Fluid Kinetic Energy Asymptotic Expansion for Two Variable Radii Moving Spherical Bubbles at Small Separation Distance
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d Kinetic Energy Asymptotic Expansion for Two Variable Radii Moving Spherical Bubbles at Small Separation Distance S. V. Sanduleanua,b,* a
Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow, 119526 Russia b Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region, 141701 Russia *e-mail: [email protected] Received September 30, 2019; revised March 2, 2020; accepted March 16, 2020
Abstract—Two spherical bubbles with changing radii are considered to be moving in ideal fluid along their center-line. The exact expression for the fluid kinetic energy is obtained. The Stokes stream function is expanded in Gegenbauer polynomials in bispherical coordinates. This expansion is used to obtain the exact series for the fluid kinetic energy quadratic form coefficients. The new series are confirmed to be correct by comparison with the known ones. The main advantage of the new kinetic energy form is the possibility to obtain asymptotic expansions at small separation distance between the bubbles. These expansions are obtained and their convergence is analyzed. The results of this work can be used to describe the bubbles approach before the contact and their coalescence in acoustic field. Keywords: bubble interaction, Bjerknes force, Stokes stream function, fluid kinetic energy, axial symmetry, asymptotic expansion DOI: 10.1134/S0015462820070083
INTRODUCTION The problem of spherical gas bubbles interaction in a pulsating pressure field is the research subject of a large number of both theoretical and experimental studies, starting with Bjerknes’ studies in the 19th century [1] and ending with the recent studies [2–5]. Bjerknes established that the interaction force between pulsating spheres located at large distances is inversely proportional to the square of the distance between them. This dependence was confirmed experimentally [6, 7]. The dynamics of spheres of variable radii at a large distance was studied analytically in [8–10]. The refinement of Bjerknes’ results is related to obtaining an expansion for the hydrodynamic interaction force in inverse powers of the distance between the spheres’ centers. The kinetic energy was found with accuracy up to terms of the order of r–3 [11, 12] and r–4 [13], and the solution itself was accurate up to r–5 [14] and to r–6 [3]. However, it was theoretically [3, 15] and experimentally [4, 16–19] shown that, when approaching the contact, this dependence is not applicable and should be found from the solution of the problem of two pulsating spheres interaction in the exact formulation. The problem of the gas bubbles in the wave’s acoustic field is conveniently studied by the method of generalized Lagrange coordinates. The main component of the Lagrange function is the kinetic energy. For spherical bubbles, the problem of calculating the kinetic energy as a function of the sphere’s radii, the distance between the sphere’s centers, and the rates of change of the radii and centers arises. Two of the methods used to construct the exact soluti
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