Random sequential packing simulations in three dimensions for aligned cubes

1989 ◽  
Vol 26 (03) ◽  
pp. 664-670 ◽  
Author(s):  
Douglas W. Cooper
Keyword(s):  
Volume Fraction ◽  
Three Dimensional ◽  
Random Packing ◽  
Limit Problem ◽  
Three Dimensions ◽  
Standard Errors ◽  
Cube Root ◽  
Random Errors ◽  
Infinite Region ◽  
The Mean

This particular three-dimensional random packing limit problem is to determine the mean fraction of a cubic space that would be occupied by aligned, fixed, equalsize cubes, placed at random locations sequentially until no more can be added. No analytical solution has yet been found for this problem. Simulation results for a finite region and finite number of attempts were extrapolated to an infinite number of attempts (N →∞) in an infinite region by multiple linear regression, using volume fraction occupied (F) as a linear combination of the ratio of the length of the small cube sides (S) to the length of the cubic region side (L) and the cube root of the ratio of the region volume to the total volume of cubes tried, (L 3/NS 3)⅓. These results for random packing in a volume with penetrable walls can be adjusted with a multiplicative correction factor to give the results for impenetrable walls. A total of N = 107 attempts at placement were made for L/S = 20/1 and N = 14 × 106 attempts were made for L/S = 10/1. The results for volume fraction packed are correlated by F = 0.430(±0.008) + 0.966(±0.072)(S/L) – 0.236(±0.029)(L 3/NS)⅓ . The numbers in parentheses are twice the standard errors of estimate of the coefficients, indicating the 95% confidence intervals due to random errors. This value for the packing density limit, 0.430 ± 0.008, is slightly larger than that given by a conjecture by Palásti [10], 0.4178. Our value is consistent with that obtained by rather different simulation methods by Jodrey and Tory [8], 0.4227 ± 0.0006, and by Blaisdell and Solomon [2], 0.4262.

Random sequential packing simulations in three dimensions for aligned cubes

1989 ◽  
Vol 26 (3) ◽  
pp. 664-670 ◽  
Author(s):  
Douglas W. Cooper
Keyword(s):  
Volume Fraction ◽  
Three Dimensional ◽  
Random Packing ◽  
Limit Problem ◽  
Three Dimensions ◽  
Standard Errors ◽  
Cube Root ◽  
Random Errors ◽  
Infinite Region ◽  
The Mean

This particular three-dimensional random packing limit problem is to determine the mean fraction of a cubic space that would be occupied by aligned, fixed, equalsize cubes, placed at random locations sequentially until no more can be added. No analytical solution has yet been found for this problem. Simulation results for a finite region and finite number of attempts were extrapolated to an infinite number of attempts (N →∞) in an infinite region by multiple linear regression, using volume fraction occupied (F) as a linear combination of the ratio of the length of the small cube sides (S) to the length of the cubic region side (L) and the cube root of the ratio of the region volume to the total volume of cubes tried, (L3/NS3)⅓. These results for random packing in a volume with penetrable walls can be adjusted with a multiplicative correction factor to give the results for impenetrable walls. A total of N = 107 attempts at placement were made for L/S = 20/1 and N = 14 × 106 attempts were made for L/S = 10/1. The results for volume fraction packed are correlated by F = 0.430(±0.008) + 0.966(±0.072)(S/L) – 0.236(±0.029)(L3/NS)⅓. The numbers in parentheses are twice the standard errors of estimate of the coefficients, indicating the 95% confidence intervals due to random errors. This value for the packing density limit, 0.430 ± 0.008, is slightly larger than that given by a conjecture by Palásti [10], 0.4178. Our value is consistent with that obtained by rather different simulation methods by Jodrey and Tory [8], 0.4227 ± 0.0006, and by Blaisdell and Solomon [2], 0.4262.

