arXiv:cond-mat/0112289v1 [cond-mat.stat-mech] 15 Dec 2001
Density Pro?le and Flow of Miscible Fluid with Dissimilar Constituent Masses
February 1, 2008
R.B. Pandey1,2 , D. Stau?er2 , R. Seyfarth2 , Luis A. Cueva2 , J.F. Gettrust1 , Warren Wood1 Naval Research Laboratory Stennis Space Center, MS 39529 Department of Physics and Astronomy University of Southern Mississippi, Hattiesburg, MS 39406-5046
Abstract A computer simulation model is used to study the density pro?le and ?ow of a miscible gaseous ?uid mixture consisting of di?ering constituent masses (mA = mB /3) through an open matrix. The density pro?le is found to decay with the height ∝ exp(?mA(B) h), consistent with the barometric height law. The ?ux density shows a power-law increase ∝ (pc ? p)? with ? ? 2.3 at the porosity 1 ? p above the pore percolation threshold 1 ? pc .
Understanding the ?ow of a complex gaseous mixtures, sedimentation, and evolution of density pro?les of its constituents in geo-marine systems and near-surface ecological environments is becoming increasingly important [1, 2]. There are a number of examples: (i) High density brines associated with salt tectonics in large salt provinces (e.g. the Gulf of Mexico) have been breaching the sea?oor and forming pools toxic to native ?ora and fauna . This involves ?ow of a mixture of ?uids with di?erent densities due to different salt content and temperature. (ii) A mixture of air and radon ?ux through unsaturated soils within the upper few meters of the land surface. Radon 222, one of the intermediate products of the decay of uranium 238 to lead 206, is an odorless, radioactive gas (with half life 3.8 days), and is common in many soils and rocks. Because radon is about 8 times more dense than air, and is relatively inert , it easily penetrates porous building materials in ground ?oors and basements, especially when pressure gradients are created by central heating systems . The U.S. Environmental Protection Agency (EPA) estimated in 1986 that 5,000 to 20,000 persons in the United States die of lung cancer each year from inhaling radioactive radon decay products in homes and buildings . (iii) Evidence of methane hydrate formation below the ocean ?oor and in sub-ocean bottom in mud-volcano involves the ?ow and sedimentation of complex gas and ?uid mixtures with dissimilar masses [6, 7, 8]. Studies of ?ow and density distribution of a mixture of miscible gas and ?uids through porous media are therefore highly desirable. Systematic studies based on the ?eld measurements of ?ow and density pro?le of gas and ?uid constituents in geomarine environment are severely limited  due to uncontrollable changes in geothermal parameters and morphological variations. Thus, computer simulation studies remain one of the most viable tools to probe such di?cult issues as ?ow [9, 10, 11] and density pro?le . Incorporating all details (multiple constituents and their characteristics) even with a coarse grained host matrix i.e. open porous media with appropriate concentration gradient and pore distributions  is a challenging issue. Lattice gas and particulate methods in general (Boltzmann, Cellular Automata, Ising (interacting), etc.) [13, 14] have been used in diverse applications of ?uid ?ow. In study of the density pro?le and ?uid ?ow of an interacting ?uid mixture through porous media, a direct application of traditional hydrodynamics approaches [15, 16] becomes intractable; apart from a major problem of enormous boundary conditions in such a porous medium 2
(percolating system), it is not clear how to include interaction or reaction between the ?uid components in hydrodynamic equations. Interacting lattice gas  may be a simple approach to initiate probing such di?cult issues. Very recently we studied the ?ow of a ?uid described by particles, say of type ”A” through an open porous medium using a computer simulation model . The porous medium is generated by a random distribution of sediment particles of concentration p on a simple cubic lattice. The bottom layer of the matrix is connected to a source of mobile ?uid particles (”A”). As soon as a bottom site becomes empty, it is immediately ?lled by a particle ”A” from the source. Particles in the bottom layer are not allowed to move below this plane due to presence of abundant source particles. On the other hand, the particles can escape the system from the top if they attempt to move to the higher layer. In this concentration driven system, the ?ux density shows a power-law decay with the porosity near percolation threshold. The steady-state density pro?le of ?uid particles depends systematically on the barrier concentration p. In this article, we extend our previous studies  to a miscible twocomponent system consisting of constituents, say ”A” and ”B” with dissimilar masses (ma and mb , mb = 3ma ). The model is described in the following section 2. We incorporate the e?ect of gravitational potential energy in moving the particles and allowing them to escape the system from the bottom layer as well. The injection probability of particles A and B at the bottom remains equal. The results are presented in section 3 with a conclusion in section 4.
Each site on a simple-cubic L × L × L lattice, with L up to 250, can be in one of four states: occupied by an A particle, occupied by a B particle, empty (0), or a barrier site. Nearest-neighbor particles interact with energy J such that A and B mix well: J(A, A) = J(B, B) = ?J(A, B) = ?J(B, A) = ?J(A, 0) = ?J(B, 0) = 1 where negative J means attraction and positive J means repulsion. The immobile barriers exert no force on the particles except to prevent them from occupying the barrier site. The energy thus is E=
where i runs over all particles, k over all neighbor sites of i, and I and K are the corresponding site variables (A, B, barrier, or 0).
