An example of dynamic fluid films.
Fluid films, such as soap films , are commonly encountered in everyday experience. A soap film can be formed by dipping a closed contour wire into a soapy solution as in the figure on the right. Alternatively, a catenoid can be formed by dipping two rings in the soapy solution and subsequently separating them while maintaining the coaxial configuration.
Stationary fluid films form surfaces of
Plateau problem
.
On the other hand, fluid films display rich
phenomena.
In the study of the
manifolds
. Then the variable thickness of the film is captured by the two dimensional density
ρ
{\displaystyle \rho }
.
The dynamics of fluid films can be described by the following
Euclidean spaces
.
The foregoing relies on the formalism of
.
The full dynamic system
Consider a thin fluid film
S
{\displaystyle S}
that spans a stationary closed contour boundary. Let
C
{\displaystyle C}
be the normal component of the
velocity field
and
V
α
{\displaystyle V^{\alpha }}
be the
contravariant components of the tangential velocity projection. Let
∇
α
{\displaystyle \nabla _{\alpha }}
be the
covariant surface derivative ,
B
α
β
{\displaystyle B_{\alpha \beta }}
be the
covariant curvature tensor ,
B
β
α
{\displaystyle B_{\beta }^{\alpha }}
be the mixed
curvature tensor and
B
α
α
{\displaystyle B_{\alpha }^{\alpha }}
be its
trace , that is
mean curvature . Furthermore, let the
internal energy density per unit mass function be
e
(
ρ
)
{\displaystyle e\left(\rho \right)}
so that the total
potential energy
E
{\displaystyle E}
is given by
E
=
∫
S
ρ
e
(
ρ
)
d
S
.
{\displaystyle E=\int _{S}\rho e\left(\rho \right)\,dS.}
This choice of
e
(
ρ
)
{\displaystyle e\left(\rho \right)}
:
e
(
ρ
)
=
σ
ρ
{\displaystyle e\left(\rho \right)={\frac {\sigma }{\rho }}}
where
σ
{\displaystyle \sigma }
is the surface energy density results in
:
E
=
σ
A
{\displaystyle E=\sigma A\,}
where A is the total area of the soap film.
The governing system reads
δ
ρ
δ
t
+
∇
α
(
ρ
V
α
)
=
ρ
C
B
α
α
ρ
(
δ
C
δ
t
+
2
V
α
∇
α
C
+
B
α
β
V
α
V
β
)
=
−
ρ
2
e
ρ
B
α
α
ρ
(
δ
V
α
δ
t
+
V
β
∇
β
V
α
−
C
∇
α
C
−
2
C
V
β
B
β
α
)
=
−
∇
α
(
ρ
2
e
ρ
)
{\displaystyle {\begin{aligned}{\frac {\delta \rho }{\delta t}}+\nabla _{\alpha }\left(\rho V^{\alpha }\right)&=\rho CB_{\alpha }^{\alpha }\\\\\rho \left({\frac {\delta C}{\delta t}}+2V^{\alpha }\nabla _{\alpha }C+B_{\alpha \beta }V^{\alpha }V^{\beta }\right)&=-\rho ^{2}e_{\rho }B_{\alpha }^{\alpha }\\\\\rho \left({\frac {\delta V^{\alpha }}{\delta t}}+V^{\beta }\nabla _{\beta }V^{\alpha }-C\nabla ^{\alpha }C-2CV^{\beta }B_{\beta }^{\alpha }\right)&=-\nabla ^{\alpha }\left(\rho ^{2}e_{\rho }\right)\end{aligned}}}
where the
δ
/
δ
t
{\displaystyle {\delta }/{\delta }t}
-derivative is the central operator, originally
due to
The Calculus of Moving Surfaces
. Note that, in compressible models, the combination
ρ
2
e
ρ
{\displaystyle \rho ^{2}e_{\rho }}
is commonly identified with pressure
p
{\displaystyle p}
. The governing system above was originally formulated in reference 1.
For the Laplace choice of surface tension
(
e
(
ρ
)
=
σ
/
ρ
)
{\displaystyle \left(e\left(\rho \right)=\sigma /\rho \right)}
the system becomes:
δ
ρ
δ
t
+
∇
α
(
ρ
V
α
)
=
ρ
C
B
α
α
ρ
(
δ
C
δ
t
+
2
V
α
∇
α
C
+
B
α
β
V
α
V
β
)
=
σ
B
α
α
δ
V
α
δ
t
+
V
β
∇
β
V
α
−
C
∇
α
C
−
2
V
β
B
β
α
=
0
{\displaystyle {\begin{aligned}{\frac {\delta \rho }{\delta t}}+\nabla _{\alpha }\left(\rho V^{\alpha }\right)&=\rho CB_{\alpha }^{\alpha }\\\\\rho \left({\frac {\delta C}{\delta t}}+2V^{\alpha }\nabla _{\alpha }C+B_{\alpha \beta }V^{\alpha }V^{\beta }\right)&=\sigma B_{\alpha }^{\alpha }\\\\{\frac {\delta V^{\alpha }}{\delta t}}+V^{\beta }\nabla _{\beta }V^{\alpha }-C\nabla ^{\alpha }C-2V^{\beta }B_{\beta }^{\alpha }&=0\end{aligned}}}
Note that on flat (
B
α
β
=
0
{\displaystyle B_{\alpha \beta }=0}
) stationary (
C
=
0
{\displaystyle C=0}
) manifolds, the
system becomes
∂
ρ
∂
t
+
∇
α
(
ρ
V
α
)
=
0
ρ
(
∂
V
α
∂
t
+
V
β
∇
β
V
α
)
=
−
∇
α
(
ρ
2
e
ρ
)
{\displaystyle {\begin{aligned}{\frac {\partial \rho }{\partial t}}+\nabla _{\alpha }\left(\rho V^{\alpha }\right)&=0\\&\\\rho \left({\frac {\partial V^{\alpha }}{\partial t}}+V^{\beta }\nabla _{\beta }V^{\alpha }\right)&=-\nabla ^{\alpha }\left(\rho ^{2}e_{\rho }\right)\end{aligned}}}
which is precisely classical Euler's equations of fluid dynamics.
A simplified system
If one disregards the tangential components of the velocity field, as frequently done in the study of thin fluid film, one arrives at the following simplified system with only two unknowns: the two dimensional density
ρ
{\displaystyle \rho }
and the normal velocity
C
{\displaystyle C}
:
δ
ρ
δ
t
=
ρ
C
B
α
α
ρ
δ
C
δ
t
=
σ
B
α
α
{\displaystyle {\begin{aligned}{\frac {\delta \rho }{\delta t}}&=\rho CB_{\alpha }^{\alpha }\\&\\\rho {\frac {\delta C}{\delta t}}&=\sigma B_{\alpha }^{\alpha }\\\end{aligned}}}
References
1. Exact nonlinear equations for fluid films and proper adaptations of conservation theorems from classical hydrodynamics P. Grinfeld, J. Geom. Sym. Phys. 16, 2009