The random packing of fibres in three dimensions

1995 ◽  
Vol 451 (1943) ◽  
pp. 737-746 ◽  
Keyword(s):  
Packing Density ◽  
Volume Fraction ◽  
Random Packing ◽  
Experimental Results ◽  
Three Dimensions ◽  
Approximate Relationship

An investigation has been carried out of the limiting packing density of an array of long straight rigid fibres distributed randomly in space as a function of the length of the fibre. We derive an approximate relationship between the limiting volume fraction V f and the slenderness λ of the fibres defined as length divided by diameter. The formula agrees well with our experimental results and those found in the literature.

scholarly journals Investigation of microvascular morphological measures for skeletal muscle tissue oxygenation by image-based modelling in three dimensions

2017 ◽  
Vol 14 (135) ◽  
pp. 20170635 ◽  
Author(s):  
B. Zeller-Plumhoff ◽  
K. R. Daly ◽  
G. F. Clough ◽  
P. Schneider ◽  
T. Roose
Keyword(s):  
Skeletal Muscle ◽  
Muscle Tissue ◽  
Cross Sections ◽  
Tissue Oxygenation ◽  
Volume Fraction ◽  
Three Dimensional ◽  
Capillary Density ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Contrast Imaging

The supply of oxygen in sufficient quantity is vital for the correct functioning of all organs in the human body, especially for skeletal muscle during exercise. Traditionally, microvascular oxygen supply capability is assessed by the analysis of morphological measures on transverse cross-sections of muscle, e.g. capillary density or capillary-to-fibre ratio. In this work, we investigate the relationship between microvascular structure and muscle tissue oxygenation in mice. Phase contrast imaging was performed using synchrotron radiation computed tomography (SR CT) to visualize red blood cells (RBCs) within the microvasculature in mouse soleus muscle. Image-based mathematical modelling of the oxygen diffusion from the RBCs into the muscle tissue was subsequently performed, as well as a morphometric analysis of the microvasculature. The mean tissue oxygenation was then compared with the morphological measures of the microvasculature. RBC volume fraction and spacing (mean distance of any point in tissue to the closest RBC) emerged as the best predictors for muscle tissue oxygenation, followed by length density (summed RBC length over muscle volume). The two-dimensional measures of capillary density and capillary-to-fibre ratio ranked last. We, therefore, conclude that, in order to assess the states of health of muscle tissue, it is advisable to rely on three-dimensional morphological measures rather than on the traditional two-dimensional measures.

Dispersion of a ball settling through a quiescent neutrally buoyant suspension

1998 ◽  
Vol 361 ◽  
pp. 309-331 ◽  
Author(s):  
JAMES R. ABBOTT ◽  
ALAN L. GRAHAM ◽  
LISA A. MONDY ◽  
HOWARD BRENNER
Keyword(s):  
Settling Velocity ◽  
Volume Fraction ◽  
Functional Dependence ◽  
Vertical Component ◽  
Transversely Isotropic ◽  
Suspended Particles ◽  
Three Dimensions ◽  
Ball Diameter ◽  
The Mean ◽  
Neutrally Buoyant

Individual falling balls were allowed to settle through otherwise quiescent well-mixed suspensions of non-colloidal neutrally buoyant spheres dispersed in a Newtonian liquid. Balls were tracked in three dimensions to determine the variances in their positions about a mean uniform vertical settling path. The primary experimental parameters investigated were the size of the falling ball and the volume fraction and size of the suspended particles. Unlike the horizontal variances, the vertical variances were found to be affected by short-time deterministic behaviour relating to the instantaneous local configurational arrangement of the suspended particles. For sufficiently long intervals between successive observations, the trajectories of the balls were observed to disperse about their mean settling paths in a random manner. This points to the existence of a Gaussian hydrodynamic dispersivity that characterizes the linear temporal growth of the variance in the position of a falling ball. The functional dependence of these horizontal and vertical dispersivities upon the parameters investigated was established.The dispersivity dyadic was observed to be transversely isotropic with respect to the direction of gravity, with the vertical component at least 25 times larger than the horizontal component. The vertical dispersivity Dˆv (made dimensionless with the diameter of the suspended spheres and the mean settling velocity) was observed to decrease with increasing falling ball diameter, but to decrease less rapidly with concentration than theoretically predicted for very dilute suspensions; moreover, for falling balls equal in size to the suspended spheres, Dˆv increased linearly with increasing volume fraction ϕ of suspended solids.In addition to the above experiments performed on suspensions of spheres, previously published settling-velocity data on the fall of balls through neutrally buoyant suspensions of rods possessing an aspect ratio of 20 were re-analysed, and vertical dispersivities calculated therefrom. (These data, taken by several of the present investigators in conjunction with other researchers, had only been grossly analysed in prior publications to extract the mean settling velocity of the ball, no attempt having been made at the time to extract dispersivity data too.) The resulting vertical dispersivities, when rendered dimensionless with the rod length and mean settling velocity, showed no statistically significant dependence upon the falling-ball diameter; moreover, all other things being equal, these dispersivities were observed to increase with increasing rod concentration.

