An algorithm is presented for the adaptive restructuring of meshes on evolving surfaces. The resolution of the relevant local length scale is maintained everywhere with prescribed accuracy through the minimization of an appropriate mesh energy function by a sequence of local restructuring operations. The resulting discretization depends on the instantaneous configuration of the surface but is insensitive to the deformation history. Application of the adaptive discretization algorithm is illustrated with three-dimensional boundary-integral simulations of deformable drops in Stokes flow. The results show that the algorithm can accurately resolve detailed features of deformed fluid interfaces, including slender filaments associated with drop breakup and dimpled regions associated with drop coalescence. Our algorithm should be useful in a variety of fields, including computational fluid dynamics, image processing, geographical information systems, and biomedical engineering problems.

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Hydrodynamic interactions in a suspension of spherical particles confined between two parallel planar walls are studied under creeping-flow conditions. The many-particle friction matrix in this system is evaluated using our novel numerical algorithm based on transformations between Cartesian and spherical representations of Stokes flow. The Cartesian representation is used to describe the interaction of the fluid with the walls and the spherical representation is used to describe the interaction with the particles. The transformations between these two representations are given in a closed form, which allows us to evaluate the coefficients in linear equations for the induced-force multipoles on particle surfaces. The friction matrix is obtained from these equations, supplemented with the superposition lubrication corrections. We have used our algorithm to evaluate the friction matrix for a single sphere, a pair of spheres, and for linear chains of spheres. The friction matrix exhibits a crossover from a quasi-two-dimensional behavior (for systems with small wall separation H) to the three-dimensional behavior (when the distance H is much larger than the interparticle distance L). The crossover is especially pronounced for a long chain moving in the direction normal to its orientation and parallel to the walls. In this configuration, a large pressure buildup occurs in front of the chain for small values of the gapwidth H, which results in a large hydrodynamic friction force. A standard wall superposition approximation does not capture this behavior.

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Relative frequencies of mechanically stable (MS) packings of frictionless bidisperse disks are studied numerically in small systems. The packings are created by successively compressing or decompressing a system of soft purely repulsive disks, followed by energy minimization, until only infinitesimal particle overlaps remain. For systems of up to 14 particles, most of the MS packings were generated. We find that the packings are not equally probable as has been assumed in recent thermodynamic descriptions of granular systems. Instead, the frequency distribution, averaged over each packing-fraction interval Deltaphi , grows exponentially with increasing phi. Moreover, within each packing-fraction interval, MS packings occur with frequencies f{k} that differ by many orders of magnitude. Also, key features of the frequency distribution do not change when we significantly alter the packing-generation algorithm; for example, frequent packings remain frequent and rare ones remain rare. These results indicate that the frequency distribution of MS packings is strongly influenced by geometrical properties of the multidimensional configuration space. By adding thermal fluctuations to a set of the MS packings, we were able to examine a number of local features of configuration space near each packing. We measured the time required for a given packing to break to a distinct one, which enabled us to estimate the energy barriers that separate one packing from another. We found a gross positive correlation between the packing frequencies and the heights of the lowest energy barriers {0}; however, there is significant scatter in the data. We also examined displacement fluctuations away from the MS packings to assess the size and shape of the local basins near each packing. The displacement modes scale as d{i} approximately epsilon{0}{gamma{i}} with gamma{i} ranging from approximately 0.6 for the largest eigenvalues to 1.0 for the smallest ones. These scalings suggest that the packing frequencies are not determined by the local volume of configuration space near each packing, which would require that the dependence of f{k} on epsilon{0} is much stronger than the dependence we observe. The scatter in our data implies that in addition to epsilon{0} there are also other, as yet undetermined variables that influence the packing probabilities.