Temperature fields in a liquid due to the thermocapillary motion of bubbles and drops

Transcription

1 Temperature fields in a liquid due to the thermocapillary motion of bubbles and drops G. Wozniak, R. Balasubramaniam, P. H. Hadland, R. S. Subramanian Experiments in Fluids ) 84±89 Ó Springer-Verlag Abstract Experiments were performed on the motion of isolated air bubbles and drops of Fluorinert FC-75 moving in a Dow-Corning silicone oil under the action of an applied temperature gradient in a reduced gravity environment aboard the Space Shuttle in orbit. The disturbance of the imposed temperature eld due to the motion of the objects was studied optically using a shearing interferometer with a Wollaston prism and the results of a typical bubble run were compared with theoretical predictions. Also, the liquid velocity eld surrounding the bubbles and drops has been qualitatively investigated in a few runs by the observation of tracer particles dispersed in the continuous phase uid. The measurement techniques are described, and the results for the temperature and ow elds are presented and discussed. 1 Introduction Bubbles and drops suspended in another uid will move when subjected to a temperature gradient due to the action Received: 27 June 2000/Accepted: 17 November 2000 G. Wozniak &) Institut fuèr Fluidmechanik und Fluidenergiemaschinen Tu Freiberg, Lampadiusstr Freiberg, Germany R. Balasubramaniam National Center for Microgravity Research on Fluids and Combustion, NASA Glenn Research Center, Mail Stop Cleveland, OH 44135, USA P. H. Hadland*, R. S. Subramanian Department of Chemical Engineering, Clarkson University Potsdam, NY , USA Present address: *Aker Offshore Partner AS, Mechanical Department P.O. Box 589, N-4001 Stavanger, Norway Presented in part at the ``Microgravity Fluid Physics and Heat Transfer Conference'' of the Engineering Foundation, Hawaii, 19±24 September The work described herein was supported by the German Space Agency DARA) through DARA Grant 50WM to G. Wozniak, and by NASA's Microgravity Sciences and Application Division, which provided nancial support for the research performed by the other three co-authors. We thank Drs. L. Zhang and X. Ma for their assistance with the evaluation of the numerical solution, A. Glathe for his assistance with the evaluation of the interferograms, and S. Younkin for his assistance with the evaluation of the tracer images. of the resulting interfacial tension gradient. This so-called thermocapillary migration can be important in the processing of materials in reduced gravity as well as in separation processes in gases and liquids in a weightless environment. The subject of thermocapillary bubble and drop motion was rst studied by Young et al. in Since then, a substantial literature has been developed in this area, which has been reviewed in Wozniak et al. 1988), Subramanian 1992), and Subramanian et al. 2001). The gravitational eld of the Earth makes it dif cult to study pure thermocapillary migration in experiments on the ground. On the Earth, buoyant forces arising from the density difference between the continuous and dispersed phase uids, which act on the bubble or drop, mask thermocapillary migration. In addition, density differences within the bulk uid, which arise naturally from temperature variations, will lead to some contribution from natural convection, which is another source of disturbance in thermocapillary experiments on the ground. Therefore, quantitative experiments which allow for comparison with theoretical predictions must be conducted in a reduced gravity environment. Previous experimental work in drop towers and sounding rockets and aboard orbiting spacecraft has been discussed in Balasubramaniam et al. 1996); in that article, we presented our results from a ight experiment conducted on board the Space Shuttle Columbia during the International Microgravity Laboratory mission IML-2) in the summer of In summer 1996, we carried out new experiments on board the Space Shuttle Columbia during the Life and Microgravity Spacelab LMS) mission, extending the parameter range of the experiments of 1994 by the use of a silicone oil of lower viscosity. These results are presented and discussed in Hadland et al. 1999). Both articles were dedicated to the dependence of the scaled bubble and drop velocities on the Marangoni number Ma. The Reynolds number, Re, and the Marangoni number, Ma, are de ned below. Re Rv 0 1 m Ma Rv 0 2 j Re and Ma re ect the relative importance of convective transport of momentum and energy, respectively, when compared with molecular transport. Here, R is the radius of the bubble or drop, m is the the kinematic viscosity of the continuous phase, and j is its thermal diffusivity. The reference velocity, v 0, is de ned as follows.