A generalized Archie’s law for n phases

Geophysics ◽  
2010 ◽  
Vol 75 (6) ◽  
pp. E247-E265 ◽  
Author(s):  
Paul W. J. Glover
Keyword(s):  
Volume Fraction ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Reservoir Rock ◽  
Archie’S Law ◽  
Phase Volume Fraction ◽  
General Law ◽  
Special Cases ◽  
The Matrix ◽  
Definition Of

Archie’s law has been the standard method for relating the conductivity of a clean reservoir rock to its porosity and the conductivity of its pore fluid for more than [Formula: see text]. However, it is applicable only when the matrix is nonconducting. A modified version that allows a conductive matrix was published in 2000. A generalized form of Archie’s law is studied for any number of phases for which the classical Archie’s law and modified Archie’s law for two phases are special cases. The generalized Archie’s law contains a phase conductivity, a phase volume fraction, and phase exponent for each of its [Formula: see text] phases. The connectedness of each of the phases is considered, and the principle of conservation of connectedness in a three-dimensional multiphase mixture is introduced. It is confirmed that the general law is formally the same as the classical Archie’s law and modified Archie’s law for one and two conducting phases, respectively. The classical second Archie’s law is compared with the generalized law, which leads to the definition of a saturation exponent for each phase. This process has enabled the derivation of relationships between the phase exponents and saturation exponents for each phase. The relationship between percolation theory and the generalized model is also considered. The generalized law is examined in detail for two and three phases and semiquantitatively for four phases. Unfortunately, the law in its most general form is very difficult to prove experimentally. Instead, numerical modeling in three dimensions is carried out to demonstrate that it behaves well for a system consisting of four interacting conducting phases.

scholarly journals Asteroidal Motion at Commensurabilities Treated in Three Dimensions

1979 ◽  
Vol 81 ◽  
pp. 207-215 ◽  
Author(s):  
Joachim Schubart
Keyword(s):  
Three Dimensional ◽  
Three Dimensions ◽  
Main Subject ◽  
Long Period ◽  
Period Effects ◽  
Special Cases ◽  
The Mean ◽  
Work Done ◽  
Asteroidal Motion

This paper consists of a review about work done on three-dimensional motion at commensurabilities of either the mean motions, or of secular periods, and of a report on the author's recent results on some special cases. Real and fictitious asteroidal orbits and the corresponding long-period effects are the main subject of interest. At first, methods are listed.

Coarsening of Cobalt Precipitates in Copper- Cobalt Alloys

1982 ◽  
Vol 21 ◽  
Author(s):  
T. Eguchi ◽  
Y. Tomokiyo ◽  
K. Oki ◽  
Y. Seno
Keyword(s):  
Aging Time ◽  
Ostwald Ripening ◽  
Volume Fraction ◽  
Early Stage ◽  
Electron Microscopic Observation ◽  
Rate Of Change ◽  
Cube Root ◽  
Electron Microscopic ◽  
The Mean ◽  
Co Alloys

ABSTRACTThe Ostwald ripening of precipitates in Cu-Co alloys was investigated by electron microscopic observation. The variation of the mean particle radius r, and the distribution of relative particle sizes, f(r/r) , were obtained as functions of the aging time at 873K . The shape of f(r/r)was rather sharp and symmetrical around r = T in the early stage. It became broad with the aging and was similar, in the late stage, to the theoretical distribution of Ardell. It appeared that the shape of f(r/r) depends on the volume fraction but the growth rate of F scarcely depends on it. The rate of change of r was proportional to the cube root of the aging time from the early stage of aging.

scholarly journals Progress on numerical simulation of nanofluids: impact of an isothermal spherical partition on the mixed convection of nanofluids within cubic enclosures

2020 ◽  
Vol 307 ◽  
pp. 01016
Author(s):  
A. BOUTRA ◽  
K. RAGUI ◽  
N. LABSI ◽  
Y.K. BENKAHLA ◽  
R BENNACER
Keyword(s):  
Lattice Boltzmann ◽  
Volume Fraction ◽  
Solid Volume Fraction ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Numerical Code ◽  
Three Dimensional Flow ◽  
Boltzmann Method ◽  
Do So

