Hele-Shaw cell

We are looking for distortions of the line of contact that are slow on the scale of the thickness h of the cell. Under these conditions, it is legitimate 

FIGURE 3.8. Horizontal Hele-Shaw cell. Before the deformation, an interface bounds a trapped drop between two horizontal plates. The deformation is the same at any height [u y) is independent of z]. 

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This structure is generated via the modified diffusion-limited aggregation (DLA) algorithm of [205] using the law p = a (m/N). Here, N = 2, 000 (the number of particles of the DLA clusters), a = 10 and ft = 0.5 are constants that determine the shape of the cluster, p is the radius of the circle in which the cluster is embedded, pc = 0.1 is the lower limit of p (always pc < p), and to is the number of particles sticking to the downstream portion of the cluster. This example corresponds to a radial Hele-Shaw cell where water has been injected radially from the central hole. Due to heterogeneity a sample cannot be used to calculate the dissolved amount at any time, i.e., an average value for the percent dissolved amount at any time does not exist. This property is characteristic of fractal objects and processes. 

Figure 5.10 is an example of the results obtained. While most investigators of VF used Hele-Shaw cells which approximate the three-dimensional bed with a pseudo-two-dimensional one, these last two groups have visualized the VF phenomenon in actual columns. 

Problem 8-7. The Hele-Shaw Cell. The Hele Shaw cell is perhaps one of the simplest constructions of a nearly unidirectional flow however, as we shall see in this problem the flow inside the cell has some remarkable properties. A Hele Shaw cell consists of two flat parallel rigid walls of vertical separation h and horizontal extent / with h/l 1. The gap between the fluids is occupied with Newtonian fluid and contains obstacles in the form of cylinders with generators normal to the walls The fluid is being driven through the cell by a steady horizontal pressure gradient Vp = G applied between the ends of the cell. 

This result implies that (at any fixed z) the flow past a cylindrical obstacle will correspond to the 2D potential flow past that obstacle. For the Hele Shaw cell the pressure p plays the role of the velocity potential velocity potential outside a circular cylinder of radius a. There is, however, one important caveat. Let us consider the circulation T defined by... 

Some excellent photgraphs of flow past obstacles in a Hele-Shaw cell are contained in An Album of Fluid Motion by M. Van Dyke (Parabolic Press, Stanford, CA, 1982). 

P. G. Saffman and G. I. Taylor, The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous fluid, Proc. R. Soc. London Ser A 245, 312-29 (1958). 

Researchers have used physical models of porous media to study flow problems for many years. For example, the Hele-Shaw cell appeared in the late 1800s (Sahimi, 1993). The first reported use of such models for two-phase systems is attributed to Chatenever and Calhoun (1952), who used Lucite and glass bead packs to view immiscible displacement of brine and crude oil (Buckley, 1991). Subsequently, etched and photo-etched glass were used to construct physical models. The use of molded resins for model construction was introduced in the 1970s (Buck-ley, 1991). 

Wooding, R.A. I960. Instability of a viscous liquid of variable density in a vertical Hele-Shaw cell. J. Fluid Mech. 7 501-515. 

R. A. Wooding, Instability of a Viscous Liquid of Variable Density in a Vertical Hele-Shaw Cell, J. Fluid Mech. (7) 501-515,1960. 

In very large Hele-Shaw cells, at a distance from the walls much larger than the cell thickness, a hexagonal order is obtained only locally, and defects destroy the long-range order. This is expected to arise from the randomness of the interactions between columns and the bottom of the cell. Once a column spans the narrow height of the cell, the solid friction with the floor and ceiling freezes the... 

Studies of bubble rise in Hele-Shaw cells demonstrated the production of an anomalous wake structure which encircles the bubble completely and yields a larger effective size of the bubble. A systematic study found that the velocity of a bubble in the presence of surfactant could be... 

Park CW, Maruvada SRK, Yoon DY (1994) The influence of surfactant on the bubble motion in hele-shaw cells. Phys Fluid 6 3267-3275... 

The first example is the so-called Hele-Shaw cells in which a fluid is confined between two parallel plates separated by a very small distance. The configuration is illustrated in Figure 3.8 [the plates are in (x, y) planes at z = 0 and z h the line is defined by the intersection of the vertical plane (x = 0) with z — Q. Again, we choose Oe — 2 for the sake of simplicity. 

Studies of bubble rise in Hele-Shaw cells demonstrated the production of an anomalous wake structure which encircles the bubble completely and yields a larger effective size of the bubble. A systematic study found that the velocity of a bubble in the presence of surfactant could be almost an order of magnitude lower than the velocity expected in the case of a clean interface. More surprisingly, drops could take on complex shapes such as oval or dumbbell shapes, or even forming tails behind them [6]. Similar drop shapes may be obtained in microfluidic situations, as shown in Fig. 3 for the case of a droplet with a tail. 

Park C, Homsy G. (1984) 2-phase displacement in Hele Shaw cells — iY covy. J Fluid Mech 139 291-308. 

Imperfect hele - Shaw cells" J. Phvs.. (Paris) 42, 1541-1546... 

Paterson A, Fermigier M, Jenifer P, Limat L (1995) Wetting on heterogeneous surfaces Experiments in an imperfect Hele-Shaw cell. Physical Review E 51-2 1291-1299... 

We now study the Rayleigh-Taylor instability in a Hele-Shaw cell with open boundaries where the general wetting results introduced above play an... 

P.G. Saffman, Viscous Fingering in Hele-Shaw cells, J. Fluid Mech. 173 73... 

With this perspective, we review our recent work where ME are presented as a convenient model system for inducing cross-diffusion-driven hydrodynamic flows in double-layer systems. We first present a general modeling and theory on buoyancy-driven convection promoted by cross-diffusion and we successively study experimentally convective patterns occurring when two ME are stratified in a Hele-Shaw cell [2,3]. 

The most convenient reactor for studying convective dynamics at the interface between two ME of different compositions consists of a vertically oriented Hele-Shaw cell, composed of two borosilicate glasses separated by a teflon spacer of 0.10 mm (see Fig. 5) [2,3,18]. Two different water-in-oil (W/0) reverse microemulsions were filled in the reactor through the inlet ports positioned at the top and the bottom of the cell. The excess of the solutions is pumped out through the cell s outlets until a flat interface between the two liquids is obtained. Each of the two microemulsions having different composition initially occupies half of the reactor height. The top and bottom solutions are prepared by using distilled... 

Shi, Y., Eckert, K. A novel Hele-Shaw cell design for the analysis of hydrodynamic instabilities in liquid systems. Chem. Eng. Sci. 63(13), 3560-3563 (2008)... 

Trevelyan, P.M.J., Ahnarcha, C., De Wit, A. Buoyancy-driven instabilities of miscible two-layer stratifications in porous media and Hele-Shaw cells. J. Fluid Mech. 670, 38-65 (2011)... 

The experiments are performed in a vertically oriented thin Hele Shaw cell having transparent 75 mm x 25 mm sapphire windows separated by a 3 mm thick plastic frame. The cell is cooled by thermoelectric elements in contact with a window face at its upper and lower ends. Observations are made at low power magnification through crossed polarizers. In the preliminary studies reported here, the cell ends were cooled at fixed power levels the lower end was cooled sufficiently to nucleate ice locally and the ice was then observed as it advanced to the upper end of the cell. The water was deionized, and the cell was open to the atmosphere for pressure relief... 

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