等热流密度加热垂直下降液膜的研究

利用等热流密度加热条件下降膜流动的三维模型方程进行线性稳定性分析和数值模拟。线性稳定性分析表明,模型方程在小到中等Reynolds数下都适用,并且流向不稳定性增长率随着Reynolds数和Marangoni数增加而增加,展向不稳定性增长率则随着Marangoni数增加而增加,随着Reynolds数增加而减小,流向和展向对扰动波数都存在一个不稳定区间。三维数值模拟表明,在等热流密度加热条件下,液膜在随机扰动的情况下最终会形成带孤立波的三维溪流状结构,液膜与气体的换热也因溪流状结构的出现而加强;在随机扰动的基础上引入占优势地位的展向最不稳定扰动会使得换热增强,液膜会提前破裂;在随机扰动的基础上引入占优势地位的流向最不稳定扰动时,液膜的换热会增强,但不会提前破裂;在随机扰动的基础上同时引入占优势地位的流向和展向最不稳定扰动时,换热会加强且液膜会提前破裂。     

Effect of an insoluble surfactant film on the stability of the concentration-driven Marangoni flow

The stability and the structure of the concentration and capillary driven Marangoni flow from a localized source is experimentally investigated in the presence of an adsorbed layer of an insoluble surfactant. It is found that the presence of the surfactant on the interface leads to the instability of the main axisymmetric flow with the result that a secondary azimuthally-periodic flow with a multivortex structure is developed. The structure of the convective motion on the interface is studied as a function of the Marangoni flow intensity and the surface density of the surfactant. The azimuthal wavenumber is shown to increase with the Marangoni number and to decrease with increase in the surface density of the surfactant. It is established that there exists a threshold value of the surface density of the surfactant at which the surface flow does not occur.     

Topological structure bifurcation of temperature field of deformable drop in Marangoni migration

The unsteady processes of the Marangoni migration of deformable liquid drops are simulated numerically in a wider range of Marangoni number (up to Ma = 500) in the present work. A steady terminal state can always be reached, and the scaled terminal velocity is a monotonic function decreasing with increasing Marangoni number, which is generally in agreement with corresponding experimental data. The topological structure of flow field in the steady terminal state does not change as the Marangoni number increases, while bifurcation of the topological structure of temperature field occurs twice at two corresponding critical Marangoni numbers. A third critical value of Marangoni number also exists, beyond which the coldest point jumps from the rear stagnation to inside the drop though the topological structure of the temperature field does not change. It is found that the inner and outer thermal boundary layers may exist along the interface both inside and outside the drop if Ma > 70. But the thickness decreases with increasing Marangoni number more slowly than the prediction of potential flow at large Marangoni and Reynolds numbers.     

Effect of vertical heat transfer on thermocapillary convection in an open shallow rectangular cavity

In order to understand the effect of the vertical heat transfer on thermocapillary convection characteristics in a differentially heated open shallow rectangular cavity, a series of two- and three-dimensional numerical simulations were carried out by means of the finite volume method. The cavity was filled with the 1cSt silicone oil (Prandtl number Pr = 13.9) and the aspect ratio ranged from 12 to 30. Results show that thermocapillary convection is stable at a small Marangoni number. With the increase of the heat flux on the bottom surface, thermocapillary convection transits to the asymmetrical bi-cellular pattern with the opposite rotation direction. The roll near the hot wall shrinks as the Marangoni number increases. At a large Marangoni number, numerical simulations predict two types of the oscillatory thermocapillary flow. One is the hydrothermal wave, which is dominant only in a thin cavity. The other appears in a deeper cavity and is characterized by oscillating multi-cellular flow. The critical Marangoni number for the onset of the oscillatory flow increases first and then decreases with the increase of the vertical heat flux. The three-dimensional numerical simulation can predict the propagating direction of the hydrothermal wave. The velocity and temperature fields obtained by three-dimensional simulation in the meridian plane are very close to those obtained by two-dimensional simulation.     

