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Axisymmetric dynamics of a bubble near a plane wall

Published online by Cambridge University Press:  02 November 2009

C. W. M. VAN DER GELD  and
J. G. M. KUERTEN
C. W. M. VAN DER GELD*
Affiliation:
Department of Mechanical Engineering, Thermo Fluids Engineering Division, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
J. G. M. KUERTEN
Affiliation:
Department of Mechanical Engineering, Thermo Fluids Engineering Division, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
*
†Email address for correspondence: [email protected]
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Abstract

Explicit expressions for the added mass tensor of a bubble in strongly nonlinear deformation and motion near a plane wall are presented. Time evolutions and interconnections of added mass components are derived analytically and analysed. Interface dynamics have been predicted with two methods, assuming that the flow is irrotational, that the fluid is perfect and with the neglect of gravity. The assumptions that gravity and viscosity are negligible are verified by investigating their effects and by quantifying their impact in some cases of strong deformation, and criteria are presented to specify the conditions of their validity. The two methods are an analytical one and the boundary element method, and good agreement is found. It is explained why a strongly deforming bubble is decelerated. The classical Rayleigh–Plesset equation is extended with terms to account for arbitrary, axisymmetric deformation and to account for the proximity of a wall. An expression for the corresponding cycle frequency that is valid in the vicinity of the wall is derived. An equation similar to the Rayleigh–Plesset equation is presented for the most important anisotropic deformation mode. Well-known expressions for the angular frequencies of some periodic solutions without a wall follow easily from the equations presented. A periodically deforming bubble without initial velocity of the centroid and without a dominating isotropic deformation component is eventually always driven towards the wall. A simplified equation of motion of the centre of a deforming bubble is presented. If desired, full deformation computations can be speeded up by selecting an artificially low value of the polytropic constant Cp/Cv.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 640 , 10 December 2009 , pp. 265 - 303
Copyright
Copyright © Cambridge University Press 2009

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