Dynamic permeability of an assemblage of soft spherical particles

Under oscillatory Stokes flow, dynamic permeability of assemblage of soft spherical particles is derived. For the bed of soft particles, the fluid‐particle system is represented as an assemblage of uniform permeable spheres fixed in space. Each sphere, with a surrounding envelope of fluid, is uncoupled from the system and considered separately. This model is popularly known as cell model. Oscillatory Stokes equations are employed inside the fluid envelope, and oscillatory Brinkman equations are used inside the porous region. Four known boundary conditions namely: Happel's, Kuwabara's, Kvashnin's, and Cunningham's are considered on the outer boundary and results are compared. The behavior of dynamic permeability is analyzed with various parameters such as Darcy number (Da), frequency parameter (?), porosity (φ), and viscosity ratio (δ). Copyright © 2013 John Wiley & Sons, Ltd.     

Limit Theorems for Supercritical Branching Random Fields

An infinite particle system in R^d is considered where the initial distribution is POISSON ian and each initial particle gives rise to a supercritical age-dependent branching process with the particles moving randomly in space. Our approach differs from the usual: instead of the point measures determined by the locations of the particles at each time, we take the particles at a “final time” and observe the past histories of their ancestry lines. A law of large numbers and a central limit theorem are proved under a space-time scaling representing high density of particles and small mean particle lifetime. The fluctuation limit is a generalized GAUSS -MARKOV process with continuous trajectories and satisfies a deterministic evolution equation with generalized random initial condition. A more precise form of the central limit theorem is obtained in the case of particles performing BROWN ian motion and having exponentially distributed lifetime.     

Separation of magnetic particles in channel flows by BEM

In-line separation of suspensions can become difficult in case of particles with comparable values of densities. For flows in micro devices in such cases gravitational settling is inefficient, and other separation techniques must be applied. In case of magneto active particles, the action of Kelvin magnetic force in a non-uniform magnetic field could be used in order to achieve a higher degree of particles separation. The contribution therefore deals with Euler-Lagrangian formulation of dilute two-phase flows. The Boundary element based computational algorithm solves the incompressible Navier-Stokes equations written in velocity-vorticity formulation. The non-uniform magnetic field is defined analytically for the case of a set of long thin wires. The particle trajectories are computed by applying the 4th order Runge-Kutta method. The computed test case consists of a narrow channel with laminar flow of suspension under Re = 1 − 10. Particle trajectories under the influence of a non-uniform magnetic field are computed for the case of magnetite and aluminium particles suspended in water. The efficiency of separation on basis of particle trajectories for different values of Re number and magnetic field strength is performed, clearly indicating superior separation of magneto active particles. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Thermocapillary motion of deformable fluid particles in tubes

The boundary integral technique is used to study the effect of deformation on the steady, creeping, thermocapillary migration of a fluid particle under conditions of axisymmetry, negligible thermal convection and an insulated tube wall. The spherical radius of the fluid particle (i.e. the radius as if the particle were a sphere, a ′= (3V _p ^′ /4π)^1/3, V _p ^′ is the particle volume) and that of the tube are denoted, respectively, by a′and b′. For small capillary numberCa = 0.05, only for a large fluid particle (a′/b′ = 0.8) is deformation significant. Fora′/b′= 0.8, hydrodynamic stresses squeeze the particle, reduce the interaction of the particle with the wall and thereby increase the terminal velocity. For small particles a′/b′< 0.8 and Ca = 0.05 the fluid particles translate as spheres, due to the fact that the fluid particle is too far away from the wall to be subject to distending hydrodynamic stresses. The deformable particle moves faster than a spherical one in the thermocapillary migration. The increase in velocity with capillary number is larger for thermocapillary motion than for buoyancy.     

