Nonextensive statistics in viscous fingering

Measurements in turbulent flows have revealed that the velocity field in nonequilibrium systems exhibits q-exponential or power-law distributions in agreement with theoretical arguments based on nonextensive statistical mechanics. Here we consider Hele–Shaw flow as simulated by the lattice Boltzmann method and find similar behavior from the analysis of velocity field measurements. For the transverse velocity, we obtain a spatial q-Gaussian profile and a power-law velocity distribution over all measured decades. To explain these results, we suggest theoretical arguments based on Darcy's law combined with the nonlinear advection–diffusion equation for the concentration field. Power-law and q-exponential distributions are the signature of nonequilibrium systems with long-range interactions and/or long-time correlations, and therefore provide insight to the mechanism of the onset of fingering processes.     

黏性指进的实验与模拟

利用简易装置观察了黏性指进现象,使用照相机拍摄多种黏性指进分形图形,测量了分形维数,并采用数值模拟中的随机模拟演示了黏性指进图形.     

Discrete scale invariance in viscous fingering patterns

We study viscous fingering patterns in a lifting Hele-Shaw cell, where a non-Newtonian fluid (oil paint) is displaced by air. The lengths of the air fingers are measured and their cumulative distribution is seen to follow a power law with log-periodic oscillations indicating the presence of discrete scale invariance. Received 21 May 1999     

Thermodynamic and scaling behaviour in finite diffusion-limited aggregation

Rényi's entropies for diffusion-limited aggregates are studied as a function of the number N of particles contained in the aggregates. It is found that Rényi's values increase with log N in a linear fashion, and that the aggregates exhibit multifractal behaviour for finite values of N. When N → ∞, the aggregate has a monofractal structure. Rényi's entropies depend on the fractal dimension of the aggregate. When the fractal dimension increases, the values of K_q decrease for q ? 1> and increase for q > 1.     

Effects of the screening breakdown in the diffusion-limited aggregation model

Several models based on the diffusion-limited aggregation (DLA) model were proposed and their scaling properties explored by computational and theoretical approaches. In this paper, we consider a new extension of the on-lattice DLA model in which the unitary random steps are replaced by random flights of fixed length. This procedure reduces the screening for particle penetration present in the original DLA model and, consequently, generates new pattern classes. The patterns have DLA-like scaling properties at small length of the random flights. However, as the flight size increases, the patterns are initially round and compact but become fractal for sufficiently large clusters. Their radius of gyration and number of particles at the cluster surface scale asymptotically as in the original DLA model. The transition between compact and fractal patterns is characterized by wavelength selection, and 1/k noise was observed far from the transition.Received: 2 March 2004, Published online: 14 December 2004PACS: 05.40.Fb Random walks and Levy flights - 05.50. + q Lattice theory and statistics (Ising, Potts, etc.) - 05.10.Ln Monte Carlo methods