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    Sergey Ramodanov

    Universitetskaya 1, Izhevsk, 426034 Russia
    Institute of Computer Science, Udmurt State University

    Senior scientist, Department of Mathematical Methods in Nonlinear Dynamics, IMM UB RAS


    Ramodanov S. M., Tenenev V. A., Treschev D. V.
    We study the system of a 2D rigid body moving in an unbounded volume of incompressible, vortex-free perfect fluid which is at rest at infinity. The body is equipped with a gyrostat and a so-called Flettner rotor. Due to the latter the body is subject to a lifting force (Magnus effect). The rotational velocities of the gyrostat and the rotor are assumed to be known functions of time (control inputs). The equations of motion are represented in the form of the Kirchhoff equations. In the case of piecewise continuous controls, the integrals of motion are indicated. Using these integrals we obtain a (reduced) system of first-order differential equations on the configuration space. An optimal control problem for several types of the inputs is then solved using genetic algorithms.
    Keywords: perfect fluid, self-propulsion, Flettner rotor
    Citation: Ramodanov S. M., Tenenev V. A., Treschev D. V.,  Self-propulsion of a body with rigid surface and variable coefficient of lift in a perfect fluid, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 4, pp.  799-813
    Sokolov S. V., Ramodanov S. M.
    We consider a system which consists of a heavy circular cylinder and a point vortex in an unbounded volume of ideal liquid. The liquid is assumed to be irrotational and at rest at infinity. The circulation about the cylinder is different from zero. The governing equations are Hamiltonian and admit an evident integral of motion — the horizontal component of the momentum. Using the integral we reduce the order and thereby obtain a system with two degrees of freedom. Most remarkable types of partial solutions of the system are presented and stability of equilibrium solutions is investigated.
    Keywords: point vortices, Hamiltonian systems, reduction, stability of equilibrium solutions
    Citation: Sokolov S. V., Ramodanov S. M.,  Falling motion of a circular cylinder interacting dynamically with a point vortex, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 3, pp.  617-628
    Ramodanov S. M., Tenenev V. A.
    In the paper we consider the motion of a rigid body in a boundless volume of liquid. The body is set in motion by redistribution of internal masses. The mathematical model employs the equations of motion for the rigid body coupled with the hydrodynamic Navier–Stokes equations. The problem is mostly dealt with numerically. Simulations have revealed that the body’s trajectory is strongly governed by viscous effects.
    Keywords: self-propulsion, Navier–Stokes equations, viscous vortical motion, numerical methods
    Citation: Ramodanov S. M., Tenenev V. A.,  Motion of a body with variable distribution of mass in a boundless viscous liquid, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 3, pp.  635-647
    Borisov A. V., Mamaev I. S., Ramodanov S. M.
    Dynamic advection
    2010, Vol. 6, No. 3, pp.  521-530
    A new concept of dynamic advection is introduced. The model of dynamic advection deals with the motion of massive particles in a 2D flow of an ideal incompressible liquid. Unlike the standard advection problem, which is widely treated in the modern literature, our equations of motion account not only for particles’ kinematics, governed by the Euler equations, but also for their dynamics (which is obviously neglected if the mass of particles is taken to be zero). A few simple model problems are considered.
    Keywords: advection, mixing, point vortex, coarse-grained impurities, bifurcation complex
    Citation: Borisov A. V., Mamaev I. S., Ramodanov S. M.,  Dynamic advection, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 3, pp.  521-530
    Borisov A. V., Mamaev I. S., Ramodanov S. M.
    The paper is concerned with a class of problems which involves the dynamical interaction of a rigid body with point vortices on the surface of a two-dimensional sphere. The general approach to the 2D hydrodynamics is further developed. The problem of motion of a dynamically symmetric circular body interacting with a single vortex is shown to be integrable. Mass vortices on $S^2$ are introduced and the related issues (such as equations of motion, integrability, partial solutions, etc.) are discussed. This paper is a natural progression of the author’s previous research on interaction of rigid bodies and point vortices in a plane.
    Keywords: hydrodynamics on a sphere, coupled body-vortex system, mass vortex, equations of motion, integrability
