On the n-fold symmetric product suspensions of a continuum

In 1979 Sam B. Nadler, Jr. defined the hyperspace suspension of a continuum. We define the n-fold symmetric product suspensions of a continuum using n-fold symmetric products. We study some properties of this hyperspace: unicoherence, local connectedness, arcwise connectedness.     

A continuum theory of dense suspensions

A continuum theory is introduced for viscous fluids carrying dense suspensions (such as blood) or emulsions of arbitrary shape and inertia. Suspended particles possess microinertia that make the mixture an anisotropic fluid whose viscosity changes with motion and orientation of suspensions. The microinertia balance law coupled with the equations of motion of an anisotropic fluid govern the ultimate outcome. By means of the second law of thermodynamics, constitutive equations are obtained in terms of the frame-independent tensors. In a special case, a theory of bar-like suspensions is obtained. The field equations, boundary and initial conditions are given for both the arbitrarily-shaped suspensions and the bar-like suspensions. The theory is demonstrated with the solution of the channel flow problem. The mean viscosity of the fluid with suspensions is determined. The motions of suspensions down flow are demonstrated.     

A continuum theory of dense suspensions

A continuum theory is introduced for viscous fluids carrying dense suspensions (such as blood) or emulsions of arbitrary shape and inertia. Suspended particles possess microinertia that make the mixture an anisotropic fluid whose viscosity changes with motion and orientation of suspensions. The microinertia balance law coupled with the equations of motion of an anisotropic fluid govern the ultimate outcome. By means of the second law of thermodynamics, constitutive equations are obtained in terms of the frame-independent tensors. In a special case, a theory of bar-like suspensions is obtained. The field equations, boundary and initial conditions are given for both the arbitrarily-shaped suspensions and the bar-like suspensions. The theory is demonstrated with the solution of the channel flow problem. The mean viscosity of the fluid with suspensions is determined. The motions of suspensions down flow are demonstrated.     

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.     

The relation between director and multipolar theories in continuum mechanics

Résumé On décrit la théorie thérmo-mécanique d'un milieu continu dans laquelle la déformation est décrite par plusieurs champs directeurs. Il est demontré qu'une des formes de la théorie est equivalente à une théorie de directeurs. Une théorie dans laquelle la déformation est décrite par plusieurs champs de déplacement est discutée et il est demontré qu'elle est equivalente à une théorie de directeurs sous une condition restrictive.     

On the dynamics of deformable ferromagnets

The paper begins with a fairly detailed presentation of a general mathematical model for the dynamics of ferromagnetic bodies undergoing arbitrarily large deformations. Next, a mathematical study is presented of the evolution problem for the magnetization field in a «soft» ferromagnetic body which is «mechanically at rest». No matter how special and simple this problem within the frame-work of the full theory, the governing equation is interesting: it is identical to the dynamic version of the harmonic-map equation usually referred to in the mathematical literature as the Gilbert form of the Landau-Lifshitz equation. Motivated by recent nonuniqueness results for the dynamic harmonicmap equation, we give a new proof of global existence of weak solutions to the Gilbert-Landau-Lifshitz equation.     

Quantum integrable systems of particles as gauge theories

We study quantum integrable systems of interacting particles from the point of view proposed by A. Gorsky and N. Nekrasov. We obtain the Sutherland system by a Hamiltonian reduction of an integrable system on the cotangent bundles to an affine algebra and show that it coincides with the Yang-Mills theory on a cylinder.We point out that there exists a tower of 2d quantum field theories. The top of this tower is the gauged G/G WZW model on a cylinder with an inserted Wilson line in an appropriate representation, which in our approach corresponds to Ruijsenaars' relativistic Calogero model. Its degeneration yields the 2d Yang-Mills theory, whose small radius limit is the Calogero model itself. We make some comments about the spectra and eigenstates of the models, which one can get from their equivalence with the field theories. Also we point out some possibilities of elliptic deformations of these constructions.Institute of Theoretical and Experimental Physics, 117259, Bol. Cheremushkinskaya 25, Moscow, Russia. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 100, No. 1, pp. 97–103, July, 1994.     

On deformable hypersurfaces in space forms

We first extend the classical Sbrana-Cartan theory of isometrically deformable euclidean hypersurfaces to the sphere and hyperbolic space. Then we construct and characterize a large family of hypersurfaces which admit a unique deformation. This is used to show, by means of explicit examples, that different types of hypersurfaces in the Sbrana-Cartan classification can be smoothly attached. Finally, among other applications, we discuss the existence of complete deformable hypersurfaces in hyperbolic space.     

