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IssuesArchive of Issues2006-3pp.45-54

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L. E. Ukrainskii, "On the motion of rigid particles in wave fields," Mech. Solids. 41 (3), 45-54 (2006)
Year 2006 Volume 41 Number 3 Pages 45-54
Title On the motion of rigid particles in wave fields
Author(s) L. E. Ukrainskii (Moscow)
Abstract Transformation of a wave fluid flow into a unidirectional motion of solid particles suspended in the fluid is considered. Equations describing this type of the particle motion relative to the fluid are obtained in two limiting cases where the dominant force in the fluid-particle interaction process is accounted for by the added mass effect and where this force is accounted for by viscous drag obeying the Stokes law. For a number of particular wave fields, the characteristics of the particle motion are obtained, and the appearance of stable and unstable quasi-equilibrium positions of the particles is established. The stable positions are due to the particle localization effect and the unstable ones are due to the formation of advective motions.

Unidirectional motions of rigid particles suspended in a carrying medium, their localization or advective motion near the saddle equilibrium positions have been repeatedly observed experimentally [1-3]. Some particular cases of such motions were studied theoretically for ideal [4, 5] and viscous [6] carrying media. In this paper, an approach originally proposed by the author in [7, 8] is described. According to this approach, the cases where the transformation of the wave fluid motion into the unidirectional motion of suspended particles occurs can be systematically established by a standard procedure based on the asymptotic methods of nonlinear mechanics [9, 10]. In the current study, this approach is applied to the simplest types of the carrying medium flow for which the particle motion can be described using the assumption of quasi-incompressibility. Using this approach, not only the mechanisms of transformation of the wave motion of the medium into the unidirectional particle motion obtained earlier by other authors could be re-established, but also some new mechanisms of such transformations have been identified, which can find practical applications. Examples have shown that in the wave fields, three particular modes of particle motion can occur: (i) unidirectional motion with respect to the carrying medium, (ii) particle localization related to the occurrence of stable quasi-equilibrium particle positions in the wave field, and (iii) advection caused by the presence of saddle equilibrium positions (when the phase space dimension exceeds 2 and doubly asymptotic Poincaré's solutions [11] of the particle motion equations are not reduced to a separatrix).
References
1.  L. Bergman, Ultrasound and its Applications in Science and Technology [Russian translation], Izd-vo Inostr. Lit-ry, Moscow, 1957.
2.  L. K. Zarembo and V. A. Krasil'nikov, Introduction to Nonlinear Acoustics [in Russian], Nauka, Moscow, 1966.
3.  L. D. Rosenberg (Editor), Physics and Engineering of Powerful Ultrasound. Volume 3. Physical Foundations of Ultrasonic Technology [Russian translation], Mir, Moscow, 1970.
4.  L. P. Gor'kov, "On the forces acting on a small particle in the acoustic field of an ideal fluid," Doklady AN SSSR, Vol. 140, No. 1, pp. 88-91, 1961.
5.  L. V. King, On the acoustic radiation pressure on sphere, Proc. Roy Soc. London, Ser. A, Vol. 147, No. 861, pp. 212-240, 1934.
6.  S. S. Dukhin, "Drift theory of an aerosol particle in a standing sound wave," Kolloidnyi Zhurnal, Vol. 22, No. 1, pp. 128-130, 1960.
7.  R. F. Ganiev and L. E. Ukrainskii, "On the motion of rigid particles suspended in an oscillating compresible medium," Prikl. Mekhanika, Vol. 11, No. 2, pp. 3-14, 1975.
8.  R. F. Ganiev and L. E. Ukrainskii, Dynamics of Particles Affected by Vibration, Kiev, Naukova Dumka, 1975.
9.  V. F. Zhuravlev and D. M. Klimov, Applied Methods in the Theory of Oscillations [in Russian], Nauka, Moscow, 1988.
10.  N. N. Bogolyubov and Yu. A. Mitropol'skii, Asymptotic Methods in the Theory of Nonlinear Oscillations [in Russian], Fizmatgiz, Moscow, 1963.
11.  A. Poincaré, Selected Works [Russian Translation], Volume 2, Nauka, Moscow, 1972.
12.  R. I. Nigmatulin, Dynamics of Multi-phase Media. Parts 1 and 2 [in Russian], Nauka, Moscow, 1987.
13.  Kh. A. Rakhmatulin, "Foundations of the dynamics of interpenetrating motions of compressible media," PMM [Applied Mathematics and Mechanics], Vol. 20, No. 1, pp. 184-195, 1956.
14.  R. Beyer, Nonlinear Acoustics. Physical Acoustics [Russian translation], Vol. 2, Part 6, pp. 266-301, Moscow, Mir, 1969.
15.  R. I. Nigmatulin, "Small-scale flows and surface effects in multiphase fluid dynamics," PMM [Applied Mathematics and Mechanics], Vol. 35, No. 3, pp. 453-463, 1971.
16.  H. Aref, "Stirring by chaotic advection," J. Fluid Mech., Vol. 143, pp. 1-21, 1984.
17.  A. V. Il'in, V. P. Kuznetsov, B. G. Novitskii, and V. M. Fridman, "Mechanism of flotation action of pulsating gas bubbles," Akust. Zhurnal, Vol. 18, No. 4, pp. 537-545, 1972.
Received 13 May 2005
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