Experimental evaluation of Marangoni stress and surfactant concentration at interface of contaminated single spherical drop using spatiotemporal filter velocimetry

Spatiotemporal filter velocimetry (SFV) was extended to Lagrangian measurements with boundary-fitted measurement areas, and was applied to flows about single spherical drops of glycerol-water solution falling in stagnant silicon oil under clean and contaminated conditions to examine its applicability to the estimation of the Marangoni stress and surfactant concentration at a moving interface. Effects of bulk concentration of surfactant on the velocity field, the Marangoni stress and the surface concentration of surfactant were discussed from the measured data. As a result, we confirmed that accurate velocity distribution in the vicinity of the interface measured by SFV enables us to evaluate interfacial velocity and interfacial shear stresses and to estimate the Marangoni stress, interfacial tension and surfactant concentration at the interface with the assumption of negligible surface viscosity. The flow inside the drop and the interfacial velocity become weak due to the Marangoni stress caused by the gradient of surfactant concentration at the interface as the bulk concentration of surfactant increases. These results demonstrate that SFV is of great use in experimental analysis of adsorption and desorption kinetics at a moving interface.     

The effect of an insoluble surfactant on the skin friction of a bubble

In this paper, we develop a novel moving mesh method suitable for solving axisymmetric free-boundary problems, including the Marangoni effect induced by surfactant or temperature variation. This method employs a body-fitted grid system where the gas–liquid interface is one line of the grid system. We model the surfactant equation of state with a non-linear Langmuir law, and, for simplicity, we limit ourselves to the situation of an insoluble surfactant. We solve complicated dynamic boundary conditions accurately on the gas–liquid interface in the framework of finite-volume methods. Our method is used to study the effect of a surfactant on the skin friction of a bubble in a uniaxial flow. For the limiting case where the surface diffusivity is zero, the effect of a tangential stress generated by the surface tension gradient, allows us to explain a new phenomenon in high concentration regimes: larger surface tension, but also larger deformation. Furthermore, this condition leads to the formation of boundary layers and flow separation at high Reynolds numbers. The influence of these complex flow patterns is examined.     

Bénard–Marangoni convection in a small circular container: influence of the Biot and Prandtl numbers on pattern dynamics and free surface deformation

We studied, experimentally, the pattern dynamics and free surface deformation in Bénard–Marangoni convection, in a circular container (aspect ratio = 6). The free surface deformation fields were visualized by interferometry and temperature fields by infrared thermography. We considered the influence of the Marangoni (up to 2,623), Biot and Prandtl numbers. More dynamics are induced by increasing the Biot number and transition to a time-dependent flow has been observed. Conversely, increasing the Prandtl number reduces the dynamics. The deformation increases as a function of the Marangoni number until it reaches asymptotic values, which are functions of the Biot and Prandtl numbers.     

Natural convection with evaporation in a vertical cylindrical cavity under the effect of temperature-dependent surface tension

The effect of surface tension on laminar natural convection in a vertical cylindrical cavity filled with a weak evaporating liquid has been analyzed numerically. The cylindrical enclosure is insulated at the bottom, heated by a constant heat flux from the side, and cooled by a non-uniform evaporative heat flux from the top free surface having temperature-dependent surface tension. Governing equations with corresponding boundary conditions formulated in dimensionless stream function, vorticity, and temperature have been solved by finite difference method of the second-order accuracy. The influence of Rayleigh number, Marangoni number, and aspect ratio on the liquid flow and heat transfer has been studied. Obtained results have revealed that the heat transfer rate at free surface decreases with Marangoni number and increases with Rayleigh number, while the average temperature inside the cavity has an opposite behavior; namely, it growths with Marangoni number and reduces with Rayleigh number.     

