A pollen grain

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

Fluid Dynamics, Ch Stokes flow, lubrication and viscous wetting

Problem

A pollen grain, modelled as a rigid sphere of radius a=15 μma = 15\,\mu\mathrm{m} and density ρp=1200 kg m−3\rho_p = 1200\,\mathrm{kg\,m^{-3}}, is carried by a brief gust to a horizontal speed U0=5.0×10−2 m s−1U_0 = 5.0\times10^{-2}\,\mathrm{m\,s^{-1}} relative to the surrounding still air (ρ=1.20 kg m−3\rho = 1.20\,\mathrm{kg\,m^{-3}}, μ=1.81×10−5 Pa s\mu = 1.81\times10^{-5}\,\mathrm{Pa\,s}). The gust then stops; the only horizontal force remaining on the grain is the Stokes drag of the air (gravity acts vertically and plays no part in this horizontal problem). ((a)) Starting from Newton's second law for the grain and Stokes's drag law, derive an expression for the horizontal speed U(t)U(t) after the gust stops, and identify the relaxation time τ\tau in terms of ρp\rho_p, aa and μ\mu. ((b)) State which property of the Stokes equations justifies inserting the instantaneous speed U(t)U(t) into the steady drag law 6πμaU6\pi\mu a U even while UU is changing, and explain why in one sentence. ((c)) Evaluate τ\tau and the total horizontal distance the grain coasts, ∫0∞U dt\int_0^{\infty}U\,\mathrm{d}t, using the numbers given, and compare that distance with the grain's radius aa.

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