1. Introduction, Notation
We consider uid systems dominated by the inuence of interfacial tension. The roles of curvature pressure
and Marangoni stress are elucidated in a variety of situtations. Particular attention will be given to the
dynamics of drops
6. More on Fluid statics
Last time, we saw that the balance of curvature and hydrostatic pressures requires
xx
g = n = (1+2 )3/2 .
x
We linearized, assuming x 1, to nd (x). Note: we can integrate directly
x xx
gx =
2
x )
g
3/2
d
dx
2
2
=
d
1
2
dx (1 + x
7. Spinning, tumbling and rolling
drops
7.1
Rotating Drops
We want to nd z = h(r) (see right). Normal stress
balance on S :
1
P + 2 r2 = n
"
2
"
curvature
centrif ugal
Nondimensionalize:
( r )2
p + 4B0 a = n,
23
entrif ugal
where p = ap , = 8 a = ccurva
5. Stress Boundary Conditions
Today:
1. Derive stress conditions at a uid-uid inter
face. Requires knowledge of T = pI + 2E
2. Consider several examples of uid statics
Recall: the curvature of a string under tension may Figure 5.1: String under tension an
3. Wetting
Puddles. What sets their size?
Knowing nothing of surface chemistry, one anticipates that Laplace pressure balances hydrostatic pressure
J
if /H gH H < c = /g = capillary length.
Note:
1. Drops with R < c remain heavily spherical
2. Large drops
4. Youngs Law with Applications
Youngs Law: what is the equilibrium contact angle e ? Horizontal force balance at contact line:
LV cos e = SV SL
SV SL
S
cos e =
=1+
(Y oung 1805)
(4.1)
LV
LV
Note:
1. When S 0, cos e 1 e undened and
spreading results.
2. V
8. Capillary Rise
Capillary rise is one of the most well-known and vivid illustrations of capillarity. It is exploited in a number
of biological processes, including drinking strategies of insects, birds and bats and plays an important role
in a number of
11. Fluid Jets
11.1
The shape of a falling uid jet
Consider a circular orice of a radius a ejecting a ux Q of uid density and kinematic viscosity
(see Fig. 11.1). The resulting jet accelerates under the inuence of gravity g z . We assume that the jet
Rey
16. More forced wetting
Some clarication notes on Wetting.
Figure 16.1: Three dierent wetting states.
Last class, we discussed the Cassie state only in the context
of drops in a Fakir state, i.e. suspended partially on a bed of
air. There is also a wet Ca
17. Coating: Dynamic Contact Lines
Last time we considered the Landau-Levich-Derjaguin Problem and deduced
V
h c C a2/3 for C a = < 103
1/3
h c C a
for C a 1.
The inuence of surfactants
Surfactants decrease which aects h slightly. But the principle eect i
15. Contact angle hysteresis, Wetting
of textured solids
Recall: In Lecture 3, we dened the equilibrium contact angle e , which is prescribed by Youngs Law:
cos e = (SV SL ) / as deduced from the horizontal force balance at the contact line.
Work done by
14. Instability of Superposed Fluids
Figure 14.1: Wind over water: A layer of uid of density + moving with relative velocity V over a layer
of uid of density .
Dene interface: h(x, y, z ) = z (x, y ) = 0 so that h = (x , y , 1).
The unit normal is given b
12. Instability Dynamics
12.1
Capillary Instability of a Fluid Coating on a Fiber
We proceed by considering the surface tension-induced
instability of a uid coating on a cylindrical ber.
Dene mean thickness
h =
1
h(x)dx
(12.1)
0
Local interfacial thicknes
9. Marangoni Flows
Marangoni ows are those driven by surface gradients. In general, surface tension depends on both the
temperature and chemical composition at the interface; consequently, Marangoni ows may be generated
by gradients in either temperature
13. Fluid Sheets
13.1
Fluid Sheets: shape and stability
The dynamics of high-speed uid sheets was rst considered by Savart (1833) after his early work on
electromagnetism with Biot, and was subsequently examined by Rayleigh (1879), then in a series of
pap
11. Fluid Jets
11.1
The shape of a falling uid jet
Consider a circular orice of a radius a ejecting a ux Q of uid density and kinematic viscosity
(see Fig. 11.1). The resulting jet accelerates under the inuence of gravity g z . We assume that the jet
Rey
15. Contact angle hysteresis, Wetting
of textured solids
Recall: In Lecture 3, we dened the equilibrium contact angle e , which is prescribed by Youngs Law:
cos e = (SV SL ) / as deduced from the horizontal force balance at the contact line.
Work done by
5. Stress Boundary Conditions
Today:
1. Derive stress conditions at a uid-uid inter
face. Requires knowledge of T = pI + 2E
2. Consider several examples of uid statics
Recall: the curvature of a string under tension may Figure 5.1: String under tension an
7. Spinning, tumbling and rolling
drops
7.1
Rotating Drops
We want to nd z = h(r) (see right). Normal stress
balance on S :
1
P + 2 r2 = n
"
2
"
curvature
centrif ugal
Nondimensionalize:
( r )2
p + 4B0 a = n,
23
entrif ugal
where p = ap , = 8 a = ccurva
10. Marangoni Flows II
10.1
Tears of Wine
The rst Marangoni ow considered was the tears of wine phenomenon (Thomson 1885 ), which actually
predates Marangonis rst published work on the subject by a decade. The tears of wine phenomenon
is readily observed
14. Instability of Superposed Fluids
Figure 14.1: Wind over water: A layer of uid of density + moving with relative velocity V over a layer
of uid of density .
Dene interface: h(x, y, z ) = z (x, y ) = 0 so that h = (x , y , 1).
The unit normal is given b
3. Wetting
Puddles. What sets their size?
Knowing nothing of surface chemistry, one anticipates that Laplace pressure balances hydrostatic pressure
J
if /H gH H < c = /g = capillary length.
Note:
1. Drops with R < c remain heavily spherical
2. Large drops
6. More on Fluid statics
Last time, we saw that the balance of curvature and hydrostatic pressures requires
xx
g = n = (1+2 )3/2 .
x
We linearized, assuming x 1, to nd (x). Note: we can integrate directly
x xx
gx =
2
x )
g
3/2
d
dx
2
2
=
d
1
2
dx (1 + x
2. Denition and Scaling of Surface
Tension
These lecture notes have been drawn from many sources, including textbooks, journal articles, and lecture
notes from courses taken by the author as a student. These notes are not intended as a complete discussion
12. Instability Dynamics
12.1
Capillary Instability of a Fluid Coating on a Fiber
We proceed by considering the surface tension-induced
instability of a uid coating on a cylindrical ber.
Dene mean thickness
h =
1
h(x)dx
(12.1)
0
Local interfacial thicknes
4. Youngs Law with Applications
Youngs Law: what is the equilibrium contact angle e ? Horizontal force balance at contact line:
LV cos e = SV SL
SV SL
S
cos e =
=1+
(Y oung 1805)
(4.1)
LV
LV
Note:
1. When S 0, cos e 1 e undened and
spreading results.
2. V
8. Capillary Rise
Capillary rise is one of the most well-known and vivid illustrations of capillarity. It is exploited in a number
of biological processes, including drinking strategies of insects, birds and bats and plays an important role
in a number of
19. Water waves
We
consider
waves
that
might
arise
from
disturbing
the
surface
of
a
pond.
We dene the normal to the surface: n =
(x ,1)
2
(1+x )1/2
xx
Curvature: n = (1+ 2 )3/2
x
We assume the uid motion is inviscid and irrota
tional: u = . Must deduce so