A thin diesel-oil slick floats on a lake; treat the oil as a deep layer of density…

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Instability, transition and turbulence · Instability, transition and turbulence

Fluid Dynamics, Ch Hydrodynamic instability: the normal-mode method

Problem

A thin diesel-oil slick floats on a lake; treat the oil as a deep layer of density ρ2=832 kg m−3\rho_2 = 832\,\mathrm{kg\,m^{-3}} with surface tension against air γ=0.025 N m−1\gamma = 0.025\,\mathrm{N\,m^{-1}}, and treat the overlying air (ρ1=1.20 kg m−3\rho_1 = 1.20\,\mathrm{kg\,m^{-3}}) as blowing uniformly and steadily over it. Take g=9.8 m s−2g = 9.8\,\mathrm{m\,s^{-2}} and model both fluids as inviscid, so the Kelvin-Helmholtz threshold derived for air over water applies with the diesel properties in place of water's. ((a)) Estimate the minimum uniform wind speed ΔUmin⁡\Delta U_{\min} needed to destabilise the flat slick surface, and compare it with the wind speeds of order 11-2 m s−12\,\mathrm{m\,s^{-1}} at which ripples are actually seen to form on such slicks. State what the uniform-wind model leaves out. ((b)) A laboratory Taylor-Couette cell has inner cylinder radius R1=4.0 cmR_1 = 4.0\,\mathrm{cm} and gap d=4.0 mmd = 4.0\,\mathrm{mm}, filled with water (μ=1.00×10−3 Pa s\mu = 1.00\times10^{-3}\,\mathrm{Pa\,s}, ρ=998 kg m−3\rho = 998\,\mathrm{kg\,m^{-3}}). Using the critical Taylor number Tac≈1.7×103Ta_c \approx 1.7\times10^{3}, estimate the inner-cylinder angular speed Ω\Omega at which Taylor vortices first appear, and check whether the gap is narrow enough for the narrow-gap scaling to be trusted, using the quoted R1=5 cmR_1 = 5\,\mathrm{cm}, d=5 mmd = 5\,\mathrm{mm} cell as the benchmark of adequacy.

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