ISSN 1063-7710, Acoustical Physics, 2013, Vol. 59, No. 6, pp. 640–643. © Pleiades Publishing, Ltd., 2013. Original Russian Text © P.V. Lebedev-Stepanov, O.V. Rudenko, 2013, published in Akusticheskii Zhurnal, 2013, Vol. 59, No. 6, pp. 693–697. 640 INTRODUCTION Slow vortex flows in a layer of viscous fluid are investigated. The flows are generated by capillary waves on a free surface of the layer. These waves are initiated by substrate vibrations. The effect is similar to the acoustic flows that arise in the process of absorp- tion of ultrasonic waves [1]. This phenomenon has been well studied and is finding application in a num- ber of technologies. In particular, review [2] is devoted to biomedical applications of acoustic flows. Flows inside thin layers and drops of fluid are used in nanotechnologies to form ordered structures of nanoparticles [3, 4]. For this purpose, the flow in a layer excited by surface waves traveling along the solid–fluid interface has been calculated in [5]. DESCRIPTION OF THE SYSTEM Let a thin layer of fluid –H < z < 0 be considered, which is located on a horizontal plate. The upper sur- face of the plate (plane z = –H) is the “bottom.” Car- tesian coordinates are introduced as shown in Fig. 1. Both motions, namely, a wave and a slow flow, can be described by a system of equations of viscous incompressible fluid dynamics: (1) Here is the fluid velocity, p is pressure, ρ is density, ν is the kinematic viscosity, and is the force exerted related to layer vibration. Equations (1) are written in ( ) () div , 0. p u u u u gt u t ∇ ∂ + ∇ =- + νΔ + = ∂ ρ       u  g  the noninertial coordinate system, in which the vibrat- ing layer is at rest. MODEL In Eqs. (1), we separate the fast oscillatory motion and slow flow, for which we set [1] (2) The prime denotes the oscillatory components. For periodic oscillations, we consider the period-averaged values to be zero: (3) By substituting (2) into system of equations (1) and by averaging, we obtain the system of equations for the slow flow: ' ' , . u u U p p P = + = +    ' 0. u p g = = =  Acoustic Flows in a Fluid Layer on a Vibrating Substrate P. V. Lebedev-Stepanov a, g and O. V. Rudenko b ,c, d, e, f a Photochemistry Center, Russian Academy of Sciences, ul. Novatorov 7a, bldg. 1, Moscow, 119421 Russia b Faculty of Physics, Moscow State University, Moscow, 119991 Russia c Prokhorov General Physics Institute, Russian Academy of Sciences, ul. Vavilova 38, Moscow, 119991 Russia d Schmidt Institute of Physics of the Earth, Russian Academy of Sciences, ul. Bol’shaya Gruzinskaya 10, Moscow, 123995 Russia e School of Engineering, Blekinge Institute of Technology, Karlskrona, 37179 Sweden f Nizhni Novgorod State University, pr. Gagarina 23, Nizhni Novgorod, 603950 Russia g National Research Nuclear University MEPhI, Kashirskoe sh. 31, Moscow, 115409 Russia e-mail: petrls@mail.ru Received May 23, 2013 Abstract—The field of radiation forces in a fluid layer on a solid substrate is calculated. This field is formed during propagation of surface capillary wave along a free surface. The wave is excited by substrate vibrations as a result of instability development. The structure of acoustic flows is studied. Their effect on small-size par- ticles and the possibilities of generating ordered structures from these particles are discussed. Keywords: capillary waves, radiation force, acoustic flows, vibrations, controlled self-assembly DOI: 10.1134/S1063771013060134 CLASSICAL PROBLEMS OF LINEAR ACOUSTICS AND WAVE THEORY x –H 0 z L Fig. 1. Layer of fluid and Cartesian coordinate system fixed to it.