Taylor instability of a fluid film on a body

The Taylor instability develops in a parallel flows when the body force acts in the direction from the heavier fluid toward the lighter [1]. It has been suggested that an increase in flow vorticity may have a stabilizing influence on the Taylor instability [2]. In studying the hydrodynamic stability of a viscous film on a body in a flow of a low-viscosity fluid [3], the author noted some stabilization of the Taylor instability with increase in Reynolds number, and suggested that cases of complete stabilization of the flow with respect to two-dimensional disturbances are possible with some increase in Reynolds number. In the present investigation, calculations revealed cases in which with increase in Reynolds number the Taylor instability goes over into a Helmholtz instability, which increases with increase in Reynolds number, and also cases in which the Taylor instability completely disappears at some value of the Reynolds number before a Helmholtz instability has developed, i.e., cases of complete stabilization of the flow with respect to two-dimensional disturbances as a result of an increase in Reynolds number.     

Taylor instability of a flat isentropic layer with isobaric boundaries

A study is made of the stability of the hydrostatic equilibrium of a layer of gas with flat boundaries in a uniform gravity field. A similar problem was considered by Plesset and Hsieh [1], who considered only the short-wavelength correction to the incompressible case, i.e., made a calculation in the approximation of small /L, and calculated the first two terms in the expansion of the growth rate with respect to the parameter /L, where is the wavelength and L the thickness of the layer. In the present paper, the growth rate of the Taylor instability is found for the case of isobaric boundaries for arbitrary ratio /L. The calculation is made for the case when there is not necessarily a vacuum at the top of the layer. An error made by Plesset and Hsieh [1] in selecting the solution at the interface with the vacuum is corrected. It is shown that in the case of an isentropic layer the Taylor mode is the only unstable one.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 158–161, January–February, 1984.I should like to thank S. I. Anisimov for constant interest in the work.     

Bénard-Marangoni instability in a viscoelastic Jeffreys' fluid layer

Coupled buoyancy (Bénard) and thermocapillary (Marangoni) convection in a thin fluid layer of a viscoelastic fluid are studied. The viscoelastic fluid is modeled by Jeffreys' constitutive equation. The lower surface of the layer is in contact with a rigid heat-conducting plate while its upper surface is subject to a temperature-dependent surface tension. The critical temperature difference between both boundaries corresponding to the onset of convection is calculated. The role of the various viscometric coefficients is discussed. In the appendix it is shown that Jeffreys' constitutive relation is easily derived from thermodynamic considerations based on extended irreversible thermodynamics.     

Instability of fluid motion in a thin spherical layer

We study flow stability in a thin spherical layer with respect to small disturbances. It is shown that for each given layer thickness there is a sequence of critical Reynolds numbers above which the motion is unstable. In its form, the critical disturbance is reminiscent of the secondary flow which develops upon loss of stability of the basic fluid flow between rotating cylinders (Taylor problem).The author wishes to thank M. I. Shliomis for his continued interest in this study.     

Hydrodynamic instability in a porous layer saturated with a heat generating fluid

Critical Rayleigh numbers determined by linear stabiliy theory are presented for porous-fluid layers of infinite horizontal extent heated internally by a uniform volumetric energy source in the fluid. The thermal coupling between the layer and its environment is represented by a general mixed boundary condition for both the conduction state and the disturbance temperature. Rigid-rigid, rigid-constant pressure, and constant pressure-rigid boundaries are considered in the computations. For a fixed ratio of upper surface Biot number to that at the lower surface, decreasing the Biot number is strictly destabilizing for values of this ratio greater than or equal to one. A layer with a rigid upper surface is generally the most stable; however, a layer with a rigid upper surface and a constant pressure lower surface exhibits the largest values of critical Rayleigh numbers for large values of Biot number.     

Thermocapillary convection in a thin layer of nonuniformly heated fluid

Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 120–128, March–April, 1989.     

Large-scale turbulence in a thin layer of nonisothermal rotating fluid

The dynamics of large-scale nonisothermal turbulence in a thin rotating layer of fluid are investigated. An hierarchical model, obtained by averaging the initial Boussinesq equations with respect to the vertical coordinates and subsequently projecting the two-dimensional equations onto a basis consisting of a system of axisymmetric spiral vortices of progressively decreasing scale, is proposed. It is shown that the presence of horizontal temperature inhomogeneities leads to a considerable increase in the turbulence decay time.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 48–55, July–August, 1988.     

Rayleigh Taylor instability of rotating compressible fluid through porous media

The hydromagnetic instability of a stratified horizontal layer of viscous compressible rotating fluid through porous media in the presence of vertical magnetic field is considered. The solution has been obtained through the use of variational principle. The dispersion relation is derived for a layer having exponential density stratification along the vertical direction. It is found that viscosity has stabilizing influence while permeability of porous medium, magnetic resistivity and coriolis forces have destabilizing influence on the system.     

Thermal instability in a rotating anisotropic porous layer saturated by a viscoelastic fluid

Linear and non-linear thermal instability in a rotating anisotropic porous medium, saturated with viscoelastic fluid, has been investigated for free-free surfaces. The linear theory is being related to the normal mode method and non-linear analysis is based on minimal representation of the truncated Fourier series analysis containing only two terms. The extended Darcy model, which includes the time derivative and Coriolis terms has been employed in the momentum equation. The criteria for both stationary and oscillatory convection is derived analytically. The rotation inhibits the onset of convection in both stationary and oscillatory modes. A weak non-linear theory based on the truncated representation of Fourier series method is used to find the thermal Nusselt number. The transient behaviour of the Nusselt number is also investigated by solving the finite amplitude equations using a numerical method. The results obtained during the analysis have been presented graphically.     

