Fluid-displacement experiments in Hele-Shaw cells filled with a viscoelastic fluid show a novel transition between a viscous fingering (VF) regime producing fractal patterns of "fingers" and a viscoelastic fracturing (VEF) regime producing fractal patterns of "cracks." VEF patterns are characterized by branching angles of 90° with respect to the main crack, behind the tip, and by a lower fractal dimension than VF. The transition is controlled by several parameters, including the Deborah number and the system deformability. PACS numbers: 46.30.Jv, 47.90.+a The intrusion under pressure of a liquid into another phase may lead either to "viscous fingering" (VF) when a high-viscosity fluid is penetrated by a low-viscosity fluid within a medium where flow is dominated by viscous friction [1] or to "hydraulic fracturing" when a solid (generally a rock) is broken open by injection of a liquid at high pressure [2]. Viscous fingering is an interfacial instability which stems from the destabilizing action of viscous forces, whereas hydraulic fracturing, which can be considered as a subdomain of the more general fracture process [3,4], is a structural instability which is driven by the release of elastic stress. In Newtonian fluids within axial or radial Hele-Shaw cells [1], VF leads to the growth of smooth fingers [5,6], whose width and splitting depend on the capillary number (ratio of viscous over capillary forces). The use of a shear-thinning nonNewtonian fluid introduces an additional instability and allows for a considerable increase of the effective capillary number [7]. This results in a drastic decrease of the wavelength of maximal growth and an increase of branching [8], leading to fractal arborescences [9-11].One can go a step further with the use of viscoelastic fluids. Viscoelastic fluids are characterized by (at least) one internal relaxation time t r , which characterizes the time scale of the structural reorganizations within the fluid [12]. It is related to the reciprocal frequencies of the motion of the colloidal particles in the fluid. If the time scale of a flow event, f n, is much shorter than / r , the medium responds essentially as an elastic body. In the opposite case, it behaves like a liquid. This can be rationalized in terms of a dimensionless number, De=f r An [13]. For De<3Cl, viscous effects dominate, whereas for De^> 1, the system behaves essentially as an elastic solid. Thus, the use of viscoelastic fluids permits one to think of a possible crossover from fractal growth by flow to fractal growth by fracture by decreasing De. We report here the results of experiments performed with concentrated dispersions of smectite clays [14][15][16] in water which clearly show the transition between a flow regime characterized by fractal patterns of "fingers" and a fracturing regime characterized by fractal patterns of "cracks." The storage and release of elastic energy in the cell is at the basis of this transition and, as anticipated, the crossover from VF to viscoelastic fracturing (VEF) occurs for a ...
We studied pattern formation in granular media in two distinct states: in the dry and non-cohesive (powder) state, on the one hand, and in the wet and cohesive (paste) state, on the other. In the first case, we have shown that gas injection in a thin layer of powder within a Hele Shaw cell leads to fractal patterns remarkably similar to viscous fingering patterns obtained with complex fluids. In the second case, we have shown that the tensile cohesive viscoplastic fracture of a layer of paste leads to self-affine rough surfaces with a Hurst exponent close to 0.88, very close to the value obtained for fragile fracture by other authors.18 Our observations reinforce the universality of two fractal growth processes and add a new element to the ambivalent nature of the granular state of matter.
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