lecture7 notes

# lecture7 notes - 7 Spinning tumbling and rolling drops 7.1...

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7. Spinning, tumbling and rolling drops 7.1 Rotating Drops We want to find z = h ( r ) (see right). Normal stress balance on S : 1 2 Δ P + Δ ρ Ω 2 r = σ ∇ · n 2 " centrifugal Nondimensionalize: ( r ) 2 Δ p + 4 B 0 = ∇ · n , a a Δ p Δ ρ Ω 2 = " curvature 3 a centrifugal where Δ p , Σ = = = = σ 8 σ curvature Rotational Bond number = const. Define surface functional: f ( r, θ ) = z h ( r ) vanishes on the surface. Thus 2 f z ˆ h r ( r r rh r r h rr n = | | ) 3 / 2 = and · n = (1+ h 2 ( r )) 1 / 2 r 2 (1+ h 2 r r Figure 7.1: The radial profile of a rotating drop. Brown + Scriven (1980) computed drop shapes and stability for B 0 > 0: 1. for Σ < 0 . 09, only axisymmetric solutions, oblate ellipsoids 2. for 0 . 09 < Σ < 0 . 31, both axisymmetric and lobed solutions possible, stable 3. for Σ > 0 . 31 no stable solution, only lobed forms Tektites : centimetric metallic ejecta formed from spinning cooling silica droplets generated by mete- orite impact. Q1 : Why are they so much bigger than raindrops?

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