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We have studied the dispersive behaviour of imbibition fronts in a porous medium by X-ray tomography. Injection velocities were varied and the porous medium was initially prewetted or not. At low velocity in the prewetted medium, the imbibition profiles are found to be distinctly hyperdispersive. The profiles are anomalously extended when compared to tracer fronts exhibiting conventional (Gaussian) dispersion. We observe a strong velocity dependence of the exponent characterizing the divergence of the dispersion coefficient for low wetting-fluid saturation. Hyperdispersion is absent at high imbibition velocities or when the medium is not prewetted.
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A theoretical study of the emergence of helices in the wake of precipitation fronts is presented. The precipitation dynamics is described by the Cahn-Hilliard equation and the fronts are obtained by quenching the system into a linearly unstable state. Confining the process onto the surface of a cylinder and using the pulled-front formalism, our analytical calculations show that there are front solutions that propagate into the unstable state and leave behind a helical structure. We find that helical patterns emerge only if the radius of the cylinder R is larger than a critical value R>R_c, in agreement with recent experiments.
Using general scaling arguments combined with mean-field theory we investigate the critical ($T simeq T_c$) and off-critical ($T e T_c$) behavior of the Casimir forces in fluid films of thickness $L$ governed by dispersion forces and exposed to long-ranged substrate potentials which are taken to be equal on both sides of the film. We study the resulting effective force acting on the confining substrates as a function of $T$ and of the chemical potential $mu$. We find that the total force is attractive both below and above $T_c$. If, however, the direct substrate-substrate contribution is subtracted, the force is repulsive everywhere except near the bulk critical point $(T_c,mu_c)$, where critical density fluctuations arise, or except at low temperatures and $(L/a) (betaDelta mu) =O(1)$, with $Delta mu=mu-mu_c <0$ and $a$ the characteristic distance between the molecules of the fluid, i.e., in the capillary condensation regime. While near the critical point the maximal amplitude of the attractive force if of order of $L^{-d}$ in the capillary condensation regime the force is much stronger with maximal amplitude decaying as $L^{-1}$. Essential deviations from the standard finite-size scaling behavior are observed within the finite-size critical region $L/xi=O(1)$ for films with thicknesses $L lesssim L_{rm crit}$, where $L_{rm crit}=xi_0^pm (16 |s|)^{ u/beta}$, with $ u$ and $beta$ as the standard bulk critical exponents and with $s=O(1)$ as the dimensionless parameter that characterizes the relative strength of the long-ranged tail of the substrate-fluid over the fluid-fluid interaction. We present the modified finite-size scaling pertinent for such a case and analyze in detail the finite-size behavior in this region.
Helical and helicoidal precipitation patterns emerging in the wake of reaction-diffusion fronts are studied. In our experiments, these chiral structures arise with well-defined probabilities P_H controlled by conditions such as e.g., the initial concentration of the reagents. We develop a model which describes the observed experimental trends. The results suggest that P_H is determined by a delicate interplay among the time and length scales related to the front and to the unstable precipitation modes and, furthermore, the noise amplitude also plays a quantifiable role.
The empirical velocity of a reaction-diffusion front, propagating into an unstable state, fluctuates because of the shot noises of the reactions and diffusion. Under certain conditions these fluctuations can be described as a diffusion process in the reference frame moving with the average velocity of the front. Here we address pushed fronts, where the front velocity in the deterministic limit is affected by higher-order reactions and is therefore larger than the linear spread velocity. For a subclass of these fronts -- strongly pushed fronts -- the effective diffusion constant $D_fsim 1/N$ of the front can be calculated, in the leading order, via a perturbation theory in $1/N ll 1$, where $Ngg 1$ is the typical number of particles in the transition region. This perturbation theory, however, overestimates the contribution of a few fast particles in the leading edge of the front. We suggest a more consistent calculation by introducing a spatial integration cutoff at a distance beyond which the average number of particles is of order 1. This leads to a non-perturbative correction to $D_f$ which even becomes dominant close to the transition point between the strongly and weakly pushed fronts. At the transition point we obtain a logarithmic correction to the $1/N$ scaling of $D_f$. We also uncover another, and quite surprising, effect of the fast particles in the leading edge of the front. Because of these particles, the position fluctuations of the front can be described as a diffusion process only on very long time intervals with a duration $Delta t gg tau_N$, where $tau_N$ scales as $N$. At intermediate times the position fluctuations of the front are anomalously large and non-diffusive. Our extensive Monte-Carlo simulations of a particular reacting lattice gas model support these conclusions.
We report forced radial imbibition of water in a porous medium in a Hele-Shaw cell. Washburns law is confirmed in our experiment. Radial imbibition follows scaling dynamics and shows anomalous roughening dynamics when the front invades the porous medium. The roughening dynamics depend on the flow rate of the injected fluid. The growth exponents increase linearly with an increase in the flow rate while the roughness exponents decrease with an increase in the flow rate. Roughening dynamics of radial imbibition is markedly different from one dimensional imbibition with a planar interface window. Such difference caused by geometric change suggests that universality class for the interface growth is not universal.
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