Unequal Spheres Research Articles

Generating, from scratch, the near-field asymptotic forms of scalar resistance functions for two unequal rigid spheres in low Reynolds number flow

The motion of rigid spherical particles suspended in a low Reynolds number fluid can be related to the forces, torques, and stresslets acting upon them by 22 scalar resistance functions, commonly notated X11A, X12A, Y11A, etc. Near-field asymptotic forms of these resistance functions were derived in Jeffrey and Onishi [“Calculation of the resistance and mobility functions for two unequal rigid spheres in low-Reynolds-number flow,” J. Fluid Mech. 139, 261–290 (1984)] and Jeffrey [“The calculation of the low Reynolds number resistance functions for two unequal spheres,” Phys. Fluids A 4, 16–29 (1992)]; these forms are now used in several numerical methods for suspension mechanics. However, the first of these important papers contains a number of small errors, which make it difficult for the reader to correctly evaluate the functions for parameters not explicitly tabulated. This short article comprehensively corrects these errors and adds formulas that were originally omitted from both papers, so that the reader can verify and implement the equations independently. The corrected expressions, rationalized and using contemporary nondimensionalization, are shown to match mid-field values of these scalars, which are calculated through an alternative method. A Python script to generate and evaluate these functions is provided.

Sedimentation of two unequal spheres in a square tube at low Reynolds numbers

The settling behaviors of two unequal spheres in a square tube at low Reynolds numbers were numerically studied through a three-dimensional lattice Boltzmann method (LBM). Due to the hydrodynamic interaction, a periodic state or a steady state is achieved according to the density ratio between the spheres. In addition, it is seen that the spheres form a staggered stable configuration at the steady state, for which both spheres rotate in the same manner. The effects of the Reynolds number were also examined.

Optimized packing unequal spheres into a multiconnected domain: mixed-integer non-linear programming approach

The problem of packing unequal spheres into a multiconnected domain (container) is considered. Given a set of spheres, the objective is to maximize the packing factor. The problem is considered as a knapsack problem and modelled as a mixed-integer non-linear programming. Characteristics of the model are indicated. We propose a new solution method based on a combination of a branch-and-bound approach and the known local optimization method. The search procedure is represented by a tree which allows handling all possible subsets of spheres. We develop a set of truncation rules to reduce the number of variants under test. The local optimization algorithm proceeds from the assumption of spheres radii being variable. A number of numerical examples are given.

Optimized Filling of a Given Cuboid with Spherical Powders for Additive Manufacturing

In additive manufacturing (also known as 3D printing), a layer-by-layer buildup process is used for manufacturing parts. Modern laser 3D printers can work with various materials including metal powders. In particular, mixing various-sized spherical powders of titanium alloys is considered most promising for the aerospace industry. To achieve desired mechanical properties of the final product, it is necessary to maintain a certain proportional ratio between different powder fractions. In this paper, a modeling approach for filling up a rectangular 3D volume by unequal spheres in a layer-by-layer manner is proposed. A relative number of spheres of a given radius (relative frequency) are known and have to be fulfilled in the final packing. A fast heuristic has been developed to solve this special packing problem. Numerical results are compared with experimental findings for titanium alloy spherical powders. The relative frequencies obtained by using the imposed algorithm are very close to those obtained by the experiment. This provides an opportunity for using a cheap numerical modeling instead of expensive experimental study.

Simulation of sedimentation of two spheres with different densities in a square tube

We simulated the sedimentations of two unequal spheres with different densities in a square tube for Galileo numbers (Ga) from 5 to 25, resulting in a Reynolds number range of based on the terminal settling velocity. The sedimentation of spheres with different densities is dynamically more complex than that of identical spheres. At high Ga the spheres oscillate in the centreline plane of the tube, where they are initially released from rest. By contrast, the spheres move to the diagonal or reverse–diagonal plane of the tube at low Ga, reaching a steady or periodic state depending on the density difference between them. A phase diagram illustrates the transitions between different sedimentation behaviours depending on Ga and the density difference. A possible mechanism for these behaviours is also presented. Furthermore, we compare two-dimensional (2-D) and three-dimensional computations for our system to attain a better understanding of the hydrodynamic interactions between two unequal spheres at low but finite Reynolds number. Comparing relative trajectories, periods of oscillation and flow features shows that 2-D circular cylinders oscillate much more strongly and frequently than spheres under the same flow conditions. In particular, spheres do not have the discontinuity in period that arises in the 2-D case from the change in rotation sign of a heavy particle.

