A thick coating of density and surface tension against air is spread on the underside…

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 thick coating of density ρ2=1400 kg m−3\rho_2 = 1400\,\mathrm{kg\,m^{-3}} and surface tension against air γ=0.070 N m−1\gamma = 0.070\,\mathrm{N\,m^{-1}} is spread on the underside of a horizontal ceiling, with air of density ρ1=1.2 kg m−3\rho_1 = 1.2\,\mathrm{kg\,m^{-3}} below it. Treat the coating and the air as inviscid, unbounded and initially at rest, so that a disturbance of the flat interface between them obeys the Rayleigh-Taylor dispersion relation s2=(ρ2−ρ1)gk−γk3ρ1+ρ2s^2 = \frac{(\rho_2-\rho_1)gk - \gamma k^3}{\rho_1+\rho_2} with g=9.8 m s−2g = 9.8\,\mathrm{m\,s^{-2}} and kk the wavenumber. ((a)) Find the cutoff wavelength λc\lambda_c above which the coating is unstable. ((b)) Find the wavelength λm\lambda_m of the fastest-growing disturbance.

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