Thermocapillary instability of the equilibrium of a flat layer containing internal heat sources

In the absence of body forces, a factor which has a strong influence on the equilibrium stability of a nonuniformly heated liquid is the dependence of the coefficient of surface tension on the temperature and the thermocapillary effect generated by it. If the equilibrium temperature gradient is sufficiently great, then the presence of the thermocapillary forces on the free surface can lead to the occurrence of convective motion. The monotonie instability of the equilibrium of a flat layer was investigated in [1–3]. Analysis of nonmonotonic disturbances [4] showed that in the case of an undeformable free surface there is no oscillatory instability. In [5] it was found that oscillatory instability is possible if there is a nonlinear dependence of the coefficient of surface tension on the temperature. The present paper is devoted to numerical investigation of the equilibrium stability of a flat layer with respect to arbitrary disturbances. It is shown that for a deformable free boundary there appears an additional neutral curve, which corresponds to monotonie capillary instability. In addition, when the capillary convection mechanism is taken into account, there appears an oscillatory instability, which becomes the most dangerous in the region of small Prandtl and wave numbers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 27–31, March–April, 1991.I thank V. K. Andreev for a helpful discussion of the work.     

Oscillatory thermocapillary instability of a liquid layer in the presence of a surfactant

The effect of capillarity and a surfactant on the stability of a liquid layer in the presence of a vertical temperature gradient is investigated. It is found that the surfactant leads to the appearance of both monotonic and oscillatory instability, the presence of a surface concentration destabilizing the equilibrium in the case of heating from below. When the free surface is heated, the surfactant stabilizes the capillary instability.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.1, pp. 6–10, January–February, 1993.     

Thermocapillary Instability of a Cylindrical Layer in the Presence of a Radial Temperature Gradient

Within a linear formulation, the thermocapillary instability of equilibrium of a cylindrical layer of heat-conducting viscous fluid in the presence of a radial temperature gradient is investigated with respect to arbitrary disturbances. It is shown that the Rayleigh instability mechanism results in the appearance of monotonous disturbances of a new type. For steady disturbances, the neutral curve is split into two separate segments, each corresponding to its own type of disturbances. For a deformable free boundary, new oscillating disturbances in the form of surface waves develop. It is found that, in the case of axial symmetry, the behavior of these disturbances completely coincides with the oscillating disturbance behavior in a plane layer.     

Thermocapillary convection in two-layer systems in the presence of a surface active agent at the interface

Convective instability in a layered system due to the thermocapillary effect was investigated in [1–5]. In these studies it was shown that the perturbations responsible for equilibrium crisis may build up either monotonically or in an oscillatory fashion. In [6] the stabilizing effect of a surface active agent (SAA) on thermocapillary instability was established for a layer with a free surface. For layers of infinite thickness the effect of SAA on thermocapillary convection was studied in [7–9]. The present investigation is concerned with thermocapillary convection in a system of two layers of finite thickness in the presence of an SAA. Convection due to the lift force is not considered. It is established that the principal result of the action of the SAA is not the stabilizing effect on the monotonic mode but the appearance of a new type of oscillatory instability.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2 pp. 3–8, March–April, 1986.In conclusion the authors wish to thank E. M. Zhukhovitskii for discussind the results.     

Convective instability of a system of horizontal layers of immiscible liquids with a deformable interface

The convective stability of equilibrium is considered for a system of two immiscible fluids which differ little in density. A generalized Boussinesq approximation is developed, making it possible to take the interface deformations properly into account. The stability of the equilibrium state of two fluids in a horizontal layer with a vertical temperature gradient is investigated. Several instability mechanisms are identified: long-wave and cellular monotonic disturbances and oscillatory disturbances. Increasing the deformability is shown to cause switching between instability mechanisms.Perm. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 32–39, March–April, 1996.     

Convection in a two-layer system due to the combined action of the Rayleigh and thermocapillary instability mechanisms

In a two-layer system loss of stability may be monotonic or oscillatory in character. Increasing oscillatory perturbations have been detected in the case of both Rayleigh [1, 2] and thermocapillary convection [3–5]; however, for many systems the minimum of the neutral curve corresponds to monotonic perturbations. In [5] an example was given of a system for which oscillatory instability is most dangerous when the thermogravitational and thermocapillary instability mechanisms are simultaneously operative. In this paper the occurrence of convection in a two-layer system due to the combined action of the Rayleigh (volume) and thermocapillary (surface) instability mechanisms is systematically investigated. It is shown that when the Rayleigh mechanism operates primarily in the upper layer of fluid, in the presence of a thermocapillary effect oscillatory instability may be the more dangerous. If thermogravitational convection is excited in the lower layer of fluid, the instability will be monotonic.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 166–170, January–February, 1987.     

