We study patterns formed by the viscous fingering instability in a porous media. When the displacing fluid preferentially wets the medium, the finger width is much larger than the pore size and, when normalized by the square root of the permeability, is found to scale with capillary number as Ca^−1/2. While traditional theories based on Hele-Shaw geometry give this dependence for the most unstable wavelength, they are unable to explain the magnitude of the finger. We consider here the effect of a velocity dependent capillary pressure in addition to the more conventional static term, and suggest that it may control the scaling of the finger width on Ca. We demonstrate the existence of this dynamic capillary pressure, which offers new insight into the basic physics of the motion of a fluid interface in porous media.

The aggregation of colloids is of substantial interest both fundamentally and practically. The level of interest has risen in recent years with the observation that colloidal aggregates are often well characterized as scale-invariant, or fractal, objects, providing a quantitative description of the structure of these random, irregular clusters.^1–4 The consequences of the scale invariance on light scattering from the clusters has been widely exploited. In static light scattering, the dependence of the scattering intensity on the scattering wave vector q allows a convenient way of determining the fractal dimension of the clusters, while Quasi-Elastic Light Scattering (QELS) has proved useful in monitoring the kinetics of the aggregation process. The combination of the scale-invariant structures of the aggregates and the power-law distributions which often occur leads to elegant scaling behavior of the dynamic light scattering.^5,6 For the large clusters (qR_g ≳1) often found in aggregation, rotational diffusion can play an important role in determining the decay of the autocorrelation of the scattered light measured in QELS. While scaling arguments have been used to account for the contribution of rotational diffusion, it is nonetheless important to determine this effect more quantitatively.

Optical scattering and absorption characteristics of fractal clusters are calculated with a self‐consistent scheme in which the multiple scattering is explicitly taken into account. Our results indicate that for optical scattering the gold colloidal aggregates with fractaldimension D=1.75 behave essentially as low dimensional objects and therefore can be accurately described by the single scattering Born approximation. However, the absorptionspectrum of the clusters is shown to be significantly affected by the proximity of the goldparticles to their nearest neighbors in the aggregate structure.

We measure the velocity dependence of the capillary pressure, ΔPc(v), between two fluids in a porous medium. At zero velocity, the interface is pinned, and a critical ΔPc must be achieved before the interface moves. For nonzero velocities, there is an additional dynamic component to ΔPc, which scales as v12 when a wetting fluid displaces a nonwetting fluid. We suggest that this dynamic component of ΔPc can stabilize viscous fingers, and obtain excellent agreement with experiment.

A Comment on the Letter by P. Wiltzius, Phys. Rev. Lett. 58, 710 (1987).

We study the influence of the fractal structure on the physical properties of colloidal gold aggregates. The optical properties are strongly influenced by both the long- and the short-range correlations of the aggregate structure, as well as the electronic plasma resonance of the gold particles. The absorption of the aggregates is dominated by the electromagnetic interaction between nearest-neighbour particles. The angular dependence of the polarized scattering reflects the long-range fractal correlations of the particles in the clusters, while that of the depolarized scattering reflects the short-range correlations of the local field corrections. The fractal structure also affects dynamic light scattering, so that rotational diffusion effects play an important role in the decay of the autocorrelation function. Finally, we show that the very tenuous nature of the fractal structure makes the aggregates quite susceptible to deformation under shear stress.

 

We study the onset of the Taylor instability for colloidal crystals in a narrow-gap Couette Cell. There is a natural competition between the radial flow required for Taylor rolls and the resistance to such motion caused by the anisotropy of the flowing solid. A new, combined Taylor–shear-melting instability is found at rotation rates above those expected for the formation of rolls, but below the shear rate required for shear melting in the absence of the Taylor instability. The control parameter for the shear melting transition appears to be a critical shear stress rather than the shear rate.

Reaction-limited cluster aggregation is modeled with the kinetic rate (Smoluchowski) equations, using a kernel determined intrinsically by the clusters fractal geometry. The kernel scales with cluster mass as M1Mλ−12 (M1≫M2), and Mλ1 (M1≊M2), with λ=1 in three dimensions, resulting in exponential kinetics and a cluster mass distribution CM∼M−τ, with τ=(3/2), in excellent accord with experiments. The singular nature of this solution forces the adjustment of the cluster fractal dimension, df, thereby determining its value.