Elimination of an isolated pore: Effect of grain size

1995 ◽  
Vol 10 (4) ◽  
pp. 1000-1015 ◽  
Author(s):  
Wan Y. Shih ◽  
Wei-Heng Shih ◽  
Ilhan A. Aksay
Keyword(s):  
Grain Size ◽  
Monte Carlo ◽  
Sample Size ◽  
Scaling Laws ◽  
Pore Diameter ◽  
Scaling Law ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Scaling Analysis ◽  
Scaling Relationships

The effect of grain size on the elimination of an isolated pore was investigated both by the Monte Carlo simulations and by a scaling analysis. The Monte Carlo statistical mechanics model for sintering was constructed by mapping microstructures onto domains of vectors of different orientations as grains and domains of vacancies as pores. The most distinctive feature of the simulations is that we allow the vacancies to move. By incorporating the outer surfaces of the sample in the simulations, sintering takes place via vacancy diffusion from the pores to the outer sample surfaces. The simulations were performed in two dimensions. The results showed that the model is capable of displaying various sintering phenomena such as evaporation and condensation, rounding of a sharp corner, pore coalescence, thermal etching, neck formation, grain growth, and growth of large pores. For the elimination of an isolated pore, the most salient result is that the scaling law of the pore elimination time tp with respect to the pore diameter dp changes as pore size changes from larger than the grains to smaller than the grains. For example, in sample-size-fixed simulations, tp ∼ d3p for dp < G and tp ∼ d2p for dp > G with the crossover pore diameter dc increasing linearly with G where G is the average grain diameter. For sample-size-scaled simulations, tp ∼ d4p for dp < G and tp ∼ d3p for dp > G. That tp has different scaling laws in different grain-size regimes is a result of grain boundaries serving as diffusion channels in a fine-grain microstructure such as those considered in the simulations. A scaling analysis is provided to explain the scaling relationships among tp, dp, and G obtained in the simulations. The scaling analysis also shows that these scaling relationships are independent of the dimensionality. Thus, the results of the two-dimensional simulations should also apply in three dimensions.

scholarly journals HYDRODYNAMIC SCALING ANALYSIS OF NUCLEAR FUSION DRIVEN BY ULTRA-INTENSE LASER-PLASMA INTERACTIONS

2012 ◽  
Vol 21 (12) ◽  
pp. 1250102 ◽  
Author(s):  
SACHIE KIMURA ◽  
ALDO BONASERA
Keyword(s):  
Laser Plasma ◽  
Laser Power ◽  
Scaling Laws ◽  
Scaling Law ◽  
Intense Laser ◽  
Scaling Analysis ◽  
Plasma Interactions ◽  
Laser Plasma Interactions ◽  
Hydrodynamic Scaling ◽  
Fusion Yield

We discuss scaling laws of fusion yields generated by laser-plasma interactions. The yields are found to scale as a function of the laser power. The origin of the scaling law in the laser driven fusion yield is derived in terms of hydrodynamic scaling. We point out that the scaling properties can be attributed to the laser power dependence of three terms: the reaction rate, the density of the plasma and the projected range of the plasma particle in the target medium. The resulting scaling relations have a predictive power that enables estimating the fusion yield for a nuclear reaction which has not been investigated by means of the laser accelerated ion beams.

Velocity distribution function for a dilute granular material in shear flow

1997 ◽  
Vol 340 ◽  
pp. 319-341 ◽  
Author(s):  
V. KUMARAN
Keyword(s):  
Distribution Function ◽  
Shear Flow ◽  
Velocity Distribution ◽  
Scaling Laws ◽  
Three Dimensional ◽  
Particle Diameter ◽  
Velocity Distribution Function ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Binary Collisions