Experimental investigation of evaporation-induced convection in water using laser based measurement techniques

Recent studies have shown that the evaporation of water can induce surface tension gradients along the water surface that ultimately lead to a surface driven flow, known as Marangoni convection. To visualize and characterize the Marangoni convection in water, this study generated evaporation driven convection in pure water with a vacuum pump to control and increase the evaporation rate of water within a rectangular cuvette that was placed within a vacuum chamber, and investigated the velocity and temperature distributions of the generated convection. The investigation was performed as the vacuum chamber pressure ranged from ∼250 Pa to ∼820 Pa. The temperature field obtained from thermocouple measurements and temperature planar laser induced fluorescence (temp-PLIF) measurements indicated that no buoyancy driven motion was generated during the investigation. Velocity vector fields captured with stereo particle image velocimetry (stereo-PIV) demonstrated a convection pattern that was strong and symmetric with the centerline of the cuvette. The strength of the convection was found to be correlated with the mean evaporation rate of water. The estimated Marangoni number exceeded the critical value typically used to characterize the onset of Marangoni convection. The convection had a similar pattern as Marangoni convection observed in volatile liquids evaporated from capillary tubes. In both cases, the convection scaled with the width of the liquid container even though the sizes of the containers differ by an order of magnitude. In addition, the size of the convection in this study was much larger than the Marangoni convection in water that was observed in previous studies.     

Flow of drag-reducing surfactant solutions in rough pipes

The paper reports the results of experimental study of the flow of hexadecyltrimethylammonium chloride (CTAC) solutions with addition of sodium salicylate (NaSal) in the rough pipes. Measurements were performed in the range of the surfactant concentration from 200 to 400 ppm at a constant molar ratio CTAC/NaSal of 1:2. Five pipes of the relative roughness k/D varying from 1.2 × 10^?2 to 5.6 × 10^?2, obtained by the covering of inner surface of the pipes with glued silicon carbide particles of different size, were studied. The roughness was observed to increase the drag of flow of CTAC/NaSal solutions already at Reynolds numbers higher than 800. With increasing relative roughness k/D, the critical value of Reynolds number, at which the drag reduction disappears, was found to decrease. However, no influence of the roughness on the critical shear stress was noted. The ratio of the critical Reynolds number for rough pipes to that of hydraulically smooth pipes was independent of the surfactant concentration. The degree of drag reduction by the flow of surfactants was greater in rough pipes than in smooth pipes.     

Convective self-oscillations near an air-bubble surface in a horizontal rectangular channel

The concentration convection in an isothermal fluid near an air bubble clamped between the vertical walls of a horizontal channel with a rectangular cross-section is studied experimentally and numerically. The channel is filled with an aqueous solution of a surfactant with a nonuniform concentration. As a result of the competition between the gravitational convection in the cavity volume and the Marangoni convection near the bubble surface, an oscillation flow regime is established. This regime is observed experimentally over several hours. In the numerical experiment, the oscillations are obtained in the presence of an initial horizontal surfactant concentration gradient. Against the background of gravitational convection, short bursts of Marangoni convection with ten times greater intensity are observed. The convective flow patterns and the oscillation periods obtained experimentally and numerically are in fairly good agreement.     

Effects of surfactant on liquid film thickness in the Bretherton problem

The effects of insoluble and soluble surfactant on the motion of a long bubble propagating through a capillary tube are investigated computationally using a finite-difference/front-tracking method. Emphasis is placed on the effects of surfactant on the liquid film thickness between the bubble and the tube wall. The numerical method is designed to solve the evolution equations of the interfacial and bulk surfactant concentrations coupled with the incompressible Navier–Stokes equations. A non-linear equation of state is used to relate surface tension coefficient to surfactant concentration at the interface. Computations are first performed for soluble cases and then repeated for the corresponding clean and insoluble cases for a wide range of governing non-dimensional parameters in order to investigate the effects of surfactant and surfactant solubility. The computed film thickness for the clean case is found to be in a good agreement with Taylor’s law indicating the accuracy of the numerical method. We found that both the insoluble and soluble surfactant generally have a thickening effect on the film thickness, which is especially pronounced at low capillary numbers. This thickening effect strengthens with increasing sensitivity of surface tension to interfacial surfactant coverage mainly due to the enhanced Marangoni stresses along the liquid film. It is also observed that film thickening shows a non-monotonic behavior for variations in Peclet number. The validity of insoluble surfactant assumption is assessed for various non-dimensional numbers and it is demonstrated that insoluble assumption is valid only when capillary number is very low, i.e., Ca ≪ 1 and when surface tension is highly sensitive to interfacial surfactant coverage, i.e., the elasticity number is large.     

