A water droplet of radius falls through still air

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

A water droplet of radius a=10 μma=10\,\mu\mathrm{m} falls through still air. Air has 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}; water has ρ=998 kg m−3\rho=998\,\mathrm{kg\,m^{-3}}, μ=1.00×10−3 Pa s\mu=1.00\times10^{-3}\,\mathrm{Pa\,s} and γ=0.0728 N m−1\gamma=0.0728\,\mathrm{N\,m^{-1}} against air. Take g=9.8 m s−2g=9.8\,\mathrm{m\,s^{-2}}. ((a)) Using the balance between surface tension and gravity for a small drop, compute the Bond number Bo=Δρ g a2/γBo=\Delta\rho\,g\,a^2/\gamma for this droplet and say whether it should be treated as spherical. ((b)) Estimate the terminal fall speed U0U_0 the droplet would have if it behaved as a rigid sphere in Stokes flow through air, then compute the Reynolds number Re=ρairU0(2a)/μairRe=\rho_{\rm air}U_0(2a)/\mu_{\rm air} and confirm that the dominant-balance argument used to derive the Stokes equations applies here. ((c)) Using the Hadamard-Rybczynski drag with air as the outer fluid and water as the drop, compute the actual terminal speed UU and explain, in terms of the viscosity ratio, why it differs so little from U0U_0.

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