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Flow transition and hydrothermal wave instability of thermocapillary-driven flow in a free rectangular liquid film
International Journal of Heat and Mass Transfer 116 (2018) 635–641
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Flow transition and hydrothermal wave instability of thermocapillarydriven flow in a free rectangular liquid film Toshiki Watanabe a,1, Yosuke Kowata a, Ichiro Ueno b,⇑ a b
Division of Mechanical Engineering, School of Science and Technology, Tokyo University of Science, 2641 Yamazaki, Noda, Chiba 278-8510, Japan Department of Mechanical Engineering, Faculty of Science and Technology, Tokyo University of Science, Japan
a r t i c l e
i n f o
Article history: Received 20 February 2017 Received in revised form 7 September 2017 Accepted 16 September 2017 Available online 21 September 2017 Keywords: Free liquid film Hydrothermal wave Thermocapillary effect Flow pattern Traveling wave
a b s t r a c t Thermocapillary-driven flow in a free liquid film, which has two gas-liquid interfaces, is experimentally investigated. Silicone oil of 5 cSt is employed as the test fluid. Two-dimensional basic flow known as ‘double-layered flow’ after Ueno and Torii (2010) is realized under small-enough Marangoni number Ma, the non-dimensional number to describe the intensity of thermocapillary effect, under the geometry considered in the present study. The flow exhibits a transition from the two-dimensional steady flow state to the three-dimensional oscillatory state when the intensity of the imposed thermocapillary effect along the free surfaces exceeds the threshold. In this oscillatory regime, two types of hydrothermal wave instabilities are observed: the traveling-wave flow and the standing-wave flow. We especially focus on the traveling-wave instability and compare it with a hydrothermal wave in a liquid layer with a single free surface investigated by other researchers. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Pettit performed a series of experiments using a liquid film that had double-free surfaces (hereafter a free liquid film) on The International Space Station in 2003 [2]. In the experiments, Pettit formed a thin free liquid film of water in a ring made by metal wire and exposed the film to non-uniform temperature distribution by placing a heated iron close to one end of the wire ring. It was realized that the fluid was driven in the film toward the heated part, that is, a flow was induced from a colder region to a hotter region in spite of the negative temperature coefficient of the surface tension. These experiments showed that the free liquid film had a potential to be one way to obtain a new kind of crystallization process of materials and/or mass transport. We were inspired by these performances and started this study. In the case of a thin liquid layer formed on a base plate, that is, a film with a single free surface (a thin liquid layer, hereafter), Smith and Davis [3] predicted thermal fluid instability in an infinite liquid layer imposed by a constant-temperature gradient along a free surface by performing a linear stability analysis. It was named ‘hydrothermal wave (HW) instability. Daviaud and Vince [4] predicted instabilities under a coupling of thermocapillary and
⇑ Corresponding author. 1
E-mail address: [email protected] (I. Ueno). Present: Konica Minolta, Inc.
http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.09.059 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.
buoyancy effects, and then a variety of experiments were conducted with a liquid film in a narrow long channel [5–7] and shallow cavity [8,9] by various research groups. Riley and Neitzel [8] realized the HW predicted by Smith and Davis [3] in an experiment for the first time in a rectangular cavity with a temperature difference between both end walls. They reported a stability diagram and indicated the criterion to realize the HW in the film as a function of the dynamic Bond number BoD as BoD 6 0:22. Shevtsova et al. [10] carried out a series of two-dimensional numerical simulations to show the stability diagram and indicated the HW criterion as BoD 6 0:25. There are few studies, on the other hand, on the thermocapillary-driven flow in a free liquid film. The longitudinal solidification of a two-dimensional single-component free liquid film was discussed by simulations [11,12]. In the case of a twosurface film, Ueno and Torii [1] indicated two types of basic flows by a series of experiments: the double-layered flow and singlelayered flow. The effect of the liquid film shape in terms of the volume ratio was experimentally indicated on the selection of the basic flow and the direction of the net flow inside the film by measuring the liquid film profiles and the surface temperature distribution in the case of steady flows by [13]. Such an effect was confirmed through a series of parabolic flight experiments [14] as well as on-orbit experiments [15]. The Pettit demonstration was reproduced numerically by Yamamoto et al. [16] and the effect of the film shape near the heated area was shown. Concerning the
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physical mechanism of the selection of basic flows in the free liquid films, a discussion has been given [17,18], but we have never obtained a comprehensive explanation for the selection of the basic flows. As for the three-dimensional oscillatory flows in the free liquid film, much less knowledge has been accumulated. Ueno and Watanabe [19] indicated the flow transitions from the timeindependent steady flow to time-dependent ‘oscillatory’ flow for both basic flows. Ueno and Torii [1] and Limsukhawat et al. [20] conducted numerical simulations on the steady and oscillatory flows in the case of the double-layered basic flow. The present study aims to illustrate the characteristics of the HW in a free liquid film. In particular, it deals with the transition processes of the induced flow patterns of the traveling wave as a function of the geometry of the free liquid film under the conditions of double-layered basic flows, by making comparisons with results in the system of a thin liquid layer formed in a rectangular cavity with a single-free surface. 2. Experimental setup The experimental geometry is shown in Fig. 1. A liquid is filled in the rectangular hole made in a flat aluminum plate. In the present geometry, two kinds of aspect ratios are defined: the horizontal aspect ratio Cx ¼ Lz =Lx and the vertical aspect ratio Cy ¼ Lx =d, where Lx is the distance between the different temperaturecontrolled end walls of the hole, d is the depth, and Lz is the distance between the side walls. The volume ratio between the liquid filled in the hole and the hole itself, V=V 0 ¼ V=ðLx dLz Þ, is kept very close to unity. One end wall is heated to maintain the temperature at T ¼ T h and another is cooled at T ¼ T c to realize a designated temperature difference DT ¼ T h T c between the end walls sustaining the free liquid film. The surfaces of the test plate, except the end and the side walls, are chemically coated with a fluorocoating agent, Marvel CoatÒ (Ryoko Chemical Co., Ltd.), to prevent the leakage of the test fluid. Silicone oil of 5 cSt (Pr ¼ 68:4 at 25 C) is employed as the test liquid. The thermo-physical properties of the test fluid [21,22] are listed in Table 1. Gold-nickel-alloy coated cross-linking acrylic particles of 15 lm in diameter are suspended in the test fluid as the tracer particles. The test fluid with particles is fully illuminated to visualize the flow patterns and monitored by a charged coupled device (CCD) camera at 30 frames per second
(fps). The surface temperature of the free liquid film is simultaneously monitored by an infrared (IR) camera at 30 fps. The intensity of the thermocapillary-induced flow is described with a nondimensional laboratory Marangoni number MaL after Riley and Neitzel [8] as follows:
;
qmj
ð1Þ
where r is the surface tension, rT (¼ @ r=@T) the temperature coefficient of surface tension, and q; m, and j are the density, kinematic viscosity, and thermal diffusivity of the test fluid, respectively. In this definition, the temperature gradient is approximated by DT=Lx . We also consider the theoretical Marangoni number [8] defined as follows:
jrT jð@T=@xÞd
2
Ma ¼
qmj
;
ð2Þ
where @T=@x indicates a temperature gradient. In the present study, we measured the surface temperature distribution using the IR camera along the center line perpendicular to the end walls and took the value in the middle region. Note that the temperature dependence of the viscosity is considered in the evaluation of both Marangoni numbers. The fluid viscosity at a temperature T [ C] is evaluated using an empirical correlation [21]:
mðTÞ 25 T ; ¼ exp 5:892 273:15 þ T m0
ð3Þ
where m0 is the kinematic viscosity of the test fluid at 25 C and T is the temperature in Celsius. The characteristic kinematic viscosity m ¼ mexp is evaluated by:
mexp ðTÞ ¼
mðT h Þ þ mðT c Þ 2
:
ð4Þ
We describe the intensity of the buoyancy effects relative to Marangoni effect by the dynamic Bond number BoD [8,10]:
BoD ¼
qgbd2 ; jrT j
ð5Þ
where b is the thermal expansion coefficient and g the gravitational acceleration. This number corresponds to the ratio between the Rayleigh number Ra ¼ ðgbðDT=Lx Þd Þ=ðmjÞ and MaL . The intensity of deformation of the free liquid film is described by the static Bond number, 4
Bo ¼
Fig. 1. Target geometry: a free liquid film exposed to a horizontal temperature gradient.
