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Viscous oscillations of a supported drop in an immiscible fluid

The small-amplitude free vibrations of a spherical drop immersed in an outer immiscible fluid and in partial contact with a solid support are considered when both fluids are assumed to be viscous and incompressible, while gravity effects are neglected. Using the normal-mode decomposition and the Green-function method, the solution of the linearized Navier-Stokes equations is reduced to the solution of an eigenvalue problem. The model includes as particular cases the viscous model for a free drop proposed by Prosperetti (1980) and the inviscid model for a supported drop previously proposed by the authors. The influence of the viscosity and of the support size are analysed both for the bubble and for the drop. At large values of the viscosity, the free drop shows significant differences with respect to the unsupported drop and a singular behaviour of the eigenvalue problem as the support size tends to zero. The comparison with the available experimental data shows a quite satisfactory agreement for both the vibration frequency and the damping constant, provided that the support angle is not too large.