periodically inhomogeneous depth
S.V. Kiyashko, A.V. Nazarovsky
The capillary ripples, parametrically generated on a liquid surface, are one of the most convenient objects for research of processes of formation of steady structures and transition to chaos. This is due to possibility to visualize easily the fields of surface waves to change the controlling parameters: supercriticality, dissipation and the length of raised waves, the spatial size of the system.
At parametrical generation of surface capillary ripples in a liquid of constant depth the spatially homogeneous structures are usually formed. In the liquid of low viscosity the square and hexahedral lattice consisting of two or three pairs of standing waves is observed, and in the layer of small depth the destruction of a regular structure in the form of dispositions and domain walls of a liquid is possible. In the liquid of strong viscosity various (roller) structures and spiral waves can exist.
In this work formation and evolution of stationary spatial roller structures and spiral waves in a layer of a liquid of low viscosity and periodic irregularity of its depth is experimentally investigated. The liquid contained fresh water with addition of liquid soap for stabilization of a surface tension was used.
To create the depth irregularity dishes with a periodic profile of a bottom were used. Average depth of a liquid was less or an order of the wavelength.
It is known that the strongest influence of periodic irregularity affects under the conditions of the Bregg’s resonance: the length of a running wave twice exceeds length of a wave of bottom irregularity. Therefore generation of structures with wavelengths of an order of two periods of bottom irregularity was investigated in the experiment. Vertical fluctuations of the dishes were fulfilled with the vibration-testing machine TV250 by TIRAvib.
In the experiments the amplitude, the frequency of oscillations of dishes were varied in the range 20-115 Hz and the depth of the liquid had the values h ~ 0,85-6 mm. The standing waves arising on a surface of a liquid were observed in reflected light. The image of the patterns formed by capillary waves, registered by means of a digital videocamera, and then was numerically processed further. Two series of the experiments differed by spatial orientation of periodic irregularities of the bottom were performed.
In the first series of experiments line (roller) structures were investigated. A study of structures was fulfilled for different frequencies and amplitudes of an external force and constant average depth of the liquid.
It is found out that in the vicinity of a resonant condition, as well as in case of a homogeneous fluid two pairs of mutually or-thogonal standing waves (a square lattice) are observed, but one of lattice directions coincides with crests of the bottom profile.
When waves are generated out of the vicinity of the resonance losses caused by the dispersion of waves increase, and thus initially one pair of waves with the wave fronts oriented normally to the crests of heterogeneity of the bottom is generated. Unlike the homogeneous fluid the linear structure of waves exists steadily at the highest values of supercriticality, and even can contain defects. These results are similar to obtained earlier by authors for a liquid with the viscosity 10 times greater than the viscosity of the fresh water – for silicone oil.
In the second series of experiments circular and spiral waves were investigated. The round dish was used where flutes of bottom roughness have been located radially in eight sectors with parallel flutes. It was revealed that similarly to the first series of experiments there is the range of parameters where only one couple of waves is generated. As this pair of waves should be perpendicular to the crests of bottom heterogeneity, thus it has the form of concentric circles (target). It is shown that for increasing supercriticality a spiral wave with different numbers of sleeves can exist steadily.
For the description of the structures appeared at parametrical excitation of capillary ripples the equations for amplitudes of standing waves are used. The free energy Lyapunov functional corresponds to these equations. Its minimum assumes possibility of a steady mode of one or several couples of standing waves. Out of a zone of resonant dispersion one couple of waves will exist as conditions of generation for slanting waves are not executed yet. Thus there are the steady structures containing one pair of standing waves focused in space and perpendicular to flutes of periodic roughness of the bottom. Therefore there are the linear structures in the square dish with the flutes of periodic bottom roughness in parallel to the dish’s walls. In case of radially focused flutes circular and spiral waves were observed.
In the conclusion it should be told that at parametrical excitation of capillary waves on a free surface of liquid of periodically non-uniform depth various roller structures and the localized spiral waves can be generated. It is found out that the main source of defects before their transition to chaos is the area with greater depth. This allows one to make conclusions on the character and the size of heterogeneity of the depth by the parameters of structures on the liquid’s surface.