jrT jðDT=Lx Þd
2
MaL ¼
qgd2 : r
ð6Þ
In our study, the experiments were conducted with five types of test plates made of aluminum with different thickness: d [mm] = 0:2; 0:4; 0:6; 1:0, and 1:2. The size of the hole in the plate is varied in the interval 1:0 6 Lx [mm] 6 6:0 and 2:0 6 Lz [mm] 6 18:0. These test plates were processed by wire electrical discharge machining with an accuracy of 0:005 mm. The dynamic and static Bond numbers under these conditions are listed in Table 2. The thermocapillary effect mainly dominates the flow induced in the film, and a little surface deformation is generated by the gravity. In addition to the series of experiments with the liquid-film holder made of aluminum, we prepared the test section made of quartz glass of 0:6 mm in thickness and (Lx ; Lz ) = (2 mm, 4 mm)
Table 1 Physical properties of the test fluid: the temperature coefficient of surface tension is referred from [21], and the others from [22].
q (kg/m3)
m (m3/s)
j (m2/s)
r (N/m)
rT (N/(m K))
b (1/K)
Pr (–)
9:12 102
5:0 106
7:31 108
19:7 103
6:37 105
1:09 103
68:4
T. Watanabe et al. / International Journal of Heat and Mass Transfer 116 (2018) 635–641 Table 2 Dynamic Bond number BoD and static Bond number Bo against film thickness d. d [mm]
BoD [–]
Bo [–]
0:2
6:4 10
3
1:8 102
0:4
2:6 102
7:3 102
0:6
5:8 102
1:6 101
1:0
1:6 101
4:5 101
1:2
2:3 101
6:5 101
as introduced by [14] in order to prove that the thermocapillary effect induces the double-layered flow under a certain condition. This plate enabled us to monitor the particles motion inside the film from the side wall to obtain a cross-sectional view. The behavior of the particles in x y plane is visualized with a laser light sheet of 0.5 mm in thickness, and detected through the side wall of the plate. We use 6-cSt silicone oil for this series of experiments in order to realize long-term observation with less evaporation of the test fluid to illustrate the particle pathlines.
3. Results and discussion 3.1. Flow transition Examples of induced flow patterns observed from above are shown in Fig. 2 by the pathlines of the suspended particles: (a) two-dimensional basic flow, (b) ‘inner’ oscillatory flow, and (c) fully ‘oscillatory’ flow. Under a small temperature difference, a two-dimensional basic flow appears (see Fig. 2(a)): the basic flow is double-layered. The particles exhibit an almost straight excursion between the end walls in x y plane. Fig. 3(a) indicates a typical example of cross-sectional in (Lx ; Lz ; d) = (2:0 mm, 4:0 mm, 0:6 mm) under V=V 0 1:0. Top image illustrates the pathlines of the particles for 2.0 s along the center line of the film (at z Lz =2), and the bottom illustrates their schematics. The circles in the top frame indicate the position measured by laser displacement sensor as indicated by Fei et al. [14]. As predicted via numerical simulation [1], there exist a pair of convection rolls in the basic flow regime. One roll lies in a top-half of the film, and the other in a bottom-half of the film. The particles flow near the free surface from the hot-end wall to the cold-end wall, and change their direction near the cold-end wall toward the center of the cold-end wall. Then the particles change their direction near the center of the
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cold-end wall, and flow back toward the hot-end wall inside the film. By increasing the temperature difference, the basic flow turns into the ‘inner’ oscillatory flow as the first transition (see Figs. 2 (b) and 3(b)). This flow pattern consists of a three-dimensional spiral roll structure in the interior region of the film in addition to the basic flow. Particles on the spiral roll travel in the span-wise direction as seen near the hot-end wall in Fig. 2(b). The cross-sectional view at z Lz =2 of this flow pattern (see Fig. 3(b)) indicates that the size of the convection rolls becomes different; the upper roll increases its size to suppress the bottom roll. It is found that a small roll structure is formed inside each rolling cell near the hot-end wall. The particles trapped in those small rolls as vortex tubes do not travel along with the basic convection roll between the hot-end and cold-end walls, then exhibit spiral motions inside the small rolls as seen in Fig. 2(b). Such a spiral movement of the particles trapped in vortex tube is also observed in different geometries [23,24]. Note that the basic flow with two convection rolls still remains in this regime. Thus, the surface temperature is still time-independent and any temporal variation of the surface temperature is hardly detected over the free surface. The internal oscillatory flow exhibits a transition to a fully three-dimensional oscillatory flow by imposing a larger temperature difference DT, see Fig. 2(c). This flow pattern is accompanied with a threedimensional time-dependent movement of suspended particles and with a temperature oscillation over the free surfaces. The transition point of this flow is defined as the critical Marangoni number, MaLc . The critical laboratory Marangoni number (MaLc ) against vertical aspect ratio Cy ð¼ Lx =dÞ is indicated in Fig. 4. Each mark indicates the averaged value out of five experimental runs and the bars indicate the maximum and minimum values of the data. MaLc becomes lower as Cy . Note that higher Cy corresponds to larger Lx . That means the flow in the film of higher Cy exhibits a transition under rather small temperature gradient. In the case of higher Cy (larger distance in the temperature gradient direction Lx and/or smaller depth d), it becomes harder for the fluid to return all the way in the interior region of the film against the viscous effect; that is, the fluid in the return flow from the cold-end wall to the hot-end wall is exposed to higher shear stress because of relatively small d. The flow in the film of higher Cy thus becomes unstable to result in smaller critical Marangoni number. Such tendency in the transition condition was also seen in the case of thin liquid layer with a single free surface [8]. It must be noted that
Fig. 2. Typical examples of the induced flow in the film of 6-cSt silicone oil of (Lx ; Lz ; d) = (2:0 mm, 4:0 mm, 0:6 mm) observed from above: (a) Ma ¼ 1:1 102 (basic flow), (b) Ma ¼ 3:9 102 and (c) Ma ¼ 10:3 102 (three-dimensional oscillatory flow). Frames (a) and (b) obtained by integrating the frames for 2.0 s, and (c) for 1.0 s. Left sides are cold wall, right sides are hot wall. Scale bar in each frame corresponds to 1.0 mm.
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Fig. 3. Typical examples of (top) pathlines of the particles and (bottom) its schematics in the cross-sectional views of induced (a) ‘double-layered’ basic flow and (b) ‘inner’ oscillatory flow under (a) DT ¼ 7 K, MaL ¼ 2:0 102 and Ma ¼ 1:4 102 , and (b) DT ¼ 10 K, MaL ¼ 5:2 102 , and Ma ¼ 3:9 102 . The flows are realized in the film of 6 cSt silicone oil in the hole of (Lx ; Lz ; d) = (2:0 mm, 4:0 mm, 0:6 mm) sustained in a glass plate. Left wall corresponds to cold wall and the right to the hot wall. The images are obtained at z Lz =2 by integrating the frames for (a) 2:0 s and (b) 0:5 s. In the frame (a) the positions of the free surfaces detected by laser displacement sensor as introduced by [14] are illustrated.
