We study experimentally and theoretically the downward vertical displacement of one miscible fluid by another lighter one in the gap of a Hele-Shaw cell at sufficiently high velocities for diffusive effects to be negligible. Under certain conditions on the viscosity ratio, M, and the normalized flow rate, U, this results in the formation of a two-dimensional tongue of the injected fluid, which is symmetric with respect to the midplane. Thresholds in flow rate and viscosity ratio exist above which the two- dimensional flow destabilizes, giving rise to a three-dimensional pattern. We describe in detail the two-dimensional regime using a kinematic wave theory similar to Yang & Yortsos (1997) and we delineate in the (M, U)-plane three different domains, characterized respectively by the absence of a shock, the presence of an internal shock and the presence of a frontal shock. Theoretical and experimental results are compared and found to be in good agreement for the first two domains, but not for the third domain, where the frontal shock is not of the contact type. An analogous treatment is also applied to the case of axisymmetric displacement in a cylindrical tube.

We study the downward miscible displacement of a fluid by a lighter and less viscous one in the gap of a Hele-Shaw cell. For sufficiently large velocities, a well-defined interface separates the two fluids. As long as the velocity or the viscosity ratio are below a critical value, the interface has the shape of a tongue symmetric across the gap. For viscosity ratios larger than a critical value, estimated at 1.5, there exists a critical velocity, above which the interface becomes unstable, leading to a new 3D pattern involving regularly spaced fingers of wavelength about 5 times the cell thickness. We delineate the stability diagram. [S0031-9007(97)04791-1] PACS numbers: 47.20.Gv, 47.20.Bp, 83.85.Pt A key issue in interface dynamics is the understanding of pattern selection. Of particular interest is the pattern selection related to the 2D Saffman-Taylor finger (ST) [1], the selection rule of which has raised a great amount of interest [1][2][3][4][5]. ST dynamics govern a variety of seemingly different physical phenomena, such as viscous fingering [2], directional solidification [3], or thermal plume [4]. We recall that the well-studied ST finger results from the displacement of a viscous fluid by an immiscible, less viscous one in a Hele-Shaw cell (consisting of two parallel plates L 3 W , separated by a small gap b). In this immiscible displacement problem, the interface in the gap consists of a nearly hemispherical meniscus, which completely spans the cell gap at the edges of the finger [6], provided that the capillary number ͑Ca hq͞g, with h the viscosity, g the surface tension, and q the fluid velocity) is sufficiently small. The patterns resulting from the balance between capillary and viscous forces are by necessity quasi-2D (namely, in the L 3 W plane). The removal of surface tension, which makes the coexistence of the fluids across the gap possible, provides the opportunity to obtain 3D patterns, also extending across the gap. The existence of such patterns has an interest of its own, as well as in connection to the 2D selection rule discussed above. 3D patterns can be achieved using two miscible fluids, provided that the displacement velocity (i.e., the Peclet number, Pe qb͞D m , where D m is the molecular diffusion coefficient) is sufficiently large for diffusion effects to be negligible [7,8], so that a sharp fluid interface can be defined. So far, only two specific experiments have addressed miscible displacements in a Hele-Shaw cell: gravity-driven flow in a rectangular geometry [9] and viscous fingering in a horizontal radial geometry (Paterson [10]). However promising Paterson's flower pattern might have been, his pioneering work was surprisingly not resumed even for the simple geometry of a rectangular cell.This Letter presents experimental results of miscible displacements in a vertical planar Hele-Shaw cell in the high Pe regime and emphasizes the role that displacement features along the gap have on the overall pattern. Depending on the velocity and viscosity ratio M (M h 1 ͞h 2 . 1, whe...

Autocatalytic reaction between reacted and unreacted species may propagate as solitary waves, namely at a constant front velocity and with a stationary concentration profile, resulting from a balance between molecular diffusion and chemical reaction. The effect of advective flow on the autocatalytic reaction between iodate and arsenous acid in cylindrical tubes and Hele-Shaw cells is analyzed experimentally and numerically using lattice BGK simulations. We do observe the existence of solitary waves with concentration profiles exhibiting a cusp and we delineate the eikonal and mixing regimes recently predicted.The motion of interfaces and the propagation of fronts resulting from chemical reactions occur in a number of different areas [1], including population dynamics [2,3] and flame propagation [4]. It is known that autocatalytic reaction fronts between two reacting species propagate as solitary waves, namely at a constant front velocity and with a stationary concentration profile [5,6]. The important issue of the selection of the front velocity was addressed earlier on, but only a few cases are well understood, such as the pioneering works of Fisher [2] and Kolmogorov-Petrovskii-Piskunov [3] on a reactiondiffusion equation with second-order kinetics [1,4,7]. The effect of advective flow (inviscid and/or turbulent) on reacting systems was analyzed extensively in the propagation of flames in the context of combustion [4,8]. On the other hand, advective effects on the behavior of autocatalytic fronts have been only recently addressed [9,10,11]. B. F. Edwards [11] studied theoretically the effect of a 2D laminar flow on an autocatalytic reaction front between two infinite planes separated by a gap b. In this geometry, the velocity profile is unidirectional in the direction z of the flow and is given by Poiseuille's equation,5 U is the maximum velocity, U is the mean velocity, ζ = 2x/b is the transverse normalized coordinate and − → z is the unit vector parallel to the flow, chosen as the direction of the front propagation in the absence of flow (see below). Consider the iodate-arsenous acid reaction described by a thirdorder autocatalytic reaction kinetics [1,5,6]:where C is the concentration of the (autocatalytic) reactant iodide, normalized by the initial concentration of iodate, D m is the molecular diffusion coefficient, and α is the reaction rate kinetic coefficient. In the absence of hydrodynamics ( − → U = − → 0 ), Eq.1 admits a well-known solitary wave solution with front velocity V 0 = αD m /2 and front width L 0 = D m /V 0 [5,6]. The use of these two quantities to normalize velocities and lengths in Eq.1, leads to two independent parameters η = b/2L 0 and ε = U /V 0 . Reference [11] investigated numerically the solitary wave solution of Eq.1, and particularly its normalized front velocity, v = V F /V 0 , as a function of ε, for different values of η. Of interest are the following asymptotic predictions:In the narrow-gap regime (η → 0 or ε → 0), it was found that v = 1 + ε. Namely, when L 0 >> b, mixing ac...