Dmitri Vainchtein

Point vortices exhibit asymmetric equilibria

The equilibrium patterns formed by interacting vortices have been puzzled over in a variety of contexts since Kelvins theory of vortex atoms was debunked by quantum mechanics. These patterns have, for example, appeared in atmospheric and oceanographic flows, and in rotating superfluid helium,. For a particular mathematical model, the point vortex equations, a catalogue of equilibria was drawn up some twenty years ago. We have revisited the point vortex model by a different approach and have discovered a host of new equilibrium states, including asymmetric patterns devoid of rotational or reflectional symmetry.

On passage through resonances in volume-preserving systems

Resonance processes are common phenomena in multiscale (slow-fast) systems. In the present paper we consider capture into resonance and scattering on resonance in 3-D volume-preserving slow-fast systems. We propose a general theory of those processes and apply it to a class of viscous Taylor-Couette flows between two counter-rotating cylinders. We describe the phenomena during a single passage through resonance and show that multiple passages lead to the chaotic advection and mixing. We calculate the width of the mixing domain and estimate a characteristic time of mixing. We show that the resulting mixing can be described using a diffusion equation with a diffusion coefficient depending on the averaged effect of the passages through resonances.Resonance processes are common phenomena in multiscale (slow-fast) systems. In the present paper we consider capture into resonance and scattering on resonance in 3D volume-preserving multiscale systems. We propose a general theory of those processes and apply it to a class of kinematic models inspired by viscous Taylor-Couette flows between two counter-rotating cylinders. We describe the phenomena during a single passage through resonance and show that multiple passages lead to the chaotic advection and mixing. We calculate the width of the mixing domain and estimate a characteristic time of mixing. We show that the resultant mixing can be described using a diffusion equation with a diffusion coefficient depending on the averaged effect of the passages through resonances.

Resonances and Particle Stochastization in Nonhomogeneous Electromagnetic Fields

Abstract In the present paper we investigate the resonant interaction between monochromatic electromagnetic waves and charged particles in configurations with magnetic field reversals (e.g., in the earth magnetotail). The smallness of certain physical parameters allows us to solve this problem using perturbation theory, reducing the problem of resonant wave–particle interaction to the analysis of slow passages of a particle through a resonance. We discuss in detail two of the most important resonant phenomena: capture into resonance and scattering on resonance. We show that these processes result in destruction of the adiabatic invariants and chaotization of particles; they also may lead to significant (almost free) acceleration of particles and may govern transport in the phase space. We calculate the characteristic times of mixing due to resonant effects and separatrix crossings, and discuss the relative importance of these phenomena.

The equation of state of a foam

The equation of state of a dry foam with ideal gas in the bubbles, proposed by Ross, is proved. This equation suggests that a foam with a free boundary will expand to a maximum volume if the external pressure is lowered at constant temperature. The same foam enclosed in a container, on the other hand, can be expanded further but, as the volume is increased, will eventually become unstable. We speculate that this instability leads to a “bubble differentiation” transition of the kind observed by Herdtle in numerical simulations of two-dimensional foam with ideal gas in the bubbles. In the simulations the foam separated into two “phases,” one with a large number of small bubbles, another with a small number of very large bubbles.

Mixing properties of steady flow in thermocapillary driven droplets

We consider mixing via chaotic advection in microdroplets suspended at the free surface of a liquid substrate and driven along a straight line using the thermocapillary effect. With the help of a model derived by Grigoriev [Phys. Fluids 17, 033601 (2005)] we show that the mixing properties of the flow inside the droplet can vary dramatically as a function of the physical properties of the fluids and the imposed temperature profile. Proper characterization of the mixing quality requires introduction of two different metrics. The first metric determines the relative volumes of the domain of chaotic streamlines and the domain of regular streamlines. The second metric describes the time for homogenization inside the chaotic domain. We compute both metrics using perturbation theory in the limit of weak temperature dependence of the surface tension coefficient at the free surface of the substrate.

Control of a vortex pair using a weak external flow

The problem of controlling the position of a pair of point vortices using a strain field or the field of a single source/sink is considered using singular control methods and dynamical system perturbation theory. We show that averaging over the fast rotation of the vortices around the centre of vorticity reduces the order of the system of evolution equations and allows one to achieve the desired control using a single control field. This article was chosen from Selected Proceedings of the 4th International Workshop on Vortex Flows and Related Numerical Methods (UC Santa-Barbara, 17-20 March 2002) ed E Meiburg, G H Cottet, A Ghoniem and P Koumoutsakos.

Limit sets for natural extensions of Schelling’s segregation model

Abstract Thomas Schelling developed an influential demographic model that illustrated how, even with relatively mild assumptions on each individual’s nearest neighbor preferences, an integrated city would likely unravel to a segregated city, even if all individuals prefer integration. Individuals in Schelling’s model cities are divided into two groups of equal number and each individual is “happy” or “unhappy” when the number of similar neighbors cross a simple threshold. In this manuscript we consider natural extensions of Schelling’s original model to allow the two groups have different sizes and to allow different notions of happiness of an individual. We observe that differences in aggregation patterns of majority and minority groups are highly sensitive to the happiness threshold; for low threshold, the differences are small, and when the threshold is raised, striking new patterns emerge. We also observe that when individuals strongly prefer to live in integrated neighborhoods, the final states exhibit a new tessellated-like structure.

Dynamics of electrons in a parabolic magnetic field perturbed by an electromagnetic wave

In this paper we study the resonance interaction between monochromatic electromagnetic waves and fully magnetized electrons in a model parabolic magnetic field (like, e.g., in the Earths magnetotail). The smallness of certain physical parameters allows us to approach this problem using perturbation theory for multiscale (slow-fast) systems: the study of the global interaction is reduced to the analysis of slow passages of particles through a resonance. At the resonance, two important phenomena occur: capture into resonance and scattering on resonance. We show that while the primary adiabatic invariant (magnetic moment or Larmor radius) remains conserved, these processes result in destruction of the second, longitudinal, adiabatic invariant. We find significant acceleration of particles by capture into resonance, while the scatterings on resonances lead to decrease in energy and chaotization of particles.

Adiabatic Invariance in Volume-Preserving Systems

We consider destruction of adiabatic invariance in volume-preserving systems due to separatrix crossings, scattering on and capture into resonances. These mechanisms result in mixing and transport in large domains of phase space. We consider several examples of systems where these phenomena occur.

Morphological transition in compressible foam

A theory is constructed to describe the morphological transition that occurs in a compressible foam when its volume is increased. The foam is observed to separate into two bubble populations or “phases,” one consisting of a large number of small bubbles, the “liquid” phase, the other consisting of a small number of large bubbles, the “gaseous” phase. First, working along lines similar to the van der Waals theory for a fluid system, approximate forms of the equation of state of the foam are derived and explored. These describe the weakly compressible range well, but fail to capture the nature of the transition. Taking a clue from the phenomenology, a theory of the “phase-separated” regime is then formulated working with the approximation that the two phases into which the foam separates are each relatively homogeneous. The successful single-phase formulas are applied to each phase, introducing an additional “order parameter” which gives the ratio of the average size of bubbles in the two phases. Approximat...