Nonlinear effects in viscoelastic drop shape oscillations

A study of axisymmetric shape oscillations of viscoelastic drops in a vacuum is conducted, using the method of weakly nonlinear analysis. The motivation is the relevance of the shape oscillations for transport processes across the drop surface, as well as fundamental interest. The study is performed for, but not limited to, the two-lobed mode of initial drop deformation. The Oldroyd-B model is used for characterizing the liquid rheological behavior. The method applied yields a set of governing equations, as well as boundary and initial conditions, for different orders of approximation. In the present paper, the equations and solutions up to second order are presented, together with the characteristic equation for the viscoelastic drop. The characteristic equation has an infinite number of roots, which determine the time dependency of the oscillations. Solutions of the characteristic equation are validated against experiments on acoustically levitated individual viscoelastic aqueous polymer solution drops. Experimental data consist of decay rate and oscillation frequency of free damped drop shape oscillations. With these data, solutions of the characteristic equation dominating the oscillations are identified. The theoretical analysis reveals nonlinear effects, such as the excess time in the prolate shape and frequency change for varying initial deformation amplitude. The influences of elasticity, measured by the stress relaxation and deformation retardation time scales, are quantified, and the effects are compared to the Newtonian case in the moderate-amplitude regime.