The velocity distribution function for the steady shear flow of disks (in two dimensions) and spheres (in three dimensions) in a channel is determined in the limit where the frequency of particle–wall collisions is large compared to particle–particle collisions. An asymptotic analysis is used in the small parameter ε, which is naL in two dimensions and n2L in three dimensions, where n is the number density of particles (per unit area in two dimensions and per unit volume in three dimensions), L is the separation of the walls of the channel and a is the particle diameter. The particle–wall collisions are inelastic, and are described by simple relations which involve coefficients of restitution et and en in the tangential and normal directions, and both elastic and inelastic binary collisions between particles are considered. In the absence of binary collisions between particles, it is found that the particle velocities converge to two constant values (ux, uy) =(±V, 0) after repeated collisions with the wall, where ux and uy are the velocities tangential and normal to the wall, V=(1−et) Vw/(1+et), and Vw and −Vw are the tangential velocities of the walls of the channel. The effect of binary collisions is included using a self-consistent calculation, and the distribution function is determined using the condition that the net collisional flux of particles at any point in velocity space is zero at steady state. Certain approximations are made regarding the velocities of particles undergoing binary collisions in order to obtain analytical results for the distribution function, and these approximations are justified analytically by showing that the error incurred decreases proportional to ε1/2 in the limit ε→0. A numerical calculation of the mean square of the difference between the exact flux and the approximate flux confirms that the error decreases proportional to ε1/2 in the limit ε→0. The moments of the velocity distribution function are evaluated, and it is found that 〈u2x〉→V2, 〈u2y〉 ∼V2ε and − 〈uxuy〉 ∼ V2εlog(ε−1) in the limit ε→0. It is found that the distribution function and the scaling laws for the velocity moments are similar for both two- and three-dimensional systems.

scholarly journals Solid-phase crystallization under continuous heating: Kinetic and microstructure scaling laws

2008 ◽  
Vol 23 (2) ◽  
pp. 418-426 ◽  
Author(s):  
J. Farjas ◽  
P. Roura
Keyword(s):  
Grain Size ◽  
Kinetic Parameters ◽  
Scaling Laws ◽  
Solid Phase ◽  
Scaling Law ◽  
Continuous Heating ◽  
Dimensionless Time ◽  
Solid Phase Crystallization ◽  
Average Grain Size ◽  
Length Scaling

The kinetics and microstructure of solid-phase crystallization under continuous heating conditions and random distribution of nuclei are analyzed. An Arrhenius temperature dependence is assumed for both nucleation and growth rates. Under these circumstances, the system has a scaling law such that the behavior of the scaled system is independent of the heating rate. Hence, the kinetics and microstructure obtained at different heating rates differ only in time and length scaling factors. Concerning the kinetics, it is shown that the extended volume evolves with time according to αex = [exp(κCt′)]m+1, where t′ is the dimensionless time. This scaled solution not only represents a significant simplification of the system description, it also provides new tools for its analysis. For instance, it has been possible to find an analytical dependence of the final average grain size on kinetic parameters. Concerning the microstructure, the existence of a length scaling factor has allowed the grain-size distribution to be numerically calculated as a function of the kinetic parameters.

scholarly journals A Hierarchical Allometric Scaling Analysis of Chinese Cities: 1991–2014

2017 ◽  
Vol 2017 ◽  
pp. 1-15 ◽  
Author(s):  
Yanguang Chen ◽  
Jian Feng
Keyword(s):  
Scaling Laws ◽  
City Planning ◽  
Allometric Scaling ◽  
Scaling Law ◽  
Urban Systems ◽  
Long Term Memory ◽  
Scaling Analysis ◽  
Scaling Exponent ◽  
Random Disturbance ◽  
Chinese Cities

The law of allometric scaling based on Zipf distributions can be employed to research hierarchies of cities in a geographical region. However, the allometric patterns are easily influenced by random disturbance from the noises in observational data. In theory, both the allometric growth law and Zipf’s law are related to the hierarchical scaling laws associated with fractal structure. In this paper, the scaling laws of hierarchies with cascade structure are used to study Chinese cities, and the method of R/S analysis is applied to analyzing the change trend of the allometric scaling exponents. The results show that the hierarchical scaling relations of Chinese cities became clearer and clearer from 1991 to 2014 year; the global allometric scaling exponent values fluctuated around 0.85, and the local scaling exponent approached 0.85. The Hurst exponent of the allometric parameter change is greater than 0.5, indicating persistence and a long-term memory of urban evolution. The main conclusions can be reached as follows: the allometric scaling law of cities represents an evolutionary order rather than an invariable rule, which emerges from self-organized process of urbanization, and the ideas from allometry and fractals can be combined to optimize spatial and hierarchical structure of urban systems in future city planning.

