Interfacial wave dynamics of a multiphase-drop system

This dissertation presents a number of mathematical formulations to investigate the interfacial wave dynamics of a bubble-laden or a particulate drop. For this pur- pose, the interfacial pulsations are described by revealing the natural frequency and decay constant spectra for the most important modes in the system. The results are obtained by constructing a solution technique based on basis function expansion of the fields governed by flow equations. This leads to a matrix formulation which yields the necessary characteristic relation for spectral quantities like frequencies, damping constants and wave-lengths. The key findings are verified by devising a few independent perturbation theories under limiting conditions. Accordingly, the dissertation research mainly concentrates on three subprojects. Firstly, we investigate the natural interfacial wave dynamics of a bubble-laden sys- tem under inviscid condition when the bubble is at an arbitrary location inside the drop. For such configuration, we observe finer deviations in the frequency of different azimuthal modes appearing in a band of clustering values. The observed behavior is akin to fine-structure split in energy levels of atomic systems. In the second prob- lem, the effects of finite viscosity on interfacial wave pulsations are estimated for a concentrically situated bubble inside a liquid drop. For such systems, the damped vibration is characterized by calculating the decay constants for the waves along with the frequencies. The results illustrate the effect of viscous dissipation in terms of finite capillary number, where its critical values are calculated for specific modes dictat- ing transformation from under-damped oscillation to over-damped one. Thirdly and finally, the analysis predicts how the frequencies and decay constants modify under small but finite capillary number limit. This problem is solved by a boundary-layer based singular perturbation method where two cases with different internal species in a concentric position are taken into account. Consequently, contrasts in asymptotic trends can be illustrated between two domains laden with either a gaseous bubble or a solid particle. Importantly, these results are also utilized to validate our general matrix formulation described in the second problem. The mathematical formulation and the comparative studies differentiate the spec- tral manifestations for systems with different internal structures. This creates a po- tential for characterization of a multiphase drop by reverse estimation even if the interior is opaque. Such capability can be potentially useful in scientific explorations of inaccessible domains like cellular entities as well as in quality control and perfor- mance enhancement of various combustive and manufacturing processes.