Convective stability of a weakly conducting liquid in an electric field

In an inhomogeneously heated weakly conductive liquid (electrical conductivity 10^–12–1 cm^–1) located in a constant electric field a volume charge is induced because of thermal inhomogeneity of electrical conductivity and dielectric permittivity. The ponderomotive forces which develop set the liquid into intense motion [1–6]. However, under certain conditions equilibrium proves possible, and in that case the question of its stability may be considered. A theoretical analysis of liquid equilibrium stability in a planar horizontal condenser was performed in [2, 4]. Critical problem parameters were found for the case where Archimedean forces are absent [2]. Charge perturbation relaxation was considered instantaneous. It was shown that instability is of an oscillatory character. In [4] only heating from above was considered. Basic results were obtained in the limiting case of disappearingly small thermal diffusivity in the liquid (infinitely high Prandtl numbers). In the present study a more general formulation will be used to examine convective stability of equilibrium of a vertical liquid layer heated from above or below and located in an electric field. For the case of a layer with free thermally insulated boundaries, an exact solution is obtained. Values of critical Rayleigh number and neutral oscillation frequency for heating from above and below are found Neutral curves are constructed. It is demonstrated that with heating from below instability of both the oscillatory and monotonic types is possible, while with heating from above the instability has an oscillatory character. Values are found for the dimensionless field parameter at which the form of instability changes for heating from below and at which instability becomes possible for heating from above.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 16–23, September–October, 1976.In conclusion, the author thanks E. M. Zhukhovitskii for this interest in the study and valuable advice.     

Occurrence of thermocapillary convection in a two-layer system in the presence of a soluble surfactant

The present study considers the effect of the solubility of a surfactant on monotonic oscillatory thermocapillary instability of equilibrium in a two-layer system. It is established that with increase in the parameter which characterizes the solubility of the surfactant a reduction takes place in the threshold of the monotonic instability; finally the monotonic perturbations become most dangerous.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 171–175, March–April, 1988.     

Development of thermogravitation convection in a two-layer system in the presence of a surface-active material on the boundary

Convective instability of equilibrium in a system of two horizontal layers of immiscible liquids, caused by the Rayleigh instability mechanism, has been studied within the framework of the linear theory in [1–5]. The present study will investigate the effect of a surface-active material (SAM), deposited on the boundary between the liquids, on the development of thermogravitation convection. Calculations were performed for two types of systems, which in the absence of a SAM show instability of a monotonic or an oscillatory character. A new type of oscillatory equilibrium instability was observed, produced by the effect of the SAM. In some region of parameter values the oscillatory instability may prove to be the more dangerous one. The action of the Marangoni effect on thermogravitation oscillations is considered.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 76–81, September–October, 1986.In conclusion, the authors express their gratitude to E. M. Zhukhovitskii for his helpful evaluation.     

Convective stability of a horizontal rotating fluid layer with distributed heat sources

The results of investigating the convective instability of a horizontal layer of rotating fluid, created by a temperature difference applied at the boundaries of the layer and by heat sources distributed according to various laws, are presented. It is shown that, when the other parameters of the problem are fixed, an increase in the internal heat release lowers the limits of both monotonic and oscillatory stability of the layer, increases the wave number and reduces the neutral oscillation frequency. An increase in source concentration towards the center of the layer intensifies the effect. As the strength of the internal heat sources and their concentration towards the center of the layer increase, the oscillating convection that develops at the stability limit when the Prandtl number is low and the rotation fairly fast is first replaced by monotonic convection and then ceases altogether.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 21–28, January–February, 1989.     

Oscillations of an ideal liquid acted upon by subface-tension forces. Case of a doubly connected free surface

Many articles have appeared on the problems of small oscillations of an ideal liquid acted upon by surface-tension forces. Oscillations of a liquid with a single free surface are treated in [1, 2]. Oscillations of an arbitrary number of immiscible liquids bounded by equilibrium surfaces on which only zero volume oscillations are assumed possible are investigated in [3], We consider below the problem of the oscillations of an ideal liquid with two free surfaces on each of which nonzero volume disturbances are kinematically possible. The disturbances satisfy the condition of constant total volume. A method of solution is presented. The problem of axisymmetric oscillations of a liquid sphere in contact with the periphery of a circular opening is considered neglecting gravity. The first two eigenfrequencies and oscillatory modes are found.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 64–71, May–June, 1976.In conclusion, the author thanks F. L. Chernous'ko for posing the problem and for his attention to the work.     

Vibrational and convective instability of a plane horizontal fluid layer at finite vibration frequencies

The vibrational-convective instability of a plane horizontal fluid layer subject to longitudinal harmonic vibrations of finite frequency and transverse stratification in a static gravity field is studied. The analysis is based on the complete convection equations in the Boussinesq approximation. It is demonstrated that in the limiting case of high-frequency vibrations the results thus obtained coincide with those obtained earlier on the basis of the averaged equations. In the limiting low-frequency case the nature of the instability is quite different being due to the instability of oscillating counterstreams.Perm'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 44–51, September–October, 1996.     

