The elastic Rayleigh drop

More than a century ago, Lord Rayleigh showed a spherical drop of inviscid liquid held by surface tension will oscillate with characteristic frequency and mode shape. This dispersion relationship has seen widespread use in multiple industries and applications. Bioprinting technologies rely on the formation of soft gel drops for printing tissue scaffolds and the dynamics of these drops can affect the process. Here we develop a model to compute the natural frequencies of a spherical drop with finite shear modulus and solid surface tension. We solve the time-dependent Navier equations of linear elasticity incorporating the solid surface tension. The motions are decomposed into i) shape oscillations and ii) rotational modes. Rotational modes are uncoupled and not affected by capillarity, whereas the frequency of shape oscillations depend upon the elastocapillary number and compressibility. For a compressible gel, there exists an infinity of radial modes for a fixed polar wavenumber and we show how these are affected by surface tension. For an incompressible gel, our results recover the Rayleigh frequency when the shear modulus is zero. Our predictions compare favorably to experiments on levitated gel drops.