2 j v 0 r TjjrT 1 jr 3 l In Eq. 3), r T is the rate of change of interfacial tension with temperature, r T 1 is the temperature gradient imposed in the continuous phase uid, and l is its dynamic viscosity. The ratios of dynamic viscosities, thermal conductivities, thermal diffusivities, and the densities of the two phases are additional parameters in the problem. In this situation, the Marangoni number plays the role of a PeÂclet number. It is the product of the Reynolds number and the Prandtl number Pr m=j. In our space experiments, we reached maximal bubble Reynolds and Marangoni numbers of Re ˆ 87.2 and Ma ˆ 5780, and in the case of drops, maximal values of Re ˆ 50.2 and Ma ˆ These values are based on the properties of the continuous phase. By replacing m and j appearing in Eqs. 1) and 2) with the corresponding properties of the drop phase, m and j, respectively, we can de ne a Reynolds number Re and a Marangoni number Ma for the drop phase. The reason for doing so is to provide an idea of the importance of convective transport of momentum and energy relative to molecular transport of these entities within the drop phase. The maximal value of Re was approximately 633 and that of Ma was approximately 9,330. In addition to the bubble and drop velocity measurements during the LMS mission, we also measured the temperature eld disturbances within the continuous phase uid due to the bubble and drop motion using a shearing interferometer. Due to unavoidable miscibility effects between the drop liquid and the continuous phase liquid, and the resulting optical disturbances, an interferogram evaluation of the temperature elds surrounding the drops was not possible. Thus, we were only able to detect the thermal elds around air bubbles. Our principal objective in this paper is to describe the techniques employed, and present and discuss typical measurements along with a comparison with theoretical predictions. Also, some information on the ow elds has been obtained using tracer particles, which is presented and discussed. The theoretical prediction of the thermal elds around a moving bubble was obtained using a numerical model described in Ma et al. 1999). It assumes steady migration of a spherical drop in a liquid in zero gravity. Ma et al. used a nite difference formulation to solve the governing equations. Further details regarding the numerical procedure can be found in that reference. 2 Experimental apparatus and procedure The experiments were performed in an apparatus labeled the BDPU bubble, drop, and particle unit), which was provided by the European Space Agency through a cooperative arrangement with the National Aeronautics and Space Administration NASA). It consists of a facility which provides power, optical diagnostics and illumination, imaging facilities including a video camera and a motion picture camera, and other support services such as heating and cooling. Within this facility, a test cell speci c to a given experiment is inserted by a crew member during the ight. Each cell is custom designed and built to meet the needs of the investigation involved. Details of the BDPU can be found in Fortezza et al. 1991). The test cells are the core of the experimental apparatus. Two rectangular test cells were available, one for the bubble experiments and one for the drop experiments, respectively. Both were of identical dimensions, measuring mm 3 in the interior, and lled with the continuous phase liquid, a DOW-Corning DC-200 series silicone oil of nominal viscosity 10 cst. A schematic of such a test cell is displayed in Fig. 1. In the long dimension, the cell was bounded by aluminum surfaces which were thermally coupled to Peltier elements. The other four walls were made of fused silica for the purpose of optical access. A typical experimental run consists of the following sequence. First, a steady temperature gradient was established along the long axis of the cavity by heating and cooling the continuous phase liquid from the ends. Then, within the liquid, a needle could be inserted by software commands from the ground, and a bubble or drop of the desired size formed at the tip of the needle could be released by rapidly withdrawing the needle. When the bubble or drop reached the hot wall, it could be removed using a tube mounted in the middle of an extraction net. After the thermal gradient had recovered from the disturbances of the bubble or drop motion, another object could be inserted in a similar way. The traverse of the objects across the silicone oil was followed using a video camera which permitted observation