A laboratory works with silicone oil of density and viscosity unless stated otherwise

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 laboratory works with silicone oil of density ρ=900 kg m−3\rho=900\,\mathrm{kg\,m^{-3}} and viscosity μ=0.5 Pa s\mu=0.5\,\mathrm{Pa\,s} unless stated otherwise. Air, when needed, has ρ=1.20 kg m−3\rho=1.20\,\mathrm{kg\,m^{-3}} and μ=1.81×10−5 Pa s\mu=1.81\times10^{-5}\,\mathrm{Pa\,s}. Take g=9.8 m s−2g=9.8\,\mathrm{m\,s^{-2}}. Work each short part using standard results for Stokes flow, the Stokeslet interaction, the Hadamard-Rybczynski drag, the Oseen criterion, the lubrication pressure scale and Washburn's law. ((a)) A steel sphere of radius a=0.50 mma=0.50\,\mathrm{mm} is towed through the oil at a steady speed U=1.0 mm s−1U=1.0\,\mathrm{mm\,s^{-1}}. Find the Stokes drag force on it. ((b)) Two identical spheres of radius a=0.50 mma=0.50\,\mathrm{mm} sediment side by side with their centres d=5.0 mmd=5.0\,\mathrm{mm} apart. Using the Stokeslet estimate for the fractional speed-up of two spheres falling side by side, ΔU/U≈3a/(4d)\Delta U/U\approx 3a/(4d), give this fraction as a percentage. ((c)) An air bubble of the same radius a=0.50 mma=0.50\,\mathrm{mm} rises through the oil with a clean (surfactant-free) surface. Taking the bubble's density as negligible and Δρ=900 kg m−3\Delta\rho=900\,\mathrm{kg\,m^{-3}}, use the Hadamard-Rybczynski drag to find its terminal rise speed. ((d)) For the sphere of part (a), moving at U=1.0 mm s−1U=1.0\,\mathrm{mm\,s^{-1}} in the oil above, compute the Reynolds number Re=ρU(2a)/μRe=\rho U(2a)/\mu and state whether an Oseen correction to the drag is needed. ((e)) A slider-bearing pad of length L=3.0 cmL=3.0\,\mathrm{cm} runs on an oil film of thickness h=15 μmh=15\,\mu\mathrm{m}, with oil viscosity μ=0.05 Pa s\mu=0.05\,\mathrm{Pa\,s} and sliding speed U=5.0 m s−1U=5.0\,\mathrm{m\,s^{-1}}. Estimate the scale of the film pressure, μUL/h2\mu UL/h^2. ((f)) Water (γ=0.0728 N m−1\gamma=0.0728\,\mathrm{N\,m^{-1}}, μ=1.00×10−3 Pa s\mu=1.00\times10^{-3}\,\mathrm{Pa\,s}, fully wetting so θc=0\theta_c=0) soaks into a bundle of parallel capillary pores of radius r=10 μmr=10\,\mu\mathrm{m}. Using Washburn's law, find the time for the wetted length to reach L=3.0 cmL=3.0\,\mathrm{cm}.

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