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
ΔP + ΔρΩ2 r2 = σ∇
" ·n
2
"
curvature
centrif ugal
Nondimensionalize:
( )2
Δp′ + 4B0 ar = ∇ · n,
2 3
ugal
ΔρΩ a
where Δp′ = aΔp
= centrif
σ , Σ =
8σ
curvature = =
Rotational Bond number = const. Define surface
functional: f (r, θ) = z − h(r) ⇒ vanishes on the
surface. Thus
ẑ−hr (r)r̂
∇f
−rhr −r 2 hrr
n = |∇|
= (1+h
2 (r))1/2 and ∇ · n = r 2 (1+h2 )3/2
r
r
Figure 7.1: The radial profile of a rotating drop.
Brown + Scriven (1980) computed drop shapes and stability for B0 > 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 V
raindrops?
σ
From raindrop scaling, we expect ℓc ∼
Δρg but
both σ, Δρ higher by a factor of 10 ⇒ large tektite
Figure 7.2: The ratio of the maximum radius to
size suggests they are not equilibrium forms, but
the unperturbed radius is indicated as a function of
froze into shape during flight.
Σ. Stable shapes are denoted by the solid line, their
Q2: Why are their shapes so different from those of metastable counterparts by dashed lines. Predicted
raindrops? Owing to high ρ of tektites, the internal 3-dimensional forms are compared to photographs
dynamics (esp. rotation) dominates the aerodynam­ of natural tektites. From Elkins-Tanton, Ausillous,
ics ⇒ drop shape set by its rotation.
Bico, Quéré and Bush (2003).
24
7.2. Rolling drops
Chapter 7. Spinning, tumbling and rolling drops
Light drops: For the case of Σ < 0, ∆ρ < 0, a spinning drop is stabilized on axis by centrifugal pressures.
For high |Σ|, it is well described by a cylinder with spherical caps. Drop energy:
E=
1 2
IΩ
2
| {z }
+
Rotational K.E.
2πrLγ
| {z }
S ur f ace energ y
∆mr 2
2
Neglecting the end caps, we write volume V = πr2 L and moment of inertia I =
= ∆ρ π2 Lr4 .
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 VE = 14 ∆ρΩ2 r2 + 2γ
r .
Minimizing with respect to r:
1/3
1/3
1
2γ
4γ
4γ
d
E
V 1/2
2
. Now r = πL
= ∆ρΩ
⇒
2
dr V = 2 ∆ρΩ r − r 2 = 0, which occurs when r = ∆ρΩ2
1
2 V 3/2
Vonnegut’s Formula: γ = 4π3/2 ∆ρΩ L
allows inference of γ from L, useful technique for small γ
as it avoids difficulties associated with fluid-solid 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 RQuere 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
MIT OpenCourseWare
http://ocw.mit.edu
357 Interfacial Phenomena
Fall 2010
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.