of one plane within the test cell. This allowed the velocities of the objects to be measured. Hadland et al. 1999) have presented and compared the results of these measurements with theoretical predictions. Their observations in the case of air bubbles are generally consistent with numerical predictions and also con rm the correctness of results from an asymptotic theory for large values of the Marangoni number. In the case of Fluorinert drops, the data on relatively small drops are consistent with numerical predictions up to about Ma 90. Details are given in Hadland et al. 1999). Another video camera from an orthogonal direction captured interferometry images to measure the thermal eld around the objects during their migration. The interferometry images were only useful at moderate temperature gradients, since high gradients cause strong refraction effects leading to optical disturbances which made an interferogram evaluation impossible. Temperatures at the inner Fig. 1. Schematic of test cell 85

3 86 surfaces of the aluminum walls were measured and controlled using thermistors, which permitted the maintenance of the hot and cold wall temperatures at a steady value within 0.1 C. In addition, temperatures at appropriate locations on the glass walls were monitored using thermistors during the experiments. No background convection that might have been caused by buoyant effects induced by the residual gravitational eld could be observed in the cell when the temperature gradient had been established. Also, the interferometry patterns indicated that the temperature eld in the test cell was nominally one-dimensional and linear, suggesting the absence of any temperature distortions that could be induced by buoyant convection in the cell. Further technical details can be found in Balasubramaniam et al. 1996) and Hadland et al. 1999). Opportunities to perform the experiments were provided at various times during the LMS mission. In each instance, two hours were used to establish a steady temperature eld within the test cell. For a stationary uid held in a container with highly conducting side walls, a three-dimensional solution of the Laplace equation yields 4,300 s as the time estimate for the temperature distribution to be at 99% of the nal result if the solution begins at the average temperature between the endwalls. This was accomplished in the rst 1,800 s by stirring with the extraction net during and after the establishment of the endwall temperatures. Several experimental runs, involving injection and subsequent extraction after the migration of each bubble or drop, were then typically performed for a period of 90±120 min. At that point, if suf cient time was available, the entire process was repeated with a different temperature gradient. Four different temperature gradients from 0.25 K/mm to 1 K/mm were applied. As mentioned earlier, a shearing interferometer also termed Schlieren or differential interferometer) was employed to measure the temperature eld around the objects during their traverse across the cell. A detailed description of this interferometer can be found in Merzkirch 1987). The core component of this optical device is the Wollaston prism providing the shear between the two beams, both of which traverse the test liquid, and then interfere with one another. Consequently, shearing interferometers are sensitive to variations in uid refractive index gradients. A constant value of this gradient would lead to an image of uniform brightness without interference fringes. The optical arrangement had a beam separation of 1.45 mm. A helium± neon laser providing light of wavelength 633 nm was used as the light source. The refractive index eld of the liquid surrounding the migrating bubble or drop is axisymmetric. This necessitates the application of an evaluation algorithm Abel integration) taking this symmetry into account, since interferometry is principally a measurement technique only for two-dimensional refractive index elds. Details of the algorithm used to evaluate the temperature elds are described in Glathe and Wozniak 1996). It is important to note that the accuracy of the interferogram evaluation depends on the fringe density. Evaluation of interferograms exhibiting only a few fringes necessitates strong interpolation. A typical evaluation procedure consists of the following steps: Digitization of video images, counting of fringe numbers, interpolation between fringes, evaluation of phase difference by Abel integration, and calculation of differential temperature eld based on the thermal refractive index