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Viscous Fingering

An outstanding problem in theoretical physics
is to understand the behavior of nonequilibrium systems, and find the universal
principles which govern their dynamics. Unlike equilibrium
dynamics, dictated by the principle of maximum entropy, there is no general
principle to constrain the dynamics of systems when they are out of
equilibrium. The complexity of the
problem led researchers to study relatively simple systems with low effective
dimension, such as interface dynamics. A large class of such systems are the
Stefan problems which describe the motion of the interface between different
thermodynamics phases, such as ice in supercooled water, or the normal and the superconducting phases
in type I superconductors when a magnetic field is suddenly turned on.

The
Saffman Taylor (ST) problem in Hele-Shaw cell constitutes a central paradigm in
this field. ** **Here
a non-viscous fluid is forced into the center of a cell made from two parallel
plates. The small gap between the plates is filled by a viscous fluid. When the
pumping rate of the non-viscous fluid is sufficiently large to exceed the
stabilizing effects of the surface tension, a fractal-like bubble of a distinctive
fingering pattern emerges, as shown left panel of Fig. 1. A similar fractal pattern is formed by
diffusion limited aggregation (DLA). The latter model, introduced by Witten and
Sander, is remarkably simple:
One
begins with an immobile seed placed at the origin of a two dimensional plane.
Then a particle launched far away from the origin diffuses until it touches the
seed. At this moment it becomes immobile, and part of the aggregate. Additional
random walkers are launched one-by-one, and accumulate once they hit the
cluster. The resulting aggregate (see right panel of figure 1) is characterized
by a fractal structure similar to that of a developed ST bubble**.**

** **

The
fractal patterns of, both DLA and the ST problem are believed to be
statistically equivalent. Yet, while DLA is more suitable for numerical computations,
the ST problem seems to be more tractable by analytical methods. The reason is
that the "idealized ST problem", where surface tension is neglected,
possesses an infinite number of conserved quantities. Unfortunately, in this
limit the problem is ill defined since a generic smooth initial interface
develops a cusp-like singularity within finite time. Surface tension hinders
the formation of cusps, but at the same time destroys the nice analytical
properties of the idealized ST problem.

In order to circumvent this problem one
may seek a different regularization procedure which maintains the analytic
structure of the problem, as well as the global statistical behavior, but may
change the local dynamics. In this respect an intriguing property the surface
tension is that it plays a role similar to: Namely, solutions of the ST problem, in a channel geometry,
have an essential singularity in the limit where surface tension approaches
zero, and the finger width is quantized to a discrete set of values.

This behavior suggests regularizing the ST
problem by quantization. There may be several quantization schemes. One of them
is based on "dispersive regularization" - a celebrated approach which goes back to
Whitham who introduced in ** **Further developments of this method were
presented in the seminal papers of Gurevich and Pitaevskii, Flaschka, Forest,
and McLaughlin, Lax and Levermore and Tsarev.** **In the context of the ST
problem, one may picture the dispersive regularization as follows [22-28]: The
interface of the ST bubble (at different stages of the evolution) is viewed as
an equipotential line of some given potential, , specified by the conserved quantities of the idealized
problem. This potential is filled by Fermions where each
one of them occupies an area that equals. In the ground state of the system, these Fermions form a
droplet whose smooth edge is associated with the Saffman-Taylor bubble. Finally
the limit and is taken, keeping the
droplet area, , fixed.

The solutions obtained by this
regularization differ from those which correspond to, since they allows for new bubbles to be formed. These new
bubbles are associated with Fermions which tunnel to distant minima of the
potential, . Thus the cusps which
appeared in the original problem are now replaced by points where two droplets,
the original and the newly formed one, merge, as demonstrated in Fig. 2.

** **

**
Figure 2**: The behavior of the generalized solutions
ST problem near a cusp.

In
reality, Hele-Shaw cells do not exhibits such a behavior, and it is an open
question whether the large scale structure generated by dispersive
regularization has the same statistical properties of DLA and ST bubbles.
Nevertheless, these solutions describe the evolution of a novel model of
viscous fingering - the so called "Gortex model", illustrated in Fig.
3. Here, the upper plate of the Hele-Shaw cell is permeable to the non-viscous
fluid (air) but not to the viscous one (oil). Air is pumped (or extracted) at
any point through this plate, such that more than one bubble may be formed, and
all of them share the same pressure.

**
Figure 3:** A schematic
illustration of the gortex model. Shaded region represents a small bubble of
air in the ambient oil. The insulator is permeable to air but not to oil.

