Stability of the wave motion on the charged interface between two immiscible fluids in the presence of a tangential velocity field discontinuity

In the second-order approximation in the dimensionless wave amplitude, the problem of nonlinear periodic capillary-gravity wave motion of the uniformly charged interface between two immiscible ideal incompressible fluids, the lower of which is perfectly electroconductive and the upper, dielectric, moves translationally at a constant velocity parallel to the interface, is solved analytically. It is shown that on the uniformly charged surface of an electroconductive ideal incompressible fluid the positions of internal nonlinear degenerate resonances depend of the medium density ratio but are independent of the upper medium velocity and the surface charge density on the interface. All resonances are realized at densities of the upper medium smaller than the density of the lower medium. In the region of Rayleigh-Taylor instability with respect to density there is no resonant wave interaction.     

Taylor instability in the discontinuity between two fluids in an electromagnetic field

The gravitational instability of the discontinuity between two compressible (or incompressible) fluids is investigated. The fluids are exposed to an electromagnetic field, and one of them is nonconducting, while the other has a finite conductivity. The magnetic Reynolds number is assumed to be small. It is shown that in contrast to the cases investigated in [1, 2], where compressible, infinitely conducting fluids were considered on both sides of the discontinuity, in the present case the electromagnetic field is not able to stabilize the discontinuity and the perturbations can propagate in fixed directions. The presence of walls inhibits the perturbation growth [2, 3], while their conductivity does not affect the instability of the discontinuity. The greatest perturbation growth is found to occur in a wave propagating along the magnetic field, when the electromagnetic field does not influence these perturbations in the case of incompressible fluids, but does influence them in the compressible case.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 24–28, September–October, 1976.The author wishes to express his appreciation to A. A. Barmin and A. G. Kulikovskii for suggesting the problem and for their continued interest in the work.     

Rayleigh-Taylor instability of the interface between conducting and nonconducting fluids in an alternating magnetic field

The influence of an alternating magnetic field on the Rayleigh-Taylor instability of a conducting fluid has been investigated [1, 2] in the limiting cases of long- and short-wave (compared with the skin-layer thickness) perturbations. Garnier [3] has reported a new mechanism of instability of the interface between conducting and nonconducting fluids which differs not only from the classical Rayleigh-Taylor instability but also from parametric excitation of perturbations. From the instability criterion obtained in [3] without restriction on the spatial scale of the perturbation a paradoxical result follows: An increase in the frequency of the field leads to instability, whereas the practical results of metal casting in an electromagnetic crystallizing tank [4] indicate the opposite effect. In the present paper, it is shown that a plane-polarized high-frequency field effectively stabilizes part of the spectrum of three-dimensional perturbations of an interface but does not completely suppress the Rayleigh-Taylor instability mechanism. The instability generated by the self-field has the nature of parametric resonance.     

Rayleigh-Taylor instability of compressible fluids in the presence of a vertical magnetic field

The influence of a vertical magnetic field on the stability of a compressible, inviscid fluid of variable density is investigated. The solution is characterized by a variational principle. Based on it an approximate solution for the fluid of exponentially varying density, confined between two planes is obtained. The magnetic field is found to have a stabilizing influence on the unstable arrangement. Also the effect of a magnetic field on the angular frequency of oscillations of the waves generated in stable arrangement is considered.     

Structure of the fluid interface in an external magnetic field in the presence of a magnetizable surfactant

The structure of the flat interface between two conventional fluids in an external magnetic field in the presence of a magnetizable surfactant is investigated with account for the dependence of the free energy of the system on the surfactant concentration gradients and the bearing phase density. The dependence of the surface tension tensor components on the magnetic field strength is determined.     

Two-dimensional instability of a tangential discontinuity in dispersive media

The instability of a tangential discontinuity in a compressible dispersive medium with respect to small two-dimensional perturbations is demonstrated. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 16–23, January–February, 1997. The work was carried out with support from the Russian Foundation for Fundamental Research (project No. 96-01-01340).     