Fig. 4. Critical laboratory Marangoni number against aspect ratio Cy .
these critical values are significantly larger when compared to the prediction in the case of the thin liquid layer [3]. This is because the temperature gradient is overestimated by DT=Lx in the definition. In order to make a direct comparison with the former works in the case of the thin liquid layer formed in a rectangular cavity, the critical condition is described in Fig. 5 in terms of the theoretical
Marangoni number, Mac , against the dynamic Bond number BoD . In this figure, the prediction by Smith and Davis [3] is plotted against BoD ¼ 0 along with the experimental data of 1-cSt silicone oil by Riley and Neitzel [8]. The critical values for the free liquid film increase as BoD increases. This trend corresponds to that observed in the case of the liquid layer, as experimentally indicated [8]. The thresholds in the case of the free liquid film are higher than those in the case of the liquid layers, especially under larger BoD . As indicated in Table 2, the films of larger BoD have larger Bo in the present system. The liquid layer is formed on the bottom plate, so that the deformation of the liquid film is only due to the convection induced in the liquid layer even under large BoD . And, even under large Bo, the liquid layer does not deform because of the bottom plate sustaining the liquid layer. The free liquid film, on the other hand, is sustained by only end- and side-walls, so that the deformation of the film due to the gravity is not avoidable under large Bo. In addition to that, the deformation of the film due to the thermocapillary-driven convection becomes more significant than that of the liquid layer because the free liquid film has two free surfaces. As BoD decreases, the present results almost converge to the predicted value Mac 368 under BoD ¼ 0 for Pr ¼ 68:4 for the single-free-surface liquid layer [3]. 3.2. Hydrothermal wave
Fig. 5. Critical theoretical Marangoni number against dynamic Bond number. Open symbols stand for the present results, cross marks for liner stability analysis in a thin liquid layer by Smith and Davis [3] (SD in the figure), and closed circles for experimental results in a liquid layer formed in a rectangular cavity by Riley and Neitzel [8] (RN). Numbers in the legend for the present results indicate the distance between the end walls (Lx ) in mm.
It has been indicated that the HW in the thin liquid layer has two types of regimes: the obliquely propagating wave [8] (or traveling wave (TW)) and the standing wave (SW). In the TW regime, thermal waves propagate steadily from cold wall to hot wall. The wave bends near the hot wall due to the vigorous temperature gradient [9]. In the SW regime, on the other hand, the two pairs of waves propagate in opposite directions with the same amplitude, wave number, and propagating velocity. The SW exhibits, as a result, the nodes of the temperature variation over the film, located at the fixed points. In the case of the free liquid film, as far as we have carried out the experiments with the present systems, the SW is rather unstable and seldom appears steadily in the free liquid film. Thus, in the following we focus on the TW regime after the onset of the HW. In the case of Cx < 2 under Cy ¼ 3:33, the wave seems to propagate in a direction almost perpendicular to the temperature gradient from the cold-end wall to the hot-end wall. In the case of Cx P 2, on the other hand, the wave propagates radially from a fixed spot very close to the cold-end wall. Such a propagation behavior was also observed in the case of the liquid layer by Bur-
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guete et al. [7]. In order to describe the characteristics of the TW, we measure the surface temperature using the IR camera and evaluated the deviation of the surface temperature. Fig. 6 illustrates a typical example of (a) surface temperature and (b) temperature deviation in the traveling HW in the film of (Lx ; Lz ; d) = (2:0 mm, 12:0 mm, 0:6 mm) under DT ¼ 12 K. Hot- and cold-end walls are illustrated at the top and bottom of each frame, respectively. Frames (1) – (5) indicate snapshots at phase (1) p=6, (2) p=3, (3) p=2, (4) 2p=3, and (5) 5p=6 of the traveling HW. From the absolute temperature variation in frame (a), it is apparently hard to detect the spatiotemporal oscillation of the temperature over the free surface. By obtaining (b) from the deviation of the surface temperature from the averaged field, one can clearly see the HW propagating from the cold end wall to the hot end wall symmetrically about the center line in span-wise direction as time elapses. We evaluate the wavelength k, the propagation angle h and the fundamental frequency f of HW as illustrated in Fig. 6(b). Fig. 7 illustrates the critical wave number k ¼ 2pd=k, nondimensional frequency F ¼ fd=U, and nondimensional phase speed C ¼ kf =U against Cx for Cy ¼ 3:33, where U indicates the thermocapillary velocity defined as:
U¼
jrT jð@T=@xÞd
qm
:
ð7Þ
These three values are almost constant in the case of Cx > 2. That is, these characteristics are defined independently of the span-wise length of the target geometry. In the case of Cx < 2, on the other hand, it is hard to find any specific tendency. The side walls might affect the wave propagation all over the film, but we have not obtained any concluding ideas yet. Fig. 8 indicates these nondimensional values against the aspect ratio Cy for Cx ¼ 9:0 > 2:0. Those three values are almost constant in the case of Cy P 2. In a film of Cy < 2, or a relatively ‘thick’ film, the wave number decreases and the phase speed increases, although the frequency remains almost the same as that in the film of Cy P 2. Note that both Bo and BoD for the film of small Cy increase, close to 1, comparing to the film of larger Cy . That is, the effect of gravity is not completely negligible. In the following, these three values are discussed by considering the gravity effect. Fig. 9 illustrates the characteristic values of the HW (discussed in Fig. 8) against BoD . The figure also indicates the results in the
Fig. 7. (a) Critical wave number, (b) nondimensional frequency, and (c) nondimensional phase speed of traveling waves just above the threshold as a function of Cx for Cy ¼ 3:33.
system of a liquid layer; the cross marks indicate the prediction of the linear stability analysis (LSA) by Smith and Davis [3] (‘SD’ in the figure) for BoD ¼ 0, the dashed line (referred as ‘RN’) and solid line (referred as ‘Sh’) indicate the threshold between the HW and oscillating multicellular flow (OMC) by Riley and Neitzel [8] and by Shevtsova et al. [10] by numerical approach, respectively. It is found that the wave number of the HW in the free liquid film almost coincides with that in the thin liquid layer against BoD
Fig. 6. Typical example of (a) surface temperature and (b) temperature deviation in the traveling HW in the film of (Lx ; Lz ; d) = (2:0 mm, 12:0 mm, 0:6 mm): Ma ¼ 1:15 103 (DT ¼ 12 K). Frames (1)–(5) indicate snapshots at phase (1) p=6, (2) p=3, (3) p=2, (4) 2p=3, and (5) 5p=6 of the traveling HW, respectively.
