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S.Dobrokhotov
, D.Minenkov (A.Ishlinskii Institute for Problems inn Mechanics Russian Academy of Sciences
, Moscow institute of Physics andn Technology
2014
We make use of the Maupertuis -- Jacobi correspondence, well known in Classical Mechanics, to simplify 2-D asymptotic formulas based on Maslovs canonical operator, when constructing Lagrangian manifolds invariant with respect to phase flows for Hamil
tonians of the form $F(x,|p|)$. As examples we consider Hamiltonians coming from the Schrodinger equation, the 2-D Dirac equation for graphene and linear water wave theory.
Guided by extensive numerical simulations, we propose a microfluidic device that can sort elastic capsules by their deformability. The device consists of a duct embedded with a semi-cylindrical obstacle, and a diffuser which further enhances the sort
ing capability. We demonstrate that the device can operate reasonably well under changes in the initial position of the the capsule. The efficiency of the device remains essentially unaltered under small changes of the obstacle shape (from semi-circular to semi-elliptic cross-section). Confinement along the direction perpendicular to the plane of the device increases its efficiency. This work is the first numerical study of cell sorting by a realistic microfluidic device.
Relative index theorems, which deal with what happens with the index of elliptic operators when cutting and pasting, are abundant in the literature. It is desirable to obtain similar theorems for other stable homotopy invariants, not the index alone.
In the spirit of noncommutative geometry, we prove a full-fledged relative index type theorem that compares certain elements of the Kasparov KK-group KK(A,B).
A numerical solution to the problem of wave scattering by many small particles is studied under the assumption k<<1, d>>a, where a is the size of the particles and d is the distance between the neighboring particles. Impedance boundary conditions are
assumed on the boundaries of small particles. The results of numerical simulation show good agreement with the theory. They open a way to numerical simulation of the method for creating materials with a desired refraction coefficient.
We compute, in topological terms, the spectral flow of an arbitrary family of self-adjoint Dirac type operators with classical (local) boundary conditions on a compact Riemannian manifold with boundary under the assumption that the initial and termin
al operators of the family are conjugate by a bundle automorphism. This result is used to study conditions for the existence of nonzero spectral flow of a family of self-adjoint Dirac type operators with local boundary conditions in a two-dimensional domain with nontrivial topology. Possible physical realizations of nonzero spectral flow are discussed.
We present direct numerical simulations of turbulent channel flow with passive Lagrangian polymers. To understand the polymer behavior we investigate the behavior of infinitesimal line elements and calculate the probability distribution function (PDF
) of finite-time Lyapunov exponents and from them the corresponding Cramers function for the channel flow. We study the statistics of polymer elongation for both the Oldroyd-B model (for Weissenberg number $Wi <1$) and the FENE model. We use the location of the minima of the Cramers function to define the Weissenberg number precisely such that we observe coil-stretch transition at $Wiapprox1$. We find agreement with earlier analytical predictions for PDF of polymer extensions made by Balkovsky, Fouxon and Lebedev [Phys. Rev. Lett., 84, 4765 (2000).] for linear polymers (Oldroyd-B model) with $Wi<1$ and by Chertkov [Phys. Rev. Lett., 84, 4761 (2000).] for nonlinear FENE-P model of polymers. For $Wi>1$ (FENE model) the polymer are significantly more stretched near the wall than at the center of the channel where the flow is closer to homogenous isotropic turbulence. Furthermore near the wall the polymers show a strong tendency to orient along the stream-wise direction of the flow but near the centerline the statistics of orientation of the polymers is consistent with analogous results obtained recently in homogeneous and isotropic flows.
In the paper taking the assumption of the slowness of the change of the parameters of the vertically stratified medium in the horizontal direction and in time, the evolution of the non-harmonic wave packages of the internal gravity waves has been ana
lyzed. The concrete form of the wave packages can be expressed through some model functions and is defined by the local behavior of the dispersive curves of the separate modes near to the corresponding special points. The solution of this problem is possible with the help of the modified variant of the special-time ray method offered by the authors (the method of geometrical optics), the basic difference of which consists that the asymptotic representation of the solution may be found in the form the series of the non-integer degrees of some small parameter. At that the exponent depends on the concrete form of representation of this package. The obvious kind of the representation is determined from the principle of the localness and the asymptotic behavior of the solution in the stationary and the horizontally-homogeneous case. The phases of the wave packages are determined from the corresponding equations of the eikonal, which can be solved numerically on the characteristics (rays). Amplitudes of the wave packages are determined from the laws of conservation of the some invariants along the characteristics (rays).
