Faraday waves in small cylinders and the sidewall non-ideality

This work is the result of a scientific inquiry into the current understanding of experimental single-mode Faraday waves, from the perspective of the linear stability theory. Given an electromechanical shaker capable of imposing vibrations of several centimeters at frequencies of up to 15 Hz, experiments were directed toward laterally "small" systems in which the cell modes are discretized and the excited wavelength was of the order of the lateral dimension. In this regime, the theoretically tractable boundary condition for the sidewalls is a stress free condition, which is a challenge to produce experimentally. In reality, the no-slip behavior of the fluid along the sidewalls and interfacial contact line effects such as capillary hysteresis introduce sidewall stresses. This marks the first attempt to match the single mode Faraday experiment to a linear theory that rigorously treats viscosity.;In these experiments it was found that the liquids FC70 and silicone oil, which, other than a slight meniscus, produced a flat interface that moved with little effort once the container was tilted. This behavior stands in contrast to that of a water-air or an immiscible silicone oil-water interface in a glass beaker, which suffer from pinning to the sidewall. Closer inspection of the FC70-oil interface at the sidewalls shows formation of a tiny oil film in a tilted cell where bulk FC70 had displaced bulk silicone oil. Wave decay experiments confirm the dominance of the bulk viscous contribution to the sidewall damping effects when measuring the overall system damping.;In interpreting the experiments, first the viscous linear stability theory of Kumar & Tuckerman[52] is presented, and modified with the stress free boundary condition to account for mode discretization. The linear theory is capable of predicting the threshold amplitude, above which the flat interface is unstable and deflection occurs. In horizontally infinite systems the well-known result is that the instability is subharmonically excited---with a frequency half that of the forcing frequency. The main implication of mode discretization is the continuum of available modes is discretized, and each available mode can be excited inside of its own frequency band. The corresponding threshold amplitudes for each band descend to a minimum amplitude near the natural frequency of the mode, and the points at which the thresholds of neighboring modes intersect are co-dimension 2 points, conditions where two modes are neutrally stable.;The theoretical concepts of the critical threshold and frequency bands at which modes appear are studied with the FC70 and silicone oil system. In a system oscillating below the instability threshold, tiny flow perturbations are seen due to emission of waves from the oscillating meniscus, an unavoidable non-ideality. Imposing vibrations above the threshold amplitude results in the gradual (or sudden) deflection and growth of the interface to some new state. Experimental thresholds are therefore marked by performing a series of trials at different amplitudes, and the lowest amplitude at which the instability is observed is marked as the threshold. Data sets are built by measuring the threshold for the experimental frequency band, and bounded with the co-dimension 2 points.;Two complete data sets of several subharmonic, harmonic, and superharmonic modes are presented for FC70 and 1.5 cSt silicone oil systems of different layer heights, and are compared to the predictions of the linear theory. While slightly higher than predicted thresholds are observed near the mode natural frequencies, the agreement is quite good as one moves toward the co-dimension 2 points. The deviation near the natural frequencies appears to be greater for modes with greater number of azimuthal nodes, suggesting an associated increase in wall damping. Considerable deviation is seen in the lower-than-predicted thresholds for the harmonic modes, most noticeably for the mode showing the same azimuthal uniformity as the sidewall meniscus, suggesting an interaction between the harmonic modes with the meniscus waves, which are also harmonic with the cell motion. Similar deviation is seen for the other harmonic modes, albeit less due to the spatial mismatch of the instability with the meniscus wave.;The power of the linear theory comes in being able to predict the mode of instability and the critical threshold at which it appears, whereas growth beyond the infinitesimal state a nonlinear theory is required. A nonlinear theory was not completed in this work and therefore the experiment was used primarily for comparison to the linear theory. Qualitative nonlinear observations, however, were numerous, and many are presented in the interest of both giving a broader sense of the experiment and insight into the parameter spaces a weakly nonlinear theory might show agreement.;The final result presented is for when the instability is excited with two frequencies, a novelty for single mode experiments. (Abstract shortened by UMI.).