Stability lecture 6

Stability lecture 6 - Lecture 6 Modifying the...

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Lecture 6 Modifying the Rayleigh-Taylor inviscid analysis to include surface tension at the interface will depress smaller scale motion. We expect that surface tension to suppress instability in the analysis – it modifies the dynamic B.C. Before 12 0 at pp z η −= = Now 11 at ppT z RR  + =   Where T is a coefficient of tension, and 1 and 2 R R are principal radii of curvature in the x and y directions. When 22 2 1, or equivalently 1 then k 2 R Rxy ηη λ +≈ + ±± and when we linearize the B.C. () 0 2 2 21 2 2 2 The final result is 0 which leads to where z t T xy A kT A kt kT At e k σ ρρ αρ ρ = ± ∂∂ + +∂  +− + =  −− ∝=  + and if which is the normal condition for instability, for a given 0 −> ,,, T αρρ for a small enough k the whole term 2 will be and there will still exist a small k (large wavelength) mode that is unstable. 0 > But for larger k , small enough wavelength, the modes will be stable ( ) 2 0 < .
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Stability lecture 6 - Lecture 6 Modifying the...

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