Array Of Rigid Spheres Research Articles

Hydrodynamic Interaction Model for Two Droplets under Large Step Shear Strains

A hydrodynamic interaction model is presented to predict the behaviors of two droplets in a medium after application of large step shear strains. The model expresses the droplets by arrays of rigid spheres. Because of retraction rate e in time differentials for the rigid spheres, they move toward the center of mass of them in each array and this motion of the rigid spheres generates flow field in the medium similar to the field caused by recovery of droplets owing to the interfacial tension. The interaction between the rigid spheres is calculated using the hydrodynamic interaction tensor. The model predicts that the distance between the droplets in deformation direction decreases with increasing time and that the shape of the droplets becomes sigmoidal. In addition, the model calculation shows change in the distance in the deformation direction is independent of initial distance between the droplets providing that the initial distance is less than a certain value. These three predictions agree with the experimental results and strongly indicate that the origin of the interaction between the droplets is the hydrodynamic interaction.

Viscosity of a Suspension with a Cubic Array of Spheres in a Shear Flow

Viscous fluid flow past an infinite periodic array of rigid spheres of the same radius is considered. A solution of the Stokes equations periodic in three variables is obtained for viscous incompressible flow with a linear velocity profile. The solution takes into account the hydrodynamic interaction of an infinite number of particles in the array. An expression for the effective viscosity of a suspension with a cubic array of particles is obtained.

Viscous flow past a periodic array of spheres

Viscous flow past an infinite periodic array of rigid spheres is considered. The hydrodynamic interaction of all the particles in the array is taken into account. An analytical solution of the problem is proposed. The forces exerted by the fluid on the array particles are calculated and an expression for the velocity of fluid filtration through the array is obtained. The results are compared with the previous theoretical and experimental results.

Hydrodynamic interactions in colloidal crystals: (II). Application to dense cubic and tetragonal arrays

Wavevector dependent friction factors, which together with the elastic properties determine the hydrodynamic damping of harmonic lattice waves in colloidal crystals, are calculated for cubic arrays of rigid spheres at low, intermediate and high sphere volume fractions, including simple cubic (SC), body-centred cubic (BCC) and face-centred cubic (FCC) lattices. Exact numerical data are presented for the rheological coefficient appropriate to pure vortex motion of particles, the so-called spin viscosity. These results for the spin viscosity are obtained for all three cubic lattices and for the whole range of volume fractions. An expression suitable for low volume fractions is derived that approximates the converged numerical data within an error of only 3%, for volume fractions up to 75% of the close packing concentration for all of the three cubic packing types. Spin viscosities of extremely dense cubic arrays are calculated on basis of lubrication theory which yields new expressions for BCC and FCC lattices, while reproducing a known result for the SC case. Constant contributions of many-particle hydrodynamic interactions are determined that yield a first correction to the lubrication expressions, appropriate for the three cubic lattices. Two types of body-centred tetragonal (BCT) arrays are considered. The first type (BCT1) consists of aligned rows of close-touching spheres, where variations of the volume fraction can be accomplished by changing the distance between these rows. The second type (BCT3) has a constant structure, such that the concentration can be varied by changing the size of the spheres. The two characteristic drag coefficients are calculated numerically for steadily sedimenting BCT arrays of both types and at all concentrations. Likewise two characteristic spin viscosities are computed for these structures. Analytical results suitable to dilute BCT3 arrays are presented for drag coefficients and spin viscosities. The expressions for the spin viscosities represent an approximation within 3%, for volume fractions up to 60% of the close packing concentration.

Sonic stop bands for cubic arrays of rigid inclusions in air

Extensive band structures have been computed for cubic arrays of rigid spheres and cubes in air. Complete stop bands are obtained for face-centered-cubic (fcc) structure; however, there is no gap for the body-centered-cubic (bcc) and simple-cubic (sc) structures. These gaps start opening up for a volume fraction of ≊54% (27%) for spherical (cubic) inclusions and tend to increase with the filling fraction, exhibiting a maximum at the close-packing. A tandem structure that allows us to achieve an ultrawideband filter for environmental or industrial noise in the desired frequency range is proposed. This work is motivated by the recent experimental observation of sound attenuation on the sculpture, by Eusebio Sempere, exhibited at the Juan March foundation in Madrid [Nature 378, 241 (1995)] and complements the corresponding theoretical work [Appl. Phys. Lett. 70, 3218 (1997)].

