Current Research Areas


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.       


                           Sharon          DLATrans

                                           Figure 1: Left panel: The Saffman-Taylor bubble (experiment); Right panel: a DLA aggregate (computer generated).



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 1965 a systematic analysis of the behavior of wave packets of nonlinear wave equations. 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:


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)






Figure 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



      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).

Text Box:  
Figure 5: The "doublon" (indicated by the arrow), a finger spilt in two by a narrow water spine, From Ref. [35]
           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].