Sessile-drop oscillations fill a symmetry-breaking periodic table

Oscillations of a sessile drop are of fundamental interest for the contact-line instabilities they can exhibit and of practical importance in a number of industrial applications. We consider the small oscillations of the inviscid sessile drop under a number of contact line conditions, including a contact-line modeled using a continuous contact-angle against speed relationship. The integro-differential equation, governing the motion of the interface, is formulated as a functional equation using inverse operators, which are parameterized by volume via the static contact angle of the drop base-state and by the mobility of the contact-line. In the symmetric limit, a hemispherical drop perturbed by a fixed contact-angle disturbance has characteristic oscillation frequencies, which are degenerate with respect to azimuthal wave-number much like the Bohr model of the atom is degenerate with respect to angular momentum quantum number. This degeneracy is broken by smoothly varying either i) the volume and/or ii) the contact line mobility. The analogy between the spectrum of these ``broken'' states and the filling order of the periodic table by energy levels both organizes and explains the hierarchy of frequencies.