Thermal instability and natural convection in a fluid layer over a porous substrate

The stability and natural convection in a system consisting of a horizontal fluid layer over a layer of porous medium saturated with the same fluid, with heating from below, are considered. The upper surface is either rigid or dynamically free with surface-tension effects allowed for. The solution is obtained using a parallel flow assumption for constant-flux thermal boundary conditions for which the onset of cellular convection corresponds to a vanishingly small wavenumber. The critical Rayleigh number and Nusselt number are found to depend on the depth ratio, the Darcy number, the viscosity ratio, the thermal conductivity ratio, and the Marangoni number. Results are given for a range of values of each of the governing parameters. The results are compared with limiting cases of the problem for standard terrestrial conditions or microgravity, and are found to be in agreement.Es werden die Stabilität und die freie Konvektion in einem System mit einer horizontalen Fluid-Schicht über einer Schicht mit einem porösen Medium, welches mit diesem Fluid gesättigt ist, untersucht. Das System wird von unten beheizt. Die Oberfläche ist entweder unelastisch oder frei von dynamischen Oberflächenspannungen. Die Lösung wird mit der Annahme einer parallelen Strömung mit konstanten thermischen Randbedingungen erhalten. Für diese Randbedingungen bedeutet das Einsetzen der zellularen Konvektion eine verschwindend kleine Wellenzahl. Die kritische Rayleigh- und Nusselt-Zahl hängen von dem Tiefenverhältnis, der Darcy-Zahl, der Viskosität, der thermischen Leitfähigkeit und von der Marangoni-Zahl ab. Für jeden Parameter werden Ergebnisse für einen bestimmten Wertebereich gegeben. Die Ergebnisse werden mit einigen Fällen mit Standard-Erdbedingungen oder Mikrogravitation verglichen und stehen mit diesen in guter Übereinstimmung.     

Electrorheological Rayleigh-Taylor instability at the interface between a porous layer and thin shell with poorly conducting couple stress fluid

This paper is concerned with the study of the Electrorheological Rayleigh-Taylor instability (ERTI) at the interface between a densely packed saturated poorly conducting couple stress porous layer accelerated by a lighter poorly conducting couple stress fluid in a thin shell in the presence of a transverse electric field and laser radiation. A simple theory based on fully developed flow approximations is used to derive the dispersion relation for the growth rate of ERTI. The cutoff and the maximum wave numbers and the corresponding maximum frequencies are obtained. It is shown that the effects of couple stress parameter and the electric field reduce the growth rate considerably compared to a non-conducting fluid in the absence of an electric field. These are favorable to control the surface instabilities in many practical applications discussed in this paper.     

Convective instability of a horizontal fluid layer with a heat-conducting permeable partition

Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 158–161, March–April, 1993.     

Vibrational-convective instability of a horizontal fluid layer with internal heat sources

The convective motion of a nonisothermal fluid in a gravity field in a vibrating cavity is caused by two mechanisms: the usual static mechanism and a vibrational mechanism. The same mechanisms are also responsible for mechanical equilibrium crisis under the conditions in which such equilibrium is possible. The research on these questions is reviewed in [1]. The problems of vibrational-convective stability examined so far relate to cases in which the nonisothermicity was created by specifying the temperature at the boundaries of the region. The present study is concerned with the vibrational-convective stability of a fluid in which the temperature nonuniformity is created by internal heat generation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 3–7, September–October, 1985.     

Convective instability of a horizontal layer of fluid between permeable membranes

The equilibrium stability is investigated of a system consisting of two semi-infinite isothermal masses of fluid divided by a horizontal layer of finite thickness of the same fluid with a vertical temperature gradient directed downwards. The transition layer is separated by thin permeable membranes. Neutral stability curves are constructed for different membrane resistances. In the case of high permeability, the equilibrium is absolutely unstable with respect to monotonic-type longwave perturbations. For low permeability membranes, instability with respect to monotonic finite-wavelength perturbations is characteristic.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 171–173, July–August, 1985.     

Rayleigh-Taylor instability of a thin liquid layer in the presence of three-dimensional perturbations

We consider the evolution of small three-dimensional perturbations of an accelerated thin liquid layer. The analytical solutions obtained correspond to various types of initial perturbations: in the form of a layer, in the initial velocities, and in the thickness of the layer. Depending on the dimensionless parameters which characterize the initial data, the perturbations can increase exponentially with time, remain bounded, and change the phase. Institute of Experimental Physics, Sarov 607200. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 1, pp. 3–9, January–February, 1999.     

Certain types of vibrationally convective instability of a two-dimensional fluid layer in zero gravity

The stability of a new equilibrium configuration possible in a two-dimensional layer of nonisothermal fluid executing high-frequency vibrations in zero gravity is investigated in the framework of the linear theory. A study is made on the basis of the equations of vibrational convection. Instability with respect to one-dimensional and two-dimensional perturbations is studied. An elementary exact solution is obtained for the one-dimensional perturbations. Vibrationally connective instability of a fluid in zero gravity has been studied in a number of papers [1-3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 4–7, September–October, 1987.The author expresses his gratitude to G. Z. Gershuni for his constant interest in my work.     

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)