gradient and the undisturbed thermal eld. Table 1 provides information on the density and viscosity of the uids used in the ight experiments along with 95% con dence intervals for the tted constants. These properties were measured at various temperatures in the range of values encountered in the experiments. In addition, the interfacial tensions at the silicone oil±air interface, and at the silicone oil±fluorinert interface, were measured as functions of temperature in the same range using a Fisher tensiometer equipped with a Du NouÈy ring, and the results were found to be adequately described by straight line ts. The tted values of r T are ) ) mn/ m K) for the silicone oil±air interface and ) ) mn/ m K) for the silicone oil±fluorinert interface, wherein we have reported 95% con dence intervals after the values. 3 Results and discussion Figure 2 shows a typical interferometric sequence of a traverse by a bubble undergoing thermocapillary migration in the silicone oil toward the warm wall and its evaluation. The diameter of the bubble is 8.2 mm, and the imposed temperature gradient is rt 1 ˆ 0:33 K=m. The mean velocity of the bubble is approximately 2.7 mm/s in the portion of the traverse shown in Fig. 2. The velocity of the bubble changed from approximately 1.8 mm/s soon after injection to approximately 2.9 mm/s just before it reached the hot wall. The traverse of the bubble lasted around 20 s. In Fig. 2, the initial distance between the bubble center and the heated wall is approximately six bubble radii. The distance between the bubble center and the side walls is around 5.5 bubble radii. Since the interference fringes Fig. 2a) represent the Abel transform of the lines of constant temperature gradient, the overall interferogram is not easy to interpret, even qualitatively, without using the algorithm described earlier. Figure 2b shows contours of the temperatures, obtained by analysis of the interferograms, from which the imposed undisturbed temperature pro le has been subtracted. This form of presentation delivers a clear impression of the extent of the thermal disturbance produced by the thermocapillary motion of the bubble. The length of the wake behind the bubble is about four bubble diameters. Due to the axisymmetry of the elds, we only show the upper half plane. The region ahead of the bubble Table 1. Constants in the best- t lines for density and viscosity as a function of temperature. Note: q = A + BT; ln l) =C + D/T; density q) is in kg/m 3 ; viscosity l) is in kg/ m s); temperature T) isink Silicone oil Fluorinert FC-75 A 1, , B ) ) C ) ) D 1, ,540 14

4 periphery exhibits a relatively high fringe concentration, representing larger temperature gradients. The evaluated interferometric sequence of Fig. 2c shows the temperature distribution in the vertical midplane of the bubble in the form of isotherms solid curves). Also, isotherms predicted by the model of Ma et al. dashed curves) are included. The numerical predictions were made for the experiment using a Marangoni number of Ma ˆ 452. In calculating this value of Ma and the corresponding Reynolds number, the properties of the continuous phase were evaluated at the average temperature of the hot and cold walls of the cell 34.5 C). The main limitations of the prediction are that constant physical properties evaluated at the above average temperature) are assumed, and that it neglects any interaction effects with the boundaries. The experimentally obtained isotherm distribution qualitatively follows the trend of the numerically predicted thermal eld. The agreement between the two elds is better in the forward half of the bubble. In the rear half and in the thermal wake region, the deviations are signi cant. Evidently, the experimentally obtained eld shows more of an in uence from convection than the theoretical prediction. Note that the predicted isotherms correctly meet the bubble surface at an angle not far from p/2. This is because the thermal conductivity of air is small compared with that of the silicone oil. However, because of the dif culty of resolving the isotherms in the immediate vicinity of the bubble surface, this behavior cannot be easily discerned from the experimentally measured isotherms. One possible reason for the overall discrepancy between the observed and predicted isotherms is the fact that the theory assumes quasi-steady migration with constant physical properties. In reality, the bubble is accelerating because the viscosity of the liquid in its neighborhood decreases as the bubble moves into warmer liquid. Also, in the theoretical