Instability of a spherical gas bubble in a variable-pressure field

Only a few studies, of which we mention [1–5], have been addressed to the problem of the stability of the accelerated motion of a spherical interface of two fluids. In the present paper we consider the problem of the stability of radial motion of the spherical boundary of a gas bubble in an incompressible inviscid liquid under the action of variable external pressure. Surface tension is not taken into account. We study the possibility of the existence of stable motions for broad classes of time dependence of the external pressure, namely for monotonic and periodic dependences. It is shown that stability is possible only for infinitely large bubble radii or for very specific assumptions concerning the initial conditions and the pressure-time dependence law.     

The stability of the motion of a liquid,due to thermocapillary forces

A study is made of the stability of the plane-parallel flow of a viscous liquid in a layer with a free boundary, under weightless conditions. The motion of the liquid is due to the dependence of the surface tension on the temperature. An exact solution for an unperturbed boundary is obtained by the same method used in [1], but with a more general boundary condition for the temperature. A study of the stability was carried out by the method of small vibrations, taking account of the perturbation of the free boundary. The article discusses the asymptotic behavior of long waves at small Reynolds numbers, and the conditions for instability are found.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 94–98, November–December, 1971.     

Evolution of finite perturbations in a viscoelastic relaxing liquid with gas bubbles

The propagation of one-dimensional perturbations in a viscoelastic relaxing liquid containing gas bubbles is investigated within the framework of the homogeneous model of the medium when the wavelength of the perturbation is much larger than the distance between the bubbles and the bubble radius. The evolution of stationary and nonstationary waves is investigated analytically and with the use of numerical integration; shock waves are also investigated. The results are compared with the behavior of perturbation waves in a Newtonian liquid with gaseous inclusions. The models of the gas-liquid medium [1, 2] are generalized to the case when the liquid phase is a viscoelastic liquid, for example, a weak aqueous solution of polymers. The propagation of longwave perturbations of finite amplitude in such a mixture is investigated using the technique developed in [3].     

Effect of a latitudinal temperature gradient on convective motions arising in a rotating spherical layer

The hydrodynamics of planetary atmospheres and Interiors are frequently directly or indirectly connected with convective motions taking place in rotating liquid spherical layers in the field of a central force. Convective stability in a spherical layer at rest, in a central gravity field, was first discussed in [1, 2]. It was shown that the critical Rayleigh number Rao at which convective instability sets in and the wave number of the critical perturbations depend essentially on the thickness of the layer. As in the plane case, the problem of the convective stability of a spherical layer is found to be degenerate, and the form of the critical perturbations cannot be determined from the linear problem. In actuality, minimization of the Rayleigh number permits establishing only the wave numberl for the spherical harmonic Y _l ^m (θ, ?), realized at the limit of stability; the parameter m remains indeterminate and thus 2l+1 independent convective modes correspond to Rao. In [3] a study was made of the convective stability of a liquid in a slowly rotating thin spherical layer. It was shown that the presence of rotation eliminates the degeneracy; at the limit of stability there arise motions corresponding to the Y _l l (θ, ?) -harmonic with a degenerate maximum at the equator, and propagating in a wave manner toward the side opposite to the rotation. In the present work a study is made of the convective stability of a flow of liquid, arising in a rotating spherical layer due to a nonuniform distribution of the temperatures at one of the boundaries of the layer. In such a statement of the problem it is possible to model large-scale motions in the atmospheres of large planets having internal sources of heat and absorbing solar radiation near the cloud cover of the atmosphere. It is established that, depending on the relationships between the parameters imparting the rotation and the inhomogeneous distribution of the temperature, there is either stabilization or destabilization of the layer in comparison with a fixed layer of the same thickness and with the same, but uniformly distributed heat flux supplied to the layer. A study is made of the form of the corresponding critical perturbations.     

