4.3
Spherical Interfaces
Generally speaking, the equilibrium shape of an interface between two immiscible fluids is determined by solving the
force balance equation (3.1) in each fluid, and then applying the YoungLaplace equation to the interface. However,
in situations in which a mass of one fluid is completely immersed in a second fluid—
e.g.
, a mist droplet in air, or a
gas bubble in water—the shape of the interface is fairly obvious. Provided that either the size of the droplet or bubble,
or the di
ff
erence in densities on the two sides of the interface, is su
ffi
ciently small, we can safely ignore the e
ff
ect of
gravity. This implies that the pressure is
uniform
in each fluid, and consequently that the pressure jump
Δ
p
is
constant
over the interface. Hence, from (4.12), the mean curvature
∇ ·
n
of the interface is also
constant
. Since a sphere is the
only closed surface which possesses a constant mean curvature, we conclude that the interface is
spherical
. This result
also follows from the argument that a stable equilibrium state is one which
minimizes
the free energy of the interface,
subject to the constraint that the enclosed volume be
constant
. In other words, the equilibrium shape of the interface
is that which has the least surface area for a given volume:
i.e.
, a sphere.
Suppose that the interface corresponds to the spherical surface
r
=
R
, where
r
is a spherical coordinate. (See
Section C.4.) It follows that
n
=
e
r

r
=
R
. (Note, for future reference, that
n
points away from the center of curvature of
the interface.) Hence, from (C.65),
∇ ·
n
=
1
r
2
∂
r
2
∂
r
vextendsingle
vextendsingle
vextendsingle
vextendsingle
vextendsingle
vextendsingle
r
=
R
=
2
R
.
(4.13)
The YoungLaplace equation, (4.12), then gives
Δ
p
=
2
γ
R
.
(4.14)
Thus, given that
Δ
p
is the pressure jump seen crossing the interface in the opposite direction to
n
, we conclude that the
pressure
inside
a droplet or bubble
exceeds
that outside by an amount proportional to the surface tension, and inversely
proportional to the droplet or bubble radius. This explains why small bubbles are louder that large ones when they
burst at a free surface:
e.g.
, champagne fizzes louder than beer. Note that soap bubbles in air have
two
interfaces
defining the inner and outer extents of the soap film. Consequently, the net pressure di
ff
erence is
twice
that across a
single interface.
4.4
Capillary Length
Consider an interface separating the atmosphere from a liquid of uniform density
ρ
that is at rest on the surface of the
Earth. Neglecting the density of air compared to that of the liquid, the pressure in the atmosphere can be regarded as
64
FLUID MECHANICS
δr
2
1
3
θ
Figure 4.2:
Interface between a liquid (1), a gas (2), and a solid (3).
constant. On the other hand, the pressure in the liquid varies as
p
=
p
0
−
ρg
z
(see Chapter 3), where
p
0
is the pressure
of the atmosphere,
g
the acceleration due to gravity, and
z
measures vertical height (relative to the equilibrium height
of the interface in the absence of surface tension). Note that
z
increases upward. In this situation, the YoungLaplace
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 Fall '12
 Fluid Dynamics, Fluid Mechanics, stress tensor, Fluid Motion