Normal Grain Growth in Three Dimensions: Monte Carlo Potts Model Simulation and Mean-Field Theory

2007 ◽  
Vol 550 ◽  
pp. 589-594 ◽  
Author(s):  
Dana Zöllner ◽  
Peter Streitenberger
Keyword(s):  
Field Theory ◽  
Grain Size ◽  
Monte Carlo ◽  
Distribution Function ◽  
Grain Growth ◽  
Mean Field Theory ◽  
Mean Field ◽  
Size Distribution Function ◽  
Three Dimensions ◽  
Self Similar

A modified Monte Carlo Potts model algorithm for single-phase normal grain growth in three dimensions in presented, which enables an extensive statistical analysis of the growth kinetics and topological properties of the microstructure within the quasi-stationary self-similar coarsening regime. From the mean-field theory an analytical grain size distribution function is derived, which is based on a quadratic approximation of the average self-similar volumetric rate of change as a function of the relative grain size as it has been determined from the simulation. The analytical size distribution function is found to be in excellent agreement with the simulation results.

Self-focusing of thin liquid jets

2007 ◽  
Vol 464 (2089) ◽  
pp. 197-206 ◽  
Author(s):  
Laurent Duchemin
Keyword(s):  
Free Surface ◽  
Time Evolution ◽  
Scaling Laws ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Liquid Jets ◽  
Self Similarity ◽  
Long Time ◽  
Self Similar ◽  
Long Time Evolution

The nonlinear evolution of an initially perturbed free surface perpendicularly accelerated, or of an initially flat free surface subject to a perturbed velocity profile, gives rise to the emergence of thin spikes of fluid. We are investigating the long-time evolution of a thin inviscid jet of this kind, subject or not to a body force acting in the direction of the jet itself. A fully nonlinear theory for the long-time evolution of the jet is given. In two dimensions, the curvature of the tip scales like t 3 , where t is time, and the peak undergoes an overshoot in acceleration which evolves like t −5 . In three dimensions, the jet evolves towards an axisymmetric shape, and the curvature and the overshoot in acceleration obey asymptotic laws in t 2 and t −4 , respectively. The asymptotic self-similar shape of the spike is found to be a hyperbola in two dimensions, a hyperboloid in three dimensions. Scaling laws and self-similarity are confronted with two-dimensional computations of the Richtmyer–Meshkov instability.

MONTE CARLO STUDIES OF THE THREE-DIMENSIONAL ISING MODEL IN EQUILIBRIUM

2001 ◽  
Vol 12 (07) ◽  
pp. 911-1009 ◽  
Author(s):  
MARTIN HASENBUSCH
Keyword(s):  
Monte Carlo ◽  
Ising Model ◽  
Monte Carlo Simulations ◽  
Three Dimensional ◽  
Finite Size ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Amplitude Ratios ◽  
Interface Tension ◽  
Monte Carlo Studies

We review Monte Carlo simulations of the Ising model and similar models in three dimensions that were performed in the last decade. Only recently, Monte Carlo simulations provide more accurate results for critical exponents than field theoretic methods, such as the ∊-expansion. These results were obtained with finite size scaling and "improved actions". In addition, we summarize Monte Carlo results for universal amplitude ratios, the interface tension, and the dimensional crossover from three to two dimensions.

