14. Non-Newtonian worms

14. Non-Newtonian Worms

A dilatant non-Newtonian fluid (e.g. a suspension of water and cornflour) is vibrated vertically in a reservoir. When certain conditions are met, wormlike structures that rise from the surface may form. Explain and investigate this phenomenon.

A cornstarch–water mixture (oobleck) placed on a vibrating speaker forms self-organizing worm-like structures that crawl, split, and rejoin. This phenomenon arises from the interplay of rheological nonlinearity and Faraday wave instability.

Power-Law Rheology (Ostwald–de Waele Model)

τ = K · γ̇ⁿ

• n < 1: shear-thinning (pseudoplastic) — toothpaste, blood
• n = 1: Newtonian — water, air
• n > 1: shear-thickening (dilatant) — cornstarch suspension

Apparent viscosity: η_app = K · γ̇^(n−1). For oobleck, n ≈ 1.5–2.5 and K ≈ 10–100 Pa·sⁿ.

Faraday Instability Threshold

Vertical vibration at amplitude A and frequency f drives subharmonic surface waves at f/2 when the dimensionless acceleration exceeds a critical value:

Γ = A·(2πf)² / g ≥ Γ_c(n, K, h₀, f)

For Newtonian fluids, Γ_c ≈ 1 (independent of viscosity at onset for inviscid limit). For shear-thickening fluids, Γ_c is larger and frequency-dependent.

Thin-Film Equation for Power-Law Fluid

∂h/∂t + ∂/∂x [K̃ · h^(2n+1) · |∂h/∂x|^(n−1) · ∂h/∂x] = f_drive(x,t)

where K̃ = K(2n+1)/(ρ(n+1)) is a nonlinear diffusion coefficient. For n>1, higher h → stronger “anti-diffusion” → fingering instability.

Worm Formation Mechanism

1. Faraday instability creates subharmonic surface waves at threshold Γ
2. Rheological nonlinearity transfers energy to higher harmonics
3. Wave-matter coupling: high-γ̇ regions solidify locally, forming ridges
4. Worm structures: narrow solidified columns on a fluidized substrate

References

• Larson, R.G. (1999). The Structure and Rheology of Complex Fluids. Oxford University Press. — Ch. 1: rheology of non-Newtonian fluids; power-law model and shear stress.
• Faraday, M. (1831). “On a peculiar class of acoustical figures.” Phil. Trans. R. Soc. Lond., 121, 299–340. — first description of Faraday waves on a vibrated fluid surface; physical basis of worm formation in vibrated cornstarch.
• Merkt, F.S., Deegan, R.D., Goldman, D.I., Rericha, E.C. & Swinney, H.L. (2004). “Persistent holes in a fluid.” Phys. Rev. Lett., 92, 184501. — persistent holes in vibrated non-Newtonian fluids; directly related to the worm-formation phenomenon.
• Waitukaitis, S.R. & Jaeger, H.M. (2012). “Impact-activated solidification of dense suspensions via dynamic jamming fronts.” Nature, 487, 205–209. — mechanism of instantaneous solidification of cornstarch under impact; explains dilatant behavior.
• Barnes, H.A. (1989). “Shear-thickening (dilatancy) in suspensions of non-aggregating solid particles in Newtonian liquids.” J. Rheol., 33(2), 329–366. — comprehensive review of dilatancy in suspensions; key reference for cornstarch physics.
• Cross, M.C. & Hohenberg, P.C. (1993). “Pattern formation outside of equilibrium.” Rev. Mod. Phys., 65, 851–1112. — general theory of pattern formation in driven systems; theoretical foundation for worm emergence.

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