Stabilization of hydrodynamic instabilities in plane geometry using high frequency pulsations

Abstract

A theoretical and experimental investigation of the Rayleigh-Taylor instability stabilization in the multi-layer liquid system using high frequency oscillations was carried out. The stabilization criterion for a layer of deep liquid, confined by air, was derived to coincide with stabilization criterion for the deep liquid above air. For a narrow layer of liquid the stabilization parameters depend only on the layer depth, but not on a wave vector. The possibility of the Saffman-Taylor instability stabilization in the Hele-Show cell using an alternating pressure was shown. The theoretical dependence of the alternating pressure amplitude on its frequency to stabilize the Saffman-Taylor instability was derived. These results may be useful for stabilization of instabilities in plasma and during mineral oil production. The problems of the Rayleigh-Taylor instability (RTI) come across in different physics fields, such as hydrodynamics, plasma physics and astrophysics. The theory of the hydrodynamic RTI is described in [1-3] for liquids in the systems of plane, cylindrical and spherical geometries under external forces and pressure gradients. The effects of liquid viscosity and spatial inhomogeneity were taken into account. The stability conditions for different cases are obtained and the nonlinear dynamics of the RTI on liquid boundaries are analyzed. The asymptotic behavior for the nonlinear stage of the RTI was investigated using the Fourier-series expansion [4,5]. A number of the experimental researches of the nonlinear and chaotic RTI dynamics [6-9] and the RTI dynamics in inertial confinement fusion [10-12] were carried out. Another example of the hydrodynamic instability is the Saffman-Taylor instability (STI). It arises on the boundary between two viscous liquids, when the less viscous one pushes the more viscous one in porous media, and consists in the development of transverse perturbations of the displacement front. The nonlinear stage of these perturbations results in the formation of so called “viscous fingers”. The problem of appearance of the viscous fingers in the Hele-Show cell is associated with the STI in porous media. The similarity between these phenomena was noticed by Paterson in [13]. The extensive theoretical, experimental and numerical investigations of the STI for different liquid systems, such as multi-layer systems or systems with mixed layer, were carried out [14-18]. In many practical problems the RTI and the STI development should be suppressed. For example, one way of the increase of mineral oil production from depleted oil field is an injection of water under the oil region and oil extrusion by the water. Since the water viscosity is less than the oil one, the STI arises on the displacement front between the water and the oil. This instability leads to the water outbreak through the oil and oil production stops. Therefore the question of its stabilization arises. There are different ways to stabilize the RTI and the STI. It is possible to use a magnetic field to suppress the RTI in plasma [19] or an oscillating magnetic field to stabilize conductive heavy liquid supported against gravity [20]. The RTI of the boundary between two liquids of different densities can be stabilized using high-frequency oscillations [21]. In that paper the theoretical model of the RTI stabilization for two “infinitely deep” ideal liquids using the Kapitza method [22] was developed as well as the number of experiments on stabilization of the RTI in the cylindrical geometry were carried out. A theory of the stabilization of the RTI is also developed using the Floquet method [2,23]. A goal of the present paper is a theoretical investigation of the stabilization of the RTI in the system of three layers of liquids in plane geometry, theoretical investigation of the stabilization of the STI in the Hele-Show cell and carrying out the experiment on stabilization the inverted vessel with liquid and layer of liquid, confined within the air. In contrast to the earlier researches on the hydrodynamic instabilities stabilizations, in which the systems with only single boundary between liquids were taken into consideration, the present paper demonstrates theoretical and experimental possibilities of the stabilization of the coupled waves systems.