Instability of the Benard–Marangoni Convection in a Porous Layer Affected by a Non-Vertical Magnetic Field

The onset of the Benard–Marangoni convection in a horizontal porous layer permeated by a magnetohydrodynamic fluid with a nonlinear magnetic permeability is examined. The porous layer is assumed to be governed by the Brinkman model; it is bounded by a rigid surface from below and by a non-deformable free surface from above and subjected to a non-vertical magnetic field. The critical effective Marangoni number and the critical Rayleigh number are obtained for different values of the effective Darcy number, Biot number, Chandrasekhar number, nonlinear magnetic parameter, and angle from the vertical axis for the cases of stationary convection and overstability. The related eigenvalue problem is solved by using the first-order Chebyshev polynomial method.     

Effect of a Magnetizable Surfactant on the Motion of a Liquid Film on the Horizontal Surface

The motion of a thin liquid film of viscous incompressible fluid on the horizontal surface in the presence of a magnetizable surfactant on the free boundary in the external inhomogeneous magnetic field is investigated. Surfactant diffusion along the free surface and the dependence of the surface tension on the magnetic field strength are taken into account. The system of evolutionary equations is derived in the lubricant approximation and steady-state film flows and their stability in the case of constant film thickness and constant surfactant number density are investigated with regard to the Marangoni effect.     

Buoyant Marangoni convection of nanofluids in square cavity

The buoyant Marangoni convection heat transfer in a differentially heated cavity is numerically studied. The cavity is filled with water-Ag, water-Cu, water-Al₂O₃, and water-TiO₂ nanofluids. The governing equations are based on the equations involving the stream function, vorticity, and temperature. The dimensionless forms of the governing equations are solved by the finite difference (FD) scheme consisting of the alternating direction implicit (ADI) method and the tri-diagonal matrix algorithm (TDMA). It is found that the increase in the nanoparticle concentration leads to the decrease in the flow rates in the secondary cells when the convective thermocapillary and the buoyancy force have similar strength. A critical Marangoni number exists, below which increasing the Marangoni number decreases the average Nusselt number, and above which increasing the Marangoni number increases the average Nusselt number. The nanoparticles play a crucial role in the critical Marangoni number.     

Effect of Internal Heat Generation on the Onset of Marangoni Convection in a Fluid Layer Overlying a Layer of an Anisotropic Porous Medium

Linear stability analysis has been performed to investigate the effect of internal heat generation on the criterion for the onset of Marangoni convection in a two-layer system comprising an incompressible fluid-saturated anisotropic porous layer over which lies a layer of the same fluid. The upper non-deformable free surface and the lower rigid surface are assumed to be insulated to temperature perturbations. The fluid flow in the porous layer is governed by the modified Darcy equation and the Beavers–Joseph empirical slip condition is employed at the interface between the two layers. The resulting eigenvalue problem is solved exactly. Besides, analytical expression for the critical Marangoni number is also obtained by using regular perturbation technique with wave number as a perturbation parameter. The effect of internal heating in the porous layer alone exhibits more stabilizing effect on the system compared to its presence in both fluid and porous layers and the system is least stable if the internal heating is in fluid layer alone. It is found that an increase in the value of mechanical anisotropy parameter is to hasten the onset of Marangoni convection while an opposite trend is noticed with increasing thermal anisotropy parameter. Besides, the possibilities of controlling (suppress or augment) Marangoni convection is discussed in detail.     