    Citation: Borisov A. V., Mamaev I. S., Ramodanov S. M.,  Coupled motion of a rigid body and point vortices on a sphere, Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 3, pp.  319-343
    Borisov A. V., Mamaev I. S., Ramodanov S. M.
    The paper develops further the algebraic-reduction method for $SO(4)$-symmetric systems on the three-dimensional sphere. Canonical variables for the reduced system are constructed both on two-dimensional and three-dimensional spheres. The method is illustrated by applying it to the two-body problem on a sphere (the bodies are assumed to interact with a potential that depends only on the geodesic distance between them) and the three-vortex problem on a two-dimensional sphere.
    Keywords: Poisson structure, Lie algebra, subalgebra, Andoyer variables
    Citation: Borisov A. V., Mamaev I. S., Ramodanov S. M.,  Algebraic reduction of systems on two- and three-dimensional spheres, Rus. J. Nonlin. Dyn., 2008, Vol. 4, No. 4, pp.  407-416
    Borisov A. V., Gazizullina L., Ramodanov S. M.
    Citation: Borisov A. V., Gazizullina L., Ramodanov S. M.,  E. Zermelo Habilitationsschrift on the vortex hydrodynamics on a sphere, Rus. J. Nonlin. Dyn., 2008, Vol. 4, No. 4, pp.  497-513
    Borisov A. V., Mamaev I. S., Ramodanov S. M.
    The paper deals with the derivation of the equations of motion for two spheres in an unbounded volume of ideal and incompressible fluid in 3D Euclidean space. Reduction of order, based on the use of new variables that form a Lie algebra, is offered. A trivial case of integrability is indicated.
    Keywords: motion of two spheres, ideal fluid, reduction, integrability
    Citation: Borisov A. V., Mamaev I. S., Ramodanov S. M.,  Motion of two spheres in ideal fluid. I. Equations o motions in the Euclidean space. First integrals and reduction, Rus. J. Nonlin. Dyn., 2007, Vol. 3, No. 4, pp.  411-422
    Ramodanov S. M.
    On the motion of two mass vortices in perfect fluid
    2006, Vol. 2, No. 4, pp.  435-443
    The system of two interacting dynamically 2D rigid circular cylinders in an infinite volume of perfect fluid was considered in [4,5], while the pioneering contribution is due to Hicks [1,2]. An allied problem, the motion of two spheres in perfect fluid, was studied by Stokes, Hicks, Carl and Vilhelm Bjerknes, Kirhhoff, and Joukowski (the references can be found in [3] and [7]). Assuming the circulations around the cylinders to be constant and making the radii of the cylinders infinitely small result in new 2D hydrodynamic objects called mass vortices [5]. The equations of motion for mass vortices expand upon the classical Kirhhoff equations governing the motion of ordinary point vortices. In this paper the motion of two mass vortices is examined in greater detail (some results have been obtained already in [5]). A reduction of order is performed; using the Poincare surfaсe-of-section technique the system is shown to be generally non-integrable. Some integrable cases are indicated. In conclusion the motion of a single mass vortex and the motion of cylinder in a half plane are briefly investigated.
    Keywords: motion of circular cylinders, mass vortices, reduction of order, vortices in a domain
    Citation: Ramodanov S. M.,  On the motion of two mass vortices in perfect fluid, Rus. J. Nonlin. Dyn., 2006, Vol. 2, No. 4, pp.  435-443
    Borisov A. V., Mamaev I. S., Ramodanov S. M.
    In this paper we consider the system of two 2D rigid circular cylinders immersed in an unbounded volume of inviscid perfect fluid. The circulations around the cylinders are assumed to be equal in magnitude and opposite in sign. Special cases of this system (the cylinders move along the line through their centers and the circulation around each cylinder is zero) are considered. A similar system of two interacting spheres was originally considered in classical works of Carl and Vilhelm Bjerknes, G. Lamb and N.E. Joukowski.

    By making the radii of the cylinders infinitesimally small, we have obtained a new mechanical system which consists of two regular point vortices but with non-zero masses. The study of this system can be reduced to the study of the motion of a particle subject to potential and gyroscopic forces. A new integrable case is found. The Hamiltonian equations of motion for this system have been generalized to the case of an arbitrary number of mass vortices with arbitrary intensities. Some first integrals have been obtained. These equations expand upon the classical Kirchhoff equations of motion for n point vortices.
    Keywords: perfect fluid, circulation, rigid body, qualitative analysis
    Citation: Borisov A. V., Mamaev I. S., Ramodanov S. M.,  Interaction of two circular cylinders in a perfect fluid, Rus. J. Nonlin. Dyn., 2005, Vol. 1, No. 1, pp.  3-21

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