Detailed finer features in spectra of interfacial waves for characterization of a bubble-laden drop

Abstract

This article describes unexplored details of the intriguing spectral manifestation of the small-amplitude waves at the surfaces of a bubble-laden drop. Its natural frequencies of interfacial pulsation reveal a non-trivial variation with the position of the cavity inside the liquid. This configurational dependence of spectra is calculated for arbitrary location of the void by using a novel approach under low capillary number and low Bond number limits. The analysis is based on expansion in two sets of basis functions where their mutual transformations are utilized to enforce interfacial boundary conditions. The obtained results quantify a few important features which have both scientific and technological significance. For a concentric geometry, the inherent azimuthal degeneracy makes the frequencies for a number of vibrational modes exactly the same. For an eccentric position of the bubble, however, this degeneracy disappears, creating small deviations in the spectral values corresponding to different azimuthal modes. Such behaviour is akin to fine-structure split in an atomic system, where different quantum numbers ensure small deviation in energy levels of the states. The formulated mathematical procedure can determine the individual frequency values for the interfacial oscillation even if these are grouped closely together in bands. The paper shows how the number of fine structures inside a band and their specific values can be exploited to predict the size and position of the cavity in an opaque drop without any direct visualization of its interior.