Small perturbations spectrum in boundary-layer flow on a flat plate

The small perturbation spectrum of a number of flows has recently been analyzed carefully [1–3]. At the same time, investigations for the boundary layer have been limited within the framework of linear perturbation theory to the neighborhood of the neutral curve although a spectrum analysis is of indubitable interest not only to find the stability criterion of a laminar stream, but also to solve a problem with initial data about the time development of an arbitrary small perturbation. In particular, the possibility of representing an arbitrary perturbation in terms of a system of basis functions is related to the question of the completeness of the system. The finiteness was proved [4] and an estimate was obtained of the domain of eigenvalue existence in an investigation of the boundary-layer stability and a deduction has been made about the finiteness of the small perturbations spectrum for boundary-layer flow on this basis. A sufficiently complete survey of the investigation of the neutral stability of a laminar boundary layer can be found in the monograph [5]. The small perturbations spectrum in a boundary layer flow is obtained in this paper by methods of the linear theory of hydrodynamic stability by using the complete boundary conditions on the outer boundary. It is shown that the small perturbations spectrum is finite for each fixed value of the wave number . Singularities in the spectrum behavior are investigated for sufficiently small .Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 112–115, July–August, 1975.The author is grateful to M. A. Gol'dshtik and V. N. Shtern for useful discussions of the results of the research.     

Stability of quasistatic equilibrium state of an inhomogeneous viscous layer on an inclined plane

A study is made of the stability against small perturbations [1] of a slow flow of an incompressible inhomogeneous linearly viscous liquid under the influence of a force of gravity on an unbounded inclined plane. Problems of such kind arise in glaciology when one estimates the stability of snow on mountain slopes or determines the catastrophic movement of a glacier; the results can also be applied to solifluction phenomena [2, 3]. Equations for perturbations of parallel flows of linearly viscous fluids in the case of a continuous variation of the viscosity and density across the flow were derived in [4]. An attempt to solve the hydrodynamic problem with allowance for a perturbation of the viscosity was made in [5]; however, in the equations for the perturbations, simplifications resulted in the neglect of terms that take into account perturbations of the viscosity. In the quasistatic formulation considered here in the case when allowance is made for perturbation of the density and viscosity, the equation for the perturbation amplitudes is an ordinary differential equation with variable coefficients; analytic solution of the equation is very difficult, even for long-wave perturbations. In this connection a study is made of the stability of a laminar model; the viscosity and density are constant within each layer. A similar hydrodynamic problem in the long-wave approximation was considered in [6]. In the present paper an exact solution is constructed in the quasistatic formulation for a two-layer model; the solution shows that the viscosity of the lower layer has an important influence on the stability. Essentially, instability is observed when the lower layer acts as a lubricant.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 20–24, November–December, 1973.     

On the spatial localization of the laminar boundary layer in a non-Newtonian dilatant fluid

In dilatant fluids the shear perturbation propagation rate is finite, in contrast to Newtonian and pseudoplastic fluids in which it is infinite [1]. Therefore, in certain dilatant fluid flows, frontal surfaces separating regions with zero and nonzero shear perturbations may be formed. Since, in a sense, the boundary layer is a “time scan” of the nonstationary shear perturbation propagation process, in dilatant fluids the boundary layer should definitely be spatially localized. This was first mentioned in [2] where, however, it was mistakenly asserted that boundary layer spatial localization does not take place in all dilatant fluids and is absent in so-called “hardening” dilatant fluids. In [3], the solutions of the laminar boundary-layer equations for speudoplastic and “hardening” dilatant fluids were investigated qualitatively. The formation of frontal surfaces in dilatant fluid flows is usually mathematically related with the existence of singular solutions of the corresponding differential equations [4]. However, since the analysis performed in [3] was inaccurate, in that study singular solutions were not found and it was incorrectly concluded that in “hardening” dilatant fluids there is no spatial boundary layer localization. The investigation performed in [5] showed that in fact in “hardening” dilatant fluids boundary layers are spatially localized, since there exist singular solutions of the corresponding differential equations. Subsequently, this result was reproduced in [6], where an attempt was also made to carry out a qualitative investigation of the solutions of the laminar boundary-layer equations for other types of dilatant fluids. The author did not find singular solutions in this case and mistakenly concluded that in these fluids there is no spatial boundary layer localization. This misunderstanding was due to the fact that in [6] it was not understood that in dilatant fluid flows the formation of frontal surfaces can be mathematically described not only in relation to the existence of singular solutions.     

