Two identical steel balls of radius fall through glycerol

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Viscous flow and dynamical similarity · Viscous flow and dynamical similarity

Fluid Dynamics, Ch Stokes flow, lubrication and viscous wetting

Problem

Two identical steel balls of radius a=0.50 mma=0.50\,\mathrm{mm} fall through glycerol, each with the isolated terminal speed U0=2.5 mm s−1U_0 = 2.5\,\mathrm{mm\,s^{-1}}. Using the Stokeslet estimate of long-range interaction, a neighbouring sphere at centre-to-centre separation dd speeds a sphere up by a fraction 34(a/d)\tfrac34(a/d) if the two lie side by side (perpendicular to the fall direction), or 32(a/d)\tfrac32(a/d) if one sits directly above the other, along the line of fall. ((a)) Find the fall speed of each ball when they sit side by side with d=5.0 mmd=5.0\,\mathrm{mm}. ((b)) Find the fall speed of the lower ball when the pair are arranged one above the other with the same separation d=5.0 mmd=5.0\,\mathrm{mm}.

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