Someone estimates the drag on a small glass marble of radius sinking at through a…

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

Fluid Dynamics, Ch Scaling, similarity and similarity solutions

Problem

Someone estimates the drag on a small glass marble of radius

a=1.0\,\mathrm{mm}

sinking at

U=5.0\,\mathrm{mm\,s^{-1}}

through a deep tank of glycerol (

\rho=1260\,\mathrm{kg\,m^{-3}}

\mu=1.41\,\mathrm{Pa\,s}

), and writes: \"The Reynolds number is

Re=\rho U(2a)/\mu=(1260)(0.0050)(0.0020)/1.41=8.9\times10^{-3}

, which is very small. Since

Re

is the ratio of the inertia term to the viscous term in the scaled Cauchy equation, a small

Re

means the inertia term is negligible and the viscous term dominates, so viscosity can be dropped and Euler's equation used instead. By d'Alembert's paradox, the inviscid drag on a sphere in steady flow is zero, so the marble should feel no drag at all.\" ((a)) Find the error in this argument: which term is actually negligible when

Re\ll1

, and which equation is wrongly dropped? ((b)) Write the correct dominant-balance (reduced) equation that applies at this

Re

, stating the pressure scale it uses.

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