Computing ECT-B-splines recursively

ECT-spline curves for sequences of multiple knots are generated from different local ECT-systems via connection matrices. Under appropriate assumptions there is a basis of the space of ECT-splines consisting of functions having minimal compact supports, normalized to form a nonnegative partition of unity. The basic functions can be defined by generalized divided differences [24]. This definition reduces to the classical one in case of a Schoenberg space. Under suitable assumptions it leads to a recursive method for computing the ECT-B-splines that reduces to the de Boor–Mansion–Cox recursion in case of ordinary polynomial splines and to Lyche's recursion in case of Tchebycheff splines. For sequences of simple knots and connection matrices that are nonsingular, lower triangular and totally positive the spline weights are identified as Neville–Aitken weights of certain generalized interpolation problems. For multiple knots they are limits of Neville–Aitken weights. In many cases the spline weights can be computed easily by recurrence. Our approach covers the case of Bézier-ECT-splines as well. They are defined by different local ECT-systems on knot intervals of a finite partition of a compact interval [a,b] connected at inner knots all of multiplicities zero by full connection matrices A ^[i] that are nonsingular, lower triangular and totally positive. In case of ordinary polynomials of order n they reduce to the classical Bézier polynomials. We also present a recursive algorithm of de Boor type computing ECT-spline curves pointwise. Examples of polynomial and rational B-splines constructed from given knot sequences and given connection matrices are added. For some of them we give explicit formulas of the spline weights, for others we display the B-splines or the B-spline curves. *Supported in part by INTAS 03-51-6637.     

Mathematical modeling of the movement of suspended particles subjected to acoustic and flow fields

When particles are subjected to an acoustic field particle trajectories depend on the particle and fluid compressibility and density values. Hence a combination of acoustic and flow fields on particles can be used to deflect and trap, or to segregate and/or fractionate fine particles in fluid suspensions. Using particle physics in an acoustic field, a mathematical model was developed to calculate trajectories of deflected particles due to the application of acoustic standing waves. The resulting second order ordinary differential equation was quite stiff and hence difficult to solve numerically and did not have a closed form solution. The analysis of the above equation showed that the basic problems with numerical solutions could not be ameliorated through the use of standard rescaling techniques. A combination of phase space and asymptotic analysis turns out to be far more useful in obtaining approximate solutions. An approximate solution was derived which enabled the calculation of the particle trajectories and concentration at collection planes in the acoustic field. Analysis of the solution showed that all the particles move toward the pressure node to which the particles are supposed to move. Particles with 2 μm diameter took approximately 20 s to reach that node. Then at the bench scale, the above technology was implemented by building a flow chamber with two transducers at opposite ends to generate an acoustic standing wave. SiC particle trajectories were tracked using captured digital images from a high-resolution microscope. The displacements of SiC particles due to an acoustic force were compared with the mathematical model predictions. For input power levels between 3.0 and 5.0 W, the experimental data were comparable to mathematical model predictions. Hence it was concluded that the proposed approximate solution was both quantitatively and qualitatively closer to experimental results than the simplified form ignoring the second order term reported in the literature.     

On the Bennequin Invariant and the Geometry of Wave Fronts

The theory of Arnold's invariants of plane curves and wave fronts is applied to the study of the geometry of wave fronts in the standard 2-sphere, in the Euclidean plane and in the hyperbolic plane. Some enumerative formulae similar to the Plücker formulae in algebraic geometry are given in order to compute the generalized Bennequin invariant J ⁺ in terms of the geometry of the front. It is shown that in fact every coefficient of the polynomial invariant of Aicardi can be computed in this way. In the case of affine wave fronts, some formulae previously announced by S.L. Tabachnikov are proved. This geometric point of view leads to a generalization to generic wave fronts of a result shown by Viro for smooth plane curves. As another application, the Fabricius-Bjerre and Weiner formulae for smooth plane and spherical curves are generalized to wave fronts.     

Lagrangian model of a massless particle on spacelike curves

We consider a model of a massless particle in a D-dimensional space with the Lagrangian proportional to the Nth extrinsic curvature of the world line. We present the Hamiltonian formulation of the system and show that its trajectories are spacelike curves satisfying the conditions k _N+a =k _N-a and k _2N =0, a=1,,N-1, where N[(D-2)/2]. The first N curvatures take arbitrary values, which is a manifestation of N+1 gauge degrees of freedom; the corresponding gauge symmetry forms an algebra of the W type. This model describes D-dimensional massless particles, whose helicity matrix has N coinciding nonzero weights, while the remaining [(D-2)/2]-N weights are zero. We show that the model can be extended to spaces with nonzero constant curvature. It is the only system with the Lagrangian dependent on the world-line extrinsic curvatures that yields irreducible representations of the Poincaré group.     

