TRANSITION TO OSCILLATORY MARANGONI CONVECTION IN LIQUID BRIDGES OF INTERMEDIATE PRANDTL NUMBER

Masahiro Kawaji^1,3, Fumiaki Otsubo², Sanja Simic¹, and Shinichi Yoda³

¹Dept. of Chemical Engineering and Applied Chemistry, University of Toronto, Toronto, Ontario M5S 3E5, Canada
²A.E.S. Co., Ltd., Tsukuba, Japan
³National Space Development Agency of Japan, 2-1-1 Sengen, Tsukuba, 305-8505, Japan

Marangoni convection experiments have been conducted with acetone (Pr = 4.3) and methanol (Pr = 6.8) liquid bridges to investigate the flow structures and temperature fields during transition from steady to oscillatory convection in a half floating zone. Although high Prandtl number fluid experiments conducted in the past have yielded a significant amount of flow and temperature field data, numerical simulations of such experiments encounter considerable difficulties due to the reduction in the thermal boundary layer thickness and its implications for finer grid resolution. On the other hand, although numerical simulations can be more readily performed for low Prandtl number fluids, such as molten metals and semi-conductors, experiments using these fluids face considerable challenges in detailed measurements of flow and thermal fields, because they are opaque and susceptible to surface oxidation at high temperatures.

Thus, the present work has involved Marangoni convection experiments with fluids of intermediate values of Prandtl number (1 < Pr < 7) for which linear stability analysis and numerical simulation results have been recently obtained by Nienhuser and Kuhlmann (2000) and Leypold et al. (2000). In the experiments, the disk diameters of 5.0 mm and 10.0 mm were used for both fluids and in order to reduce the surface evaporation rate, a quartz tube was fitted around the liquid bridge as shown in Figure 1. In the experiments, the liquid bridge height was changed from 1.5 mm to 4.5 mm, covering aspect ratios from 0.3 up to 1.6.The liquid bridge shape data were obtained for both fluids covering a wide range of volume ratios for each aspect ratio and disk diameter.

Figure 1 Schematic of Marangoni convection experiment apparatus

In order to understand the physical mechanism responsible for transition from steady to oscillatory Marangoni convection in a half floating zone as well as to verify linear stability analysis and three-dimensional numerical simulation predictions, various advanced measurement techniques were employed to yield detailed experimental data. The disk temperatures were measured using fine thermocouples. For DT, a pair of thermocouples attached to the disks were connected in series and the cold junctions were enclosed in a thick-walled aluminum box to directly measure the temperature difference. Liquid temperature fluctuations were measured using a micro-thermocouple with a wire size of 25 mm inserted into the liquid bridge from the side through a hole drilled on the quartz tube wall. The thermocouple junction was positioned just inside the liquid surface between the upper disk and mid-height. All the thermocouples were calibrated using a platinum resistance thermometer sensor (Pt-100) with < 0.03 K error. The uncertainty in DT measurement was estimated to be ±0.02^oC.

For flow visualization, the Particle Image Velocimetry and photochromic dye activation techniques were used to measure the bulk flow pattern and surface velocity in acetone liquid bridges, respectively. With PIV, either a vertical or horizontally oriented laser sheet illumination and silver-coated particles were used. The vertical light sheet allowed measurements of toroidal vortices expanding and contracting with time at the onset of oscillatory convection. The horizontal light sheet allowed detection of rotational motion in the radial plane as well as the mode of oscillation for different aspect ratios. The surface velocities on acetone bridges were measured using a photochromic dye activation technique, which has been previously used in various gas-liquid flow experiments by the authors (Kawaji, 1998). It uses a UV pulse laser to activate photochromic dye (TNSB) molecules dissolved in the working liquid at a concentration of 100 ~ 500 ppm. The acetone-dye solution is transparent, but changes to a purple color upon UV irradiation. The purple liquid acts as a tracer, the motion of which can be monitored by a color video camera to obtain the liquid surface velocity as shown in Fig. 2.

Figure 2 Photochromic dye activation set-up for surface velocity measurement

The bridge shape data and volume reduction rates were first obtained from video images of liquid bridges for both fluids. A typical video image of the acetone bridge for a 10 mm disk diameter and 2.8 mm bridge height is shown in Figure 3. From similar images, acetone and methanol bridge shape data could be easily obtained by capturing the video images and digitizing the shapes using an image analysis program on a PC. Due to the evaporation effect, the volume ratio for both fluids decreased with time. For example, acetone bridge's volume ratio varied from 100 ~ 108% to 85 ~ 95% over a period of 74 ~ 610 seconds. By using the volume ratios obtained at the beginning (VR > 100%) and at the end (VR < 100%) and the time elapsed, the average evaporation rate could be quantified in terms of the volume reduction rate per unit time. As shown in Table 1, the evaporation rate decreased strongly with the liquid bridge diameter for both fluids. The methanol had a smaller evaporation rate, approximately 1/4 ~ 1/5 of acetone.

Figure 3 A typical image of an acetone bridge (D = 10.0mm, H = 3.0 mm)

Table 1 Volume Reduction Rates for Acetone and Methanol Liquid Bridges

Fluid	Diameter	Volume Reduction Rate (%Vol./sec)
Acetone	5.0	0.12 ~ 0.19
Acetone	10.0	0.03 ~ 0.06
Methanol	5.0	~0.05
Methanol	10.0	~0.01

In the experiment, the disk temperature difference, DT, was varied to either increase or decrease past the critical value while the volume ratio of the liquid bridge was allowed to slowly decrease from > 100% to < 100% due to evaporation. For each disk diameter and aspect ratio, fluid and disk temperature measurements were made several times to check the reproducibility of the critical temperature difference data at the onset of oscillatory Marangoni convection.

