The role of a deep wavy wall on the natural frequencies of a fluid interface: theory and experiments *

The natural frequency of an active fluid whose free surface is exposed to a passive fluid is known to dominantly depend on the density difference of the fluids, the surface tension and the wave number of the disturbances that cause flow oscillations. In addition to these factors, patterned walls are expected to influence the natural frequency. To analyse the role of a patterned substrate with arbitrarily deep indentations a reduced order model is developed in conjunction with a Weighted Residual Integral Boundary Layer (WRIBL) technique. From the analysis, it is learned that the natural frequency generally but not always decreases as the amplitude of the wall pattern increases from a flat wall and that there are exceptional zones of disturbances. These exceptional zones appear when the wave number of the disturbance is twice that of the wall pattern provided the wall has a trigonometric cosine wave pattern and not a trigonometric sine wave. These results are substantiated both by a perturbation series analysis about the flat wall case. The numerical calculations are validated by detailed physical experiments. As a consequence in the shift of the natural frequency, the physical experiments also show a shift in the instability regions when fluid is subject to Faraday excitation.