On the minimum ropelength of knots and links

The ropelength of a knot is the quotient of its length by its thickness, the radius of the largest embedded normal tube around the knot. We prove existence and regularity for ropelength minimizers in any knot or link type; these are C ^1,1 curves, but need not be smoother. We improve the lower bound for the ropelength of a nontrivial knot, and establish new ropelength bounds for small knots and links, including some which are sharp. Oblatum 11-IV-2001 & 20-III-2002?Published online: 17 June 2002     

Modeling and simulation of the motion of nanoparticles in cylindrical capillaries allowing particle‐to‐wall interactions

The process of transporting nanoparticles at the blood vessels level stumbles upon various physical and physiological obstacles; therefore, a Mathematical modeling will provide a valuable means through which to understand better this complexity. In this paper, we consider the motion of nanoparticles in capillaries having cylindrical shapes (i.e., tubes of finite size). Under the assumption that these particles have spherical shapes, the motion of these particles reduces to the motion of their centers. Under these conditions, we derive the mathematical model, to describe the motion of these centers, from the equilibrium of the gravitational force, the hemodynamic force and the van der Waals interaction forces. We distinguish between the interaction between the particles and the interaction between each particle and the walls of the tube. Assuming that the minimum distance between the particles is large compared with the maximum radius R of the particles and hence neglecting the interactions between the particles, we derive simpler models for each particle taking into account the particles‐to‐wall interactions. At an error of order O(R) or O(R³)(depending if the particles are 'near' or 'very near' to the walls), we show that the horizontal component of each particle's displacement is solution of a nonlinear integral equation that we can solve via the fixed point theory. The vertical components of the displacement are computable in a straightforward manner as soon as the horizontal components are estimated. Finally, we support this theory with several numerical tests. Copyright © 2016 John Wiley & Sons, Ltd.     

Asymptotics of particle trajectories in infinite one-dimensional systems with collisions

We investigate the asymptotic behavior of the trajectories of a tagged particle (tp) in an infinite one-dimensional system of point particles. The particles move independently when not in contact: the only interactions are Harris type generalized elastic collisions which prevent crossings. This is achieved by relabeling the independent trajectories when they cross. When these trajectories are differentiable, as in particles with velocities undergoing Ornstein-Uhlenbeck processes, collisions correspond to exchange of velocities. We prove very generally that the suitably scaled tp trajectory converges (weakly) to a simple Gaussian process. This extends the results of Spitzer for New tonian particles to very general non-crossing processes. The proof is based on the consideration of the simpler process which counts the crossings of the origin by the independent trajectories.     

Balance equation and the quasitemperature of channeled particles in equilibrium

Using the methods of nonequilibrium statistical thermodynamics, we obtain the equation for the transverse energy and momentum balance for fast atomic particles moving in the planar channeling regime. Based on the solution of this equation, we obtain an expression for the transverse quasitemperature in the quasiequilibrium in terms of the basic parameters of the theory. We show that the equilibrium quasitemperature of channeled particles is established because of particle diffusion in the space of transverse energies (subsystem “heating”), the dissipative process (“cooling”), and the anharmonic effects of particle oscillations between the channel walls (the redistribution of energies over the oscillatory degrees of freedom is the internal thermalization of the subsystem). According to the estimates for particles with an energy of the order of 1 MeV, the quasitemperature values are in the characteristic temperature range for a low-temperature plasma.     

Duality properties of indicatrices of knots

The bridge index and superbridge index of a knot are important invariants in knot theory. We define the bridge map of a knot conformation, which is closely related to these two invariants, and interpret it in terms of the tangent indicatrix of the knot conformation. Using the concepts of dual and derivative curves of spherical curves as introduced by Arnold, we show that the graph of the bridge map is the union of the binormal indicatrix, its antipodal curve, and some number of great circles. Similarly, we define the inflection map of a knot conformation, interpret it in terms of the binormal indicatrix, and express its graph in terms of the tangent indicatrix. This duality relationship is also studied for another dual pair of curves, the normal and Darboux indicatrices of a knot conformation. The analogous concepts are defined and results are derived for stick knots.     