At the onset of oscillatory convection, the following phenomena were observed to occur.

• Periodic fluid temperature fluctuations
• Surface velocity fluctuations with an appearance of azimuthal component
• Vortex expansion and contraction in PIV images
• Variations of radial temperature and flow patterns in cross-sectional views of IR image and tracer particle motion

For acetone, the transition from steady to oscillatory convection was found to occur at disk temperature differences of 1.6 ~ 2.3 K, and liquid temperature fluctuations with amplitudes of 0.02 ~ 0.1 K were detected, as shown in Figure 4. Similar appearances of a periodic temperature fluctuation with an increasing amplitude were also observed in methanol experiments at disk temperature differences ranging from 1.6 to 4.3 K.

Figure 4 Variation of acetone temperature fluctuations at the onset of oscillatory convection (D = 5.0 mm, H = 2.5 mm)

For acetone, the transition to oscillatory convection was also accompanied by the appearance of an azimuthal component of surface velocity. The dye tracer motion changed from a steady one-dimensional (downward) trajectory to a two-dimensional trajectory with a fluctuating azimuthal displacement as well. Although the same technique has been used previously to measure the velocity of a liquid layer surface in an open boat under Marangoni convection (Tudose and Kawaji, 1999 and 2000), to our knowledge, this work presents the first direct measurement of surface velocity fluctuations at the onset of oscillatory Marangoni convection in a half floating zone.

At the same time, the toroidal vortex started to expand and contract in size, and the radial flow field started to rotate. The surface oscillations could be detected for acetone at DT values well above the critical temperature difference, but the amplitude of surface oscillations was too small to be detected at the onset of oscillatory convection.

The critical Marangoni numbers for acetone were calculated using the following definition previously used by Velten et al. (1991) for NaNO₃ (Pr = 7) and KCl (Pr = 1).

       (1)

The critical Marangoni number data for both acetone and methanol varied strongly with the disk diameter and aspect ratio, as has been previously observed for higher Prandtl number fluids. The critical Marangoni numbers for acetone were, however, found to be significantly higher than Velten et al.'s(1991) data for Pr = 1 and 7 as shown in Figure 5, due to the effects of strong surface evaporation and heat loss. On the other hand, the critical Marangoni numbers for methanol with much smaller rates of surface evaporation showed good agreement with Velten et al.'s (1991) data for NaNO₃ (Pr = 7).

Although previous linear stability analyses by Kuhlmann et al. (1999) and Wanschura et al. (1997) also predicted significantly smaller values of Ma_cr for Pr = 4 without any significant heat loss from the liquid bridge surface, a recent analysis by Nienhuser and Kuhlmann (2000) using the actual properties of acetone and a simple model of heat loss from the bridge surface due to evaporation, clarified the effects of surface evaporation. Their predictions of the critical Reynolds number (or critical Marangoni number divided by Prandtl number) indicated excellent agreement with the present acetone data. Their stability analysis also predicted that an acetone bridge with a large rate of heat loss from the free surface due to evaporation, would behave in a manner similar to that of low Prandtl number fluids rather than high Prandtl number fluids. If there were no evaporation heat loss, the behavior would be similar to that of high Prandtl number fluids. This is an interesting finding that implies an opportunity exists for investigating the dynamics of low Prandtl number Marangoni flow using an intermediate Prandtl number fluids like acetone. Such an investigation would be advantageous from a flow visualization point of view, as low Prandtl number fluids are opaque and prevent direct flow visualization, and also present other experimental difficulties at high temperatures such as surface oxidation.

Figure 5 Comparison of critical Marangoni numbers, Ma_R2_/H, for acetone with Velten et al. s (1991) data

Recently, Leypold et al. (2000) performed three-dimensional numerical simulations of Marangoni convection in intermediate Prandtl number fluids, Pr = 4 and 7. They predicted that the frequency of flow and temperature oscillations would increase linearly with the disk temperature differences above DT_cr. The power spectral density data for both acetone and methanol temperature fluctuations showed linear variations with the excess disk temperature difference as shown in Figure 6, in qualitative agreement with Leypold et al.'s predictions.

The Marangoni convection experiments using intermediate Prandtl number fluids described above have thus yielded data useful for understanding the mechanism of the onset of oscillatory convection and verification of some of the predictions of linear stability analysis and numerical simulations. In the future, the surface evaporation rate should be better quantified so that a more rigorous model of surface evaporation can be developed and incorporated into both the linear stability analysis and numerical simulation models. The dependence of surface evaporation rate on the liquid temperature, surrounding temperature and pressure, partial pressure and latent heat need to be further investigated for both acetone and methanol.

Figure 6 Variation of acetone temperature oscillation frequency with disk temperature difference (D= 5.0 mm, H= 2.5mm, Aspect ratio = 1.0)

Finally, internal temperature distributions in acetone bridges under steady and oscillatory convection should be determined using a traversing thermocouple. An infrared imager should also be used to measure the temperature distributions on the free surface, so that these results can be compared with the linear stability analysis and numerical simulation predictions.

ACKNOWLEDGEMENTS: Authors gratefully acknowledge National Space Development Agency of Japan (NASDA) for this work. Administrative support from Mr. T. Arai and Dr. K. Takagi of NASDA Space Utilization Research Center is also gratefully acknowledged.

REFERENCES

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Leypold, J., Kuhlmann, H.C. and Rath, H.J.: "Three-dimensional numerical simulation of thermocapillary flows in cylindrical liquid bridges", accepted for publication in J. Fluid Mechanics (2000).

Nienhuser, Ch. and Kuhlmann, H.C.: "Floating zone stability I. Cylinder: Medium Prandtl numbers, II. Deformed surfaces", presented at the NASDA Meeting on Marangoni convection research, Tokyo, March 6, 2000.

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