Boundary integral equations as applied to an oscillating bubble near a fluid-fluid interface E. Klaseboer, B. C. Khoo Abstract A new method is presented to describe the behaviour of an oscillating bubble near a fluid-fluid interface. Such a situation can be found for example in underwater explosions (near muddy bottoms) or in bubbles generated near two (biological) fluids separated by a membrane. The Laplace equation is assumed to be valid in both fluids. The fluids can have different density ratios. A relationship between the two velocity potentials just above and below the fluid-fluid interface can be used to update the co-ordinates of the new interface at the next time step. The boundary integral method is then used for both fluids. With the resulting equations the normal velocities on the interface and the bubble are obtained. Depending on initial distances of the bubble from the fluid-fluid interface and density ratios, the bubbles can develop jets towards or away from this interface. Gravity can be important for bubbles with larger dimensions. Keywords Oscillating bubbles, Fluid-fluid interface, Boundary integral method, Jet impact, Oscillation time, Laplace equation 1 Introduction The study of oscillating bubbles has a long history. Lord Rayleigh (1917) was perhaps the first to investigate cavitation bubbles in the early 20th century and the Rayleigh-Plesset equation is still being used to investigate one-dimensional spherical bubble behaviour today. Since then much work has been done in this area: bubble dynamics can be found in fields ranging from cavitation on ship propellers and pumps; Plesset and Chapman (1971), underwater explosions; Cole (1948), and even in the recent claim of cold fusion in sonoluminiscence; Taleyarkhan et al. (2002). Two stages can be distinguished during the bubble oscillation process (assuming the bubble is spher- ically symmetric at the start). First of all, the expansion phase where the bubble still remains more or less spherical, this is to be followed by the collapse phase. If the collapse occurs near an interface, the bubble almost always seems to develop a high-speed jet. The jet can be directed towards this interface, if the interface is a solid boundary, Zhang et al. (2001). On the other hand, if the bubble is located near a free interface, a jet directed away from the free interface can be observed, Wang et al. (1996A). A bubble in a gravity field can also exhibit a jet behaviour in the collapse phase with the jet directed upwards in opposite direction to the gravity vector (pointing downwards). The combination of gravity and the presence of a surface can give rise to several effects. Depending on the parameters, the influence of the surface and gravity can eliminate each other resulting in no jet, or reinforce each other to make the jet even stronger. Fully three-dimensional models exist to account for more complex jet formation and toroidal shape configuration for non axisymmetrical cases (for example an oscillating bubble near a vertical solid wall with gravity pointing downwards), Zhang et al. (2001). Among the various numerical methods, for example Chan et al. (2000), one way to simulate and represent the bubble is via the boundary integral method (BIM); Wang (1998), Wang et al. (1996 A and B), Zhang et al. (2001). BIM has the distinct advantage of reducing the dimension of the problem by one, thus may lead to potential savings in computer memory and time (even though the system of equations will lead to a full matrix). For a review on the use of boundary integral methods in multi-component fluids see Hou et al. (2001), where much emphasis is put on free surface flows and inclusion of surface tension. Several studies on the dynamics of bubbles near solid boundaries; Chan et al. (2000), Wang (1998), near free surfaces: Blake and Gibson (1981), Robinson et al. (2001), Wang et al. (1996A and B), Zhang et al. (2001) or even in confined spaces; Yuan et al. (1999), can be found. To the authors best knowledge, no research seems to have been carried out for the behaviour of a bubble near a fluid-fluid interface of (imposed) different properties. Such a study is critical to evaluate the dynamic characteristics of bubbles near a biological surface for example (modelled as a thin mem- brane separating another fluid medium of vastly dissimilar property). This paper deals with the behaviour of an oscillating bubble near such a fluid-fluid interface, with emphasis on the oscillation times, jet directions and jet velocities. Computational Mechanics 33 (2004) 129–138 Ó Springer-Verlag 2003 DOI 10.1007/s00466-003-0508-2 Received: 18 February 2003 / Accepted: 9 September 2003 Published online: 20 November 2003 E. Klaseboer Institute of High Performance Computing, 1 Science Park Road #01–01 The Capricorn, Singapore 117528 B. C. Khoo (&) Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260 e-mail: mpekbc@nus.edu.sg [or] Singapore – MIT Alliance, 4 Engineering Drive 3, Singapore 117576 129