Sonic stop-bands for cubic arrays of rigid inclusions in air

Extensive band structures have been computed for cubic arrays of rigid spheres and cubes in air. Complete stop bands are obtained for the face-centered-cubic (fcc) structure; however, there is no gap for the body-centered-cubic (bcc) and simple-cubic (sc) structures. These gaps start opening up for a filling fraction of $$ \approx 54\% $$ (27%) for spherical (cubic) inclusions and tend to increase with the filling fraction, exhibiting a maximum at the close-packing. We also propose a tandem structure that allows the achievement of an ultrawideband filter for environmental or industrial noise in the desired frequency range. This work is motivated by the recent experimental measurement of sound attenuation on the sculpture, by Eusebio Sempere, exhibited at the Juan March Foundation in Madrid (Nature 378, 241 (1995)) and complements the corresponding theoretical work (Appl. Phys. Lett. 70, 3218 (1997)).

Scattering of incoherently processed FM signals from arrays of rigid spheres

The peak target strength for arrays of rigid spherical scatterers insonified by FM signals which are incoherently processed is investigated for frequency, bandwidth, and spatial dependence. Examples of backscattering are given with two scatterers and line arrays of scatterers. Multiple interactions are ignored. For a fixed pulse length, increasing the bandwidth of an incoherently processed FM signal results in greater peak target strength due to increased opportunities for constructive interference from the scatterers. In the Rayleigh region, the ka dependence of an individual scatterer further enhances peak target strength as the bandwidth increases. Increased bandwidth also smooths the aspect dependence of the scattering pattern. Decreasing the spacing between the spheres decreases the number of ripples in the scattering patterns.

Plastic flow in ductile materials containing a cubic array of rigid spheres

The results of three-dimensional finite element calculations are presented for a cubic array of rigid spherical inclusions embedded in an elastic perfectly plastic matrix. The analysis examines the strengthening effect of the inclusions under macroscopic loads of uniaxial tension and pure shear. At low volume fractions, the fractional increase in strength is more modest than the volume fraction of particles, and only becomes comparable at around 40% loading. The local stress and deformation fields in the region of the inclusion are presented. The numerical results show that for volume fractions of inclusions below about 25% the surface tractions on the inclusion have the same order of magnitude as the matrix material's tensile yield stress. Volume fractions of 40% particles are necessary before the interface tractions are approximately double the yield strength.

Low frequency coupling in the planar rectangular lattice

The solution for the three−dimensional problem of multiple scattering by a doubly periodic planar array of bounded obstacles is specialized to consider low frequency coupling effects for spheres. For all of the usual boundary conditions, we derive closed form approximations for the transmission and reflection amplitudes explicitly in terms of the physical parameters of the isolated scatterers, lattice spacings, and direction of incidence. The results are compared with those obtained earlier for the grating of parallel cylinders and for the periodic line (and planar random distribution) of spheres. We show, for example, that the pseudo−Brewster effects are anisotropic for the rectangular lattice cell and that the leading effects of close packing for a square array of rigid spheres may be approximated by single scattering by an equivalent spheroid representing elongation in the plane of the array and contraction along the normal. For dense packing at very low frequencies, we obtain explicit closed form approximations including up to octupole−octupole coupling effects.

THE ABSORPTION OF HIGH-ENERGY RADIATION, A MONTE CARLO STUDY: III. DENSITY AND TRANSPARENCY DISTRIBUTIONS

Random spherical arrays of rigid spheres are being studied as a model for a nucleus under high-energy bombardment. This paper reports on the distribution of sphere centers, the matter distribution, and the transparency for such arrays. Low (12.5%) and high (42.2%) nominal densities are treated for arrays numbering 27 and 216 spheres. At high densities a strong surface correlation of geometric origin is observed. Analytic formulas are presented which in general adequately represent the Monte Carlo data.