Stability of flow of a thin layer of viscous magnetic liquid

The laminar flow of a thin layer of heavy viscous magnetic liquid down an inclined wall is examined. The stability and control of the flow of an ordinary liquid are affected only by alteration of the angle of inclination of the solid wall and the velocity of the adjacent gas flow. When magnetic liquids are used [1, 2], an effective method of flow control may be control of the magnetic field. By using magnetic fields of various configurations it is possible to control the flow of a thin film of viscous liquid, modify the stability of laminar film flow, and change the shape of the free surface of the laminarly flowing thin film, a factor which plays a role in mass transfer, whose rate depends on the phase contact surface area. The magnetic field significantly affects the shape of the free surface of a magnetic liquid [3, 4]. In this paper the velocity profile of a layer of viscous magnetic liquid adjoining a gas flow and flowing down an inclined solid wall in a uniform magnetic field is found. It is shown that the flow can be controlled by the magnetic field. The problem of stability of the flow is solved in a linear formulation in which perturbations of the magnetic field are taken into account. The stability condition is found. The flow stability is affected by the nonuniform nature of the field and also by its direction.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 59–65, September–October, 1977.     

Effect of gas compressibility on the stability of a boundary layer above a permeable surface at subsonic velocities

In the present study we investigate the stability of a boundary layer for the condition that the velocity perturbations at the permeable surface are nonzero. The stability for the boundary layer of an incompressible liquid in such a formulation was considered in [1]. For the case of subsonic velocities the effect of compressibility on the flow inside the boundary layer is weak, and in the present article this effect was neglected. The unsteady flow in narrow pores of a permeable covering depends strongly on the compressibility of the gas. Therefore, in the derivation of the relation connecting the pressure oscillations at the permeable surface with the oscillations of the flow through it, the effect of the compressibility was taken into consideration. It is shown that the boundary conditions, and therefore also the stability of the boundary layer at the permeable surface, depend considerably on the Mach number, even for a subsonic exterior flow.     

Convection development in a liquid layer with a free boundary

In certain calculations of the critical Rayleigh number for a liquid layer with free boundary which is heated from below, the linearization method has been used and it has been assumed that the temperature perturbations disappear at the undisturbed free boundary.Proper linearization shows that the temperature perturbation is proportional to the free surface perturbation, and the latter is proportional to the normal stress perturbation with the proportionality factor F=²/gh³ (g is the free-fall acceleration, is the kinematic viscosity, h is the liquid layer thickness). In §1 we present a formulation of the problem with account for the parameter F; in §2 we consider the linearized equations and the existence of a stability threshold is proved-a positive eigenvalue-and it is established that with an increase in the parameter F/P (P is the Prandtl number) the value of the critical Rayleigh number Ra_* decreases; §3 presents the results of a numerical calculation of Ra as a function of the parameter F/P.Convection development in a liquid layer with a free surface on which a given temperature is maintained was studied in [1, 2]. The value R_*=1100 found for the critical Rayleigh number agrees well with the experimental value. In the calculations made in [1, 2] the linearization method is used, and it is assumed that the temperature perturbations disappear at the undisturbed free boundary. Strictly speaking, this assumption is not correct.Correct linearization shows that the temperature perturbation is proportional to the perturbation of the free boundary, and the latter is proportional to the normal stress perturbation (see below (2.3)).The problem formulation is presented in §1; §2 deals with the linearized equations and the existence (Theorem 2.1) is demonstrated of a stability threshold—which is a simple positive eigenvalue; §3 presents the results of a numerical calculation of R_* as a function of the parameter =F/P.     

Self-similar solutions of nonstationary equations of a plane laminar magnetohydrodynamic boundary layer

Self-similar solutions of nonstationary equations of the boundary layer in ordinary hydrodynamics are discussed in [1, 2]. In this paper self-similar solutions of nonstationary equations of a plane magnetohydrodynarnic boundary layer are sought. In this case, a transformation to curvilinear coordinates of a certain special form is employed. Its choice is determined by the requirements essential to reducing the equations of the boundary layer to a system of ordinary equations. H. Weyl's iterative method is used to solve the equations describing the flow over a plate suddenly set in motion.     

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.     

