A layer of light oil ( ) floats on a much deeper layer of water ( )

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 layer of light oil (ρ1=800 kg m−3\rho_1 = 800\,\mathrm{kg\,m^{-3}}) floats on a much deeper layer of water (ρ2=1000 kg m−3\rho_2 = 1000\,\mathrm{kg\,m^{-3}}), both at rest, and surface tension is ignored. A disturbance of wavenumber k=50 rad m−1k = 50\,\mathrm{rad\,m^{-1}} is imposed on the interface, and g=9.8 m s−2g = 9.8\,\mathrm{m\,s^{-2}}. Using s2=At gks^2 = At\,gk with At=(ρtop−ρbottom)/(ρtop+ρbottom)At = (\rho_{\text{top}}-\rho_{\text{bottom}})/(\rho_{\text{top}}+\rho_{\text{bottom}}), a working finds s2=800−1000800+1000×9.8×50=−54.4 s−2.s^2 = \frac{800-1000}{800+1000}\times9.8\times50 = -54.4\ \mathrm{s^{-2}}. Someone argues: "Since s2s^2 is negative, s=±−54.4=±7.4i s−1s = \pm\sqrt{-54.4} = \pm7.4i\ \mathrm{s^{-1}} is imaginary, and an imaginary ss inside este^{st} makes the disturbance oscillate with ever-growing swings, so the interface is unstable." ((a)) Find the error in this reasoning. ((b)) State the correct stability verdict and describe what the interface actually does.

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