Oscillations of a soft viscoelastic drop

A soft viscoelastic drop has dynamics governed by the balance between surface tension, viscosity, and elasticity, with the material rheology often being frequency dependent. These properties are utilized in bioprinting technologies for tissue engineering and drop deposition processes for splash suppression. We study the free and forced oscillations of a soft viscoelastic drop deriving 1) the dispersion relationship for free oscillations, and 2) the frequency response for forced oscillations, of a soft material with arbitrary rheology. We illustrate the results for the classical cases of a Kelvin- Voigt and Maxwell model, which are relevant to soft gels and polymer fluids, respectively. We compute the complex frequencies, which are characterized by an oscillation frequency and decay rate, as they depend upon the dimensionless elastocapillary and Deborah numbers and map the boundary between regions of underdamped and overdamped motions. We show how the frequency response of the droplet due to forced oscillation changes with viscoelasticity and how it can potentially be used to measure the rheology.