On periodic motions of an orbital dumbbell-shaped body with a cabin-elevator

The motion of a dumbbell-shaped body (a pair of massive points connected with each other by a weightless rod along which the elevator, i.e., a third point, is moving according to a given law) in an attractive Newtonian central field is considered. In particular, such a mechanical system can be considered as a simplified model of an orbital cable system equipped with an elevator. The practically most interesting case where the cabin performs periodic ??shuttle??motions is studied. Under the assumption that the elevator mass is small compared with the dumbbell mass, the Poincaré theory is used to determine the conditions for the existence of families of system periodic motions analytically depending on the arising small parameter and passing into some stable radial steady-state motion of the unperturbed problem as the small parameter tends to zero. It is also proved that, for sufficiently small parameter values, each of the radial relative equilibria generates exactly one family of such periodic motions. The stability of the obtained periodic solutions is studied in the linear approximation, and these solutions themselves are calculated up to terms of the firstorder in the small parameter. The contemporary studies of the motion of orbital dumbbell systems apparently originated in Okunev??s papers [1, 2]. These studies were continued in [3], where plane motions of an orbit tether (represented as a dumbbell-shaped satellite) in a circular orbit were considered in the satellite approximation. In [4], in the case of equal masses and in the unbounded statement, the energy-momentum method was used to perform the dynamic reduction of the problem and analyze the stability of relative equilibria. A similar technique was used in [5], where, in contrast to the above-mentioned problems, the massive points were connected by an elastic spring resisting to compression and forming a dumbbell with elastic properties. Under such assumptions, the stability of radial configurations was investigated in that paper. The bifurcations and stability of steady-state configurations of a deformable elastic dumbbell were also studied in [6]. Various obstacles arising in the construction of orbital cable systems, in particular, the strong deformability of known materials, were discussed in [7]. In [8], the problem of orbital motion of a pair of massive points connected by an inextensible weightless cable was considered in the exact statement. In other words, it was assumed that a unilateral constraint is imposed on themassive points. The conditions of stability of vertical positions of the relative equilibria of the cable system, which were obtained in [8], can be used for any ratio of the subsatellite and station masses. In turn, these results agree well with the results obtained earlier in the studies of stability of vertical configurations in the case of equal masses of the system end bodies [3, 4]. One of the basic papers in the dynamics of three-body orbital cable systems is the paper [9]. The steady-state motions and their bifurcations and stability were studied depending on the elevator cabin position in [10].     

Shapes of a rotating rheonomic rod

The case of a rotating fluid mass is one of the classical fields of mechanics [1]. In particular, the solution of creep problems for a rotating mass is actual in geophysics in connection with Earth gravity force simulation on rotating samples under laboratory conditions [2]. A special case of a rheonomic rod in a potential field was studied in [3], where it was shown that the main problem about the rod shapes is the problem of determining the relations between the Lagrangian and Euler coordinates in the creep process.In what follows, we show how this problem can be solved for a rotating rod.     

Linear stability of viscous incompressible flow in a circular viscoelastic tube

Flow stability in rigid tubes has been the subject of much research [1]. The overwhelming majority of authors of both theoretical and experimental studies now conclude that Poiseuille flow in a circular rigid tube is linearly stable. However, real tubes all possess elastic properties, the influence of which has not been investigated in such detail. For certain selected values of the parameters characterizing an elastic tube it has been shown that with respect to infinitesimal axisymmetric perturbations Poiseuille flow in the tube can be unstable [2]. In this case boundary conditions that did not take into account the fairly large velocity gradient of the undisturbed flow near the tube wall were used. The present paper reports the results of a numerical investigation of the linear stability of Poiseuille flow in a circular elastic tube with respect to three-dimensional perturbations in the form of traveling waves propagated along the system (azimuthal perturbation modes with numbers 0, 1, 2, 3, 4, and 5 are considered). It is shown that the elastic properties of the tube can have an important influence on the linear stability spectrum. In the case of axisymmetric perturbations it is possible to detect an instability which, at Reynolds numbers of more than 200, exists only for tubes whose modulus of elasticity is substantially less than that of materials in common use. The instability to perturbations of the second azimuthal mode is different in character, inasmuch as at Reynolds numbers greater than unity it occurs in stiffer tubes. Moreover, as the Reynolds number increases it can also occur in tubes of greater stiffness. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 126–134, November–December, 1986.     

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.