model, the liquid is assumed to be unbounded. In practice, the walls of the test cell will have some in uence on both the velocity and temperature elds. It is worth mentioning that in the last panel of Fig. 2 the bubble has reached the wall of the test cell. This results in the development of a new ow eld, and a consequent alteration of the structure of the isotherms near the surface of the bubble. Figure 3 shows an interferogram of the bubble of Fig. 2 after its traverse, when it remains motionless at the center of the extraction net. The bubble is driving two symmetric vortices in the vertical mid-plane, a feature that is already well represented by the axisymmetric interferogram itself. Figure 3b presents the evaluated quantitative information. The upper plot of Fig. 3b again shows equally spaced contours of the temperature difference between the actual and the undisturbed thermal eld. The other plot shows isotherms directly. All temperatures or temperature differences are given in degrees Celsius. Since thermocapil- 87 c Fig. 2a±c. Bubble migration run Ma ˆ 452); a interferograms; b temperature difference from the undisturbed eld; c isotherms, isotherm number and corresponding temperatures in C) are indicated at the bottom

5 88 larity drives the ow toward the colder part of the surface of the bubble, the direction of motion of the upper vortex is clockwise. It is interesting to note that the extent of the thermal eld disturbance of a stationary bubble is much larger than that due to a moving bubble. This is because the hydrodynamic force on the moving bubble is zero, and the disturbance velocity from the motion of the bubbles decays relatively rapidly away from the bubble surface. In contrast, when the bubble is held xed, the hydrodynamic force is not zero. Since the Reynolds number of the ow is not very large, the disturbance ow penetrates fairly deeply into the continuous phase uid. As a result, the thermal eld is altered at greater distances from the bubble when compared with the thermal eld for the case of the moving bubble, shown in Fig. 2c. It is worthy of note that the liquid near a moving bubble is cooler than that far away at the same z location; on the contrary, the liquid near a stationary bubble is warmer than that far away from it. Figure 4 displays the interferogram of the refractive index eld around a Fluorinert drop undergoing thermocapillary migration in the silicone oil. The applied temperature gradient is 0.25 K/mm. The diameter of the drop is 5.9 mm, and its average velocity is approximately 0.36 mm/s. The direction of motion is from the left to the right warm) side. The long ``Schlieren-like'' trace behind the drop indicates its path across the cell. Curiously, the trace behind the drop is not straight, indicating that the drop has followed a sinuous trajectory for which we have no explanation yet. The disturbance trace ahead of the drop results from some previous drops that had passed some minutes before. The traces offer clear evidence of Fig. 4. Interferogram showing the disturbance of the thermal eld around a moving drop Fig. 3a, b. Bubble stationary near the center of the extraction net: a interferogram; b temperature differences and isotherms Fig. 5. a Streak picture of tracer trajectory in the silicone oil near a bubble. b Velocity vector plot evaluated from these pictures

6 mass transfer between the Fluorinert and the silicone oil even though no change in the size of the drop was detected during the traverse; these uids are not completely immiscible with each other. No ``immiscible'' liquid pair can be considered as completely immiscible, however, because there is always some mass transfer between liquid phases. Although in our case the dissolution is very slight, the sensitivity of the interferometer is such that this small effect in uences the interferogram considerably as Fig. 4 demonstrates. Therefore, it is not possible to extract the thermal eld disturbance from the interferogram. We note that measured drop speeds were compared with those predicted from a theoretical model in Hadland et al. 1999). As mentioned earlier, we introduced some tracer particles into the cell for visualization of the ow in the continuous phase uid around bubbles. We had to choose relatively large particles, approximately 50 lm in diameter, which consisted of hollow glass spheres coated with silver. This restriction results from the power limitation of the light source used to illuminate the tracers in the midplane of the cell via a light sheet technique. Pre- ight illumination tests showed that smaller tracers