An
experimental realization of the Gortex model seems to be out of reach. However,
it turns out that Steve
Lipson’s group at the Technion devised a very similar system. It consists
of a clean mica substrate covered by a thin water film. Under proper conditions
of temperature and pressure, two films of different thickness, approximately 2
and 12 nm, can coexist. This behavior is a consequence of the competing van der
Waals and polar surface forces between the water and the substrate. A first
order phase transition between the film heights is induced by changing the
vapor pressure, and thus the dynamics of evaporation, in this system, manifests
itself through the motion of the interface which separates the phases. Thus no
additional plate is needed in order to confine the fluid height. Evaporating thin films constitute a new model
of viscous fingering which we proposed to study. Click here and here in order to see movies
showing the experimental recording of the evaporation dynamics (curtsey of
Steve Lipson)

In
order to highlight the differences and similarities between the dynamics of
volatile thin films and the ST problem, let us write down the equations of
motion describing the evolution of idealized volatile films. These consist of
(a) Darcy's law:

(1)

where is the (two dimensional) velocity field, is the film height, is its viscosity, and is the pressure, and (b)
the assumption of uniform evaporation within the regions of large film
thickness:

, (2)

**F****igure 4:** Examples
for the evolution of evaporating droplets. Left panel shows snapshots of the
contour of a droplet with a fivefold symmetry. Middle panel shows a dry patch
penetrating, from left to right, into the liquid domain, and right panel
demonstrates the evolution of droplet pinching during evaporation.

where is the two dimensional
fluid density, and is the evaporation
rate (in units of mass per unit time per
unit area). The fluid dynamics, in
regions of small film thickness, may be ignored since the velocity is
approximately zero, and evaporation is negligible due to the strong adhesion
force between the substrate and the fluid. Combining the above equations one
concludes that, inside the region of high film thickness, , henceforth referred as the "droplet", the
pressure satisfies Poisson's equation, while outside it is constant.
Without loss of generality one may set this constant to be zero, and thus

(3)

where . The above equation describes the pressure away from the
interface. Near the interface, additional degrees of freedom, such as the film
height and curvature, come into play. The local behavior near the droplet's
edge sets the boundary conditions for the above equations. In the idealized
case where the surface tension energy associated with the interface can be
neglected, the pressure is continuous and therefore the boundary condition for
the above equation is

for ,
(4)

where denotes the interface
contour, and is a complex
coordinate on the mathematical complex plane where the droplet resides. This
boundary condition describes the approximate behavior of the system over time
intervals where the effect of surface tension can be neglected. The
experimental data (see e.g. [35]) indicate that these are rather extensive time
intervals since very narrow fluid spines exist for rather long time.

There are two
differences between the above equations and those of the idealized ST problem:
Firstly, in the ST problem the interior domain is occupied by the less viscous
fluid, while here the droplet represents a domain occupied by viscous fluid. In
other words the interior and the exterior domains switch parts. Secondly, in
the ST problem the sink is located at infinity and the pressure satisfies
Laplace equation, while here it is uniformly distributed within the droplet as
manifested by Poisson's equation (3).

Eqs. (3) and (4) are, in fact, also the
governing equations of another problem known as the "stamping
problem". The latter describes the evolution of a droplet in Hele-Shaw cell when the gap
between the two parallel plates changes in time. The stamping problem has been
studied by Entov and Étingorf [38], and later by Gustafsson [39]. Similar to
the idealized ST problem it possesses an infinite number of conserved
quantities which determine the droplet shape [38]. Thus one may construct a
large set of exact solutions describing the evolution of the system. Examples
for such solutions, which have been obtained using conformal maps [37], are
depicted in Fig. 4.

Although the idealized equations of volatile thin films and the stamping
problem seem to be equivalent, they differ by two important ingredients. First, in volatile thin films dry patches may
nucleate inside the droplet; while a similar situation cannot occur in the
stamping problem. Second, and perhaps
more important, is that the local dynamics near the droplet's interface is very
different: For the stamping problem it may be accounted by adding a simple
surface tension term, but in the case of volatile thin films the situation is
more complicated. It has been shown
that, here, an additional instability takes over near the interface [34]. It
leads to an inhomogeneous distribution of the liquid along the droplet edge,
and hence to a particular set of solutions, very different from those of the
stamping problem. A common interesting
pattern that appears in volatile films is the so called "doublon",
shown in Fig. 5. It is a finger spilt in two by a narrow water spine (which
eventually brakes into small droplets due to surface tension), see also the
middle panel of Fig. 4. The doublon is similar to a soliton, in the sense that
it moves large distances without changing its general shape. Similar types of
fingers have been found in several numerical studies of related problems
[40-42].