Surface tension:
Capillary pressure, scaling, wetting
John W. M. Bush
Department of Mathematics MIT
18.357: Lecture 3
Surface tension:
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σ =forcelength
analogous to a negative surface pressure
P = F/A
! gradients in surface tension necessarily drive surface motion
! measure the force required to withdraw a plate from a free surface
A simple way to measure surface tension
NOTES
r · n̂ =1
R1+
1
R2
Curvature
where R1 , R2 are the principal radii of curvature
n̂
R1
R2
Curvature:
Which way does the air go?
r · n̂ =1
R1+
1
R2for a sphere= 2/R
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R
Who cares about surface tension?
Capillary pressures in biology
1 mm
Ostwald ripening
The scaling of surface tension
Fundamental Concept
The laws of Nature cannot depend on arbitrarily chosen system of units.A system is most succinctly described in terms of dimensionless variables.
DIMENSIONAL ANALYSIS
Deduction of Dimensionless groups: Buckingham’s Theorem
For a system with M physical variables (e.g. density, speed, length, viscosity)describable in terms of N fundamental units (e.g. mass, length, time, temperature),there are M - N dimensionless groups that govern the system.
E.g. Translation of a sphere
Physical variables:
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U , a,ν , ρ ,D ⇒
Fundamental units: M , L , T
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⇒M = 5N = 3
Ua
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ρ ,νD
M - N = 2 dimensionless groups:
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Cd =DρU 2 , Re =
U aν
System uniquely determined by a single relation:
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Cd = F(Re)
The scaling of surface tension g
U
water
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ρ ,νvacuum
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Bo =ρ ga2
σ=
GRAVITYCURVATURE
= Bond number
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We =ρU 2aσ
=INERTIA
CURVATURE= Weber number
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Ca =ρνUσ
=VISCOSITYCURVATURE
= Capillary number
a
Note: is dominant relative to gravity when
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σ
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Bo < 1
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a < σρg⎛
⎝ ⎜
⎞
⎠ ⎟
1/ 2
= ℓ c = capillary length ~ 2mm for air-wateri.e.
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σ
When is surface tension important relative to gravity?
• when curvature pressures are large relative to hydrostatic:
a ρ
σgBo ! 1
Bo > 1
i.e. for drops small relative to the capillary length:
Bond number:
2 mm for air-watera < lc =
(
σ
ρg
)1/2
∼
Bo =ρga
σ/a=
ρga2
σ< 1
(σ = 70 dynes/cm)
Surface tension dominates the world of insects - and of microfluidics.
Falling rain drops
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a < ℓ c = σ /ρg ≈ 2mm
If a drop is small relative tothe capillary length
maintains it against the destabilizing influence ofaerodynamic stresses.
,
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ρU 2a2 ~ M g = 43 π a
3 ρ g
Small drops
Force balance:
Fall speed:
Drop integrity requires:
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σ
U ∼
√
ρ ga
ρa
a
ρaU2∼ ρga < σ/a
Drops larger than the capillarylength
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a > ℓ c ≈ 2mm
,
break up under the influence ofaerodynamic stresses.
The break-up yields drops with size of order:
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ℓ c ≈ 2mm
Big drops
Puddles
Wetting
Who cares about wetting?
The world’s smallest lizard: the Brazilian Pygmy Gecko
Partial wetting
Total wetting
Water on glass