Mathematical Analysis of an Approximation Model for a Spherical Cloud of Cavitation Bubbles
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Mathematical Analysis of an Approximation Model for a Spherical Cloud of Cavitation Bubbles Rostislav Vodák1 · Pavel Ženˇcák1
Received: 17 June 2015 / Accepted: 1 November 2017 / Published online: 7 November 2017 © Springer Science+Business Media B.V., part of Springer Nature 2017
Abstract In the paper we introduce a new approximation scheme for modelling a spherical cloud of cavitation bubbles based upon a model developed in (Wang and Brennen in J. Fluids Eng. 121(4):872–880, 1999) which consists of fully nonlinear continuum mixture equations coupled with the Rayleigh-Plesset equation for dynamics of the bubbles. We prove existence of a unique, local-in-time solution to the equations using the Banach fixed-point theorem which also provides us with the convergent scheme for a numerical simulation of the solution. We further demonstrate acquired numerical results. Keywords Cavitation · Rayleigh-Plesset equation · Approximation scheme · Mathematical analysis · Numerical solution Mathematics Subject Classification 76E30 · 76M25 · 34C99
1 Introduction Cavitation is responsible for reduced performance, efficiency, lifetime, reliability and safety of many hydraulic machines and devices. Much effort was devoted to experimental studies which demonstrate intensive noise and damage potential associated with the collapse of a cavitating cloud of bubbles [3, 4, 10, 12, 17, 21, 24, 25] or which try to assess the cavitation erosion risk [18]. Attention was also paid to the coherent collapse of cavitation bubbles clouds which results in greater material damage [27] and greater noise generation [22]. Recent rapid progress of computer science has encouraged many authors to examine unsteady cavitation phenomena using continuum mixture models and numerical tools. The linear dynamics of clouds of bubbles was extensively studied and we refer the reader to Supported by the Grant No. 13-23550S of Czech Science Foundation (GACR).
B R. Vodák
[email protected] P. Ženˇcák [email protected]
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Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacky University, tˇr. 17 listopadu 12, 772 00 Olomouc, Czech Republic
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R. Vodák, P. Ženˇcák
[29] for references. Cavitation is, however, a very unstable and nonlinear phenomenon and thus the nonlinear terms in corresponding equations must be retained. The first attempt to understand its highly nonlinear effects can be found in [14, 15] and [16] but the nonlinear and chaotic behavior of periodically driven bubble clouds was examined earlier in [26] and [6]. Dynamics driven interaction between the bubbles in a cavitation cluster was studied in [5] via applying methods of chaos physics. Collective oscillations of a bubble cloud in an acoustic field are theoretically analyzed with concepts and techniques of condensed matter physics in [31]. In [1] we can find several finite difference schemes applied to the RayleighPlesset equation. In papers [23] and [29] numerical studies of the nonlinear dynamics of finite clouds of cavitation bubbles was perfor
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