are hardly visible. In order to minimize the mechanical in uence of the tracers on the velocity eld, and interference from their possible lodging on the interface, we kept the tracer concentration relatively low. From the experimental observations, there was no evidence that the tracers showed any preference for the interface versus the bulk. Because of the low concentration of tracer particles, however, it was dif- cult to nd a suf cient number of tracer particles in the eld of view to permit adequate velocity resolution to reconstruct the velocity eld quantitatively. Therefore, only very limited qualitative ow velocity information could be obtained, and we present a typical result in Fig. 5. Figure 5a displays a streak picture of the ow in the continuous phase, around a bubble that is held at the tip of the needle. In this experiment, the left wall is at 25 C and the right wall at 45 C. The diameter of the bubble is 5.3 mm. The image represents the movement of the tracers in approximately 4 min. One clearly recognizes two symmetric vortices in the midplane around the bubble. It is also evident that the maximal velocities appear at the periphery of the bubble. This indicates that the bubble surface drives the ow. The ow direction at the bubble surface is toward the cold wall, since the surface tension increases in this direction. Figure 5b shows a plot of vectors interpolated from raw vectors obtained by analysis of the tracer images using particle image tracking software. A least squares interpolation technique was used. Since we only obtained a limited number of raw vectors, the error in the interpolation is relatively large. This is also the reason why the stagnation points above and below the bubbles are not exactly lined up. Clearly, the velocity eld shown in Fig. 5b is of a qualitative nature. 4 Concluding remarks Experiments on the thermocapillary migration of bubbles and drops performed aboard the Space Shuttle Columbia during the LMS mission have provided us the opportunity to study the temperature eld around the objects interferometrically. The observed isotherm structure is in qualitative agreement with that predicted from a numerical solution. Deviations between numerical predictions and experimental data result from the fact that theory assumes quasi-steady migration with constant physical properties and an unbounded continuous phase contrary to the experimental situation. Thermal elds around drops could not be evaluated due to optical disturbances resulting from mass transfer between the drop and the continuous phase. Also, some qualitative data on the velocity eld have been obtained by the observation of the motion of tracer particles. A typical tracer result shows the expected thermocapillary ow caused by a stationary bubble held at the tip of the injection needle. Quantitative measurements of the thermocapillary velocity elds will require an optimized optical set up and smaller tracers. References Balasubramaniam R; Lacy CE; Wozniak G; Subramanian RS 1996) Thermocapillary migration of bubbles and drops at moderate values of the Marangoni number in reduced gravity. Phys Fluids 8: 872±880 Fortezza R; Di Palermo P; Dewandre T; Gonfalone A 1991) The bubble drop particle unit BDPU). IML-2. Investigators Working Group Meeting Proceedings. ESTEC, Noordwijk, The Netherlands Glathe A; Wozniak G 1996) On a novel approach to evaluate Schlieren interferograms of axisymmetric refractive index elds. Flow Meas Instrum 7 3/4): 281±286 Hadland PH; Balasubramaniam R; Wozniak G; Subramanian RS 1999) Thermocapillary migration of bubbles and drops at moderate to large Marangoni number and moderate Reynolds number in reduced gravity. Exp Fluids 26: 240±248 Ma X; Balasubramaniam R; Subramanian RS 1999) Numerical simulation of thermocapillary drop motion with internal circulation. Numer Heat Transfer A 35: 291±309 Merzkirch W 1987) Flow visualization, 2nd edn. Academic Press, New York Subramanian RS 1992) The motion of bubbles and drops in reduced gravity. In: Chhabra R, de Kee D eds) Transport processes in bubbles, drops, and particles. Hemisphere, New York, pp 1±42 Subramanian RS; Balasubramaniam R; Wozniak G 2001) Fluid mechanics of bubbles and drops. In: Monti R ed) Physics of uids in microgravity. Gordon & Breach, Amsterdam to appear) Wozniak G; Siekmann J; Srulijes J 1988) Thermocapillary bubble and drop dynamics under reduced gravity ± survey and prospects. Z Flugwiss Weltraumforsch 12: 137±144 Young NO; Goldstein JS; Block MJ 1959) The motion of bubbles in a vertical temperature gradient. J Fluid Mech 6: 350±356 89