Article pubs.acs.org/Langmuir
Formation, Evolution, and Extinction of Standing Waves in Evaporation from Pores Cosimo Buffone*,† and Khellil Sefiane†,‡ †
School of Engineering, The University of Edinburgh, Mayfield Road, Edinburgh EH9 3JL, United Kingdom Tianjin Key Laboratory of Refrigeration Technology, Tianjin University of Commerce, Tianjin 300134, People’s Republic of China
‡
ABSTRACT: We report on the formation, evolution, and extinction of standing waves (SWs) detected by infrared measurements at the upper region of a curved meniscus interface pinned at the mouth of a horizontally positioned capillary pore. The SWs are clear and strong in acetone but absent in ethanol for both tube sizes investigated (1−2 mm diameter). Dependent upon the tube size and the initial liquid filling ratio, the SWs start sooner for a lower filling ratio. The intriguing experimental observation is that the SWs disappear at a specified liquid length between the receding meniscus and the one pinned at the tube mouth, which seems to depend strongly upon the tube size and independent of the initial liquid filling ratio. The origin of the SWs could be due to the strong interaction between surface tension and gravity, which also generates oscillatory periodic Marangoni flow in the meniscus liquid phase.
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liquid films), Benard−Marangoni convection dominates, whereas for R > lC (thick liquid films), Rayleigh−Benard convection dominates, as reported by Zeytounian.6 Later on, a different configuration was investigated where thermocapillary convection was generated by a lateral temperature gradient existing along the liquid−vapor interface created by imposing a temperature difference between the two end walls of a shallow liquid layer as reported in Smith and Davis,7,8 Davis,9 and Riley.10 When a liquid meniscus is formed as in evaporation, condensation, or boiling, there are three important regions, namely, macroregion, microregion, and adsorbed layer. Much work was performed in the 1970s by Wayner and co-workers in understanding the mechanism involved in the heat and mass transfer from evaporating thin liquid films.11,12 The subject was however still not fully understood, as demonstrated by numerous other works that have appeared in the area during the 1990s.13−15 It is worth mentioning that most of the early works on Marangoni convection did not deal with evaporative cooling. More recently, it has been clearly experimentally
INTRODUCTION Marangoni or thermocapillary-driven convection has been investigated as early as the middle of the 19th century as a result of the recognition that surface tension becomes an important parameter in small-scale processes when an interface is present. The first researcher to have reported surface tension effects was Thomson;1 he reported on the spreading of drops of alcohol on water, the so-called “tears of wine”. Benard2 investigated the case of a liquid layer heated from below and observed convective patterns. However, Benard attributed the convection to buoyancy; it was only half a century later that Pearson3 proposed a different explanation, attributing the cause of the flow to surface tension and introducing what later would have been called the Marangoni number by Scriven and Sternling.4 Surface tension is really important in defining the interface at microscale as demonstrated by Levich and Krylov.5 For such problems, a dimensionless number is introduced, which represents the ratio Ma/Ra = ((∂σ/∂T)/ρgβ)(1/R2) (where Ma and Ra are the Marangoni and Rayleigh numbers, respectively, σ is the surface tension, ρ is the liquid density, g is the gravitational acceleration, β is the volumetric expansion coefficient, and R is the characteristic length); when introducing the capillary length lC = (σ/ρg)1/2, the ratio becomes Ma/Ra = (1/β)(lC/R)2. Therefore, for R < lC (thin © XXXX American Chemical Society
Received: August 10, 2016 Revised: October 1, 2016 Published: November 3, 2016 A
DOI: 10.1021/acs.langmuir.6b02970 Langmuir XXXX, XXX, XXX−XXX
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Figure 1. This IR time sequence shows the meniscus interfacial temperature during the passing of a SW for 2 mm internal diameter and acetone. The SW can be noted from frames a.2 to a.5 with a varying red area in the upper part of the meniscus.
demonstrated16−18 that the evaporation peaks indeed in the microregion, which is typically a few micrometers in length. The triple-line region in the evaporation of liquids is shown to be of paramount importance because of very high heat flux present near the contact line, as shown by Hohmann and Stephan.17 Pratt and Hallinan19 and Pratt et al.20 have investigated the stability of a volatile pentane meniscus in a tube where a heat source is applied at the edge of the tube. The main unsolved issue by these previous authors is the hydrodynamics in the liquid phase of the meniscus. This study has been partially performed instead by Buffone et al.21,22 and, more recently, Minetti and Buffone.23
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ethanol and dried in air. The IR experimental setup is the same as reported by Buffone and Sefiane.24 The IR camera was directed at the curved meniscus interface at the tube mouth in the axial direction. The IR camera used in this study is the FLIR ThermaCAM SC3000 that has a thermal sensitivity of 20 mK at 30 °C and an accuracy of 1% or 1 °C for temperatures up to 150 °C. The GaAs, Quantum Well Infrared Photon FPA detector has a spectral range of 8−9 μm centered in one of the two atmospheric “‘windows’” with a resolution of 320 × 240 pixels and is Stirling cooled to 70 K. The field of view at a minimum focus distance (26 mm) is 10 × 7.5 mm. A continuous electronic zoom (1−4 times) is also provided. The IR camera is calibrated annually by FLIR Systems, and the error found during the last calibration is within the accuracy stated above. We used to acquire images at 50 Hz. At this recording speed, we recorded around 180 s per cycle before we stored the images on the hard drive and a new recording was made; between recordings, there has been a gap of around 10−15 s, where no recording was made. The images grabbed were transferred to a dedicated personal computer (PC) with installed ThermaCAM research software (by FLIR System). The image spatial resolution of the present camera is 31.25 μm for a focal distance of 26 mm. It is worth mentioning that ethanol and acetone, as used in this study, are a semi-transparent fluid to IR at the camera wavelengths of 8−9 μm. The emissivity of ethanol and acetone depends upon the liquid thickness, as clearly shown also by Brutin et al.25 for drops. Therefore, the IR measurements of the present investigation give a good indication of the temperature distribution of the liquid close to the meniscus interface but not of the interface itself.
