Imperfect Hele - Shaw cells

2014 The covering surfaces of a Hele-Shaw cell may be roughened mechanically, or modified by random chemical patches which change the wetting properties. We discuss first the quasi-static injection of a single fluid (with a finite contact angle 03B8e) in an imperfect cell. The shape of the injected region depends critically on a dimensionless number i (ratio of the boundary energy 03C4 over the capillary energy fluctuations in one patch). At 03C4 ~ 1 we expect a « soft » penetration in the form of percolation clusters. At 03C4 ~ 1 we expect a « hard » behaviour; penetration should occur only above a certain coercive pressure, and the boundary is much more stiff. In the soft case, we also consider one dimensional flows, where the fluid advances with a small but finite velocity U (or capillary number Ca). The width l of the front between dry and wet regions should scale like Ca-1. The size 03BE of the largest clusters sprouting from the interface is smaller than l and scales with a different exponent. J. Physique 47 (1986) 1 541 -i546SEPTEMBRE 1986, 1 Classification Physics Abstracts 05.20 47.15 68.45G 75.60 1. Two types of disorder. An incompressible fluid moving in a narrow gap (thickness h) between two parallel plates, gives rise to simple potential flows [1]. When the fluid is injected in an empty cell, the interface is stable, but when the injected fluid pushes out a second liquid of higher viscosity, a classic instability sets in [2]. It has recently been noticed that modifications of the plate surfaces may then affect the interfacial pattern [3]. In the present note, we analyse some elementary effects of two types of randomness : (A) Geometrical disorder : the upper plate, for instance, instead of being planar (z = h), is defined by an irregular surface : (B) Chemical disorder : the interfacial tensions YS1 (YS2) between the upper surface S and liquid 1 (2) are modulated : 2. Static equilibrium. Consider the interface between a fluid (1) at uniform pressure p and a fluid (2) at pressure 0. The equilibrium conditions correspond to a. force balance along a horizontal direction n(n, ny) normal to the interface (Fig. 1). For case A this gives : Here y is the 1-2 interfacial tension, and po is the equilibrium pressure for a smooth cell : (Je being the contact angle : the second equality expresses the classical Young condition for (Jeo Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019860047090154100