Finite-size Spheres Research Articles

Analytical solution of the Sommerfeld–Page equation

The Sommerfeld–Page equation describes the non-relativistic dynamics of a classical electron modeled by a sphere of finite size with a uniform surface charge density. It is a delay differential equation, and almost no exact solution of this equation was known until recently. However, progress has been made, and an analytical solution was recently found for an almost identical delay differential equation, which arose in the context of the mathematical modeling of the COVID-19 epidemics. Inspired by this research, we offer a pedagogical exposition of how one can find an analytical solution of the Sommerfeld–Page equation.

Spheres and fibers in turbulent flows at various Reynolds numbers

We perform fully coupled numerical simulations using immersed boundary methods of finite-size spheres and fibers suspended in a turbulent flow for a range of Taylor-Reynolds numbers 12.8<Reλ<442 and solid mass fractions 0≤M≤1. Both spheres and fibers reduce the turbulence intensity with respect to the single-phase flow at all Reynolds numbers, with fibers causing a more significant reduction than the spheres. The particles' effect on the anomalous dissipation tends to vanish as Re→∞. A scale-by-scale analysis shows that both particle shapes provide a “spectral shortcut” to the flow, but the shortcut extends further into the dissipative range in the case of fibers. Multifractal spectra of the near-particle dissipation show that spheres enhance dissipation in two-dimensional sheets, and fibers enhance the dissipation in structures with a dimension greater than one and less than two. In addition, we show that spheres suppress vortical flow structures, whereas fibers produce structures which completely overcome the turbulent vortex stretching behavior in their vicinity. Published by the American Physical Society 2024

Motion of finite-size spheres released in a turbulent boundary layer

Individual magnetic wax spheres with specific gravities of 1.006, 1.054 and 1.152 were released from rest on a smooth wall in water at friction Reynolds numbers, Reτ=680 and 1320 (sphere diameters d+=58 and 122 viscous units, respectively). Three-dimensional tracking was conducted to understand the effects of turbulence and wall friction on sphere motions. Spheres subjected to sufficient mean shear initially lifted off of the wall before descending back towards it. These lifting spheres translated with the fluid above the wall, undergoing saltation or resuspension, with minimal rotations about any axis. By contrast, spheres that did not lift off upon release mainly slid along the wall. These denser spheres lagged the fluid more significantly due to greater wall friction. As they slid downstream, they began to roll forward after which small repeated lift-off events occurred. These spheres also rotated about both the streamwise and wall-normal axes. In all cases, the sphere trajectories were limited to the buffer and logarithmic regions, and all wall collisions were completely inelastic. In the plane parallel to the wall, the spheres migrated in the spanwise direction about 12% of the streamwise distance traveled suggesting that spanwise forces are important. Variations in sphere kinematics in individual runs were likely induced by high and low momentum zones in the boundary layer, vortex shedding in the sphere wakes, and wall friction. The repeated lift-offs of the forward rolling denser spheres were attributed to a Magnus lift.

Voronoï tessellation analysis of sets of randomly placed finite-size spheres

The purpose of this note is to clarify the effect of the finite size of spherical particles upon the characteristics of their spatial distribution through a random Poisson process (RPP). This information is of special interest when using RPP data as a reference for the analysis of the spatial structure of a given (non-RPP) particulate system, in which case ignoring finite-size effects upon the former may yield misleading conclusions. We perform Monte Carlo simulations in triply-periodic spatial domains, and then analyze the particle-centered Voronoï tessellations for solid volume fractions ranging from 10−5 to 0.3. We show that the standard-deviation of these volumes decreases with the solid volume fraction, the deviation from the value of point sets being reasonably approximated by an exponential function. As can be expected, the domain size for which the random assemblies of finite-size particles are generated has a constraining effect if the number of particles per realization is chosen too small. This effect is quantified, and recommendations are given. We have also revisited the case of random point sets (i.e. the limit of vanishing particle diameter), for which we have confirmed the accuracy of the earlier data by Tanemura (2003).