Two-phase flow simulation for distinguishing deformable particles with a LiMCA system

In the metallurgical industry, Liquid Metal Cleanliness Analyser (LiMCA) commercial equipment cannot distinguish between hard particles (e.g., oxides, borides) and deformable particles (e.g., bubbles, molten salts). Therefore, hard particle concentrations can sometimes be grossly overestimated, which reduces the measurement accuracy. However, the method could potentially discriminate between deformable particles and hard particles by evaluating the particle's ability to deform. In this work, the coupled multiphysics problem of a particle deforming within current-carrying aluminium metal passing through the electric sensing zone (ESZ) is simulated using the conservative level-set (CLS) method. An emphasis is placed on understanding the transient deformation history, and the effect of the capillary number, Reynolds number, and confinement ratio on deformation are studied. Furthermore, a computational basis is given to estimate the influence of particle deformation on electrical resistance pulses (ERP). It is found that ERP features of deformation particles, including the peak magnitude and the pulse width, are different from those of hard particles. Based on the results, the effect of a particle's deformation and the feasibility to discriminate it from non-deformable particles in the LiMCA system is evaluated.     

On the statistical mechanics of deformable media

Random stress and strain fields are examined in relation to machine parts made of materials of the randomly reinforced fiberglass type subjected to random external loads. The conditions of existence of the principal axes of the fourth-order tensor representing the second-order central moments of a plane random stress or strain field are determined. A method of determining the principal axes and principal values is described. A numerical example is worked.V. A. Steklov Mathematical Institute, Sverdlovsk Branch, Academy of Sciences of the USSR. S. M. Kirov Ural Polytechnic Institute, Sverdlovsk. Translated from Mechanika Polimerov, Vol. 4, No. 4, pp. 618–623, July–August, 1968.     

A boundary integral formulation for elastically deformable particles in a viscous fluid

This paper reports a formulation and implementation of a mixed (both direct and indirect) boundary element method using the double layer and its adjoint in a form suitable for solving Stokes flow problems involving elastically deformable particles. The formulation is essentially the Completed Double Layer Boundary Element Method for solving an exterior traction problem for the surrounding fluid or solid phase, followed by an interior displacement, and a mobility problem (if required) for the elastic particles. At the heart of the method is a deflation procedure that allows iterative solution strategies to be adopted, effectively opens the way for large-scale simulations of suspensions of deformable particles to be performed. Several problems are considered, to illustrate and benchmark the method. In particular, an analytical solution for an elastic sphere in an elongational flow is derived. The stresslet calculations for an elastic sphere in shear and elongational flows indicate that elasticity of the inclusions can potentially lead to positive second normal stress difference in shear flow, and an increase in the tensile resistance in elongational flow.This work is supported by a grant from the Australian Research Grant Council. X-J F wishes to acknowledge the support of the National Natural Science Foundation of China.     

On omega-categorical simple theories

In the present paper we shall prove that countable ω-categorical simple CM-trivial theories and countable ω-categorical simple theories with strong stable forking are low. In addition, we observe that simple theories of bounded finite weight are low.     

On almost categorical theories

Siberian Mathematical Journal -     

On theories of random variables

We study theories of spaces of random variables: first, we consider random variables with values in the interval [0, 1], then with values in an arbitrary metric structure, generalising Keisler’s randomisation of classical structures. We prove preservation and non-preservation results for model theoretic properties under this construction:

1. The randomisation of a stable structure is stable.
2. The randomisation of a simple unstable structure is not simple.

We also prove that in the randomised structure, every type is a Lascar type.     

Asymptotic behavior of the wave function of three particles in a continuum

We study the wave function of a system of three particles in a continuum. The Faddeev equations are used to explicitly identify the singularities of the wave function in the momentum space. We obtain the asymptotic behavior of the wave function in the configuration space by calculating the asymptotic behavior of the Fourier transform of the wave function in the momentum space. Our attention is focused on configurations in which two particles are at a relatively small distance from each other while the third particle is significantly remote from the center of mass of the pair. We show that the coordinate asymptotic form of the wave function for such a configuration contains scattered waves of a new type in addition to the standard terms. We use the obtained exact data concerning the coordinate asymptotic form of the wave function to critically analyze the multiplicative ansatz used in several works to describe systems of three particles in a continuum.