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.     

Thermocapillary instability of a deformable liquid film

The instability of a plane liquid film with a uniform transverse temperature gradient under conditions of weightlessness is considered. The surface tension is assumed to depend linearly on the temperature. On the basis of an exact solution of the neutral perturbation problem for a layer with deformable boundaries, the instability domains, the dispersion curves, and the shape of the perturbations are determined. It is shown that on the interval of low Prandtl numbers both thermocapillary waves with predominantly longitudinal flow and capillary waves, supported by the thermocapillary effect, with intense transverse liquid flow can develop on the film.Perm'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 30–36, September–October, 1996.     

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.     

Jet Formation in Gravitational Flow of a Heated Wavy Liquid Film

Jet formation was studied in the region of two-dimensional and three-dimensional waves in a heated liquid film flowing down a vertical surface. Jet-to-jet spacings were measured versus the film Reynolds number and the heat flow density. Three-dimensional waves on the film surface were formed naturally or by artificial perturbations. In addition to the thermocapillary mechanism of jet formation, a thermocapillary–wavy mechanism was found to exist.     

Stability of thermocapillary advective flow in a slowly rotating liquid layer under microgravity conditions

The stability of thermocapillary flow developed in a slowly rotating fluid layer under microgravity conditions is investigated. Both boundaries of the layer are free and assumed to be plane. The tangential thermocapillary Marangoni force exerts on the boundaries, where heat transfer takes place in accordance with the Newton law, the temperature of the medium in the neighborhood of the boundaries being a linear function of the coordinates. The axis of rotation is perpendicular to the liquid layer, rotation is weak so that the centrifugal force can be neglected. Being the solution of the Navier-Stokes equations, the thermocapillary flow in question can be described analytically. The neutral curves which describe the wavenumber dependence of the critical Marangoni number for various Taylor numbers and various directions of the horizontal temperature gradient on the layer boundaries are obtained within the framework of the linear stability theory. The behavior of finite-amplitude perturbations beyond the stability threshold is studied numerically.     

Stability of plane magnetohydrodynamic couette flow with asymmetrical velocity profile

The hydrodynamic stability of plane magnetohydrodynamic Couette flow with asymmetrical velocity profile formed by a transverse magnetic field is investigated within the framework of the linear theory. The complete spectrum of the small perturbations is studied for the characteristic Hartmann numbers. The perturbations are classified in accordance with their phase velocity at large wave numbers. It is established that the stability of the flow is controlled by only one type of perturbations. The critical parameters of the problem are determined. The instability in question recalls the instability of Hartmann flow against asymmetrical perturbations.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 12–18, May–June, 1971.The author thanks M. A. Gol'dshtik for interest in the work and V. A. Sapozhnikov and V. N. Shtern for useful discussions.     

Stability of a nonisothermal boundary layer in the presence of periodic perturbations of the exterior flow velocity

The stability of a laminar boundary layer in the presence of high-frequency time-periodic perturbations of the exterior flow velocity, in particular, acoustic vibrations, is investigated in a series of papers which are reviewed in detail in monograph [1]. The mechanisms by which such perturbations influence the stability and transition to the turbulent flow regime can vary. For example, they can lead to the deformation of the averaged field of the basic flow. However, there was good reason not to discuss the effect of this type of perturbation earlier, as it was considered that the change in the basic flow was very small even for perturbations of great amplitude. The aim of the present paper is to demonstrate how perturbations or pulsations in the exterior flow velocity can, by changing the basic flow, have a strong influence on the stability of the laminar boundary layer of a gas under appreciably nonisothermal conditions. Examples of calculations that support this assertion are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 183–186, September–October, 1984.The authors wish to thank A. A. Maslov for his help with the calculations.     

Locality properties of the problem of hydrodynamic stability

Locality properties are formulated for short-wave length disturbances in the problem of hydrodynamic stability, which together with global flow stability enable us to study the stability of particular sections of the stream, e.g., the flow core or the zone next to the wall. The locality properties are illustrated in the spectrum of small perturbations of plane Poiseuille flow and flows which are obtained by deforming a small section of the Poiseuille parabola. Such a deformation produces points of inflection which lead to the appearance of growing perturbations with wavelength of the order of the deformation zone. It is shown that discontinuities in the velocity profile leads to the loss of stability for high enough Reynolds' numbers.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 11, No. 2, pp. 56–61, March–April, 1970.     

Nonlinear stability of two-dimensional steady flows of an ideal fluid with constant vorticity and their axisymmetric analogs

Flows with constant vorticity are widely used as local models of more complicated flows [1]. In many cases, such flows are stable against finite two-dimensional perturbations. In particular, the inviscid plane-parallel Couette flow has the property of nonlinear stability. Similar treatment of a class of axisymmetric flows yields nonlinear stability of a spherical Hill vortex and inviscid Poiseuille flow in a circular tube with respect to axisymmetric perturbations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 16–21, October–December, 1981.     

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.     

Oscillating thermocapillary convection regimes driven by a point heat source

The stability of an axisymmetric thermocapillary flow driven by a point heat source located in the neighborhood of the free surface of a fluid filling a deep tank is investigated theoretically and experimentally. It is shown that for certain values of the depth and power of the heat source thermocapillary convection becomes unstable with respect to oscillating perturbations of the surface shape. Perm’, Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 92–103, March–April, 2000.     

Stability of the boundary layer of a non-Newtonian liquid obeying a power-type rheological law

The stability of the stationary (steady-state) laminar boundary layer of a non-Newtonian liquid obeying a power-type rheological law at a semiinfinite plate situated in a longitudinal flow is analyzed. An approximate formula is derived for estimating the minimum Reynolds number at which the flow loses stability with respect to slight two-dimensional perturbations. Calculations of the point of stability loss for aqueous solutions of carboxyl methyl cellulose are presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 121–124, March–April, 1971.     