Covering a body using unequal spheres and the problem of finding covering holes

Covering a body using unequal spheres and the problem of finding covering holes

Efficiency limits of the three-sphere swimmer

We consider a swimmer consisting of a collinear assembly of three spheres connected by two slender rods. This swimmer can propel itself forward by varying the lengths of the rods in a way that is not invariant under time reversal. Although any non-reciprocal strokes of the arms can lead to a net displacement, the energetic efficiency of the swimmer is strongly dependent on the details and sequences of these strokes, and also the sizes of the spheres. We define the efficiency of the swimmer using Lighthill's criterion, i.e., the power that is needed to pull the swimmer by an external force at a certain speed, divided by the power needed for active swimming with the same average speed. Here, we determine numerically the optimal stroke sequences and the optimal size ratio of the spheres, while limiting the maximum extension of the rods. Our calculation takes into account both far-field and near-field hydrodynamic interactions. We show that, surprisingly, the three-sphere swimmer with unequal spheres can be more efficient than the equally-sized case. We also show that the variations of efficiency with size ratio is not monotonic and there exists a specific size ratio at which the swimmer has the highest efficiency. We find that the swimming efficiency initially rises by increasing the maximum allowable extension of the rods, and then converges to a maximum value. We calculate this upper limit analytically and report the highest value of efficiency that the three-sphere swimmer can reach.

Dynamic fracture behaviour and evolution mechanism of soft coal with different porosities and water contents

Determining the effect of water content and porosity on the dynamic mechanics of soft coal is a key problem in reinforcing soft coal seams by water injection. In this study, soft coal specimens with different water contents and porosities were successfully prepared and tested using a Split Hopkinson Pressure Bar (SHPB). The results show that the effect of water content on the strength of soft coal can be divided into two stages based on the critical point of optimum water content. With the effects of increasing porosity, the optimum water content gradually increased, while the peak strength corresponding to the optimum water content gradually decreased. In addition, based on the analysis of the dynamic failure process of the coal recorded using a high-speed camera, the dynamic mechanical model varies with the distance, and the liquid bridge volume between particles was established according to the water-bearing unequal spheres models. Moreover, the mechanism of the effects of porosity and water content on the fracture expansion and dynamic strength of soft coal were further discussed.

Transition in swimming direction in a model self-propelled inertial swimmer

We propose a reciprocal, self-propelled model swimmer at intermediate Reynolds numbers ($Re$). Our swimmer consists of two unequal spheres that oscillate in antiphase generating nonlinear steady streaming (SS) flows. We show computationally that the SS flows enable the swimmer to propel itself, and also switch direction as $Re$ increases. We quantify the transition in the swimming direction by collapsing our data on a critical $Re$ and show that the transition in swimming directions corresponds to the reversal of the SS flows. Based on our findings, we propose that SS can be an important physical mechanism for motility at intermediate $Re$.

Mobilities of polydisperse hard spheres near a no-slip wall

We have derived an analytical formulation for far-field hydrodynamic interactions among unequal size hard spheres near a no-slip wall and have implemented the formulation in a Stokesian dynamics model to simulate a suspension of polydisperse particles in a semi-bounded domain. The formulation is based on the multipole expansion of the boundary integral of Stokes flow, and the mobility tensors are deduced from Fáxen’s law together with Green’s function for Stokes flow near a no-slip wall. Lubrication approximation is incorporated to account for the close-distance interactions between any two particles and also between the particles and the wall. The implementation is validated against previous formulations for equal size particles and against a boundary-element code for unequal size particles. The code can be used to simulate the Stokesian or Brownian interaction of unequal particles in presence of a no-slip wall. In the current study, we applied the Stokesian dynamics model to investigate the trajectories of two unequal hard spheres and their redistribution during sedimentation parallel to a wall. We further used it to demonstrate the cases of many-particle sedimentation toward or parallel to a wall.

