On a toroidal method to solve the sessile drop oscillation problem

Abstract

The natural oscillation of a drop is a classical fluid mechanics problem. Analytical expressions for the simple case of free, spherical drops were obtained by Rayleigh, Lamb, Chandrasekhar and others using spherical coordinate system. In recent times, the focus on this problem has shifted towards a sessile drop supported on a flat substrate, as evident through some recent works. The majority of these are computational in nature. In this talk, I will present an alternative new mathematical framework, the toroidal coordinate system, to solve this long-standing problem analytically for small drops (Bond number << 1) with pinned contact lines. I start with the governing hydrodynamic equations and boundary conditions, write them in terms of the toroidal coordinate system and then obtain solutions by reducing them to an eigenmode problem. Resonant frequencies are identified for zonal, sectoral and tesseral vibration modes and compared with results presented in the literature and by other models. The impact of viscous dissipation in the bulk liquid, at the contact line, and contact line mobility is discussed qualitatively. I conclude with a discussion of the importance of conformal mapping for solving axisymmetric physical problems with complicated geometries.