Linear oscillations of sessile bubbles

The resonant behavior of bubbles in contact with a partially-wetting substrate are important for heat transfer nucleate boiling and solder joint quality in ultrasonic-assisted soldering. A theoretical analysis was carried out to investigate the natural oscillations of a compressible sessile bubble on a planar solid support in an ambient liquid. We consider small disturbances to the spherical-cap interface whose three phase contact line is either (i) pinned or (ii) moves freely with a fixed contact angle. The governing equations for the inviscid incompressible ambient liquid are reduced by a normal-mode expansion to a functional eigenvalue problem on linear operators using a boundary integral approach. We use inverse operators to construct Green’s function for these two contact line conditions, and the Rayleigh-Ritz method to find the characteristic oscillation frequencies and related modal shapes, which are characterized by the mode number pair (l,m). We show how the characteristic oscillation frequencies, modal shape and the flow field depend on the contact angle (α) and dimensionless equilibrium bubble pressure (Π).  Instabilities of the breathing mode (0,0) and the Noether mode (1,1) will be highlighted.