials require finding the plug flow region at the center of the pipe (where stresses are below τ_y) surrounded by yielded flowing material near the walls.

Viscoelastic fluids add another layer of complexity: they have both viscous dissipation and elastic energy storage. Polymer melts and solutions fall into this category. When you shear them, they both flow and deform elastically — and when the stress is removed, they partially recover. This produces striking phenomena like the Weissenberg effect (a polymer solution climbs a rotating rod rather than being flung outward) and die swell (extruded polymer expands upon leaving a nozzle because stored elastic energy relaxes). The Deborah number De = relaxation time / flow time scale characterizes whether elastic or viscous behavior dominates. At De ≪ 1, the fluid appears viscous; at De ≫ 1, it behaves elastically. Understanding these behaviors is essential for polymer processing, food engineering, and biological fluid mechanics — domains where assuming Newtonian behavior would give wildly incorrect predictions.