Questions: Non-Newtonian Fluids and Power-Law Models

Question 1

You tap a ketchup bottle gently and nothing flows out. You shake it hard and ketchup flows freely. Which power-law behavior does this demonstrate, and what is the value of n relative to 1?

• A. Shear-thickening (dilatant), n > 1 — harder shaking increases viscosity, eventually overcoming friction
• B. Shear-thinning (pseudoplastic), n < 1 — higher shear rate reduces apparent viscosity, allowing flow
• C. Newtonian, n = 1 — the viscosity is constant, but the applied force must exceed a yield stress
• D. Shear-thickening, n < 1 — n less than 1 means more resistance at higher shear

Answer: (B)

Ketchup is a classic shear-thinning fluid. At rest (low shear rate), polymer chains and particles are randomly arranged and entangled, producing high apparent viscosity — ketchup won't pour. Under high shear (hard shaking), chains align with the flow direction and disentangle, dramatically reducing apparent viscosity — ketchup flows easily. In the power-law model τ = K(dV/dy)ⁿ, the apparent viscosity is τ/(dV/dy) = K(dV/dy)^(n-1). For n < 1, this decreases as shear rate increases — shear-thinning. Option D has the logic backwards: n < 1 means shear-thinning, not shear-thickening.

Question 2

A cornstarch-water slurry (oobleck) behaves as a nearly solid surface when struck quickly but flows like a liquid when touched gently. What does this tell you about n in the power-law model?

• A. n < 1, because the fluid resists fast motion more than slow motion
• B. n = 1, because the fluid has a single well-defined viscosity
• C. n > 1, because apparent viscosity increases with shear rate — the fluid becomes more resistant as you push harder or faster
• D. n = 0, because the shear stress is independent of shear rate

Answer: (C)

Oobleck is shear-thickening (dilatant): resistance increases with shear rate. In the power-law model, apparent viscosity = K(dV/dy)^(n-1). For n > 1, this quantity increases with shear rate — you encounter more resistance the faster you try to move through the fluid. The physical mechanism is particle jamming: at high shear rates, the fluid lubricating particles breaks down and they jam together, increasing resistance. At low shear rates, particles remain dispersed and lubricated, so the fluid flows readily. This is the opposite of ketchup's behavior.

Question 3

A shear-thinning fluid has a lower apparent viscosity at high shear rates than at low shear rates.

• T. True
• F. False

Answer: True

This is the defining characteristic of shear-thinning (pseudoplastic) behavior, corresponding to n < 1 in the power-law model. Apparent viscosity, defined as the ratio τ/(dV/dy) at a given shear rate, equals K(dV/dy)^(n-1). When n < 1, the exponent (n-1) is negative, so apparent viscosity decreases as shear rate increases. This is why polymer solutions, paints, and blood are easier to pump at high flow rates than at low flow rates — a practically important property for processing and circulatory design.

Question 4

In the power-law model τ = K(dV/dy)^n, a Newtonian fluid corresponds to n = 0, and the consistency index K equals the dynamic viscosity.

• T. True
• F. False

Answer: False

Newtonian behavior corresponds to n = 1, not n = 0. When n = 1, the power-law model reduces to τ = K(dV/dy)^1 = K(dV/dy), which is identical to Newton's law of viscosity τ = μ(dV/dy) with μ = K. At n = 0, the shear stress would equal K regardless of shear rate — that would describe a perfectly rigid solid or a Bingham-plastic at yield, not a Newtonian fluid. The consistency index K has units that depend on n (not the standard Pa·s of dynamic viscosity unless n = 1), so equating K to viscosity is only valid at n = 1.

Question 5

What physical mechanism causes polymer solutions to be shear-thinning, and how does this differ from the mechanism causing cornstarch-water suspensions to be shear-thickening?

Model answer: In polymer solutions, long chain molecules are randomly coiled and entangled at rest, creating high resistance. Under shear, chains uncoil and align with the flow direction, reducing entanglement and lowering apparent viscosity — shear-thinning results. This process is largely reversible: remove the shear and chains re-coil. In cornstarch-water (and dense particle suspensions generally), particles are normally separated by a thin lubricating fluid layer. At high shear rates, this lubrication breaks down and particles come into direct frictional contact or jam together, dramatically increasing resistance — shear-thickening results. Both mechanisms are reversible when shear is removed, but they arise from completely different microstructural physics: molecular conformation change (polymers) versus particle contact forces (suspensions).

Understanding the mechanism matters for engineering applications. A shear-thinning polymer solution becomes easier to pump at higher flow rates — an advantage. A shear-thickening suspension becomes nearly impossible to pump if shear rate exceeds a critical threshold — a hazard. Knowing the mechanism also predicts temperature sensitivity, time-dependence (thixotropy vs. rheopexy), and how to modify the behavior through formulation changes.