TY - JOUR

T1 - Axisymmetric dynamics of a bubble near a plane wall

AU - Geld, van der, C.W.M.

AU - Kuerten, J.G.M.

PY - 2009

Y1 - 2009

N2 - Explicit expressions for the added mass tensor of a bubble in strongly nonlineardeformation and motion near a plane wall are presented. Time evolutions andinterconnections of added mass components are derived analytically and analyzed.Interface dynamics have been predicted with two methods, assuming that the flowis irrotational, that the fluid is perfect and with the neglect of gravity. The assumptionsthat gravity and viscosity are negligible are verified by investigatingtheir 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 methodsare an analytical one and the boundary element method, and good agreementis found. It is explained why a strongly deforming bubble is decelerated. The classicalRayleigh-Plesset equation is extended with terms to account for arbitrary,axi-symmetric deformation and to account for the proximity of a wall. An expressionfor the corresponding cycle frequency that is valid in the vicinity of thewall is derived. An equation similar to the Rayleigh-Plesset equation is presentedfor the most important anisotropic deformation mode. Well-known expressionsfor the angular frequencies of some periodic solutions without a wall follow easilyfrom the equations presented. A periodically deforming bubble without initial velocityof the centroid and without a dominating isotropic deformation componentis eventually always driven towards the wall. A simplified equation of motion ofthe center of a deforming bubble is presented. If desired, full deformation computationscan be speeded up by selecting an artificially low value of the polytropicconstant Cp/Cv.

AB - Explicit expressions for the added mass tensor of a bubble in strongly nonlineardeformation and motion near a plane wall are presented. Time evolutions andinterconnections of added mass components are derived analytically and analyzed.Interface dynamics have been predicted with two methods, assuming that the flowis irrotational, that the fluid is perfect and with the neglect of gravity. The assumptionsthat gravity and viscosity are negligible are verified by investigatingtheir 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 methodsare an analytical one and the boundary element method, and good agreementis found. It is explained why a strongly deforming bubble is decelerated. The classicalRayleigh-Plesset equation is extended with terms to account for arbitrary,axi-symmetric deformation and to account for the proximity of a wall. An expressionfor the corresponding cycle frequency that is valid in the vicinity of thewall is derived. An equation similar to the Rayleigh-Plesset equation is presentedfor the most important anisotropic deformation mode. Well-known expressionsfor the angular frequencies of some periodic solutions without a wall follow easilyfrom the equations presented. A periodically deforming bubble without initial velocityof the centroid and without a dominating isotropic deformation componentis eventually always driven towards the wall. A simplified equation of motion ofthe center of a deforming bubble is presented. If desired, full deformation computationscan be speeded up by selecting an artificially low value of the polytropicconstant Cp/Cv.

U2 - 10.1017/S0022112009991340

DO - 10.1017/S0022112009991340

M3 - Article

SN - 0022-1120

VL - 640

SP - 265

EP - 303

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

ER -