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Fig. 8. (a) Critical wave number, (b) nondimensional frequency, and (c) nondimensional phase speed of traveling waves just above the threshold as a function of Cy for Cx ¼ 9:0.
by the LSA (k 2:63 for Pr ¼ 68:4) and by experiment (k 2:5 for Pr ¼ 13:9). Ueno and Torii [1] indicated that the wave number is 2:75 for Pr ¼ 10:0 using three-dimensional numerical simulations as well. The frequency and phase speed in the free liquid film, on the other hand, show higher values than those in the thin liquid layer. These results indicate that the HW in the free liquid film propagates at a speed three times faster than that in the thin liquid layer; nonetheless, the wave number is almost the same. There exists a significant difference in the boundary condition between these two geometries. The free liquid film has two free surfaces and no solid wall parallel to the free surfaces, so that the basic flow consists of two roll structures between the free surfaces as shown by Ueno and Torii [1] and in Fig. 3 in this report. The thin liquid layer, on the other hand, has a single free surface and is laid on the solid surface as a non-slip boundary, so that the basic flow consists of a single roll structure between the free surface and the solid surface as shown by Smith and Davis [3], Riley and Neitzel [8], Shevtsova et al. [10], and Ueno and Torii [1]. Although Ueno and Torii [1] showed that the structure of the HW in the free liquid film is affected by the geometry, we have not reached any conclusive elucidation to explain such differences of the fundamental frequency and phase speed. Further research is needed in order to discuss the physical mechanisms of HW in the free liquid film. It is noteworthy that a unique flow pattern is realized in a ‘thick’ free liquid film, where the static deformation of the film is not avoidable under the normal gravity condition. Fig. 10 illustrates the spatiotemporal variation of the temperature deviation over (a) the top and (b) bottom surfaces of the film of (Lx ; Lz ; d) = (4:0 mm, 2:0 mm, 1:2 mm) under MaL ¼ 5:4 103 . The variation of the surface temperature is monitored along the center line at z Lz =2. Over the top surface, the traveling HW propagating from
Fig. 9. (a) Critical wave number, (b) nondimensional frequency, and (c) nondimensional phase speed as a function of the dynamic Bond number. Open diamonds indicate the present results in a free liquid film (Cx ¼ 9:00), cross marks for LSA in a thin liquid layer by Smith and Davis [3] (SD in figure), and closed circles for experimental results in a liquid layer formed in a rectangular cavity by Riley and Neitzel [8] (RN). The dashed line (referred as ‘RN’) and solid line (referred as ‘Sh’) indicate the threshold between the HW and OMC by Riley and Neitzel [8] and by Shevtsova et al. [10], respectively.
Fig. 10. Spatio-temporal evolution of the TW and SW regime at DT ¼ 53:0 K (MaL ¼ 5:4 103 ; ¼ 0:17) in the film of (Lx ; Lz ; d) = (4:0 mm, 2:0 mm, 1:2 mm); at (a) top surface and (b) bottom surface.
T. Watanabe et al. / International Journal of Heat and Mass Transfer 116 (2018) 635–641
the cold-end wall to the hot-end wall at a constant speed with constant wave number is clearly recognized. The HW bents its angle near the hot-end wall, which is similar to the behavior of the HW realized in the thin liquid layer [9] as aforementioned. Generally, we have observed a similar structure over the bottom surface as well. That is, when the TW (SW) is observed over the top surface, the TW (SW) is also observed over the bottom surface. In the present case, on the other hand, spots of the HW emerge intermittently along the center line at a constant frequency over the bottom surface, which is a typical behavior of the standingwave-type HW. It is found that the traveling HW is realized in the top region of the film, and the standing HW in the bottom region. It is quite seldom that the flow field exhibits multiple oscillatory patterns simultaneously in closed systems such as thin liquid layers and liquid bridges as well as free liquid films, to the best of our knowledge. We need information about the threedimensional flow structure inside the free liquid film in order to elucidate this unique pattern. 4. Concluding remarks An experimental study on the thermocapillary-driven flow in a thin free liquid film of high-Prandtl-number fluid was conducted. The free liquid film was formed in a rectangular hole in an aluminum plate whose thickness was of the order of 0.1–1 mm and was exposed to designated temperature differences between the end walls. We especially focused on time-dependent flow fields due to the hydrothermal wave (HW) instability, so that the target geometry was limited to the thin liquid film in which ‘doublelayered’ basic flow, as defined by Ueno and Torii [1], was realized. We indicated the transition condition from the twodimensional time-independent (‘steady’) flow to the threedimensional time-dependent (‘oscillatory’) flow in terms of the critical Marangoni number. Through the observation of the variation of the surface temperature over the free surfaces, the characteristic values of the HWs such as the wave number, fundamental frequency, and phase speed were indicated against the aspect ratios and dynamic Bond number. Then, we compared the characteristic values of the HW with those in a thin liquid layer that had a single free surface. It was indicated that the nondimensional wave number of the HW in the free liquid film almost coincided with that in the thin liquid layer, and that the nondimensional fundamental frequency and phase speed for the free liquid film were higher than those in the thin liquid layer. Physical explanations for such trends between the free liquid film and the liquid layer were not delivered, and further research is necessary. The unique flow pattern realized in the free liquid film was also introduced; traveling and standing hydrothermal waves coexisted in the free liquid film in a rather ‘thick’ film. Acknowledgement We would like to acknowledge Dr. Donald R. Pettit, from the National Aeronautics and Space Administration (NASA), for fruitful discussion. Mr. Takeshi Katsuta, Ms. Natsuki Ishikawa, and Mr. Linhao Fei, former students of the Graduate School of Tokyo University of Science, are acknowledged for their invaluable helps
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in carrying out the series of experiments. This work was partially supported by Grant-in-Aid for Challenging Exploratory Research from Japan Society for the Promotion of Science (JSPS) (Grant No.: 16K14176). The author IU acknowledges the support by Fund for Strategic Research Areas from Tokyo University of Science. References [1] I. Ueno, T. Torii, Thermocapillary-driven flow in a thin liquid film sustained in a rectangular hole with temperature gradient, Acta Astronaut. 66 (7–8) (2010) 1017–1021. [2] D.R. Pettit, Saturday Morning Science Video, 2003. (cited 2017 Sept. 7th). [3] M.K. Smith, S.H. Davis, Instabilities of dynamic thermocapillary liquid layers. Part 1. Convective instabilities, J. Fluid Mech. 132 (1983) 119–144. [4] F. Daviaud, J.M. Vince, Traveling waves in a fluid layer subjected to a horizontal temperature gradient, Phys. Rev. E 48 (6) (1993) 4432–4436. [5] J.F. Mercier, C. Normand, Buoyant-thermocapillary instabilities of differentially heated liquid layers, Phys. Fluids 8 (6) (1996) 1433–1445. [6] N. Garnier, A. Chiffaudel, Nonlinear transition to a global mode for travelingwave instability in a finite box, Phys. Rev. Lett. 86 (1) (2001) 75–78. [7] J. Burguete, N. Mukolobwiez, F. Daviaud, N. Garnier, A. Chiffaudel, Buoyantthermocapillary instabilities in extended liquid layers subjected to a horizontal temperature gradient, Phys. Fluids 13 (10) (2001) 2773–2787. [8] R.J. Riley, G.P. Neitzel, Instability of thermocapillary-buoyancy convection in shallow layers. Part 1. Characterization of steady and oscillatory instabilities, J. Fluid Mech. 359 (1998) 143–164. [9] H. Kawamura, E. Tagaya, Y. Hoshino, A consideration on the relation between the oscillatory thermocapillary flow in a liquid bridge and the hydrothermal wave in a thin liquid layer, Int. J. Heat Mass Transf. 50 (2007) 1263–1268. [10] V.M. Shevtsova, A.A. Nepomnyashchy, J.C. Legros, Themocapillary-buoyancy convection in a shallow cavity heated from the side, Phys. Rev. E 67 (2003) 066308. [11] A.M. Anderson, S.H. Davis, Solidification of free liquid films, J. Fluid Mech. 617 (2008) 87–106. [12] H. Kaneko, K. Koba, I. Ueno, Numerical simulation on effect of melt convection upon solidification process of pure material by phase-field method (in japanese), Int. J. Microgr. Sci. Appl. 26 (3) (2009) 237–243. [13] I. Ueno, T. Katsuta, T. Watanabe, Flow patterns induced by thermocapillary effect in thin free liquid film accompanying with static/dynamic deformations, in: Proc. 2nd European Conf. Microfluidics (Microfluidics 2010) (Toulouse, France, 8th–10th Dec. 2010) CD-ROM (2010) microFLU10-203. [14] L. Fei, K. Ikebukuro, T. Katsuta, T. Kaneko, I. Ueno, D.R. Pettit, Effect of static deformation on basic flow patterns in thermocapillary-driven free liquid film, Microgr. Sci. Technol. 