Thermocapillary instability of a plane layer with a vertical temperature gradient

The oscillating disturbances in a plane layer with a temperature gradient are analyzed. It is shown that for heating from below taking the deformability of the free surface into account leads to the appearance of short-wave oscillatory instability, which becomes the most dangerous mode. Moreover, the interaction of the capillary and thermocapillary instability mechanisms results in the appearance of oscillating disturbances of a new type, which lead to equilibrium crisis at high Marangoni numbers. It is established that when the free boundary is heated, the onset of convection is possible only with respect to oscillatory disturbances.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 19–23, May–June, 1992.     

Estimating the Hydraulic Conductivity of Two-Dimensional Fracture Networks Using Network Geometric Properties

Based on a modified Darcy–Brinkman–Maxwell model, a linear stability analysis of a Maxwell fluid in a horizontal porous layer heated from below by a constant flux is carried out. The non-oscillatory instability and oscillatory instability with different hydrodynamic boundaries such as rigid and free surfaces at the bottom are studied. Compared with the rigid surface cases, onset of fluid motion due to non-oscillatory instability and oscillatory instability is found to occur both more easily for the system with a free bottom surface. The critical Rayleigh number for onset of fluid motion due to non-oscillatory instability is lower with a constant flux heating bottom than with an isothermal heating bottom, but the result due to oscillatory instability is in contrast. The effects of the Darcy number, the relaxation time, and the Prandtl number on the critical Rayleigh number are also discussed.     

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.     

The instability mechanism of single and multilayer Newtonian and viscoelastic flows down an inclined plane

The instability mechanism of single and multilayer flow of Newtonian and viscoelastic fluids down an inclined plane has been examined based on a rigorous energy analysis as well as careful examination of the eigenfunctions. These analyses demonstrate that the free surface instability in single and multilayer flows in the limit of longwave disturbances (i.e., the most dangerous disturbances) arise due to the perturbation shear stresses at the free surface. Specifically, for viscoelastic flows, the elastic forces are destabilizing and the main driving force for the instability is the coupling between the base flow and the perturbation velocity and stresses and their gradient at the free surface. For Newtonian flows at finite Re, the driving force for the interfacial instability in the limit of longwaves depends on the placement of the less viscous fluid. If the less viscous fluid is adjacent to the solid surface then the main driving force for the instability is interfacial friction, otherwise the bulk contribution of Reynolds stresses drives the instability. For viscoelastic fluids in the limit of vanishingly small Re, the driving force for the instability is the coupling of the base flow and perturbation velocity and stresses and their gradients across the interface. In the limit of shortwaves the interfacial stability mechanism of flow down inclined plane is the same as plane Poiseuille flows (Ganpule and Khomami 1998, 1999a, b). Received: 20 October 2000/Accepted: 11 January 2001     

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.     

Experimental Investigation of Thermocapillary Convection Induced by a Local Temperature Inhomogeneity near the Liquid Surface. 1. Solid Source of Heat

The structure and stability of a thermocapillary flow from a concentrated source of heat located near the free surface of the liquid filling a deep reservoir are experimentally studied. For a certain power of the heat source, oscillatory instability leading to formation of surface waves is observed. Possible mechanisms of the observed instability are discussed.     

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.     

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.     

Effects of an axial flow component on the Honji instability

This paper is concerned with a numerical study of the three-dimensional Honji instability that can arise in an oscillatory flow impinging on a circular cylinder. It is well known that when the fluid motion far from the cylinder is perpendicular to its axis then the flow is liable to a three-dimensional breakdown via this instability which initially appears as an axially periodic mushroom-like structure attached to the surface of the cylinder. Here the focus is on examining the Honji instability under an oblique inflow. The obliqueness of the free stream is represented by an angle of attack through introducing an axial flow component. It is found that the Honji mode is suppressed by increasing the axial flow component, and when this component is sufficiently large the instability mechanism is no longer operative so that all that remains is a featureless two-dimensional columnar flow. At smaller values of angle of attack, though the Honji structure remains, it is deformed by the axial flow component. The developed two-layer near-cylinder vortical structures can be related to the energy and momentum transfer between the two layers.     