Time development for one 250^3 lattice; t=10 (+), 100 (x), 1000 (*), 5000 (full sq.), 10000 (open sq.) 1
0.01 A-density 0.001 0.0001 1e-05 0 50 100 height Time development for one 250^3 lattice; t=10 (+), 100 (x), 1000 (*), 5000 (full sq.), 10000 (open sq.) 1 150 200 250
0.01 B-density 0.001 0.0001 1e-05 0 50 100 height 150 200 250
Figure 1: Crossover from constant to exponential density pro?les, with increasing times t as given in headline, for A-particles (part a) and B-particles (part b). No sites are barriers in this simulation. A and B particles can move with a Metropolis probability exp(??E/kB T ) to neighboring sites; here ?E is the energy change associated with this move and kB T = 5 is the thermal energy. If a particle on the lowest plane moves upwards or horizontally, we inject a new particle A or B (with equal probability) onto the vacated site. If a particle on the lowest plane wants to move 4
downward, it drops out. A particle moving upward from the highest plane (relatively a rare event) is lost, without any injection from the top. Periodic boundary conditions are applied in the horizontal directions. In this way, an ∞ × ∞ × L plate is approximated, with new material injected from the bottom at a nearly constant rate. Gravity pulls down the particles through an energy mkB T where the lattice constant and the gravitational constant are incorporated into the dimensionless mass m = 0.1 for A and m = 0.3 for B particles. Thus the barometric height law gives an equilibrium density ∝ exp(?mh) as a function of height h, 1 < h < L. One time step is an attempt to move each particle once (on average) through random sequential updating; it does not matter much if instead we enforce exactly one attempt per particle for each time step. For L = 30 we used up to t = 106 time steps, without seeing any long-time e?ects; for larger L (50 to 250) typically t = 104 to 105 gave equilibrium. Initially, the lattice is occupied homogeneously with a low concentration of particles, half A and half B. Our computer program allows many more choices for interactions and boundary e?ects and is developed to investigate many di?erent systems (details are available from email@example.com). One di?usion attempt, without barrier sites, took about 0.5 microseconds on one Cray-T3E processor.
Fig.1 shows for our largest lattices how the initial constant density pro?les change with time, starting from the two boundaries, into a roughly exponential decay for intermediate heights. The equilibrium density pro?les, Fig.2, at intermediate densities are consistent with the barometric height law shown as straight lines in these semilogarithmic plots, both with and without barrier sites. The nearest-neighbor correlation functions (not shown), i.e. the number of A or B particles surrounding a particle at height h, decay qualitatively similar to the density pro?les. Fig.3 shows as a function of the barrier concentration p the system’s permeability, de?ned as the net ?ow (per unit time and unit cross-sectional area) at the top or bottom surfaces divided by the injection rate at the lowest plane. The straight line in this log-log plot suggests a power law similar to that of the electrical conductivity in random resistor networks. Fig.4 shows 5
Density of A (+) and B (x) particles, 64 lattices 100^3, t = 10^5, barrier concentration 1/2
Density of A(+) and B(x) particles, 64 lattices 100^3, t = 20000, no barriers 1
1e-06 0 10 20 30 40 50 height 60 70 80 90 100
Figure 2: Equilibrium density pro?les with (part a) and without (part b) half the sites occupied by barriers. The straight lines are ∝ exp(?mh). the density pro?les of the B particles for the same simulations. We see there an exponential decay, followed by a plateau whose value strongly increases with increasing p, suggesting a trapping of particles by the barrier sites. We have also looked at the velocity and instantaneous velocity pro?les of the two constituents which are consistent with our expectations, i.e., more mobility toward the top.
(Flux density)/(injection rate); 10 lattices 50^3 measured at top (+) and bottom (x) plane 0.001
0.1 concentration difference
Figure 3: Double-logarithmic plot of permeability versus pc ? p where pc = 0.6884 is the percolation threshold and p the barrier concentration. For p > pc there is no in?nite connected set of ?uid sites between the barriers and thus the permeability vanishes. The straight line has a slope 2.3.
A computer simulation model is used to study the ?ow rate, sedimentation, and density pro?le of a mixture of miscible particle systems for a range of porosities above the pore percolation threshold in an open porous medium. In our concentration driven system, the steady-state density pro?les for both A and B ?uid are reached. Both density pro?les show the well-known exponential decay ∝ exp(?mA(B) h) with height h. The e?ect of mass di?erence is vividly illustrated in the density pro?les: while for our lattice sizes the density of A particles continues to decay up to the top plane the density of B particles already saturates at some low value. The saturated density of B particles increases on decreasing the porosity 1 ? p - we speculate this saturation is due to trapping of particles in the pores.
Barrier concentrations 0.0 (+), 0.1 (x), 0.2 (*), 0.3 (open sq.), 0.4 (full sq.), 0.5 (open o), 0.6 (full o)
0.01 B-density 0.001 0.0001
Figure 4: Equilibrium density pro?les of B-particles for various barrier concentrations (same simulations as for Fig.3). The ?ux of particles A at the bottom becomes equal to the outward ?ux from the top in steady state. The ?ux density of particles A decays with porosity with a power-law ∝ (pc ? p)? with ? ? 2. Acknowledgments: We acknowledge partial supports from ONR PE# 0602435N and DOEEPSCoR grant. This work was supported in part by grants of computer time from the DOD High Performance Computing Modernization Program at the Major Shared Resource Center (MSRC), NAVO, Stennis Space Center, and from the Julich supercomputer center.
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