Correlations for heat transfer coefficient and pressure drop in the annulus of concentric helical coils

Heat transfer and pressure drop characteristics in the annulus of concentric helical coils heat exchangers were experimentally investigated. The effects of coil curvature ratio, flow configuration, number of turns and addition of surfactant were investigated. Five test coils were designed and manufactured to study the effect of different parameters on heat transfer and pressure drop. The liquids used in the present study were water and oleyl-dihydroxy-etheyl-amine-oxide (ODEAO, C₂₂H₄₅NO₃ = 371) non-ionic aqua surfactant solution flowing through the annulus side. The inner side Reynolds number range 11,000–27,000 and the annulus side range 5,000–19,000. The results showed that the annulus Nusselt number decreases as annulus curvature ratio increases and increases when number of turns decrease. Moreover, the friction factor increases with the curvature ratio and also increases as number of turns decreases. Both Nusselt number and friction factor decrease when ODEAO concentration increases.     

Surface activity and bubble motion

This paper reviews recent progress in the theories of the surface boundary conditions of adsorbed solutes in liquids, and of the effects of those solutes on the steady motion of a bubble or drop in the liquid. Both singular perturbation theory and numerical solutions have useful roles in this problem, and their relationship is explored. In addition, analytical solutions are given to two problems concerning a spherical bubble rising steadily at low Reynolds number in a viscous fluid. One of these is displacement of the internal vortex centre from its position in the absence of surface activity when there is a small stagnant cap of surfactant at the rear. The results agree with experimental data in the direction of that displacement but give only about half its amount. The other problem is the velocity perturbation all round the surface caused by a very dilute solution of a weak surfactant at high Péclet number. This compares quite well with the numerical solution for a Péclet number of 60, having relative errors of order (60)^–1/2 as would be expected.     

Effect of span length and temperature on the 3-D confined flow around a vortex promoter

This article presents a numerical study on the influence of span length and wall temperature on the 3-D flow pattern around a square section vortex promoter located inside a micro-channel in the low Reynolds number regime. The first objective of the work is to quantify the critical Reynolds number that defines the onset of vortex shedding and to identify the different regimes that appear as a function of the channel aspect ratio (span to height ratio). We found that the critical Reynolds number for the onset of the Karman street regime increases as the aspect ratio decreases. In particular, for the aspect ratio of 1/2 the critical Reynolds number is nearly six times the critical Reynolds number of the 2-D problem. An intermediate oscillating regime between the steady and the Karman street solutions was also found to exist within a rather wide range of Reynolds numbers for small channel aspect ratios. The second objective was to investigate the influence of the vortex promoter wall temperature on both vortex shedding and flow pattern. This has practical engineering implications because the working fluid considered in the article is water that has a viscosity that depends significantly on temperature and promotes a strong coupling between the momentum and energy equations that influences the system behaviour. Results indicate that high surface temperature on the prism promotes the onset of the Karman street, suggesting design guidelines for micro-channel based heat sinks that make use of vortex promoters.     

Effect of span length and temperature on the 3-D confined flow around a vortex promoter

This article presents a numerical study on the influence of span length and wall temperature on the 3-D flow pattern around a square section vortex promoter located inside a micro-channel in the low Reynolds number regime. The first objective of the work is to quantify the critical Reynolds number that defines the onset of vortex shedding and to identify the different regimes that appear as a function of the channel aspect ratio (span to height ratio). We found that the critical Reynolds number for the onset of the Karman street regime increases as the aspect ratio decreases. In particular, for the aspect ratio of 1/2 the critical Reynolds number is nearly six times the critical Reynolds number of the 2-D problem. An intermediate oscillating regime between the steady and the Karman street solutions was also found to exist within a rather wide range of Reynolds numbers for small channel aspect ratios. The second objective was to investigate the influence of the vortex promoter wall temperature on both vortex shedding and flow pattern. This has practical engineering implications because the working fluid considered in the article is water that has a viscosity that depends significantly on temperature and promotes a strong coupling between the momentum and energy equations that influences the system behaviour. Results indicate that high surface temperature on the prism promotes the onset of the Karman street, suggesting design guidelines for micro-channel based heat sinks that make use of vortex promoters.