Stability of convective motion in a vertical layer in the presence of longitudinal vibrations

The influence of vibrations of a cavity containing a fluid on the convective stability of the equilibrium has been investigated on a number of occasions [1]. The stability of convective flows in a modulated gravity field has not hitherto been studied systematically. There is only the paper of Baxi, Arpaci, and Vest [2], which contains fragmentary data corresponding to various values of the determining parameters of the problem. The present paper investigates the linear stability of convective flow in a vertical plane layer with walls at different temperatures in the presence of longitudinal harmonic vibrations of the cavity containing the fluid. It is assumed that the frequency of the vibrations is fairly high; the motion is described by the equations of the averaged convective motion. The stability boundaries of the flow with respect to monotonic perturbations in the region of Prandtl numbers 0 ? P ? 10 are determined. It is found that high-frequency vibrations have a destabilizing influence on the convective motion. At sufficiently large values of the vibration parameter, the flow becomes unstable at arbitrarily small values of the Grashof number, this being due to the mechanism of vibrational convection, which leads to instability even under conditions of weightlessness, when the main flow is absent [3, 4].     

Secondary convective motions in a plane vertical layer

Secondary plane-parallel motion in a vertical layer between isothermal planes heated to different temperatures is unstable at low and moderate values of the Prandtl number with respect to monotonically increasing disturbances [1]. The results of numerical experiments carried out by the method of networks [2, 3] indicate that this instability leads to the development of stationary secondary motions; the secondary motions have also been investigated in [4] by averaging the original equations. In the present paper we consider plane and three-dimensional stationary spatially periodic secondary motions near the threshold at which the motions develop. We make use of the methods of branching theory which were used earlier for the investigation of isothermal flows [5–9]. We determine the regions of “soft∝ and “hard∝ instability of the plane-parallel motion and the region of stability of the secondary motions. We give the results obtained by calculation of the basic characteristics of the secondary motions.     

Stability of a plane-parallel convective flow of a liquid in a horizontal layer

We consider the stationary plane-parallel convective flow, studied in [1], which appears in a two-dimensional horizontal layer of a liquid in the presence of a longitudinal temperature gradient. In the present paper we examine the stability of this flow relative to small perturbations. To solve the spectral amplitude problem and to determine the stability boundaries we apply a version of the Galerkin method, which was used earlier for studying the stability of convective flows in vertical and inclined layers in the presence of a transverse temperature difference or of internal heat sources (see [2]). A horizontal plane-parallel flow is found to be unstable relative to two critical modes of perturbations. For small Prandtl numbers the instability has a hydrodynamic character and is associated with the development of vortices on the boundary of counterflows. For moderate and for large Prandtl numbers the instability has a Rayleigh character and is due to a thermal stratification arising in the stationary flow.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 95–100, January–February, 1974.     

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.     

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.     

Stability of steady plane-parallel convective motion of a chemically active medium

A study is made of a vertical plane layer of reacting fluid whose boundaries are maintained at constant equal temperatures. As a result of heating due to a chemical reaction of zeroth order taking place in the fluid a steady plane-parallel convective flow develops in the layer, and if the internal heat release is sufficiently intense this can become unstable. The linear stability of this motion has hitherto been considered only in the hydro-dynamic formulation [1], in which one can ignore the thermal perturbations and their influence on the development of the hydrodynamic perturbations (the region of small Prandtl numbers). In the present paper, the stability boundary is determined for arbitrary values of the Prandtl number and the Frank-Kamenetskii parameter FK characterizing the steady plane-parallel regime. An important difference between this flow and the types of convective motion hitherto studied [2] is that the basic planeparallel flow of the reacting medium is possible only in a definite range of the parameter FK: At values of the parameter exceeding a critical value, there is a thermal explosion — abrupt strong heating of the fluid. This is due to the essentially nonlinear dependence of the heat release of a chemical reaction on the temperature.     

Effect of a Standing Acoustic Wave on the Development of Large-Scale Convection in a Horizontal Layer

The effect of a standing acoustic wave on the development of long-wave convective perturbations in a horizontal layer with thermally insulated boundaries is investigated. The main two-dimensional flow is determined. A nonlinear amplitude equation with spatially-periodic coefficients is derived for investigating the stability of the main flow and secondary convection flows in the neighborhood of the stability threshold. The intensity of the acoustic field is assumed to be low. It is shown that the acoustic action leads to destabilization of the layer. Plane and three-dimensional perturbations are critical at large and small Prandtl numbers, respectively. Nonlinear one-dimensional steady-state solutions of the amplitude equation are obtained and their stability is investigated.     

