Conductivity of N-Dimensional Composites Containing Hyperspherical Inclusion

A problem of determining the macroscopic or effective thermal conductivity of an N-dimensional composite medium containing N-dimensional nonoverlapping hyperspherical inclusions is considered. Since the macroscopic conductivity is expected to become less sensitive to the detailed spatial distribution of the inclusions for N ≥ 4, only the special case of periodic arrangement of the inclusions is considered. An expression for the macroscopic conductivity correct to O(χ3N + 8), χ being the ratio of diameter of the inclusions to the spacing between them, is derived and the numerical results for the conductivity are presented as a function of χ and N for the two special cases of perfectly conducting and insulating inclusions. The effective conductivity of the composite is found to approach that of the continuous matrix in higher dimensions

The Planar Singular Solutions of Stokes and Laplace Equations and their Application to Transport Processes Near Porous Surfaces

The planar singular solutions of Stokes and Laplace equations are derived and applied to a number of transport problems associated with porous surfaces. The velocity, pressure, concentration, and temperature slip coefficients are determined exactly for the semi-infinite periodic arrays of spheres and these are compared with the predictions of two approximate continuum theories formulated by Brinkman [Appl. Sci. Res. Sect. A I, 27 (1947)] and Chang and Acrivos [J. Appl. Phys. 59, 375 (1986)] to assess the utility of such theories in accurately predicting various overall properties related to the porous surfaces. It is found that in general these theories provide fairly accurate estimates of these properties even when the length scales based on the relevant macroscopic properties such as the permeability are much smaller than the length scales characterizing the microstructure of the porous media

A Method for Computing Stokes Flow Interactions Among Spherical Objects and its Application to Suspensions of Drops and Porous Particles

A method for computing Stokes flow interactions in suspensions of spherical objects is described in detail and applied to the suspensions of porous particles, drops, and bubbles to determine their hydrodynamic transport coefficients

Inclusion of Lubrication Forces in Dynamic Simulations

A new method is described for incorporating close-field, lubrication forces between pairs of particles into the multiparticle Stokes flow calculations. The method is applied to the suspensions of both spherical as well as cyliridrical particles, and results computed by the method are shown to be in excellent agreement with the exact known results available in the literature

Nusselt Number for Flow Perpendicular to Arrays of Cylinders in the Limit of Small Reynolds and Large Peclet Numbers

The problem of determining the Nusselt number N, the nondimensional rate of heat or mass transfer, from an array of cylindrical particles to the surrounding fluid is examined in the limit of small Reynolds number Re and large Peclet number Pe. N in this limit can be determined from the details of flow in the immediate vicinity of the particles. These are determined accurately using a method of multipole expansions for both ordered and random arrays of cylinders. The results for N/Pe^1/3 are presented for the complete range of the area fraction of cylinders. The results of numerical simulations for random arrays are compared with those predicted using effective-medium approximations, and a good agreement between the two is found. A simple formula is given for relating the Nusselt number and the Darcy permeability of the arrays. Although the formula is obtained by fitting the results of numerical simulations for arrays of cylindrical particles, it is shown to yield a surprisingly accurate relationship between the two even for the arrays of spherical particles for which several known results exist in the literature suggesting thereby that this relationship may be relatively insensitive to the shape of the particles

Mass Transfer Coefficients for Laminar Longitudinal Flow in Hollow-Fibre Contactors

We consider the problem of predicting the rate of mass transfer to a fluid flowing parallel to the axes of randomly placed aligned tubes, a model of hollow-fibre contactors. The analysis is carried out for the limiting cases of short contactors, for which the concentration boundary layers remain thin compared with the radius of the tubes, and for the fully developed case corresponding to very long tubes. Numerical simulations for random arrays are carried out for N randomly placed tubes within a unit cell of a periodic array. It is shown that the mass transfer coefficient for the fully developed case is vanishingly small in the limit N to infinity. This suggests that the mass transfer coefficient for a random array of tubes of radius a enclosed in a shell of radius S will vanish logarithmically as the ratio S/a is increased. This behaviour arises due to the logarithmically divergent nature of concentration disturbances caused by each tube in the plane normal to its axis. A theory is developed for determining conditionally averaged velocity and concentration fields and its predictions are shown to compare very well with the results of rigorous numerical computations. The predictions of the theory are also shown to compare well with the measurements of the mass transfer coefficients in hollow-fibre contactors reported in the literature

An O(N) Algorithm for Stokes and Laplace Interactions of Particles

A method for computing Laplace and Stokes interactions among N spherical particles arbitrarily placed in a unit cell of a periodic array is described. The method is based on an algorithm by Greengard and Rokhlin [J. Comput. Phys. 73, 325 (1987)] for rapidly summing the Laplace interactions among particles by organizing the particles into a number of different groups of varying sizes. The far-field induced by each group of particles is expressed by a multipole expansion technique into an equivalent field with its singularities at the center of the group. The resulting computational effort increases only linearly with N. The method is applied to a number of problems in suspension mechanics with the goal of assessing the efficiency and the potential usefulness of the method in studying dynamics of large systems. It is shown that reasonably accurate results for the interaction forces are obtained in most cases even with relatively low-order multipole expansions

Transport Processes in Random Arrays of Cylinders. I. Thermal Conduction

A numerical method is developed that takes into account the many particle interactions in a rigorous manner to determine the effective thermal conductivity of Km of a composite medium consisting of parallel circular cylinders of thermal conductivity ak suspended in a matrix of conductivity k. Numerical results for Km are presented for a wide rane of a and o, the area fraction of the cylinders, after averaging over several computer-generated random arrays of cylinders. The results obtained via this exact method are compared with those of various approximate analystical methods to assess their utility in predicting Km

Transport Processes in Random Arrays of Cylinders. II. Viscous Flow

A numerical method is developed that takes into account the many-particle interactions in a rigorous manner to determine the effective thermal conductivity Km of a composite medium consisting of parallel circular cylinders of thermal conductivity ak suspended in a matrix of conductivity k. Numerical results for Km are presented for a wide range of a and o, the area fraction of the cylinders, after averaging over several computer-generated random arrays of cylinders. The results obtained via this exact method are compared with those of various approximate analytical methods to assess their utility in predicting Km

Numerical Simulation of a Gas–Liquid Flow in a Fixed Bed

A countercurrent gas–liquid flow through a fixed bed of spherical particles is examined numerically by solving the particle-scale equations governing the gas and liquid flows. The liquid is assumed to flow along the surface of the particles forming a thin film. The case of small gas flow rates is examined in detail first. In this limit the presence of the liquid film increases the gas pressure drop over its value for a dry bed by three mechanisms: The liquid film makes the apparent size of the particles larger, decreases the pore space for the gas flow, and, with its velocity pointing opposite to the mean gas flow, increases the apparent velocity of the gas compared with the particle surface. The excess pressure drop is determined for both periodic and random arrangements of particles. Next, the case of high gas flow rates where the traction exerted by the gas at the gas–liquid interface is comparable to the weight of the liquid film is examined. In this regime the liquid holdup increases with the gas flow rate and the pressure drop-gas velocity relation is nonlinear. The results of numerical simulations are compared with approximate models and it is shown that a simple capillary model yields reasonably accurate predictions for the liquid holdup and gas pressure drop