Parametrically driven surface waves in viscoelastic liquids

Abstract

When a container of liquid is subject to vertical sinusoidal oscillation, the free surface becomes unstable at a critical driving acceleration and gives rise to standing waves. Here, we consider containers of finite depth, neglect the influence of lateral boundaries, and perform a linear stability analysis for viscoelastic liquids. Floquet theory is applied to transform the linearized governing equations into a recursion relation for the temporal modes of the free surface deformation. In the absence of external forcing, the recursion relation yields the dispersion equation for free surface waves on viscoelastic liquids. In the presence of external forcing, the recursion relation is studied both numerically and analytically for the case where the polymer stresses are described by a single-mode Maxwell model. When surface tension forces are sufficiently strong relative to elastic forces, the numerical results show that the standing waves respond subharmonically to the driving frequency. However, the instability threshold increases less rapidly with the driving frequency than that for a Newtonian liquid of the same zero-shear viscosity. When elastic forces become sufficiently strong relative to surface tension forces, the standing waves can respond harmonically within certain ranges of the driving frequency if the product of the driving frequency and the liquid relaxation time is not too large or too small. Dramatic changes are seen in the behavior of the neutral stability curves, dispersion relations, and instability thresholds. In the case where the viscous boundary layer thickness is much less than the disturbance wavelength, the viscoelastic recursion relation is simplified to yield a Mathieu equation which is nonlocal in time. The method of multiple scales is then applied to determine the instability threshold analytically. The results of this study indicate that the behavior of parametrically driven surface waves is very sensitive to the liquid relaxation time, and suggest that such waves may serve as a useful tool for the measurement of rheological properties.