Nonlinear oscillations and collapse of elongated bubbles subject to weak viscous effects: Effect of internal overpressure

Abstract

The details of nonlinear oscillations and collapse of elongated bubbles, subject to large internal overpressure, are studied by a boundary integral method. Weak viscous effects on the liquid side are accounted for by integrating the equations of motion across the boundary layer that is formed adjacent to the interface. For relatively large bubbles with initial radius R-0 on the order of millimeters, P-St=P-St(')/(2 sigma/R-0)similar to 300 and Oh=mu/(sigma R-0 rho)(1/2)similar to 200, and an almost spherical initial shape, S similar to 1, Rayleigh-Taylor instability prevails and the bubble breaks up as a result of growth of higher modes and the development of regions of very small radius of curvature; sigma, rho, mu, and P-St(') denote the surface tension, density, viscosity, and dimensional static pressure in the host liquid while S is the ratio between the length of the minor semiaxis of the bubble, taken as an axisymmetric ellipsoid, and its equivalent radius R-0. For finite initial elongations, 0.5 <= S < 1, the bubble collapses either via two jets that counterpropagate along the axis of symmetry and eventually coalesce at the equatorial plane, or in the form of a sink flow approaching the center of the bubble along the equatorial plane. This pattern persists for the above range of initial elongations examined and large internal overpressure amplitudes, epsilon(B)>= 1, irrespective of Oh. It is largely due to the phase in the growth of the second Legendre mode during the after-bounce of the oscillating bubble, during which it acquires large enough positive accelerations for collapse to take place. For smaller bubbles with initial radius on the order of micrometers, P-St similar to 4 and Oh similar to 20, and small initial elongations, 0.75 < S <= 1, viscosity counteracts P-2 growth and subsequent jet motion, thus giving rise to a critical value of Oh(-1) below which the bubble eventually returns to its equilibrium spherical shape, whereas above it collapse via jet impact or sink flow is obtained. For moderate elongations, 0.5 <= S <= 0.75, and large overpressures, epsilon(B)>= 0.2, jet propagation and impact along the axis of symmetry prevails irrespective of Oh. For very large elongations, S < 0.5, and above a certain threshold value of Oh the counterpropagating jets pinch the contracting bubble sidewalls in an off-centered fashion. (c) 2007 American Institute of Physics.