Packing Unequal Spheres into Various Containers

The paper considers an optimization problem of packing different solid spheres into containers of the following types: a cuboid, a sphere, a right circular cylinder, an annular cylinder, and a spherical layer. The radii of spheres are assumed to vary. It allows us to propose a new way to derive starting points belonging to the feasible domain of the problem, as well as to carry out a non-exhaustive search of local extrema, using a modification of the jump algorithm (JA), which implements a continuous transition from one local minimum to another with a better value of the objective function. A reduction of the solution space dimension and rearrangements of sphere pairs allow improving the objective function value. The results obtained are compared with benchmark ones.

The Precariousness of Public Health: On Tuberculosis Control in Contemporary France

ABSTRACTThrough an ethnographic exploration of tuberculosis control in one of France’s poorest regions, Seine-Saint-Denis, I interrogate the relationships between public health planning and interventions in conditions of multiple precarity. I show that the encounter between the feasible and the fantastic in the realm of public health generates feelings of absurdity and futility among medical professionals, characteristic of disease control in the precarious present. Precarity is neither a social and economic condition per se, nor is the link between disease and precarity static. It is a dynamic process of political inclusions, exclusions, and inequalities, which differ substantially within the unequal spheres of precarious lives. The contradictions in tuberculosis control that I describe are thus not only characteristic of French public health but of global public health today, where illusions in disease control encounter the exclusionary realities of social life.

Stokesian simulation of two unequal spheres in a pressure-driven creeping flow through a cylinder

In this paper, a novel numerical technique is generalized for analysis of mutual flow interactions between unequal particles in cylinder-bound pressure-driven flow of highly viscous fluid. The methodology is general enough to account for any polydisperse many-sphere system confined in a circular channel. The scheme is especially useful for the geometric configurations where both interparticle and particle-wall hydrodynamic interactions are significant. In such situation, particulate dynamics is governed by strong spatial variation of mobility tensors relating motion-inducing quantities (like force and torque on the bodies along with the pressure-drop in the vessel) to motion-defining ones (like translational and rotational velocities of each sphere as well as the flow-rate). Accordingly, the mobility tensors are evaluated as spatial functions to solve two different problems. First, we compute force and torque due to incoming fluid on two particles when these are fixed at a specific position inside the flow-domain. Then, we assume both to be freely suspended, and determine their motion driven by the transporting medium. In both problems, the additional pressure-loss in the channel due to the presence of the particles is also calculated representing the increase in channel-resistivity and effective viscosity, respectively.

A dichotomous search-based heuristic for the three-dimensional sphere packing problem

In this paper, the three-dimensional sphere packing problem is solved by using a dichotomous search-based heuristic. An instance of the problem is defined by a set of n unequal spheres and an object of fixed width and height and, unlimited length. Each sphere is characterized by its radius and the aim of the problem is to optimize the length of the object containing all spheres without overlapping. The proposed method is based upon beam search, in which three complementary phases are combined: (i) a greedy selection phase which determines a series of eligible search subspace, (ii) a truncated tree search, using a width-beam search, that explores some promising paths, and (iii) a dichotomous search that diversifies the search. The performance of the proposed method is evaluated on benchmark instances taken from the literature where its obtained results are compared to those reached by some recent methods of the literature. The proposed method is competitive and it yields promising results.

Packing fraction of trimodal spheres with small size ratio: An analytical expression

In previous papers analytical expressions were derived and validated for the packing fraction of bimodal hard spheres with small size ratio, applicable to ordered (crystalline) [H. J. H. Brouwers, Phys. Rev. E 76, 041304 (2007);H. J. H. Brouwers, Phys. Rev. E 78, 011303 (2008)] and disordered (random) packings [H. J. H. Brouwers, Phys. Rev. E 87, 032202 (2013)]. In the present paper the underlying statistical approach, based on counting the occurrences of uneven pairs, i.e., the fraction of contacts between unequal spheres, is applied to trimodal discretely sized spheres. The packing of such ternary packings can be described by the same type of closed-form equation as the bimodal case. This equation contains the mean volume of the spheres and of the elementary cluster formed by these spheres; for crystalline arrangements this corresponds to the unit cell volume. The obtained compact analytical expression is compared with empirical packing data concerning random close packing of spheres, taken from the literature, comprising ternary binomial and geometric packings; good agreement is obtained. The presented approach is generalized to ordered and disordered packings of multimodal mixes.

