Damped interfacial oscillation of a particle-embedded viscous drop

This article describes how a drop with an embedded particle exhibits interfacial waves with transient decay due to the interplay between capillary and viscous effects. To reveal the damped oscillation of the system properly, the deformation and pressure fields inside the domain are described in terms of complete sets of basis functions. Such representation leads to a matrix formulation which enforces no-slip condition at the solid-liquid interface and ensures correct discontinuity in normal stress due to surface tension at the drop periphery. The resulting characteristic equation involving the natural frequencies and the decay constants is solved numerically to determine these quantities. The matrix expression implies a block-diagonalized structure with two uncoupled blocks corresponding to two distinctly different dynamics. The first of these is related to pure rotational velocities on spherical surfaces which monotonically decay in time without any fluctuation in the absence of any peripheral deformation. By contrast, the second block is associated with the undulation in shape. Due to the restoring features of surface tension, the latter can exhibit underdamped oscillatory modes, if the capillary number Ca is below a critical value. However, even these waves would become overdamped if the critical number is exceeded. These values of Ca for a few most relevant modes are plotted in this paper as functions of particle-to-drop size ratio. Also, the natural frequencies for the underdamped cases as well as the damping constants for all considered modes are presented for different size ratios and capillary numbers. The findings are verified by matching the computed results to a novel boundary layer theory under low capillary number limit. Under the limiting condition, both sets of independent calculations yield the same decay constants and natural frequencies providing mutual validations.