29 (2017) 29–36. [15] D.R. Pettit, Episode 3: Thin Film Physics, 2012. (cited 2017 Sept. 7th). [16] T. Yamamoto, Y. Takagi, Y. Okano, S. Dost, Numerical investigation for the effect of the liquid film volume on thermocapillary flow direction in a thin circular liquid film, Phys. Fluids 25 (2013) 082108. [17] B. Messmer, T. Lemee, K. Ikebukuro, I. Ueno, R. Narayanan, Confined thermocapillary flows in a double free-surface film with small Marangoni numbers, Int. J. Heat Mass Transf. 78 (2014) 1060–1067. [18] H.C. Kuhlmann, Large-scale liquid motion in free thermocapillary films, Microgr. Sci. Technol. 26 (2014) 397–400. [19] I. Ueno, T. Watanabe, Flow transition in a free rectangular liquid film under a temperature gradient, Int. J. Transp. Phenom. 12 (2011) 301–306. [20] D. Limsukhawat, Y. Dekio, K. Ikebukuro, C. Hong, I. Ueno, Flow patterns induced by thermocapillary effect in a thin free liquid film of a high Prandtl number fluid, Prog. Comput. Fluid Dynam., An Int. J. (PCFD) 13 (2013) 133– 144. [21] L. Shin-Etsu Chemical Co., Technical Note: Silicone Oil KF96, Tech. Rep., ShinEtsu Chemical Co., Ltd., 2011 (in Japanese). [22] I. Ueno, S. Tanaka, H. Kawamura, Oscillatory and chaotic thermocapillary convection in a half-zone liquid bridge, Phys. Fluids 15 (2) (2003) 408–416. [23] D. Vigolo, S. Radi, H.A. Stone, Unexpected trapping of particles at a t junction, Proc. Natl. Acad. Sci. USA 111 (2014) 4770–4775. [24] J.T. Ault, A. Fani, K.K. Chen, S. Shin, F. Gallaire, H.A. Stone, Vortex-breakdowninduced particle capture in branching junctions, Phys. Rev. Lett. 117 (2016) 084501.
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Flow transition and hydrothermal wave instability of thermocapillarydriven flow in a free rectangular liquid film Toshiki Watanabe a,1, Yosuke Kowata a, Ichiro Ueno b,⇑ a b
Division of Mechanical Engineering, School of Science and Technology, Tokyo University of Science, 2641 Yamazaki, Noda, Chiba 278-8510, Japan Department of Mechanical Engineering, Faculty of Science and Technology, Tokyo University of Science, Japan
a r t i c l e
i n f o
Article history: Received 20 February 2017 Received in revised form 7 September 2017 Accepted 16 September 2017 Available online 21 September 2017 Keywords: Free liquid film Hydrothermal wave Thermocapillary effect Flow pattern Traveling wave
a b s t r a c t Thermocapillary-driven flow in a free liquid film, which has two gas-liquid interfaces, is experimentally investigated. Silicone oil of 5 cSt is employed as the test fluid. Two-dimensional basic flow known as ‘double-layered flow’ after Ueno and Torii (2010) is realized under small-enough Marangoni number Ma, the non-dimensional number to describe the intensity of thermocapillary effect, under the geometry considered in the present study. The flow exhibits a transition from the two-dimensional steady flow state to the three-dimensional oscillatory state when the intensity of the imposed thermocapillary effect along the free surfaces exceeds the threshold. In this oscillatory regime, two types of hydrothermal wave instabilities are observed: the traveling-wave flow and the standing-wave flow. We especially focus on the traveling-wave instability and compare it with a hydrothermal wave in a liquid layer with a single free surface investigated by other researchers. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Pettit performed a series of experiments using a liquid film that had double-free surfaces (hereafter a free liquid film) on The International Space Station in 2003 [2]. In the experiments, Pettit formed a thin free liquid film of water in a ring made by metal wire and exposed the film to non-uniform temperature distribution by placing a heated iron close to one end of the wire ring. It was realized that the fluid was driven in the film toward the heated part, that is, a flow was induced from a colder region to a hotter region in spite of the negative temperature coefficient of the surface tension. These experiments showed that the free liquid film had a potential to be one way to obtain a new kind of crystallization process of materials and/or mass transport. We were inspired by these performances and started this study. In the case of a thin liquid layer formed on a base plate, that is, a film with a single free surface (a thin liquid layer, hereafter), Smith and Davis [3] predicted thermal fluid instability in an infinite liquid layer imposed by a constant-temperature gradient along a free surface by performing a linear stability analysis. It was named ‘hydrothermal wave (HW) instability. Daviaud and Vince [4] predicted instabilities under a coupling of thermocapillary and
⇑ Corresponding author. 1
E-mail address: [email protected] (I. Ueno). Present: Konica Minolta, Inc.
http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.09.059 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.
buoyancy effects, and then a variety of experiments were conducted with a liquid film in a narrow long channel [5–7] and shallow cavity [8,9] by various research groups. Riley and Neitzel [8] realized the HW predicted by Smith and Davis [3] in an experiment for the first time in a rectangular cavity with a temperature difference between both end walls. They reported a stability diagram and indicated the criterion to realize the HW in the film as a function of the dynamic Bond number BoD as BoD 6 0:22. Shevtsova et al. [10] carried out a series of two-dimensional numerical simulations to show the stability diagram and indicated the HW criterion as BoD 6 0:25. There are few studies, on the other hand, on the thermocapillary-driven flow in a free liquid film. The longitudinal solidification of a two-dimensional single-component free liquid film was discussed by simulations [11,12]. In the case of a twosurface film, Ueno and Torii [1] indicated two types of basic flows by a series of experiments: the double-layered flow and singlelayered flow. The effect of the liquid film shape in terms of the volume ratio was experimentally indicated on the selection of the basic flow and the direction of the net flow inside the film by measuring the liquid film profiles and the surface temperature distribution in the case of steady flows by [13]. Such an effect was confirmed through a series of parabolic flight experiments [14] as well as on-orbit experiments [15]. The Pettit demonstration was reproduced numerically by Yamamoto et al. [16] and the effect of the film shape near the heated area was shown. Concerning the
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physical mechanism of the selection of basic flows in the free liquid films, a discussion has been given [17,18], but we have never obtained a comprehensive explanation for the selection of the basic flows. As for the three-dimensional oscillatory flows in the free liquid film, much less knowledge has been accumulated. Ueno and Watanabe [19] indicated the flow transitions from the timeindependent steady flow to time-dependent ‘oscillatory’ flow for both basic flows. Ueno and Torii [1] and Limsukhawat et al. [20] conducted numerical simulations on the steady and oscillatory flows in the case of the double-layered basic flow. The present study aims to illustrate the characteristics of the HW in a free liquid film. In particular, it deals with the transition processes of the induced flow patterns of the traveling wave as a function of the geometry of the free liquid film under the conditions of double-layered basic flows, by making comparisons with results in the system of a thin liquid layer formed in a rectangular cavity with a single-free surface. 2. Experimental setup The experimental geometry is shown in Fig. 1. A liquid is filled in the rectangular hole made in a flat aluminum plate. In the present geometry, two kinds of aspect ratios are defined: the horizontal aspect ratio Cx ¼ Lz =Lx and the vertical aspect ratio Cy ¼ Lx =d, where Lx is the distance between the different temperaturecontrolled end walls of the hole, d is the depth, and Lz is the distance between the side walls. The volume ratio between the liquid filled in the hole and the hole itself, V=V 0 ¼ V=ðLx dLz Þ, is kept very close to unity. One end wall is heated to maintain the temperature at T ¼ T h and another is cooled at T ¼ T c to realize a designated temperature difference DT ¼ T h T c between the end walls sustaining the free liquid film. The surfaces of the test plate, except the end and the side walls, are chemically coated with a fluorocoating agent, Marvel CoatÒ (Ryoko Chemical Co., Ltd.), to prevent the leakage of the test fluid. Silicone oil of 5 cSt (Pr ¼ 68:4 at 25 C) is employed as the test liquid. The thermo-physical properties of the test fluid [21,22] are listed in Table 1. Gold-nickel-alloy coated cross-linking acrylic particles of 15 lm in diameter are suspended in the test fluid as the tracer particles. The test fluid with particles is fully illuminated to visualize the flow patterns and monitored by a charged coupled device (CCD) camera at 30 frames per second