The deformation of an interface between two fluids in a porous medium

Summary The interface between two moving fluids in a porous medium will, in general, deform under the influence of gravity and drag forces. An example of some importance is the formation of so-called gravity tongues in oil reservoirs. This paper deals with the displacement of oil by water in a homogeneous non-horizontal oil stratum. The deformation of such an interface can be deduced by numerical procedures based upon exact methods. The use of these methods is limited, however, owing to the fact that in oil reservoirs the dip is usually smaller than 10 to 20 degrees. In such cases, where the interface is initially horizontal, the computation of the form of the interface as a function of time becomes so enormous, even when a fast electronic computer is used, that an approximative method is more useful. In this paper two approximate solutions are presented. The first one is obtained by using a simplified form of the dynamic interface condition, in which the flow velocity component perpendicular to the dip direction of the reservoir is neglected. This simplification has previously been used by Dietz, who gave a first-order approximation with respect to time. More complicated results are obtained by using the second approximation where, in accordance with the dynamic boundary condition, this velocity component is more or less taken into account. In both methods, the form of the interface as a function of time is expressed in a parametric representation. Moreover, the amount of water that has passed a given cross-section and the flow of water at this section are obtained as a function of time and the parameter used. Results of both methods are compared with each other and with those obtained by an exact method. Both approximations are found to be good in those cases where the dip of the reservoir is not too high, but this is precisely when exact methods are impracticable.Nomenclature d thickness of the idealised reservoir (see fig. 1) - f function of y as given by (2.7) - f, f, f first, second and third derivative of f with respect to y - F(y, ) function of y and as given in the appendix - G dimensionless quantity - G* dimensionless quantity {= G cos /(1–G sin )} - H(y, ) function of y and as given in the appendix - M dimensionless quantity _{2 1}/ _{1 2} - p pressure - q _w the flow of water at a given cross-section - Q _w the total amount of water that has already passed a given cross-section at a certain time - S ₀ oil saturation in the oil region - S _w water saturation in the water region - r integration variable - s the co-ordinate along the interface (positive direction as given in fig. 1) - t time - t _w time at which water breaks through at a given cross-section - u ₁ mean velocity component of fluid 1 in x-direction in the pores of the porous medium (water) - u ₂ mean velocity component of fluid 2 in x-direction in the pores of the porous medium (oil) - U _r the relative deformation velocity of the interface {=(x _i –W ₀ t)/t} _y - the mean fluid velocity vector in the pores of the porous medium - v ₁ mean velocity component of fluid 1 in y-direction in the pores of the porous medium (water) - v ₂ mean velocity component of fluid 2 in y-direction in the pores of the porous medium (oil) - v _n mean velocity component of the fluids normal to the interface (positive direction from fluid 1 to fluid 2) - W ₀ mean velocity of fluid 1 (water) when x –, where the velocity component in y-direction is equal to zero - x co-ordinate, parallel to the boundaries of the reservoir (see fig. 1) - x _e value of x for a given cross-section - x _i , y _i values of the x and y co-ordinates corresponding to the points of the interface - x ₀(y) initial value of the x co-ordinate of the points of the interface (at t=0) - y co-ordinate, perpendicular to the boundaries of the reservoir (see fig. 1) - y _e (t) time-dependent value of the y co-ordinate of the interface if the value of the x co-ordinate is equal to x _e - y _i , x _i values of the y and x co-ordinates corresponding to the points of the interface - z vertical co-ordinate (positive direction as given in fig. 1) - the angle between the horizon and the boundaries of the reservoir (see fig. 1) - the angle between the x axis and the normal to the interface (see fig. 1) - _e the angle if the value of x _i is equal to x _e - ₀(y) initial value of the angle (at t=0) - effective permeability of the porous medium divided by the product of the porosity and fluid saturation - ₁ effective permeability of the porous medium to fluid 1 divided by the product of the porosity and the saturation of fluid 1 - ₂ effective permeability of the porous medium to fluid 2 divided by the product of the porosity and the saturation of fluid 2 - fluid viscosity - ₁ viscosity of fluid 1 (water) - ₂ viscosity of fluid 2 (oil) - fluid density - ₁ density of fluid 1 (water) - ₂ density of fluid 2 (oil) - porosity of the porous medium Formerly with Koninklijke/Shell Exploratie en Produktie Laboratorium, Rijswijk, The Netherlands.     

Surface waves and stability of tangential velocity discontinuity on a solid-fluid boundary

Solutions of the Rayleigh-wave type on the boundary of an elastic half-space and a moving layer of ideal fluid are obtained. The limiting cases of zero flow velocity and a tangential velocity discontinuity in the fluid were investigated in [1–3]. In [4] the order of magnitude of the critical flow velocity was estimated. An increase in the velocity scales used in engineering and experimental practice (see [5], for instance) has aroused interest in a more thorough analysis of the effect.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 43–46, May–June, 1981.     

Instability of a plane interface between two immiscible conducting viscous fluids in a normal electrostatic field

The dispersion relation for motions of a charged plane interface between two viscous incompressible immiscible conducting fluids is analyzed numerically for finite values of all the parameters involved. It is shown that in addition to the well-known aperiodic (Tonkes-Frenkel’ type) instability for certain values of the physical parameters an oscillatory instability with periodically growing amplitude may be realized in the system. Yaroslavl’. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 116–123, November–December, 1998.     

Convective instability of a current-carrying conductive liquid in a long channel with an external magnetic field

The stability of equilibrium of an electrically conductive liquid in a long channel in the presence of electric and magnetic fields is considered, under the condition that there exist gradients in temperature and liquid conductivity along the channel.     

