Vakhtang Putkaradze

Braiding patterns on an inclined plane

A jet of fluid flowing down a partially wetting, inclined plane usually meanders but — by maintaining a constant flow rate — meandering can be suppressed, leading to the emergence of a beautiful braided structure. Here we show that this flow pattern can be explained by the interplay between surface tension, which tends to narrow the jet, and fluid inertia, which drives the jet to widen. These observations dispel misconceptions about the relationship between braiding and meandering that have persisted for over 20 years.

Euler-alpha and vortex blob regularization of vortex filament and vortex sheet motion

The Euler-alpha and the vortex blob model are two different regularizations of incom- pressible ideal fluid flow. Here, a regularization is a smoothing operation which controls the fluid velocity in a stronger norm than

Theory and experiment for one-dimensional directed self-assembly of nanoparticles

L^2

Boundary Effects on Exact Solutions of the Lagrangian-Averaged Navier–Stokes-α Equations

. The Euler-alpha model is the inviscid version of the Lagrangian averaged Navier–Stokes-alpha turbulence model. The vortex blob model was introduced to regularize vortex flows. This paper presents both models within one general framework, and compares the results when applied to planar and axisymmetric vortex filaments and sheets. By certain measures, the Euler-alpha model is closer to the unregularized flow than the vortex blob model. The differences that result in circular vortex filament motion, vortex sheet linear stability properties, and core dynamics of spiral vortex sheet roll-up are discussed.

Double-bracket dissipation in kinetic theory for particles with anisotropic interactions

A promising method of particle self-assembly using patterned surfaces is described. A set of long (order of millimeters), nanoscale-width grooves is etched into a substrate, and an aqueous solution containing particles of ∼50- or 80-nm diameter is deposited on the surface. Upon the evaporation of the solution, the particles are dragged into the grooves by the receding contact line. A partial differential equation, incorporating screening, is constructed to investigate the final distribution of particles in the grooves. A complete analysis of the stationary states for the density equation in one and two dimensions is performed. Additionally, the nonlinear evolution of the density is studied numerically and the results compare well with both the analytic results and the experiments.

Geometric dissipation in kinetic equations

In order to clarify the behavior of solutions of the Lagrangian-averaged Navier–Stokes-α (LANS-α) equations in the presence of solid walls, we identify a variety of exact solutions of the full equations and their boundary layer approximations. The solutions demonstrate that boundary conditions suggested for the LANS-α equations in the literature(1) for a bounded domain do not apply in a semi-infinite domain. The convergence to solutions of the Navier–Stokes equations as α → 0 is elucidated for infinite-energy solutions in a semi-infinite domain, and non-uniqueness of these solutions is discussed. We also study the boundary layer approximation of LANS-α equations, denoted the Prandtl-α equations, and report solutions for turbulent jets and wakes. Our version of the Prandtl-α equations includes an extra term necessary to conserve linear momentum and corrects an earlier result of Cheskidov.(2)

On Flexible Tubes Conveying Fluid: Geometric Nonlinear Theory, Stability and Dynamics

We derive equations of motion for the dynamics of anisotropic particles directly from the dissipative Vlasov kinetic equations, with the dissipation given by the double-bracket approach (double-bracket Vlasov, or DBV). The moments of the DBV equation lead to a non-local form of Darcy’s law for the mass density. Next, kinetic equations for particles with anisotropic interaction are considered and also cast into the DBV form. The moment dynamics for these double-bracket kinetic equations is expressed as Lie–Darcy continuum equations for densities of mass and orientation. We also show how to obtain a Smoluchowski model from a cold plasma-like moment closure of DBV. Thus, the double-bracket kinetic framework serves as a unifying method for deriving different types of dynamics, from density-orientation to Smoluchowski equations. Extensions for more general physical systems are also discussed.

Kinetic models of oriented self-assembly

Abstract A new symplectic variational approach is developed for modeling dissipation in kinetic equations. This approach yields a double bracket structure in phase space which generates kinetic equations representing coadjoint motion under canonical transformations. The Vlasov example admits measure-valued single-particle solutions. Such solutions are reversible. The total entropy is a Casimir, and thus it is preserved. To cite this article: D.D. Holm et al., C. R. Acad. Sci. Paris, Ser. I 345 (2007).

Rotating concentric circular peakons

We derive a fully three-dimensional, geometrically exact theory for flexible tubes conveying fluid. The theory also incorporates the change of the cross section available to the fluid motion during the dynamics. Our approach is based on the symmetry-reduced, exact geometric description for elastic rods, coupled with the fluid transport and subject to the volume conservation constraint for the fluid. We first derive the equations of motion directly, by using an Euler–Poincaré variational principle. We then justify this derivation with a more general theory elucidating the interesting mathematical concepts appearing in this problem, such as partial left (elastic) and right (fluid) invariance of the system, with the added holonomic constraint (volume). We analyze the fully nonlinear behavior of the model when the axis of the tube remains straight. We then proceed to the linear stability analysis and show that our theory introduces important corrections to previously derived results, both in the consistency at all wavelength and in the effects arising from the dynamical change of the cross section. Finally, we derive and analyze several analytical, fully nonlinear solutions of traveling wave type in two dimensions.

Helical states of nonlocally interacting molecules and their linear stability: a geometric approach

We suggest kinetic models of dissipation for an ensemble of interacting oriented particles, for example, moving magnetized particles. This is achieved by introducing a double bracket dissipation in kinetic equations using an oriented Poisson bracket, and employing the moment method to derive continuum equations for magnetization and density evolution. We show how our continuum equations generalize the Debye-Hückel equations for attracting round particles, and Landau-Lifshitz-Gilbert equations for spin waves in magnetized media. We also show formation of singular solutions that are clumps of aligned particles (orientons) starting from random initial conditions. Finally, we extend our theory to the dissipative motion of self-interacting curves.New kinetic models of dissipation are proposed for the dynamics of an ensemble of interacting oriented particles, for example, moving magnetized nano-particles. This is achieved by introducing double-bracket dissipation into kinetic equations by using an oriented Poisson bracket and employing the moment method to derive continuum equations for the evolution of magnetization and mass density. These continuum equations generalize the Debye–Huckel equations for attracting round particles, and Landau–Lifshitz–Gilbert equations for spin waves in magnetized media. The dynamics of self-assembly is investigated as the emergent concentration into singular clumps of aligned particles (orientons) starting from random initial conditions. Finally, the theory is extended to describe the dissipative motion of self-interacting curved filaments.