Basic flows of a micropolar liquid

Steady plane shear flow, rotation flow, plane wave disturbances, oscillatory boundary layers, and free surface waves in a micropolar liquid are investigated.     

Three-dimensional numerical simulation of thermocapillary flow in a shallow cylindrical pool

In order to understand the characteristics of thermocapillary flow, we conducted a series of unsteady three-dimensional (3D) numerical simulations of thermocapillary flow for fluids with Prandtl number Pr = 0.011 and 1.0 in a shallow cylindrical pool with an azimuthal temperature non-uniform, an adiabatic solid bottom and free surface. The simulation results indicate that thermocapillary flow is steady 3D flow at the small temperature non-uniform. When temperature non-uniform exceeds some critical value, the flow will undergo a transition to 3D oscillatory flow, which is characterized by traveling oscillatory wave reversely along the flow direction on the free surface and diffusing to two sides at the same time. Further, the critical conditions for the flow transition are determined.     

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 and diffusive surfactant transfer in multiphase liquid systems

Laser interferometry was used to investigate diffusive and convective mass transfer in a multicomponent fluid mixture with a liquid–liquid or liquid–gas interface. For this purpose, an immobile gas bubble or insoluble fluid droplet, having the shape of a short cylinder with a free lateral surface, was inserted into a thin liquid layer. In the case of non-uniform distribution of the dissolved surfactant component, the Marangoni convection near the drop/bubble was initiated by the surface tension inhomogeneities, depending on the surfactant concentration. The applied experimental techniques allowed us to study the structure and evolution of the convective flows and concentration fields in a liquid layer, which due to its small thickness were nearly two-dimensional. Making use of both the vertical and horizontal orientation of the liquid layer, we investigated the mass transfer process at different levels of the interaction between gravity and capillary forces. During the experiments, we detected new solutocapillary phenomena, which were found to be caused by oscillatory regimes of solutal convection occurring around air bubbles and chlorobenzene drops in heterogeneous aqueous solutions of alcohol with a vertical surfactant concentration gradient. The role of the oscillatory instability in the processes of drop saturation by the surfactant from its water solution and an inverse process of surfactant extraction from the drop into the surrounding homogeneous fluid (water) was determined. A reasonable explanation for the driving mechanisms of the discovered effects has been proposed.     

Stress-driven interfacial instability of a bilayer structure: The interaction between free surface and interface

The morphological evolution of strained films is of technological importance to microelectronics and nanotechnology. The morphological instability of a bilayer system is analyzed, consisting of an elastic film and an elastic substrate with a misfit strain on the coherent interface. A kinetic model is derived by considering the morphological fluctuations of different perturbation amplitudes along both the free surface and the interface and the coupling effect between the film and the substrate. The couplings include the misfit strain, surface/interface energy, and surface/interface diffusion, which determine the morphological instability of the system. A quadratic dispersion relationship is established for the growth rate of the longitudinal surface and interfacial perturbations along the free surface and the interface, respectively. The propagation of the surface perturbations is revealed from the free surface to the interface, and the characteristic frequencies are identified for the initiation of the morphological instability.     

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.     

Experimental and numerical investigation of nonlinear thermocapillary oscillations in an annular geometry

The present work investigates thermocapillary flow in a cylindrical configuration using large Prandtl number liquids. The flow is studied using coordinated simulations and experimental optical methods such as PIV and flow visualization. In this way, properties of the oscillatory state can be obtained in great detail. Considerable attention is given to the search for the parameters influencing the onset of the instability. It is found that the onset of oscillations can be correlated using a thermocapillary Reynolds number. The oscillations take the form of a standing wave close to the onset, which is replaced by a travelling wave for stronger forcing. The selection of azimuthal wave number of the oscillatory mode is determined from geometrical parameters, and resembles the wave number selection in vortex ring instabilities. Throughout we obtain good agreement between experiments and simulations using a mathematical model with an undeformed adiabatic free surface.     

