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.     

Secondary convective motions in a plane inclined layer

A linear theory of stability of a plane-parallel convective flow between infinite isothermal planes heated to different temperature was developed in [1–6]. At moderate Pr values the instability is monotonic and leads to the development of steady secondary motions. These motions for the case of a vertical layer have been investigated by the net [7, 8] and small-parameter [9] methods. In this paper steady secondary motions in an inclined layer are investigated. The small-parameter and net methods are used. The hard nature of excitation of secondary motions in a defined range of tilt angles is established. There are two types of secondary motions, whose regions of existence overlap — vortices at the boundary of countercurrent streams and convection rolls; the hard instability is due to the development of convection rolls. The analog of the Squire transformation obtained in [4] for infinitely small disturbances of a plane-parallel convective flow is extended to secondary motions of finite amplitude.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 3–9, May–June, 1977.I thank G. Z. Gershumi, E. M. Zhukhovitskii, and E. L. Tarunin for interest in the work and valuable discussion.     

Secondary vibrational convective motions in a flat vertical layer of liquid

Investigations of the stability of steady-state plane-parallel convective motion between vertical planes heated to different temperatures [1–5] have shown that this motion, depending on the value of the Prandtl number P, exhibits instability of two types. With small and moderate Prandtl numbers, the instability is of a hydrodynamic nature. It is brought about by monotonic perturbations which, in the supercritical region, develop into a periodic, with respect to the vertical, system of steady-state vortices at the interface between the opposing convective flows. Articles [6, 7] are devoted to the numerical investigation of nonlinear secondary steady-state flows. If P>11.4, there appears a new mode of instability, i.e., running thermal waves increasing in the flow; with P>12, this mode becomes more dangerous [4]. This instability is connected with the development of vibrational perturbations, and it can be considered that in the supercritical region the perturbations lead to the establishment of steady-state vibrations. Linear theory has made it possible to determine the boundaries of the regions of stability. In the present article a numerical investigation is made of nonlinear supercritical conditions developing as a result of a loss of stability of the steady-state flow with respect to vibrational perturbations.     

Particle-image-velocimetry applied to thermocapillary convection

Thermocapillary convection is studied experimentally using particle-image-velocimetry for flow visualization and analysis. This method offers the advantage of measuring the entire flow field (velocity field, streamlines etc.) in a selected plane within the fluid at a given instant of time in contrast to point by point methods like laser-Doppler-velocimetry (LDV). The paper describes the method and presents quantitative results for different Marangoni numbers.Presented in part at the VIIth European Symposium on Materials and Fluid Sciences in Microgravity, Oxford University, UK, September 10–15, 1989     

Droplet interactions in thermocapillary motion

The dynamics of two spherical droplets, whose movement is driven by variations in interfacial tension caused by a temperature gradient, are analyzed using a method of reflections. Both droplets have the same internal fluid properties but may differ in size, and the configuration of the droplets is arbitrary relative to the direction of the undisturbed temperature gradient. The method of reflections is based on an analysis of the thermal and hydrodynamic disturbances produced by a single droplet placed in an arbitrarily varying temperature field. The results for two-droplet interactions are correct to 0(r⁻⁶) where r is the center-to-center distance between the droplets. For the specific case of bubbles oriented parallel with the undisturbed temperature gradient, my results agree very well with numerical calculations found in the literature. The results for two, identical droplets are used to find the 0(ø) effect on mean droplet velocity of a bounded suspension, where ø is the volume fraction of the droplets. In general, the interactions between droplets moving by thermocapillary effects are much weaker than in the case of sedimentation.     

Experimental study on thermocapillary convection

In the present paper, the experimental studies on thermocapillary convection are reviewed. The author‘s interest is mainly focused on the onset of oscillatory thermocapillary convection,the features of oscillatory flow pattern, and the critical Marangoni number related with temperature and free surface oscillation. The coordinated measurement in a microgravity environment of a drops haft is also addressed.     

Universal motions for a class of viscoelastic materials of differential type

Universal quasi-static motions for a class of incompressible, viscoelastic materials of differential type are examined. These time dependent motions are similar to corresponding static universal deformations well-known for incompressible, isotropic elastic materials. General details are illustrated for the pure torsion problem, and specific results and physical effects are provided for the viscoelastic Mooney-Rivlin model.     

A new type of atomic force microscope based on chaotic motions

Local flow variation (LFV) method of non-linear time series analysis is applied to develop a chaotic motion-based atomic force microscope (AFM). The method is validated by analyzing time series from a simple numerical model of a tapping mode AFM. For both calibration and measurement procedures the simulated motions of the AFM are nominally chaotic. However, the distance between a tip of the AFM and a sample surface is still measured accurately. The LFV approach is independent of any particular model of the system and is expected to be applicable to other micro-electro-mechanical system sensors where chaotic motions are observed or can be introduced.     

Characteristics of surface oscillation in thermocapillary convection

The characteristics of surface oscillation in a rectangular pool of silicone oil have been investigated experimentally. The horizontal cross-section of the pool is 52 mm × 36 mm, and the depth of the silicone oil layer is in the range of 1.1-4.8 mm. The applied temperature difference between the two sidewalls leads to shear flow along the free surface from hot to cold and a back flow in the underlying layer. With the increase of the temperature difference, the original steady flow will become unstable to unsteady flow. A CCD laser displacement-sensor with high resolution is used to measure the position of the liquid surface dynamically. And the Hilbert-Huang transform is chosen to analyze the experiment data which is nonlinear and non-stationary. The characteristics of surface oscillation have been obtained. And the relationship of the characteristics with the temperature difference and liquid layer depth has been discussed in details.     