(fps). The surface temperature of the free liquid film is simultaneously monitored by an infrared (IR) camera at 30 fps. The intensity of the thermocapillary-induced flow is described with a nondimensional laboratory Marangoni number MaL after Riley and Neitzel [8] as follows:
;
qmj
ð1Þ
where r is the surface tension, rT (¼ @ r=@T) the temperature coefficient of surface tension, and q; m, and j are the density, kinematic viscosity, and thermal diffusivity of the test fluid, respectively. In this definition, the temperature gradient is approximated by DT=Lx . We also consider the theoretical Marangoni number [8] defined as follows:
jrT jð@T=@xÞd
2
Ma ¼
qmj
;
ð2Þ
where @T=@x indicates a temperature gradient. In the present study, we measured the surface temperature distribution using the IR camera along the center line perpendicular to the end walls and took the value in the middle region. Note that the temperature dependence of the viscosity is considered in the evaluation of both Marangoni numbers. The fluid viscosity at a temperature T [ C] is evaluated using an empirical correlation [21]:
mðTÞ 25 T ; ¼ exp 5:892 273:15 þ T m0
ð3Þ
where m0 is the kinematic viscosity of the test fluid at 25 C and T is the temperature in Celsius. The characteristic kinematic viscosity m ¼ mexp is evaluated by:
mexp ðTÞ ¼
mðT h Þ þ mðT c Þ 2
:
ð4Þ
We describe the intensity of the buoyancy effects relative to Marangoni effect by the dynamic Bond number BoD [8,10]:
BoD ¼
qgbd2 ; jrT j
ð5Þ
where b is the thermal expansion coefficient and g the gravitational acceleration. This number corresponds to the ratio between the Rayleigh number Ra ¼ ðgbðDT=Lx Þd Þ=ðmjÞ and MaL . The intensity of deformation of the free liquid film is described by the static Bond number, 4
Bo ¼
Fig. 1. Target geometry: a free liquid film exposed to a horizontal temperature gradient.
jrT jðDT=Lx Þd
2
MaL ¼
qgd2 : r
ð6Þ
In our study, the experiments were conducted with five types of test plates made of aluminum with different thickness: d [mm] = 0:2; 0:4; 0:6; 1:0, and 1:2. The size of the hole in the plate is varied in the interval 1:0 6 Lx [mm] 6 6:0 and 2:0 6 Lz [mm] 6 18:0. These test plates were processed by wire electrical discharge machining with an accuracy of 0:005 mm. The dynamic and static Bond numbers under these conditions are listed in Table 2. The thermocapillary effect mainly dominates the flow induced in the film, and a little surface deformation is generated by the gravity. In addition to the series of experiments with the liquid-film holder made of aluminum, we prepared the test section made of quartz glass of 0:6 mm in thickness and (Lx ; Lz ) = (2 mm, 4 mm)
Table 1 Physical properties of the test fluid: the temperature coefficient of surface tension is referred from [21], and the others from [22].
q (kg/m3)
m (m3/s)
j (m2/s)
r (N/m)
rT (N/(m K))
b (1/K)
Pr (–)
9:12 102
5:0 106
7:31 108
19:7 103
6:37 105
1:09 103
68:4
T. Watanabe et al. / International Journal of Heat and Mass Transfer 116 (2018) 635–641 Table 2 Dynamic Bond number BoD and static Bond number Bo against film thickness d. d [mm]
BoD [–]
Bo [–]
0:2
6:4 10
3
1:8 102
0:4
2:6 102
7:3 102
0:6
5:8 102
1:6 101
1:0
1:6 101
4:5 101
1:2
2:3 101
6:5 101
as introduced by [14] in order to prove that the thermocapillary effect induces the double-layered flow under a certain condition. This plate enabled us to monitor the particles motion inside the film from the side wall to obtain a cross-sectional view. The behavior of the particles in x y plane is visualized with a laser light sheet of 0.5 mm in thickness, and detected through the side wall of the plate. We use 6-cSt silicone oil for this series of experiments in order to realize long-term observation with less evaporation of the test fluid to illustrate the particle pathlines.
3. Results and discussion 3.1. Flow transition Examples of induced flow patterns observed from above are shown in Fig. 2 by the pathlines of the suspended particles: (a) two-dimensional basic flow, (b) ‘inner’ oscillatory flow, and (c) fully ‘oscillatory’ flow. Under a small temperature difference, a two-dimensional basic flow appears (see Fig. 2(a)): the basic flow is double-layered. The particles exhibit an almost straight excursion between the end walls in x y plane. Fig. 3(a) indicates a typical example of cross-sectional in (Lx ; Lz ; d) = (2:0 mm, 4:0 mm, 0:6 mm) under V=V 0 1:0. Top image illustrates the pathlines of the particles for 2.0 s along the center line of the film (at z Lz =2), and the bottom illustrates their schematics. The circles in the top frame indicate the position measured by laser displacement sensor as indicated by Fei et al. [14]. As predicted via numerical simulation [1], there exist a pair of convection rolls in the basic flow regime. One roll lies in a top-half of the film, and the other in a bottom-half of the film. The particles flow near the free surface from the hot-end wall to the cold-end wall, and change their direction near the cold-end wall toward the center of the cold-end wall. Then the particles change their direction near the center of the
637
cold-end wall, and flow back toward the hot-end wall inside the film. By increasing the temperature difference, the basic flow turns into the ‘inner’ oscillatory flow as the first transition (see Figs. 2 (b) and 3(b)). This flow pattern consists of a three-dimensional spiral roll structure in the interior region of the film in addition to the basic flow. Particles on the spiral roll travel in the span-wise direction as seen near the hot-end wall in Fig. 2(b). The cross-sectional view at z Lz =2 of this flow pattern (see Fig. 3(b)) indicates that the size of the convection rolls becomes different; the upper roll increases its size to suppress the bottom roll. It is found that a small roll structure is formed inside each rolling cell near the hot-end wall. The particles trapped in those small rolls as vortex tubes do not travel along with the basic convection roll between the hot-end and cold-end walls, then exhibit spiral motions inside the small rolls as seen in Fig. 2(b). Such a spiral movement of the particles trapped in vortex tube is also observed in different geometries [23,24]. Note that the basic flow with two convection rolls still remains in this regime. Thus, the surface temperature is still time-independent and any temporal variation of the surface temperature is hardly detected over the free surface. The internal oscillatory flow exhibits a transition to a fully three-dimensional oscillatory flow by imposing a larger temperature difference DT, see Fig. 2(c). This flow pattern is accompanied with a threedimensional time-dependent movement of suspended particles and with a temperature oscillation over the free surfaces. The transition point of this flow is defined as the critical Marangoni number, MaLc . The critical laboratory Marangoni number (MaLc ) against vertical aspect ratio Cy ð¼ Lx =dÞ is indicated in Fig. 4. Each mark indicates the averaged value out of five experimental runs and the bars indicate the maximum and minimum values of the data. MaLc becomes lower as Cy . Note that higher Cy corresponds to larger Lx . That means the flow in the film of higher Cy exhibits a transition under rather small temperature gradient. In the case of higher Cy (larger distance in the temperature gradient direction Lx and/or smaller depth d), it becomes harder for the fluid to return all the way in the interior region of the film against the viscous effect; that is, the fluid in the return flow from the cold-end wall to the hot-end wall is exposed to higher shear stress because of relatively small d. The flow in the film of higher Cy thus becomes unstable to result in smaller critical Marangoni number. Such tendency in the transition condition was also seen in the case of thin liquid layer with a single free surface [8]. It must be noted that
Fig. 2. Typical examples of the induced flow in the film of 6-cSt silicone oil of (Lx ; Lz ; d) = (2:0 mm, 4:0 mm, 0:6 mm) observed from above: (a) Ma ¼ 1:1 102 (basic flow), (b) Ma ¼ 3:9 102 and (c) Ma ¼ 10:3 102 (three-dimensional oscillatory flow). Frames (a) and (b) obtained by integrating the frames for 2.0 s, and (c) for 1.0 s. Left sides are cold wall, right sides are hot wall. Scale bar in each frame corresponds to 1.0 mm.