Steady convective motions in a plane horizontal fluid layer with permeable boundaries

In a plane horizontal fluid layer bounded by permeable plane surfaces which are heated to different temperatures and between which transverse flow takes place with uniform velocity, convection occurs at a definite critical Rayieigh number. The study of the disturbance spectrum and the convective stability, made within the framework of linear theory in [1], showed that convective instability in the layer with permeable boundaries, just as in the case of the Rayieigh problem, is associated with the development of monotonie disturbances. It turns out that the transverse motion in the layer leads to a considerable increase of the Rayieigh number. Linear theory does not permit analysis of the development of the disturbances in the supercritical region. Analysis of the developed nonlinear motion can be made only on the basis of the complete nonlinear convection equations.In this investigation we made a numerical study of nonlinear motions in the supercritical region. Calculations were made on a computer via the grid method. Solutions are obtained for the nonlinear equations of motion over a wide range of Rayieigh numbers for different values of the Peclet number, whichdefines the intensity of the transverse motion in the layer.The author wishes to thank E. M. Zhukovitskii for his guidance, and G. Z. Gershuni and E. L. Tarunin for their interest and assistance in the study.     

Stability of convective flow in a vibration field with respect to three-dimensional perturbations

A plane-parallel convective flow in a vertical layer between boundaries maintained at different temperatures becomes unstable when the Grashof number reaches a critical value (see [1]). In [2, 3] the effect of high-frequency harmonic vibration in the vertical direction on the stability of this flow was investigated. The presence of vibration in a nonisothermal fluid leads to the appearance of a new instability mechanism which operates even under conditions of total weightlessness [4]. As shown in [2, 3], the interaction of the usual instability mechanisms in a static gravity field and the vibration mechanism has an important influence on the stability of convective flow. In this paper the flow stability is considered for an arbitrary direction of the vibration axis in the plane of the layer and the stability characteristics with respect to three-dimensional normal perturbations are determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 116–122, March–April, 1988.     

Numerical study of convection of a liquid heated from below

The equilibrium of a liquid heated from below is stable only for small values of the vertical temperature gradient. With increase of the temperature gradient a critical equilibrium situation occurs, as a result of which convection develops. If the liquid fills a closed cavity, then there is a discrete sequence of critical temperature gradients (Rayleigh numbers) for which the equilibrium loses stability with respect to small characteristic disturbances. This sequence of critical gradients and motions may be found from the solution of the linear problem of equilibrium stability relative to small disturbances. If the temperature gradient exceeds the lower critical value, then (for steady-state heating conditions) there is established in the liquid a steady convective motion of a definite amplitude which depends on the magnitude of the temperature gradient. Naturally, the amplitude of the steady convective motion cannot be determined from linear stability theory; to find this amplitude we must solve the problem of convection with heating from below in the nonlinear formulation. A nonlinear study of the steady motion of a liquid in a closed cavity with heating from below was made in [1]. In that study it was shown that for Rayleigh numbers R which are less than the lower critical value R_c steady-state motions of the liquid are not possible. With R>R_c a steady convection arises, whose amplitude near the threshold is small and proportional to (R–R_c)^1/2 (the so-called soft instability)-this is in complete agreement with the results of the phenom-enological theory of Landau [2, 3].Primarily, various versions of the method of expansion in powers of the amplitude [4–8] have been used, and, consequently, the results obtained in those studies are valid only for values of R which are close to R_c, i. e., near the convection threshold.It is apparent that the study of developed convective motion far from the threshold can be carried out only numerically, with the use of digital computers. In [9, 10] the numerical methods have been successfully used for the study of developed convection in an infinite plane horizontal liquid layer.The present paper undertakes the numerical study of plane convective motions of a liquid in a closed cavity of square section. The complete nonlinear system of convection equations is solved by the method of finite differences on a digital computer for various values of the Rayleigh number, the maximal value exceeding by a factor of 40 the minimal critical value R_c. The numerical solution permits following the development of the steady motion which arises with R>R_c in the course of increase of the Rayleigh number and permits study of the oscillatory motions which occur at some value of the parameter R. The heat transfer through the cavity is studied. The corresponding linear problem on equilibrium stability is solved approximately by the Galerkin method.     

Spontaneous swirling in axisymmetric MHD flows of an ideally conducting fluid with closed streamlines

The region of instability of the Hill-Shafranov viscous MHD vortex with respect to azimuthal axisymmetric perturbations of the velocity field is determined numerically as a function of the Reynolds number and magnetization in a linear formulation. An approximate formulation of the linear stability problem for MHD flows with circular streamlines is considered. The further evolution of the perturbations in the supercritical region is studied using a nonlinear analog model (a simplified initial system of equations that takes into account some important properties of the basic equations). For this model, the secondary flows resulting from the instability are determined. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 40–50, May–June, 2007.     