Parametric instability of a nonuniformly heated horizontal layer of liquid dielectric in a variable electric field

The parametric instability of a nonuniformly heated horizontal layer of liquid dielectric with free isothermal boundaries in a transverse electric field is studied analytically. An instability map is obtained. It is shown that instability can develop at some critical electric field strength which depends on the frequency and is several times greater than the critical strength of the constant electric field.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 184–186, September–October, 1993.     

Stability of a nonstationary round jet of an ideal incompressible fluid

The stability of transient flow in a cylinder of an ideal incompressible fluid with a free boundary is studied. There are 20 different cases of the behavior of small disturbances as a function of the parameters of the problem. In particular, if surface tension is not taken into account a round jet is stable with respect to axially symmetrical disturbances, but the introduction of capillary forces leads to a strong instability.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 80–84, July–August, 1972.In conclusion the author thanks V. V. Pukhnachev for formulation of the problem and valuable advice.     

Development and instability of steady convective flows in a square cavity heated from below and a field of vertically directed vibrational forces

The present paper is devoted to numerical investigation of the spatial structure and stability of secondary vibrational convective flows resulting from instability of the equilibrium of a fluid heated from below. Vibrations parallel to the vector of the gravitational force (vertical vibrations) are considered. As in earlier work [7–9], a region of finite size is used — a square cavity heated from below. It is shown that enhancement of the vibrational disturbance of the natural convective flow may either stabilize or destabilize flows with different spatial structures; it may also stabilize certain solutions of the system of convection equations that are unstable in the absence of vibrational forces. In addition, increase of the vibrational Rayleigh number can lead to a change of the mechanisms responsible for equilibrium instability and oscillatory instability of the secondary steady flows.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 9–18, March–April, 1991.I thank G. Z. Gershuni for assistance and extremely fruitful discussions of the results of the paper.     

Theoretical analysis of acoustic instability of a hypersonic shock layer on a porous wall

The possibility of controlling the laminar-turbulent transition in hypersonic shock layers by means of porous coatings is considered. The linear stability of the shock layer to acoustic disturbances is analyzed. A dispersion relation is derived in an analytical form and analyzed for different characteristic values of porosity of the wall, which allows one to study the spectrum of acoustic disturbances in the shock layer. Analytical expressions for the growth rate of instability of acoustic disturbances are presented as functions of the reflection factor. Their structure indicates that the porous coating effectively decreases acoustic instability of the shock layer.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 1, pp. 44–54, January–February, 2005.     

Thermocapillary instability of a liquid layer in the presence of a soluble surfactant

The effect of capillarity on the stability of a plane layer of viscous heat-conducting liquid in the presence of a soluble surfactant is investigated. It is found that an increase in surfactant solubility has a stabilizing effect on equilibrium. Monotonic instability is the most dangerous mode in the case of long-wave perturbations, whereas in the short-wave region loss of stability is induced by oscillatory perturbations.Krasnoyarsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 3–8, January–February, 1996.     

Connective instability of a two-layer system with thermally insulated boundaries

The investigation of the convective stability of mechanical equilibrium of two horizontal layers of immiscible fluids has revealed the characteristics of such systems [1–3]. In particular, it has been found that, as distinct from a homogeneous horizontal layer, under certain conditions two-layer systems experience convective instability when uniformly heated from above and, moreover, oscillatory instability when heated from below. In [1–3] the problem was solved for a system with isothermal outer boundaries. In this paper the stability of equilibrium of two-layer systems is investigated for thermally insulated outer boundaries. Special attention is given to the study of the long wave instability mode.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 22–28, March–April, 1986.The authors wish to thank O. V. Kustova for assisting with the computations.     

On thermal instabilities in a viscoelastic fluid

The Bénard-Marangoni thermal instability problem for a viscoelastic Jeffreys’ fluid layer bounded above by a realistic free deformable surface and by a plane surface below is investigated using a linear stability analysis. It has been shown that both the relaxation time and surface deflection have a destabilizing effect unlike the retardation time. A point of codimension 2 has been identified which means that instability here takes the form of a competition between stationary and oscillatory convections. When the lower boundary is free but plane, an analytic treatment has identified an oscillatory disturbance with zero critical wavenumber which is not found in the absence of surface deformation.     

Natural oscillations arising from loss of stability in parallel flows of a viscous liquid under long-wavelength periodic disturbances

It was shown in [1] that a parallel flow with an arbitrary nonconstant velocity profile is unstable for long-wavelength spatially periodic disturbances along the flow. The present paper shows that this instability leads to a supercritical natural oscillation mode of the simple wave type. This mode is calculated using the Lyapunov-Schmidt method in the form given in [2], along with the asymptotic curve of the wavelengths [1]. If the long wavelength disturbances are the most dangerous (this occurs, for example, when there is a sinusoidal velocity profile), then the natural oscillation mode is stable for spatially periodic disturbances having the same wavelength.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 32–35, January–February, 1973.