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Fig. 3. Typical examples of (top) pathlines of the particles and (bottom) its schematics in the cross-sectional views of induced (a) ‘double-layered’ basic flow and (b) ‘inner’ oscillatory flow under (a) DT ¼ 7 K, MaL ¼ 2:0 102 and Ma ¼ 1:4 102 , and (b) DT ¼ 10 K, MaL ¼ 5:2 102 , and Ma ¼ 3:9 102 . The flows are realized in the film of 6 cSt silicone oil in the hole of (Lx ; Lz ; d) = (2:0 mm, 4:0 mm, 0:6 mm) sustained in a glass plate. Left wall corresponds to cold wall and the right to the hot wall. The images are obtained at z Lz =2 by integrating the frames for (a) 2:0 s and (b) 0:5 s. In the frame (a) the positions of the free surfaces detected by laser displacement sensor as introduced by [14] are illustrated.
Fig. 4. Critical laboratory Marangoni number against aspect ratio Cy .
these critical values are significantly larger when compared to the prediction in the case of the thin liquid layer [3]. This is because the temperature gradient is overestimated by DT=Lx in the definition. In order to make a direct comparison with the former works in the case of the thin liquid layer formed in a rectangular cavity, the critical condition is described in Fig. 5 in terms of the theoretical
Marangoni number, Mac , against the dynamic Bond number BoD . In this figure, the prediction by Smith and Davis [3] is plotted against BoD ¼ 0 along with the experimental data of 1-cSt silicone oil by Riley and Neitzel [8]. The critical values for the free liquid film increase as BoD increases. This trend corresponds to that observed in the case of the liquid layer, as experimentally indicated [8]. The thresholds in the case of the free liquid film are higher than those in the case of the liquid layers, especially under larger BoD . As indicated in Table 2, the films of larger BoD have larger Bo in the present system. The liquid layer is formed on the bottom plate, so that the deformation of the liquid film is only due to the convection induced in the liquid layer even under large BoD . And, even under large Bo, the liquid layer does not deform because of the bottom plate sustaining the liquid layer. The free liquid film, on the other hand, is sustained by only end- and side-walls, so that the deformation of the film due to the gravity is not avoidable under large Bo. In addition to that, the deformation of the film due to the thermocapillary-driven convection becomes more significant than that of the liquid layer because the free liquid film has two free surfaces. As BoD decreases, the present results almost converge to the predicted value Mac 368 under BoD ¼ 0 for Pr ¼ 68:4 for the single-free-surface liquid layer [3]. 3.2. Hydrothermal wave
Fig. 5. Critical theoretical Marangoni number against dynamic Bond number. Open symbols stand for the present results, cross marks for liner stability analysis in a thin liquid layer by Smith and Davis [3] (SD in the figure), and closed circles for experimental results in a liquid layer formed in a rectangular cavity by Riley and Neitzel [8] (RN). Numbers in the legend for the present results indicate the distance between the end walls (Lx ) in mm.
It has been indicated that the HW in the thin liquid layer has two types of regimes: the obliquely propagating wave [8] (or traveling wave (TW)) and the standing wave (SW). In the TW regime, thermal waves propagate steadily from cold wall to hot wall. The wave bends near the hot wall due to the vigorous temperature gradient [9]. In the SW regime, on the other hand, the two pairs of waves propagate in opposite directions with the same amplitude, wave number, and propagating velocity. The SW exhibits, as a result, the nodes of the temperature variation over the film, located at the fixed points. In the case of the free liquid film, as far as we have carried out the experiments with the present systems, the SW is rather unstable and seldom appears steadily in the free liquid film. Thus, in the following we focus on the TW regime after the onset of the HW. In the case of Cx < 2 under Cy ¼ 3:33, the wave seems to propagate in a direction almost perpendicular to the temperature gradient from the cold-end wall to the hot-end wall. In the case of Cx P 2, on the other hand, the wave propagates radially from a fixed spot very close to the cold-end wall. Such a propagation behavior was also observed in the case of the liquid layer by Bur-
T. Watanabe et al. / International Journal of Heat and Mass Transfer 116 (2018) 635–641
639
guete et al. [7]. In order to describe the characteristics of the TW, we measure the surface temperature using the IR camera and evaluated the deviation of the surface temperature. Fig. 6 illustrates a typical example of (a) surface temperature and (b) temperature deviation in the traveling HW in the film of (Lx ; Lz ; d) = (2:0 mm, 12:0 mm, 0:6 mm) under DT ¼ 12 K. Hot- and cold-end walls are illustrated at the top and bottom of each frame, respectively. Frames (1) – (5) indicate snapshots at phase (1) p=6, (2) p=3, (3) p=2, (4) 2p=3, and (5) 5p=6 of the traveling HW. From the absolute temperature variation in frame (a), it is apparently hard to detect the spatiotemporal oscillation of the temperature over the free surface. By obtaining (b) from the deviation of the surface temperature from the averaged field, one can clearly see the HW propagating from the cold end wall to the hot end wall symmetrically about the center line in span-wise direction as time elapses. We evaluate the wavelength k, the propagation angle h and the fundamental frequency f of HW as illustrated in Fig. 6(b). Fig. 7 illustrates the critical wave number k ¼ 2pd=k, nondimensional frequency F ¼ fd=U, and nondimensional phase speed C ¼ kf =U against Cx for Cy ¼ 3:33, where U indicates the thermocapillary velocity defined as:
U¼
jrT jð@T=@xÞd
qm
:
ð7Þ
These three values are almost constant in the case of Cx > 2. That is, these characteristics are defined independently of the span-wise length of the target geometry. In the case of Cx < 2, on the other hand, it is hard to find any specific tendency. The side walls might affect the wave propagation all over the film, but we have not obtained any concluding ideas yet. Fig. 8 indicates these nondimensional values against the aspect ratio Cy for Cx ¼ 9:0 > 2:0. Those three values are almost constant in the case of Cy P 2. In a film of Cy < 2, or a relatively ‘thick’ film, the wave number decreases and the phase speed increases, although the frequency remains almost the same as that in the film of Cy P 2. Note that both Bo and BoD for the film of small Cy increase, close to 1, comparing to the film of larger Cy . That is, the effect of gravity is not completely negligible. In the following, these three values are discussed by considering the gravity effect. Fig. 9 illustrates the characteristic values of the HW (discussed in Fig. 8) against BoD . The figure also indicates the results in the
Fig. 7. (a) Critical wave number, (b) nondimensional frequency, and (c) nondimensional phase speed of traveling waves just above the threshold as a function of Cx for Cy ¼ 3:33.