An Algorithm to Packing Unequal Spheres in a Larger Sphere

We consider the problem of packing unequal spheres into a larger sphere as dense as possible. This problem is seldom researched so far as we know, but its importance and real world applications have been widely revealed. In this paper, we present a method using the techniques of Tabu search, Quasi-human basin-hopping strategy, and BFGS method as its components to resolve this problem, and get some solutions for a serial of thoroughly selected instances.

The efficient cluster packing model – An atomic structural model for metallic glasses

A structural model is described for metallic glasses based on a new sphere packing scheme – the efficient filling of space by solute-centered clusters. This model combines random positioning of solvent atoms with atomic order of solutes. It shows that metallic glasses contain ⩽4 topologically distinct species and that solutes possess specific sizes relative to solvent atoms to produce efficient atomic packing. Validation is achieved by quantitative predictions of nearest-neighbor partial coordination numbers, medium-range solute ordering, density and metallic glass topologies. Good agreement is achieved in each of these areas. This model is able to reproduce compositions for a broad range of metallic glasses, provides specific guidance for the exploration of new bulk metallic glasses and may give new insights into other metallic glass studies. The new scheme introduced here for the efficient filling of space in extended systems of unequal spheres may have relevance to other fields.

A new resistance function for two rigid spheres in a uniform compressible low-Reynolds-number flow

The pressure moment of a rigid particle is defined as the trace of the first moment of the surface stress acting on the particle. We calculate the pressure moments of two unequal rigid spheres immersed in a uniform compressible linear flow, using twin multipole expansions and lubrication theory. Following the practice established in previous studies on two-body hydrodynamic interactions at low Reynolds number, the results are expressed in terms of a new (stresslet) resistance function.

Unequal-sphere packing model for the structural arrangement of the well-ordered adsorbate-substrate system

In order to understand the well-ordered adsorbate-substrate systems at atomic level, a method is developed based on the simulation of packing arrangements for layers of unequal spheres, in three-dimensional space. The model, based on geometrical principles, is developed for fcc structure consisting of two hexagonal ordered layers. During simulation, adsorbate spheres were accommodated in different positions, forming a great variety of structures, in dependence of the intersphere distance of the upper layer spheres. Using the average height of the adsorbate layer on the flat substrate as a determinant parameter, several specific structures have been selected as the most probable: $(\ensuremath{\surd}3\ifmmode\times\else\texttimes\fi{}\ensuremath{\surd}3)\mathrm{R}30\ifmmode^\circ\else\textdegree\fi{}$, $(\ensuremath{\surd}7\ifmmode\times\else\texttimes\fi{}\ensuremath{\surd}7)\mathrm{R}19.1\ifmmode^\circ\else\textdegree\fi{}$, and $(3\ifmmode\times\else\texttimes\fi{}3)$. Indeed, they correspond to typical accommodations of the iodine adatoms on the Pt(111) surface, earlier found in experimental studies, which clearly supports the validity of our model. The model developed in our study could completely and satisfactorily describe the accommodation process of the iodine adlayer on the Pt(111) surface. This methodology could be of great help for interpretation of scanning tunneling microscopy images, better understanding of adlayer structures, and design of adsorbate-substrate systems with exciting properties.

Stokes flow for the axisymmetric motion of several spherical particles perpendicular to a plane wall

The creeping flow around several spherical particles moving on a line perpendicular to a plane wall is calculated numerically using the boundary integral method. The locations of the point forces on the surfaces of the spheres are chosen so as to describe precisely the lubrication regions when the surfaces are close to one another. Earlier results are recovered for the cases of a single sphere and a wall and of two equal spheres far from a wall. New results are presented for two (equal or unequal) spheres close to a plane wall and several equal spheres far from a wall.