The main objective of our work is to light out the three-dimensional flow of an Ag-water nanofluid within a lid-driven cubical space which equipped with a spherical heater into its center. Due to its crucial role in the characterization of the main transfer within such configurations, impact of some parameters is widely inspected. It consists the Richardson value (0,05 to 50), the solid volume fraction (0% to 10%), as well as the heater geometry (10% ≤ d ≤ 25%). To do so, a numerical code based on the Lattice-Boltzmann method, coupled with a finite difference one, is used. The latter has been validated after comparison between the present results and those of the literature. It is to note that the three dimensions D3Q19 model is adopted based on a cubic Lattice, where each pattern of the latter is characterized by nineteen discrete speeds.

Propagation of Low Frequency Elastic Disturbances in a Three-Dimensional Composite Material

1975 ◽  
Vol 42 (1) ◽  
pp. 159-164 ◽  
Author(s):  
W. Kohn
Keyword(s):  
Secular Equation ◽  
Displacement Vector ◽  
Three Dimensional ◽  
Low Frequency ◽  
Three Dimensions ◽  
Low Frequencies ◽  
One Dimensional ◽  
Coupled Partial Differential Equations ◽  
Constant Coefficients ◽  
The Mean

This paper is a generalization to three dimensions of an earlier study for one-dimensional composites. We show here that in the limit of low frequencies the displacement vector ui(r,t) can be written in the form ui (r,t) = (∂ij + vijl (r) ∂/∂xl + …) Uj (r,t). Here Uj (r,t) is a slowly varying vector function of r and t which describes the mean displacement of each cell of the composite. Its components satisfy a set of three coupled partial differential equations with constant coefficients. These coefficients are obtainable from the three-by-three secular equation which yields the low-lying normal mode frequencies, ω(k). Information about local strains is contained in the function vijl(r), which is characteristic of static deformations, and is discussed in detail. Among applications of this method is the structure of the head of a pulse propagating in an arbitrary direction.

Vortical structures in the near wake of tabs with various geometries

2017 ◽  
Vol 825 ◽  
pp. 167-188 ◽  
Author(s):  
A. M. Hamed ◽  
A. Pagan-Vazquez ◽  
D. Khovalyg ◽  
Z. Zhang ◽  
L. P. Chamorro
Keyword(s):  
Large Scale ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Incoming Flow ◽  
Mean Flow ◽  
Near Wake ◽  
Hairpin Vortices ◽  
Vortical Structures ◽  
The Mean ◽  
Hairpin Structures

The vortical structures and turbulence statistics in the near wake of rectangular, trapezoidal, triangular and ellipsoidal tabs were experimentally studied in a refractive-index-matching channel. The tabs share the same bulk dimensions, including a 17 mm height, a 28 mm base width and a $24.5^{\circ }$ inclination angle. Measurements were performed at two Reynolds numbers based on the tab height, $Re_{h}\simeq 2000$ (laminar incoming flow) and 13 000 (turbulent incoming flow). Three-dimensional, three-component particle image velocimetry (PIV) was used to study the mean flow distribution and dominant large-scale vortices, while complementary high-spatial-resolution planar PIV measurements were used to quantify high-order statistics. Instantaneous three-dimensional fields revealed the coexistence of a coherent counter-rotating vortex pair (CVP) and hairpin structures. The CVP and hairpin vortices (the primary structures) exhibit distinctive characteristics and strength across $Re_{h}$ and tab geometries. The CVP is coherently present in the mean flow field and grows in strength over a significantly longer distance at the low $Re_{h}$ due to the lower turbulence levels and the delayed shedding of the hairpin vortices. These features at the low $Re_{h}$ are associated with the presence of Kelvin–Helmholtz instability that develops over three tab heights downstream of the trailing edge. Moreover, a secondary CVP with an opposite sense of rotation resides below the primary one for the four tabs at the low $Re_{h}$. The interaction between the hairpin structures and the primary CVP is experimentally measured in three dimensions and shows complex coexistence. Although the CVP undergoes deformation and splitting at times, it maintains its presence and leads to significant mean spanwise and wall-normal flows.

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