system of a liquid layer; the cross marks indicate the prediction of the linear stability analysis (LSA) by Smith and Davis [3] (‘SD’ in the figure) for BoD ¼ 0, the dashed line (referred as ‘RN’) and solid line (referred as ‘Sh’) indicate the threshold between the HW and oscillating multicellular flow (OMC) by Riley and Neitzel [8] and by Shevtsova et al. [10] by numerical approach, respectively. It is found that the wave number of the HW in the free liquid film almost coincides with that in the thin liquid layer against BoD
Fig. 6. Typical example of (a) surface temperature and (b) temperature deviation in the traveling HW in the film of (Lx ; Lz ; d) = (2:0 mm, 12:0 mm, 0:6 mm): Ma ¼ 1:15 103 (DT ¼ 12 K). Frames (1)–(5) indicate snapshots at phase (1) p=6, (2) p=3, (3) p=2, (4) 2p=3, and (5) 5p=6 of the traveling HW, respectively.
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Fig. 8. (a) Critical wave number, (b) nondimensional frequency, and (c) nondimensional phase speed of traveling waves just above the threshold as a function of Cy for Cx ¼ 9:0.
by the LSA (k 2:63 for Pr ¼ 68:4) and by experiment (k 2:5 for Pr ¼ 13:9). Ueno and Torii [1] indicated that the wave number is 2:75 for Pr ¼ 10:0 using three-dimensional numerical simulations as well. The frequency and phase speed in the free liquid film, on the other hand, show higher values than those in the thin liquid layer. These results indicate that the HW in the free liquid film propagates at a speed three times faster than that in the thin liquid layer; nonetheless, the wave number is almost the same. There exists a significant difference in the boundary condition between these two geometries. The free liquid film has two free surfaces and no solid wall parallel to the free surfaces, so that the basic flow consists of two roll structures between the free surfaces as shown by Ueno and Torii [1] and in Fig. 3 in this report. The thin liquid layer, on the other hand, has a single free surface and is laid on the solid surface as a non-slip boundary, so that the basic flow consists of a single roll structure between the free surface and the solid surface as shown by Smith and Davis [3], Riley and Neitzel [8], Shevtsova et al. [10], and Ueno and Torii [1]. Although Ueno and Torii [1] showed that the structure of the HW in the free liquid film is affected by the geometry, we have not reached any conclusive elucidation to explain such differences of the fundamental frequency and phase speed. Further research is needed in order to discuss the physical mechanisms of HW in the free liquid film. It is noteworthy that a unique flow pattern is realized in a ‘thick’ free liquid film, where the static deformation of the film is not avoidable under the normal gravity condition. Fig. 10 illustrates the spatiotemporal variation of the temperature deviation over (a) the top and (b) bottom surfaces of the film of (Lx ; Lz ; d) = (4:0 mm, 2:0 mm, 1:2 mm) under MaL ¼ 5:4 103 . The variation of the surface temperature is monitored along the center line at z Lz =2. Over the top surface, the traveling HW propagating from
Fig. 9. (a) Critical wave number, (b) nondimensional frequency, and (c) nondimensional phase speed as a function of the dynamic Bond number. Open diamonds indicate the present results in a free liquid film (Cx ¼ 9:00), cross marks for LSA in a thin liquid layer by Smith and Davis [3] (SD in figure), and closed circles for experimental results in a liquid layer formed in a rectangular cavity by Riley and Neitzel [8] (RN). The dashed line (referred as ‘RN’) and solid line (referred as ‘Sh’) indicate the threshold between the HW and OMC by Riley and Neitzel [8] and by Shevtsova et al. [10], respectively.
Fig. 10. Spatio-temporal evolution of the TW and SW regime at DT ¼ 53:0 K (MaL ¼ 5:4 103 ; ¼ 0:17) in the film of (Lx ; Lz ; d) = (4:0 mm, 2:0 mm, 1:2 mm); at (a) top surface and (b) bottom surface.
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the cold-end wall to the hot-end wall at a constant speed with constant wave number is clearly recognized. The HW bents its angle near the hot-end wall, which is similar to the behavior of the HW realized in the thin liquid layer [9] as aforementioned. Generally, we have observed a similar structure over the bottom surface as well. That is, when the TW (SW) is observed over the top surface, the TW (SW) is also observed over the bottom surface. In the present case, on the other hand, spots of the HW emerge intermittently along the center line at a constant frequency over the bottom surface, which is a typical behavior of the standingwave-type HW. It is found that the traveling HW is realized in the top region of the film, and the standing HW in the bottom region. It is quite seldom that the flow field exhibits multiple oscillatory patterns simultaneously in closed systems such as thin liquid layers and liquid bridges as well as free liquid films, to the best of our knowledge. We need information about the threedimensional flow structure inside the free liquid film in order to elucidate this unique pattern. 4. Concluding remarks An experimental study on the thermocapillary-driven flow in a thin free liquid film of high-Prandtl-number fluid was conducted. The free liquid film was formed in a rectangular hole in an aluminum plate whose thickness was of the order of 0.1–1 mm and was exposed to designated temperature differences between the end walls. We especially focused on time-dependent flow fields due to the hydrothermal wave (HW) instability, so that the target geometry was limited to the thin liquid film in which ‘doublelayered’ basic flow, as defined by Ueno and Torii [1], was realized. We indicated the transition condition from the twodimensional time-independent (‘steady’) flow to the threedimensional time-dependent (‘oscillatory’) flow in terms of the critical Marangoni number. Through the observation of the variation of the surface temperature over the free surfaces, the characteristic values of the HWs such as the wave number, fundamental frequency, and phase speed were indicated against the aspect ratios and dynamic Bond number. Then, we compared the characteristic values of the HW with those in a thin liquid layer that had a single free surface. It was indicated that the nondimensional wave number of the HW in the free liquid film almost coincided with that in the thin liquid layer, and that the nondimensional fundamental frequency and phase speed for the free liquid film were higher than those in the thin liquid layer. Physical explanations for such trends between the free liquid film and the liquid layer were not delivered, and further research is necessary. The unique flow pattern realized in the free liquid film was also introduced; traveling and standing hydrothermal waves coexisted in the free liquid film in a rather ‘thick’ film. Acknowledgement We would like to acknowledge Dr. Donald R. Pettit, from the National Aeronautics and Space Administration (NASA), for fruitful discussion. Mr. Takeshi Katsuta, Ms. Natsuki Ishikawa, and Mr. Linhao Fei, former students of the Graduate School of Tokyo University of Science, are acknowledged for their invaluable helps
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in carrying out the series of experiments. This work was partially supported by Grant-in-Aid for Challenging Exploratory Research from Japan Society for the Promotion of Science (JSPS) (Grant No.: 16K14176). The author IU acknowledges the support by Fund for Strategic Research Areas from Tokyo University of Science. References [1] I. Ueno, T. Torii, Thermocapillary-driven flow in a thin liquid film sustained in a rectangular hole with temperature gradient, Acta Astronaut. 66 (7–8) (2010) 1017–1021. [2] D.R. Pettit, Saturday Morning Science Video, 2003.