Investigation of thermogravitational-thermocapillary steady-state convective flow stability at low Prandtl numbers

The stability of gravitational-capillary flow in a square cavity with isothermal vertical and adiabatic horizontal boundaries is investigated. The region of stable regimes in the Grashof number-Marangoni number plane is determined for a fluid with a Prandtl number equal to 0.02. In [1] the stability of steady-state thermogravitational convection regimes in a laterally heated square cavity was numerically investigated. The Galerkin method with a system of coordinate functions constructed as proposed in [2] was used to solve the system of equations of free convection in the Oberbeck-Boussinesq approximation. Below, the variant of the Galerkin method described in [2] is used to investigate the stability of steady-state regimes of free convection flow developing under the combined influence of thermogravitational and thermocapillary forces.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 8–13, March–April, 1990.     

Thermocapillary and thermogravitational convection in a two-layer system with curved interface

In an earlier study [1], the present authors used the complete nonlinear hydrodynamic equations to investigate thermocapillary convection in a two-layer system. Oscillatory instability of the equilibrium was established for some ratios of the parameters. In the present paper, a study is made of the influence on the thermocapillary convective motions of two different factors — curvature of the interface and gravity. It is established that curvature of the interface can lead to significant changes in the flow structure and hysteresis transitions between convection regimes. In the case of the joint influence of the thermogravitational and thermocapillary instability mechanisms, many different flow regimes are found: steady motions with different directions of rotation of the vortices and periodic and nonperiodic oscillatory motions with different spatial structures.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 175–179, May–June, 1984.We thank E. M. Zhukovitskii for discussing the results.     

Numerical modeling of thermocapillary convection in a liquid layer with a nonlinear dependence of the surface tension on temperature

The principal characteristics of thermocapillary convection in a rectangular channel with one of the boundaries heated to a temperature higher and the other to a temperature lower than T₀ are investigated numerically on the basis of the Navier-Stokes equations. Certain convection characteristics corresponding to normal and anomalous thermocapillary effects are qualitatively compared. The conditions under which self-similar solutions of the type obtained in [10] can be used to describe the flow in a bounded region are determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 138–143, January–February, 1991.     

Rayleigh stability of shear flow in relaxing media

The stability of steady-state flow is considered in a medium with a nonlocal coupling between pressure and density. The equations for perturbations in such a medium are derived in the linear approximation. The results of numerical integration are given for shear motion. The stability of parallel layered flow in an inviscid homogeneous fluid has been studied for a hundred years. The mathematics for investigating an inviscid instability has been developed, and it has been given a physical interpretation. The first important results in flow stability of an incompressible fluid were obtained in the papers of Helmholtz, Rayleigh, and Kelvin [1] in the last century. Heisenberg [2] worked on this problem in the 1920's, and a series of interesting papers by Tollmien [3] appeared subsequently. Apparently one of the first problems in the stability of a compressible fluid was solved by Landau [4]. The first investigations on the boundary-layer stability of an ideal gas were carried out by Lees and Lin [5], and Dunn and Lin [6]. Mention should be made of a series of papers which have appeared quite recently [7–9]. In all the papers mentioned flow stability is investigated in the framework of classical single-phase hydrodynamics. Meanwhile, in recent years, the processes by which perturbations propagate in media with relaxation have been intensively studied [10–12].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 87–93, May–June, 1976.     

Stability of pipe flow with blowing

The stability of self-similar flows in a porous circular pipe [1–4] with respect to classical and self-similar perturbations of axisymmetric and nonaxisymmetric form is investigated. The case of blowing through the porous lateral surface is examined. Two formulations of the linear stability problem are considered and stability in the sense of self-similar evolution is also investigated. The limiting stability situations are analyzed. Relations for the critical values of the blowing rate parameters are presented for all the types of perturbations investigated. It is shown that nonaxisymmetric classical perturbations are the most dangerous.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 63–71, January–February, 1989.     

Thermocapillary Vortices Induced by a Light Beam near a Bubble Surface in a Hele-Shaw Cell

This paper studies thermocapillary vortices induced by local heating of a bubble surface in a Hele-Shaw cell by a light beam. It is found that the vortex rotation frequency and its depth depend on the distance from the light-beam projection onto the layer to the bubble boundary. The surface velocity of the thermocapillary flow is calculated using the balance of the near-surface and return flows of the thermocapillary vortex and the equality of capillary and dynamic pressures. It is shown that a decrease in the surface velocity and the vortex rotation frequency with increase in the distance from the light beam to the bubble surface is due to a decrease in the temperature gradient between the illuminated and cold poles of the bubble.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 5, pp. 93–99, September–October, 2005.     

Stability of nonisothermal couette flow

A study is made of the loss of stability of the stationary flow of a viscous fluid between two heated rotating cylinders. The linearized stability problem is studied in the Boussinesq approximation. The perturbations are assumed to be periodic in the time, and also in the axial and azimuthal directions. The neutral curves are calculated numerically. The ranges of variation of the parameters of the problem are found, and in them the most dangerous perturbations (in the class of spatially periodic perturbations) are those without rotational symmetry.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 167–170, January–February, 1980.I thank V. I. Yudovich for constant interest in the work.     

Convective instability of Marangoni-Poiseuille flow under a longitudinal temperature gradient

An exact solution is obtained for the problem of steady flow in a system of two horizontal layers of immiscible fluids with a common interface. The stability of the flow is studied by a linearization method. It is shown that the occurrence of instabilities is due to the different governing parameters of the fluids (thickness, heating conditions, viscous and thermal conductivity of the fluids). It is found that under constant gravity conditions, the perturbations are monotonic, and in zero gravity, oscillatory thermocapillary instability occurs.