Development of a steady relief at the interface of fluids in a vibrational field

The vibrations of a vessel strongly influence the behavior of the interface of the fluids in it. Thus, vertical vibrations can lead both to the parametric excitation of waves (Faraday ripples) and to the suppression of the Rayleigh-Taylor instability [1–2]. At the present time, the influence of vertical vibrations on the behavior of a fluid surface have been studied in sufficient detail (see, for example, review [3]). The behavior of an interface of fluids in the case of horizontal vibrations has been studied less. An interesting phenomenon has been revealed in the experimental papers [4, 5]: in the case of fairly strong horizontal vibrations of a vessel containing a fluid with a free surface, the fluid collects near one of the vertical vessel walls, the free surface being practically plane and stationary with respect to the vessel, while its angle of inclination to the horizon depends on the vibration rate. But if there is a system of immiscible fluids with comparable but different densities in the vessel, horizontal vibrations lead to the formation of a steady wave relief at the interface. An explanation of the behavior of a fluid with a free boundary was given in [6] on the basis of averaged equations of fluid motion in a vibrational field. The present paper is devoted to an analysis of the behavior of the interface of fluids with comparable densities in a high-frequency vibrational field. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 8–13, November–December, 1986.     

Experimental studies on the dynamic stability of liquid in a spherical tank covered with diaphragm under vertical excitation

Experimental studies are conducted on the liquid sloshing characteristics in a spherical tank covered with a flexible diaphragm. A spherical acrylic tank with 145.2 mm radius is used as a test tank, and is filled with water. Silicon diaphragms, plane or hemispherical type, with 0.2 mm thickness are used as test diaphragms. The test tank is harmonically excited in the vertical direction by an electro-dynamic exciter. During the test, vibrations due to parametric instability occur when the excitation frequency is twice the natural frequency. Parametric instability regions for some natural modes are measured and are presented in the excitation frequency–excitation acceleration diagram for three cases: liquid surface is uncovered (i.e., free surface), covered with a plane diaphragm, and covered with a hemispherical diaphragm, with the volume of filling water being changed appropriately.     

Solitary waves on the surface of a viscoelastic fluid running down an inclined plane

In this paper, we study the existence and the role of solitary waves in the finite amplitude instability of a layer of a second-order fluid flowing down an inclined plane. The layer becomes unstable for disturbances of large wavelength for a critical value of Reynolds number which decreases with increase in the viscoelastic parameter M. The long-term evolution of a disturbance with an initial cosinusoidal profile as a result of this instability reveals the existence of a train of solitary waves propagating on the free surface. A novel result of this study is that the number of solitary waves decreases with in crease in M. When surface tension is large, we use dynamical system theory to describe solitary waves in a moving frame by homoclinic trajectories of an associated ordinary differential equation.     

Unsteady laminar flow developing in a curved duct

The flow developing in a tightly curved U-bend of square cross section has been investigated experimentally and via numerical simulation. Both long-time averages and time histories of the longitudinal (streamwise) component of velocity were measured using a laser-Doppler velocimeter. The Reynolds number investigated was Re = 1400. The data were obtained at different bend angles, θ, and were confined to the symmetry plane of the bend. At Re = 1400, the flow entering the bend is steady, but by θ = 90° it develops an oscillatory component of motion along the outer-radius wall. Autocorrelations and energy spectra derived from the time histories yield a base frequency of approximately 0.1 Hz for these oscillations. Flow-visualization studies showed that the proximity of the outer-radius wall served to damp the amplitude of the spanwise oscillations.

Numerical simulations of the flow were performed using both steady and unsteady version of the finite-difference elliptic calculation procedure of Humphrey et al. (1977). Although the unsteadiness observed experimentally does not arise spontaneously in the calculations, numerical experiments involving the imposition of a periodic time-dependent perturbation at the inlet plane suggest that the U-bend acts upon the incoming flow so as to damp the amplitude of the imposed oscillation while altering its frequency.

The oscillations observed experimentally, and numerically as a result of the periodic perturbation, have been linked to the formation of Goertler-type vortices of the outer-radius wall in the developing flow. The vortices, which develop as a result of the centrifugal instability of the flow on the outer-radius wall, undergo a further transition to an unsteady regime at higher flow rates.