Unformatted text preview: the end caps, we write volume V = πr2 L and moment of inertia I = = ∆ρ π Lr4 .
2 Figure 7.3: A bubble or a drop suspended in a denser ﬂuid, spinning with angular speed Ω.
The energy per unit drop volume is thus
Minimizing with respect to r:
d
dr E
V 1
= 2 ∆ρΩ2 r − 2γ
r2 E
V = 1 ∆ρΩ2 r2 +
4 = 0, which occurs when r =
1
∆ρΩ2
4π 3/2 2γ
r. 4γ
∆ρΩ2 1/3 . Now r = V 1/2
πL = 4γ
∆ρΩ2 1/3 ⇒ V 3/2
L Vonnegut’s Formula: γ =
allows inference of γ from L, useful technique for small γ
as it avoids diﬃculties associated with ﬂuidsolid contact.
Note: r grows with σ and decreases with Ω. 7.2 Rolling drops Figure 7.4: A liquid drop rolling down an inclined plane.
(Aussillous and Quere 2003 ) Energetics: for steady descent at speed V, M gV sin θ =Rate of viscous
dissipation= 2µ Vd (∇u)2 dV , where Vd is the dissipation zone, so this sets V ⇒ Ω = V /R is the angular
speed. Stability characteristics diﬀerent: bioconcave oblate ellipsoids now stable. MIT OCW: 18.357 Interfacial Phenomena 25 Prof. John W. M. Bush...
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This note was uploaded on 01/23/2014 for the course MATH 18.357 taught by Professor Johnw.m.bush during the Fall '10 term at MIT.
 Fall '10
 JohnW.M.Bush

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