On spherical representations of quivers and generalized complexes

For any quiverQ we consider spherical representationsV ofQ such that the isomorphism class ofV is a spherical variety. We suggest an approach for classifying such representations for anyQ and obtain a classification forQ being an equioriented Dynkin diagramA _n. In particular, all complexes are spherical representations. We introduce a category of representations that we call generalized complexes and classify spherical generalized complexes. For the quivers that we call crumbly we prove that any spherical generalized complex has a polynomial algebra of covariants on the closure of its isomorphism class.Partially supported by INTAS-OPEN grant 97-1570 and RFFI grant 98-01-00598.     

Magnetic trajectories in three-dimensional Lie groups

We study magnetic trajectories in Lie groups equipped with bi-invariant Riemannian metric. We define the Lorentz force of a magnetic field in a Lie group G, and then, we give the Lorentz force equation for the associated magnetic trajectories that are curves in G. When the manifold is a Lie group G equipped with bi-invariant Riemannian metric, we derive differential equation system that characterizes magnetic flow associated with the Killing magnetic curves with regard to the Lie reduction of the curve γ in G.     

Space curves with positive torsion

Summary Space curves may be classified under various kinds of deformation. The following six kinds of deformation have been of special interest; namely first, second and third order homotopy and isotopy. (We say the deformation is k-th order if the first k derivatives remain independent during the deformation.) The first order homotopy classification of space curves may be accomplished using well-known methods of Whitney; there is only one class. The second and third order homotopy classification was done by Feldman[1] and Little[6], respectively. The first order isotopy classification of space curves is knot theory; a subject of its own. The second order isotopy classification has been done by W. F. Pohl (unpublished). Thus, aside from knot theory, the only remaining problem is the third order isotopy problem. In this paper we give a partial answer. Our result is partial because we must restrict the class of curves with which we are dealing; namely to curves with a ? twist ?. But it may well be that every curve does have a twist, in which case our restricted class of curves would be all curves and the classification would be complete. In addition we construct a curve of positive torsion with any preassigned self-linking number in any preassigned knot class; a question raised by W. F. Pohl. Entrata in Redazione il 4 settembre 1976.     

Local structure of ideal shapes of knots

Relatively extremal knots are the relative minima of the ropelength functional in the C¹ topology. They are the relative maxima of the thickness (normal injectivity radius) functional on the set of curves of fixed length, and they include the ideal knots. We prove that a C1,1 relatively extremal knot in Rn either has constant maximal (generalized) curvature, or its thickness is equal to half of the double critical self distance. This local result also applies to the links. Our main approach is to show that the shortest curves with bounded curvature and C¹ boundary conditions in Rn contain CLC (circle-line-circle) curves, if they do not have constant maximal curvature.     

On diagrammatic bounds of knot volumes and spectral invariants

In recent years, several families of hyperbolic knots have been shown to have both volume and λ₁ (first eigenvalue of the Laplacian) bounded in terms of the twist number of a diagram, while other families of knots have volume bounded by a generalized twist number. We show that for general knots, neither the twist number nor the generalized twist number of a diagram can provide two-sided bounds on either the volume or λ₁. We do so by studying the geometry of a family of hyperbolic knots that we call double coil knots, and finding two-sided bounds in terms of the knot diagrams on both the volume and on λ₁. We also extend a result of Lackenby to show that a collection of double coil knot complements forms an expanding family iff their volume is bounded.     

On Invariants of Virtual Links

We propose a new method of generalizing classical link invariants for the case of virtual links. In particular, we have generalized the knot quandle, the knot fundamental group, the Alexander module, and the coloring invariants. The virtual Alexander module leads to a definition of VA-polynomial that has no analogue in the classical case (i.e. vanishes on classical links).     

Chemical waves and localized chaos in a model of heterogeneous catalytic reaction in a porous particles

A mathematical model of an oscillatory chemical reaction in a porous catalyst particle is considered. The model describes an oscillatory medium uniformly distributed throughout the volume of a spherical particle. The dynamical interaction of the reaction with the diffusive flow of the gaseous reagent inside the pores generates nonstationary dissipative structures in the oscillatory medium on the surface of the catalyst. Depending on the pressure in the gaseous phase, the model produces specific chemical waves and localized spatio-temporal chaos. The study was partially supported by the Russian Foundation for Basic Research (grant No. 96-03-34427a). Translated from Chislennye Metody i Vychislitel'nyi Eksperiment, Moscow State University, pp. 31–43, 1998.