Characteristics of thermocapillary convection in two-component fluids

The linear problem of convective instability near the surface of a stratified two-component liquid medium (for example, saline sea water) is considered. The specificity of the problem consists in need to take into account both background stratifications and a difference in the transfer coefficients and boundary conditions for two substances (heat and admixture concentration), as well as the thermocapillary effect. It is shown that there is a vast region of monotonic instability in the medium stable in accordance with all criteria previously known. The two-component nature of the medium makes the development of anomalously intense perturbations deeply penetrating into the hydrostatically stable medium possible. A dimensionless instability criterion is formulated and neutral curves are calculated.     

On periodic motions of lattices of Toda type via critical point theory

Communicated by P. Rabinowitz     

Spiral thermocapillary flows in two-layer systems

Parallel thermocapillary flows in infinite layers can be calculated easily. However, in reality thermocapillary flows occur in channels or closed cavities, and they are three-dimensional. In the present paper, a two-layer fluid system filling a channel with a rectangular cross section is considered. A numerical investigation of three-dimensional spiral thermocapillary flows generated by a temperature gradient imposed along the channel has been performed. Both the case of a zero longitudinal pressure gradient (a through flow in the channel) and that of a zero longitudinal fluid flux (a flow in the closed cavity) are investigated. Steady and oscillatory thermocapillary motions, and transitions between them are studied.     

双液滴热毛细迁移的实验研究

在地面实验中,研究在竖直向上温度梯度流场中,双液滴的热毛细迁移运动和它们间的相互 作用. 豆油和硅油分别作为实验系统的母液和液滴. 液滴迁移轨迹被记录和分析. 实验结果 表明,大小液滴的实验迁移速度均小于理论迁移速度;两个液滴之间的相互作用对小液滴的 影响更为明显,使之出现倾斜``8'字的迁移轨迹;两个液滴的运动速度都出现了振荡现象, 小液滴可能出现短时反向迁移. 实验结果与数值模拟结果较为相似,但尚有差异,需要进一 步进行研究.     

Interaction between thermocapillary and buoyancy driven convection

Convective flows driven by the variation of surface tension due to a radial temperature gradient along a liquid-gas interface were studied. Three liquids of different viscosities were applied, so that a wide range of Marangoni numbers was encountered. Light sheet technique and differential interferometry were taken to analyse the thermal flows. The mechanism of stationary thermocapillary convection, the influence of the radial temperature gradient and the kinematic viscosity on the Marangoni boundary layer thickness are discussed. Transitions from the steady to the oscillatory Marangoni convection are discovered and the oscillations are visualized with differential interferometry.List of symbols a thermal diffusivity - D cell diameter - f tangential stress - H cell height - Mg Marangoni number, Mg = U · R/a - Pr Prandtl number, Pr = v/a - r radial coordinate tangential to the interface - R cell radius - Re Reynolds number, Re = UR/v - T temperature - T _b, T_m temperature at the boundary and in the centre of the cell, respectively - T temperature difference, T — T _b — T_m - U reference velocity, U = ¦d/dT¦(T/R) R/ - v _r radial stream velocity - v _x velocity at the interface - z axial coordinate normal to the interface - dynamic viscosity - kinematic viscosity - surface tension - d/dT thermal coefficient of surface tension A version of this paper was presented at the 7th Physico-Chemical Hydrodynamics, PCH Conference, June 25–29, 1989, Cambridge, MA, USA     

Interaction of the thermovibrational and thermocapillary convection mechanisms

Thermal convection in microgravity induced by the combined action of thermocapillary forces and vibrations is considered. In the high-frequency approximation the thermovibrational forces are written for the case in which isothermal mixing of the fluid is unimportant. The authors' own work and new results of investigating the effect of vibration on thermocapillary instability, of numerically modeling the heat and mass transfer in open cavities and of experiments on the vibrational suppression of surface deformation are reviewed. The prospects for the practical application of the results obtained are discussed.     

On the regularity of some thermocapillary convection models

The classical model of confined thermocapillary convection is analyzed. Its vorticity singularity, independent of the contact angle, leads to infinite pressure values at contact lines, forbidding any numerical use of the Laplace equation to calculate free surface shapes. Four models are explored to overcome this difficulty: an explicit polynomial filtering, a Navier slip at the solid boundaries, an interface viscosity model and the combination of slip and interface viscosity. Regular solutions are obtained with the first and last approaches. Only the last one is based on physical considerations and, by the introduction of physical length scales, avoids infinite pressure values at the contact line.     

Stability of thermocapillary motion in a cylindrical layer

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 50–52, July–August, 1989.     

Effect of thermocapillary forces on free-surface fluid motion

The investigation of thermocapillary convection in fluids with a parabolic temperature dependence of the surface tension is continued within the context of the problem formulated in [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 3–7, September–October, 1989.     

Three-dimensional numerical simulation of thermocapillary instabilities in floating zones

Instability of thermocapillary convection in liquid bridges of low Prandtl number fluids is investigated by direct three-dimensional and time-dependent simulation of the problem. The field equations are numerically solved explicit in time and with finite difference methods in a staggered cylindrical grid. The numerical results are analyzed and interpreted in the general context of the bifurcation's theory. According to recent stability analyses the computations show that for semiconductor melts the first bifurcation is characterized by the loss of spatial symmetry rather than by the onset of oscillatory flow. When the basic axisymmetric flow field becomes unstable, after a short transient, a three-dimensional supercritical steady state is obtained. It is shown that the flow field organization, depending on the critical wave number, is related to the geometrical aspect ratio of the liquid bridge and that lower is the aspect ratio, higher is the critical wave number and more complex the thermofluid-dynamic field structure.