Graphics Reference
In-Depth Information
6.4.2 Surface Tension
Another subject we'll only briefly touch on is adding surface-tension forces
for small-scale water phenomena. This is, it should be warned, a subject
of ongoing research both within graphics and scientific computing. For ex-
ample, the test case of running a sphere of water at rest in zero-G with
surface tension (which should result in surface-tension forces exactly can-
celing pressure forces, so velocity remains zero) is surprisingly hard to get
right.
The physical chemistry of surface tension is conceptually simple. Water
molecules are more attracted to other water molecules than to air molecules,
and vice versa. Thus, water molecules near the surface tend to be pulled
in towards the rest of the water molecules and vice versa. In a way they
are seeking to minimize the area exposed to the other fluid, bunching up
around their own type. The simplest linear model of surface tension can
in fact be phrased in terms of a potential energy equal to a surface-tension
coecient
γ
times the surface area between the two fluids (
γ
for water and
air at normal conditions is approximately 0
.
073
J/
m
2
). The force seeks to
minimize the surface area.
The surface area of the fluid is simply the integral of 1 on the boundary:
A
=
∂
Ω
1
.
Remembering our signed distance properties, this is the same as
A
=
∂
Ω
∇
φ
·
n.
Now we use the divergence theorem in reverse to turn this into a volume
integral:
A
=
Ω
∇·∇
φ
Consider a virtual infinitesimal displacement of the surface,
δx
.Th s
changes the volume integral by adding or subtracting infinitesimal amounts
of the integrand along the boundary. The resulting infinitesimal change in
surface area is
δA
=
∂
Ω
(
∇·∇
φ
)
δx
·
n.
Thus, the variational derivative of surface area is (
φ
)
n
. Our surface-
tension force is proportional to this, and since it is in the normal direction
we can think of it in terms of a pressure jump at the air-water interface (as
∇·∇