Thermocapillary instability of the interface between two immiscible liquids in the presence of volume heat sources

The problem of the stability of the interface between two infinite layers of different immiscible liquids is considered. It is assumed that within the liquid a distributed volume heat source, simulating Joule heating, is given. The stability of the rest state with respect to small unsteady disturbances is investigated. The investigation is carried out using the real boundary conditions at the interface between the two liquids rather than the model boundary conditions usually employed in such problems [5]. The problem considered is related to the practical question of the stability of electrolyzer processes. In the present case a possible threshold mechanism of development of oscillations of the electrolyte-aluminum interface is examined. A numerical example with liquid parameters that coincide with those of the electrolyte and aluminum shows that the thermocapillary instability mechanism can, in fact, be the source of surface waves at the electrolyte-aluminum interface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 156–160, September–October, 1990.     

Interaction between a fast magnetohydrodynamic shock wave and a tangential discontinuity

The self-similar problem of the oblique interaction between a fast shock wave and a tangential discontinuity is solved within the framework of the ideal magnetohydrodynamic model. The constraints on the initial parameters necessary for the existence of a regular solution are found. Various feasible wave flow patterns are found. In the space of the governing parameters boundaries between the solutions of various types are constructed. The basic features of the developing flows and their dependence on the initial data are clarified.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 159–168, March–April, 1995.     

Natural oscillations of a liquid of finite electrical conduction in the presence of an external magnetic field

The effect of the finite electrical conduction (finiteness of the magnetic Reynolds number), which is considered a dissipative factor, on small natural oscillations of an ideal heavy liquid of finite depth whose free surface borders on vacuum is studied. A constant external horizontal magnetic field is applied to the liquid. The energy-balance equation is derived, and the theorem of wave attenuation with time is proved. Numerical calculations and the resulting asymptotic formulas give a complete pattern of the spectrum, including its continuous part. The amplitude-frequency characteristics of the wave modes are presented. Rostov State University, Rostov-on-Don 344007. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 2, pp. 3–10, March–April, 2000.     

Linear instability of the electrified free interface between two cylindrical shells of viscoelastic fluids through porous media

In this paper, we have discussed the linear stability analysis of the electrified surface separating two coaxial Oldroyd-B fluid layers confined between two impermeable rigid cylinders in the presence of both interfacial insoluble surfactant and surface charge through porous media. The case of long waves interfacial stability has been studied. The dispersion relation is solved numerically and hence the effects of various parameters are illustrated graphically. Our results reveal that the influence of the physicochemical parameter β is to shrink the instability region of the surface and reduce the growth rate of the unstable normal modes. Such important effects of the surfactant on the shape of interfacial structures are more sensitive to the variation of the β corresponding to non-Newtonian fluids-model compared with the Newtonian fluids model. In the case of long wave limit, it is demonstrated that increasing β, has a dual role in-fluence (de-stabilizing effects) depending on the viscosity of the core fluid. It has a destabilizing effect at the large values of the core fluid viscosity coefficient, while this role is exchanged to a regularly stabilizing influence at small values of such coefficient.     

Nonlinear gravity-capillary waves instability in superposed magnetic fluids influenced by a vertical magnetic field and time-dependent acceleration

This paper deals with the effect of a periodic forcing on nonlinear modulation of interfacial gravity-capillary waves propagating between two magnetic fluids of infinite depth under the influence of a constant vertical magnetic field. Based on the method of multiple scales expansion for a small amplitude of periodic force, two parametric nonlinear Schrödinger equations with explicit expressions of coefficients are derived in the resonance case. A classical nonlinear Schrödinger equation is derived in the non-resonance case. The stability of the uniform time-dependent solution is analyzed. Theoretical analysis and numerical calculations show that the resonance point is affected by the magnetic field and the applied frequency. The linear stability shows that the periodic force has a destabilizing influence in the stability criterion. It is observed that the vertical field plays the same role, and that the acceleration frequency plays a dual role in the nonlinear stability criterion. Instability was revealed in the system for large values of the applied magnetic field, but the small values of the field redistribute the stable areas.     

Convective instability in a rapidly rotating fluid layer in the presence of a non-uniform magnetic field

Magnetoconvective instabilities in a rapidly rotating, electrically conducting fluid layer heated from below in the presence of a non-uniform, horizontal magnetic field are investigated. It was first shown by Chandrasekhar that an overall minimum of the Rayleigh number may be reached at the onset of magnetoconvection when a uniform basic magnetic field is imposed. In this paper, we show that the properties of instability can be quite different when a non-uniform basic magnetic field is applied. It is shown that there is an optimum value of the Elsasser number provided that the basic magnetic field is a monotonically decreasing or increasing function of the vertical coordinate. However, there exist no optimum values of the Elsasser number that can give rise to an overall minimum of the Rayleigh number at the onset of magnetoconvection if the imposed basic magnetic field has an inflexion point. The project supported by the National Natural Science Foundation of China (40174026 and 40074041)