A cornflour-and-water suspension is a shear-thickening power-law fluid,

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Viscous flow and dynamical similarity · Solid continuum mechanics and complex fluids

Bending, Ringing, Breaking and Flowing, Ch Non-Newtonian and viscoelastic fluids

Problem

A cornflour-and-water suspension is a shear-thickening power-law fluid, τ=Kγ˙n\tau = K\dot\gamma^n, with K=2.0 Pa s1.5K = 2.0\,\mathrm{Pa\,s^{1.5}} and n=1.5n = 1.5. It is pumped in steady, fully developed laminar flow along a straight pipe of radius R=15 mmR = 15\,\mathrm{mm} and length L=3.0 mL = 3.0\,\mathrm{m}; gravity does not contribute to the driving pressure gradient and end effects are ignored. For a power-law fluid in this pipe geometry the wall shear rate γ˙w\dot\gamma_w (the shear rate at r=Rr=R) and the volumetric flow rate V˙\dot V are related by V˙=nπR33n+1γ˙w,\dot V = \frac{n\pi R^3}{3n+1}\dot\gamma_w, once γ˙w\dot\gamma_w is known, while γ˙w\dot\gamma_w is set by the wall shear stress through τw=Kγ˙w n\tau_w = K\dot\gamma_w^{\,n} together with the force balance τw=RΔp/(2L)\tau_w = R\Delta p/(2L). ((a)) Find the pressure gradient Δp/L\Delta p/L needed to give a wall shear rate γ˙w=50 s−1\dot\gamma_w = 50\,\mathrm{s^{-1}}, and the resulting flow rate V˙1\dot V_1. ((b)) The pressure gradient is then doubled. Find the new wall shear rate γ˙w2\dot\gamma_{w2} and the new flow rate V˙2\dot V_2. ((c)) State the factor by which the flow rate increased in going from part (a) to part (b), and say whether it is more or less than the factor of 22 a Newtonian fluid would give for the same doubling of the pressure gradient, explaining the difference in terms of shear-thickening.

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