Q-COLOR PROBLEM IN D DIMENSIONS

1988 ◽  
Vol 02 (06) ◽  
pp. 1503-1511 ◽  
Author(s):  
C. Y. PAN ◽  
X. Y. CHEN
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Cell Growth ◽  
Recursion Formula ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Four Dimensions ◽  
Color Problem ◽  
Hypercubic Lattice ◽  
Growth Method

Two methods are introduced to deal with the general q-color problem on a d-dimensional hypercubic lattice: one is the cell-growth method which gives an approximative solution of the problem in two and three dimensions, another is to combine the cell-growth method with the Monte Carlo simulation which leads to a recursion formula for the problem in d-dimensions. The results of both methods are in excellent agreement with Lieb's exact solution for the case of q = 3 in two dimensions and support Mattis' conjecture in the case of q > d, improve it in the case of 2 < q ≲ d. The validity of the recursion formula is supported by a Monte Carlo simulation up to four-dimensions.

scholarly journals Nonlinear mushy-layer convection with chimneys: stability and optimal solute fluxes

2013 ◽  
Vol 716 ◽  
pp. 203-227 ◽  
Author(s):  
Andrew J. Wells ◽  
J. S. Wettlaufer ◽  
Steven A. Orszag
Keyword(s):  
Rayleigh Number ◽  
Scaling Laws ◽  
Finite Amplitude ◽  
Two Dimensions ◽  
Solute Flux ◽  
Scaling Analysis ◽  
Mushy Layer ◽  
Similar System ◽  
Solute Fluxes ◽  
Rayleigh Numbers

AbstractWe model buoyancy-driven convection with chimneys – channels of zero solid fraction – in a mushy layer formed during directional solidification of a binary alloy in two dimensions. A large suite of numerical simulations is combined with scaling analysis in order to study the parametric dependence of the flow. Stability boundaries are calculated for states of finite-amplitude convection with chimneys, which for a narrow domain can be interpreted in terms of a modified Rayleigh number criterion based on the domain width and mushy-layer permeability. For solidification in a wide domain with multiple chimneys, it has previously been hypothesized that the chimney spacing will adjust to optimize the rate of removal of potential energy from the system. For a wide variety of initial liquid concentration conditions, we consider the detailed flow structure in this optimal state and derive scaling laws for how the flow evolves as the strength of convection increases. For moderate mushy-layer Rayleigh numbers these flow properties support a solute flux that increases linearly with Rayleigh number. This behaviour does not persist indefinitely, however, with porosity-dependent flow saturation resulting in sublinear growth of the solute flux for sufficiently large Rayleigh numbers. Finally, we consider the influence of the porosity dependence of permeability, with a cubic function and a Carman–Kozeny permeability yielding qualitatively similar system dynamics and flow profiles for the optimal states.

SCALING LAW OF SUPERFLUID–INSULATOR TRANSITION IN THE 1D BOSE–HUBBARD MODEL

2012 ◽  
Vol 26 (04) ◽  
pp. 1250014 ◽  
Author(s):  
SHI-JIAN GU ◽  
JUNPENG CAO ◽  
SHU CHEN ◽  
HAI-QING LIN
Keyword(s):  
Hubbard Model ◽  
Critical Point ◽  
Scaling Laws ◽  
Scaling Law ◽  
Finite Size ◽  
Superfluid Density ◽  
Insulator Transition ◽  
Scaling Analysis ◽  
Finite Size Scaling ◽  
Size Scaling

The finite size scaling behavior of superfluid–insulator transition in the one-dimensional Bose–Hubbard model is studied. It is shown that the superfluid density of the system with finite size has a maximum at a certain interaction Um and the derivative of superfluid density has a minimum at a certain interaction Ud. The critical point Uc can be quantified by the scaling analysis of either Um or Ud. The transition point Um tends to the critical point Uc from the region of U < Uc, while the Ud tends to the Uc from the region of U > Uc. The transition points Um and Ud satisfy different finite size scaling laws and have the different critical exponents. The divergence speed of the superfluid density is much smaller than that of its derivative at the critical point.

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