Effect of high-frequency vibration on the development of secondary convective conditions

An investigation is made of the development of convective flows of a viscous incompressible liquid, subjected to high-frequency vibration. The nonlinear equations of convection are used in the Boussinesq approximation, averaged in time. The amplitude of the perturbations is assumed to be small, but finite. For a horizontal layer with solid walls the existence of both subcritical and supercritical stable secondary conditions is established. In a linear statement, the problem of stability in the presence of a modulation has been discussed in [1–3]. Articles [4–6] were devoted to investigation of the nonlinear problem. In [4], the method of grids was used to study secondary conditions in a cavity of square cross section. In the case of a horizontal layer with free boundaries [5, 6], the character of the branching is established by the method of a small parameter.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 90–96, March–April, 1976.The authors thank I. B. Simonenko for his useful evaluation of the work.     

Vibrational Convection in an Inclined Fluid Layer Heated from Below

The stability of mechanical equilibrium of an inclined fluid layer with respect to three-dimensional perturbations under the action of high-frequency vibration is studied. It is shown that under heating from below the spiral perturbations are always the most dangerous for vibration transverse to the layer. For vertical vibration the stability limit is determined by three-dimensional perturbations whose shape depends in a complicated way on the angle of inclination of the layer and the vibrational Rayleigh number. In the limiting case of a thin vertical layer supercritical vibrational-convective motions are calculated numerically and analytically and scenarios of transition from quasi-equilibrium to irregular motions are studied.     

Stability of stationary convective flow of a mixture in a vertical porous layer

In contrast to the corresponding viscous flow, the convective flow of a homogeneous liquid in a planar vertical layer whose boundaries are maintained at different temperatures is stable [1]. When a porous layer is saturated with a binary mixture, in the presence of potentially stable stratification one must expect an instability of thermal-concentration nature to be manifested. This instability mechanism is associated with the difference between the temperature and concentration relaxation times, which leads to a buoyancy force when an element of the fluid is displaced horizontally. In viscous binary mixtures, the thermal-concentration instability is the origin of the formation of layered flows, which have been studied in detail in recent years [2–4]. The convective instability of the equilibrium of a binary mixture in a porous medium was considered earlier by the present authors in [5]. In the present paper, the stability of stationary convective flow of a binary mixture in a planar vertical porous layer is studied. It is shown that in the presence of sufficient longitudinal stratification the flow becomes unstable against thermal-concentration perturbations; the stability boundary is determined as a function of the parameters of the problem.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 150–157, January–February, 1980.     

Transition to instability of a phase interface in a porous medium in the isothermal approximation

The stability of the phase interface in geothermal systems is considered in the isothermal approximation with allowance for capillary effects. The dispersion relation is obtained and the domains of stability and instability of steady-state vertical flows are found. Possible types of transition to instability, namely, transitions with the most unstable mode corresponding to zero and infinite wavenumbers or to all wavenumbers simultaneously, are described. In the first case the nonlinear Kolmogorov-Petrovskii-Piskunov equation describing the evolution of a narrow strip of weakly unstable modes on the stability threshold is derived. The effect of the parameters of the system on its stability is investigated.     

Stability of plane self-similar flows in expanding regions

The possibility of applying geometrical acoustics to the investigation of the stability of flows in expanding regions was pointed out by Galin and Kulikovskii [1], who investigated the stability of homogeneous gas flows separated by discontinuity surfaces. Eckhoff [2] applied geometrical acoustics to the analysis of the stability of solutions of symmetric hyperbolic systems whose coefficients do not depend explicitly on the time. The treatment was given for unbounded regions in the case when acoustic points are absent. The stability of gas-dynamic flows satisfying these restrictions was considered by Eckhoff and Storesletten [3, 4]. The present paper is devoted to the question of the stability of plane self-similar flows in expanding regions [5] with respect to weak two-dimensional perturbations. Propagation of perturbations through the gas is described in the approximation of geometrical acoustics [6–8]. The intensity of the perturbations is characterized by the total energy E of a wave packet, whose behavior as t → ∞ is chosen as the criterion of stability of the considered flow. It is shown that E → 0 with the time in problems of a strong explosion and a decelerated piston. In the problem of an accelerated piston, the total energy of weak perturbations increases unboundedly with the time.     