Slip-shear and inertial migration of finite-size spheres in plane Poiseuille flow

The effect of the slip-shear as well as the rotation on the inertial migration of finite-size spheres with blockage ratio (ratio of the sphere diameter to the channel gap) 0.2 are investigated in a plane Poiseuille flow at Re=100–500 using dissipative particle dynamics simulations, by means of changing the driving force, the sphere density, and the fluid-sphere boundary slip condition, respectively. Neutrally buoyant spheres with equal density with the fluid and with no-slip boundary condition are observed to equilibrated at lateral positions around midway between the channel centerline and the wall, and the positions move closer to the channel centerline with increasing Re. The spheres always lag behind the surrounding fluid. A correction to the slip is proposed by considering the effect of the distance between the sphere and the wall. A forward driving force or a larger density facilitates the forward motion of the spheres and leads to their leading relative to the surrounding fluid, which drives the spheres to lateral equilibrium positions close to the wall; and vice versa. All spheres rotate with angular velocities approximate to but a bit less than the half of the local fluid shear rate at the sphere centroid, no mater the spheres are neutrally or non-neutrally buoyant, expect the spheres much close to the wall. The lift induced by rotation is found one to two orders of magnitude less than the lift induced by slip-shear due to the small angular slip velocity between the sphere and the surrounding fluid; while the slip-shear-induced lift is found critical to determine the lateral equilibrium positions of finite-size spheres cooperating with the wall-induced lift and the shear-gradient-induced lift as Re varies. An empirical theory considering the last three lifts above mentioned is built, which interprets the relation between the slip-shear and the equilibrium positions of the spheres. Above relation holds for spheres with no-slip boundary condition, whilst for neutrally buoyant spheres with full-slip boundary condition, though they always lag behind the surrounding fluid, they migrate towards the wall as Re increases, as their hydrodynamic interactions with the fluid are changed.

Buoyancy-Driven Flow through a Bed of Solid Particles Produces a New Form of Rayleigh-Taylor Turbulence

Rayleigh-Taylor (RT) fluid turbulence through a bed of rigid, finite-size spheres is investigated by means of high-resolution direct numerical simulations, fully coupling the fluid and the solid phase via a state-of-the-art immersed boundary method. The porous character of the medium reveals a totally different physics for the mixing process when compared to the well-known phenomenology of classical RT mixing. For sufficiently small porosity, the growth rate of the mixing layer is linear in time (instead of quadratical) and the velocity fluctuations tend to saturate to a constant value (instead of linearly growing). We propose an effective continuum model to fully explain these results where porosity originated by the finite-size spheres is parametrized by a friction coefficient.

The effect of polydispersity in a turbulent channel flow laden with finite-size particles

We study turbulent channel flows of monodisperse and polydisperse suspensions of finite-size spheres by means of Direct Numerical Simulations using an immersed boundary method to account for the dispersed phase. Suspensions with 3 different Gaussian distributions of particle radii are considered (i.e. 3 different standard deviations). The distributions are centered on the reference particle radius of the monodisperse suspension. In the most extreme case, the radius of the largest particles is 4 times that of the smaller particles. We consider two different solid volume fractions, 2% and 10%. We find that for all polydisperse cases, both fluid and particles statistics are not substantially altered with respect to those of the monodisperse case. Mean streamwise fluid and particle velocity profiles are almost perfectly overlapping. Slightly larger differences are found for particle velocity fluctuations. These increase close to the wall and decrease towards the centerline as the standard deviation of the distribution is increased. Hence, the behavior of the suspension is mostly governed by excluded volume effects regardless of particle size distribution (at least for the radii here studied). Due to turbulent mixing, particles are uniformly distributed across the channel. However, smaller particles can penetrate more into the viscous and buffer layer and velocity fluctuations are therein altered. Non trivial results are presented for particle-pair statistics.

Sedimentation of finite-size spheres in quiescent and turbulent environments

Sedimentation of a dispersed solid phase is widely encountered in applications and environmental flows, yet little is known about the behaviour of finite-size particles in homogeneous isotropic turbulence. To fill this gap, we perform direct numerical simulations of sedimentation in quiescent and turbulent environments using an immersed boundary method to account for the dispersed rigid spherical particles. The solid volume fractions considered are ${\it\phi}=0.5{-}1\,\%$, while the solid to fluid density ratio ${\it\rho}_{p}/{\it\rho}_{f}=1.02$. The particle radius is chosen to be approximately six Kolmogorov length scales. The results show that the mean settling velocity is lower in an already turbulent flow than in a quiescent fluid. The reductions with respect to a single particle in quiescent fluid are approximately 12 % and 14 % for the two volume fractions investigated. The probability density function of the particle velocity is almost Gaussian in a turbulent flow, whereas it displays large positive tails in quiescent fluid. These tails are associated with the intermittent fast sedimentation of particle pairs in drafting–kissing–tumbling motions. The particle lateral dispersion is higher in a turbulent flow, whereas the vertical one is, surprisingly, of comparable magnitude as a consequence of the highly intermittent behaviour observed in the quiescent fluid. Using the concept of mean relative velocity we estimate the mean drag coefficient from empirical formulae and show that non-stationary effects, related to vortex shedding, explain the increased reduction in mean settling velocity in a turbulent environment.