The Onset of Double-Diffusive Convection in a Superposed Fluid and Porous Layer Under High-Frequency and Small-Amplitude Vibrations

We numerically simulate the initiation of an average convective flow in a system composed of a horizontal binary fluid layer overlying a homogeneous porous layer saturated with the same fluid under gravitational field and vibration. In the layers, fixed equilibrium temperature and concentration gradients are set. The layers execute high-frequency oscillations in the vertical direction. The vibration period is small compared with characteristic timescales of the problem. The averaging method is applied to obtain vibrational convection equations. Using for computation the shooting method, a numerical investigation is carried out for an aqueous ammonium chloride solution and packed glass spheres saturated with the solution. The instability threshold is determined under two heating conditions—on heating from below and from above. When the solution is heated from below, the instability character changes abruptly with increasing solutal Rayleigh number, i.e., there is a jump-wise transition from the most dangerous shortwave perturbations localized in the fluid layer to the long-wave perturbations covering both layers. The perturbation wavelength increases by almost 10 times. Vibrations significantly stabilize the fluid equilibrium state and lead to an increase in the wavelength of its perturbations. When the fluid with the stabilizing concentration gradient is heated from below, convection can occur not only in a monotonous manner but also in an oscillatory manner. The frequency of critical oscillatory perturbations decreases by 10 times, when the long-wave instability replaces the shortwave instability. When the fluid is heated from above, only stationary convection is excited over the entire range of the examined parameters. A lower monotonic instability level is associated with the development of perturbations with longer wavelength even at a relatively large fluid layer thickness. Vibrations speed up the stationary convection onset and lead to a decrease in the wavelength of most dangerous perturbations of the motionless equilibrium state. In this case, high enough amplitudes of vibration are needed for a remarkable change in the stability threshold. The results of numerical simulation show good agreement with the data of earlier works in the limiting case of zero fluid layer thickness.     

Stability of plane wave solutions of the two-space-dimensional nonlinear Schrödinger equation

The stability of plane, periodic solutions of the two-dimensional nonlinear Schrödinger equation to infinitesimal, two-dimensional perturbation has been calculated and verified numerically. For standing wave disturbances, instability is found for both odd and even modes; as the period of the unperturbed solution increases, the instability associated with the odd modes remains but that associated with the even mode disappears, which is consistent with the results of Zakharov and Rubenchik [8], Saffman and Yuen [4] and Ablowitz and Segur [1] on the stability of solitons. In addition, we have identified travelling wave instabilities for the even mode perturbations which are absent in the long-wave limit. Extrapolation to the case of an unperturbed solution with infinite period suggests that these instabilities]may also be present for the soliton. In other words, the soliton is unstable to odd, standing-wave perturbations, and very likely also to even, travelling-wave perturbations.     

Stability of wavy conditions in a film flowing down an inclined plane

The nonlinear theory of motion in a film of liquid flowing down an inclined plane predicts the existence of an interval k₀m, inside of which the wave number of periodic wave motion may lie [1]. The condition of the stability of experimentally attained motions imposes a limitation on their wave numbers. In [2] a numerical investigation of the stability of wavy motions was made; in the investigated range of change in the Galileo number and the wave number all the motions were found to be unstable; however, the fastest growing were perturbations imposed on a motion with a determined wave number (“optimal” conditions). In [3] the instability of motions with a wavelength exceeding some limiting value was established in a long-wave approximation. In the present work, within the framework of the two-dimensional problem, an investigation was made of the stability of periodic wavy motions, based on expansion in terms of the small parameter k_m. It is established that, within the interval k₀m, there lies a finite subinterval of wave numbers for which wavy motions are stable. The narrowness of this interval (δk≈0.07 k_m) may be the reason why, in the experiment, with not too great Galileo numbers for fully established periodic wavy motions, no substantial differences in the wave-length are observed [4].     

Convection currents in a vertical layer with a space-modulated temperature distribution on the boundaries

Steady convective motions in a plane vertical fluid layer are investigated. The temperature along the boundaries of the layer varies harmonically and has different average values on each of the boundaries. Thus space-period modulation of the temperature of the walls is assigned along with average lateral heating of the layer. The form of the plane steady motions and regions of existence of through currents and currents of cellular structure are found for various values of the parameters of the problem by the finite difference grid-point method. The dependence of the main characteristics of fluid motion on the Grashof number is determined. The results presented in the article pertain to the case when the period of modulation of the temperature of the boundaries coincides with the wavelength of the critical mode of a plane-parallel current. A numerical investigation of supercritical motions in a vertical layer with plane isothermal boundaries heated to a different temperature was carried out in [1–3]. The effect of a space-periodic inhomogeneity due to curvature of walls on the form and stability of convective motions in a vertical layer with lateral heating was examined in [4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 20–25, September–October, 1978.The author thanks E. M. Zhukhovitskii for formulating the problem and supervising the work and G. Z. Gershuni for discussions and useful comments.