Coherent acoustic wave propagation in media with pair-correlated spheres

Propagation of plane compressional waves in a non-viscous fluid with a dense distribution of identical spherical scatterers is investigated. The analysis is based on the multiple scattering approach proposed by Fikioris and Waterman, and is generalized to include arbitrary choice of the pair-correlation functions used to represent the distribution of the scatterers. A closed form solution for the effective wavenumber as a function of the concentration of pair-correlated finite-size spheres is derived up to the second order. In the limit of uncorrelated point-scatterers, this solution is identical to that obtained by Lloyd and Berry. Different pair-correlation functions are exemplified and compared, and the resulting differences discussed.

Off-shell mean-field electromagnetic T-matrix of finite-size spheres and fuzzy scatterers

We introduce a method for computing simple energy-conserving model T-matrices intended for use in multiple-scattering formalisms. Our method is to make separable approximations to the physical scattering potentials. Such a `mean-field' approximation permits a ready analytic evaluation of the T-matrix for a variety of scatterers, even `fuzzy' scatterers (e.g. fractal aggregates). In the small-size limit, our method reduces to the point-like resonant T-matrix introduced by Nieuwenhuizen et al in 1992 thereby demonstrating the intrinsic mean-field nature of their approximation. We discuss causality in both models and show that the separable potential model possesses a wider range of applicability than the point-like model does.

Basis-set expansion method for relativistic atoms in the atomic-nucleus model of a finite sphere of constant electric field

Use of a new atomic-nucleus model is proposed for relativistic atomic and molecular calculaitons; the atomic nucleus has a sphere of finite size and the electric field in the sphere is assumed constant. It is shown that for the basis-set expansion method based on this model, the Slater-type functions of integral principal quantum numbers can be appropriately adopted and the Slater-type function of “ n=l” quantum numbers (e.g., 1p, 2d,3f, etc.) appearing in the point-nucleus model are not needed even if we use the scheme of kinetic balance. Test calculations on one-electron hydrogenic atoms imply the applicability of the proposed atomic-nucleus model to many-electron systems.

Sedimentation and dispersion of non-neutrally buoyant Brownian particles in cellular circulatory flows simulating local fluid agitation.

Spatially periodic fluid ``vortical'' velocity fields are utilized to model the role of microscale convection upon the sedimentation and dispersion of isolated spherical particles. In particular, generalized Taylor dispersion theory is employed to calculate the mean Stokes sedimentation velocity vector and dispersivity dyadic for a finite-size spherical Brownian particle settling in a laminar, spatially periodic fluid velocity field with zero mean flow (but nonzero ``circulation''). Surprisingly, such local cellular vortexlike motions are shown to have no effect whatsoever upon the long-time mean sedimentation velocity of the sphere, which is identical to its steady-state Stokes-law value occurring in a quiescent fluid. This conclusion is, inter alia, independent of the sphere size a relative to the spatial period l of the cellular flow (provided that a/l\ensuremath{\ll}1), as well as of the ratio of the particle settling velocity to the magnitude of the cellular fluid velocity or circulation. On the other hand, local convection does affect particle dispersion relative to the mean particle velocity in these systems, consistent with recent calculations for a particular cellular velocity field (Mofatt [Rep. Prog. Phys. 46, 621 (1983)], Sagues and Horsthemke [Phys. Rev. A 34, 4136 (1986)], Rosenbluth et al. [Phys. Fluids 30, 2636 (1987)]). We obtain a general expression for the dispersion coefficient for sedimenting, finite-size spheres in the limit of low Pecl\'et number; this general result is consistent with the specific formula derived by the latter authors for the particular cellular ``Taylor'' flow analyzed by them, applicable to the case of a nonsedimenting, point-size tracer particle.