EXPERIMENTAL SECTION
In this study, we investigate the spontaneous evaporative cooling effect. We have investigated two borosilicate capillary tubes of 1 and 2 mm internal diameter and used ethanol and acetone as evaporating fluids. Because of the low partial pressure of these liquids in air, the liquid evaporates spontaneously inside the tubes. We have positioned the capillary tubes with their axis horizontal. We have filled the tubes from the opposite side to the one where the primary meniscus was positioned for infrared (IR) measurements. A different procedure from the one described by Buffone and Sefiane24 was employed, namely, we pushed the meniscus at the primary tube mouth after we started recording IR images; this allowed us to capture the transient during which the meniscus reaches the tube mouth and its temperature drops dramatically below ambient. When the primary meniscus is pinned at the tube mouth, the second meniscus recedes inside the tube approaching the primary meniscus, because of the mass lost during evaporation, which mainly happens at the primary meniscus. In the present study, we are interested in the nature of standing waves (SWs) present at the meniscus interface during evaporative cooling. The liquids were used as received from the manufacturers. The borosilicate tubes were also used as received, after being rinsed in
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RESULTS AND DISCUSSION Figure 1 presents a typical IR image sequence during the passing of a SW for the borosilicate capillary tube with an internal diameter of 2 mm, where we indicate with dashed circles the tube wall and with a line in the bottom left corner a 1 mm scale. The passing of a SW can be noted in Figure 1 in the B
DOI: 10.1021/acs.langmuir.6b02970 Langmuir XXXX, XXX, XXX−XXX
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Langmuir upper part of the meniscus with a red region that become more intense in frames a.3 and a.4. Figure 2 shows the temperature evolution of the “up” marker of the meniscus interface (as in Figure 1) for ethanol and 2 mm
Figure 3. Temperature evolution of the “up” marker for the 2 mm internal diameter tube and acetone. The initial and end transients between which there is no temperature oscillations are highlighted by the vertical dashed lines. Figure 2. Temperature evolution of the meniscus interface for a 2 mm internal diameter capillary tube and ethanol as liquid. The Roman numbers (from I° to V°) denote when we stopped IR recording to store the sequence before a new sequence was recorded.
is worth noting how the initial and ending transients of the SWs unravel between two extreme temperature values between which the meniscus temperature oscillates. For the 2 mm internal diameter tube, the meniscus temperature during the initial transient becomes the lower of the two extreme temperatures (as in Figure 3) between which the meniscus temperature oscillates; when the SW stops, the meniscus temperature becomes the higher value of the two extreme temperatures. The difference is the transient of the 1 mm internal diameter tube case with the oscillations starting from the higher value of the two extreme temperatures (see Figure 4) during oscillations; however, when the SW stops, also for the 1 mm internal diameter case, the temperature becomes the higher value. In Figure 5, we perform the fast Fourier transform of the temperature evolution of Figures 3 and 4 when the SW is operating. It can be seen that some main frequencies appear at well below 1 Hz for the 1 mm tube (as also reported by Buffone at el.26), but no such distinct frequencies can be spotted for the 2 mm tube. We believe that, with these SWs being a sort of instability in a highly three-dimensional oscillatory periodic flow, there is not a single distinct frequency at which the temperature oscillations happen. The experimental evidence shows that, the smaller the tube size, the stronger the driving force (surface tension) and the more distinct the oscillation frequency. We have conducted experiments with acetone and a different initial filling ratio and two tube sizes (1 and 2 mm internal diameter) to ascertain if the SWs start and stop at the same location and if they are always present during evaporation. The results of this investigation are reported in Figure 6 in terms of the L/D ratio. For the initial filling ratio, “short” stands for a length less than the capillary rise of acetone inside the tube; “capillary” stands for a length equal to the capillary rise (h = 4σ/ρgD, where h is the capillary height, σ is the surface tension, ρ is the density, and D is the tube diameter); and “long” stands for a length longer than the capillary rise. The vertical dashed lines represent when the SWs start appearing, and the vertical
internal diameter tube from before the meniscus becomes pinned at the tube mouth until all liquid is evaporated inside the capillary tube (and the menisci have disappeared). The Roman numbers from I° to V° indicate when we stopped recording with the IR camera to save the data before a new sequence was acquired. Clearly, for ethanol, there is no occurrence of SWs and the temperature variations in each recorded sequence of around 180 s in the graph are a maximum of 3 times the accuracy of the IR camera, are larger than the apparent temperature “jump” during the beginning and ending of recording cycle, and can be considered as noise. Figure 3 reports the temperature evolution of the “up” marker for a 2 mm internal diameter tube and acetone. The first part of the curve (delimited on the right side by the first vertical dashed line) in this figure shows that there is an initial transient during which the acetone temperature drops considerably (more than 9 °C) when the meniscus approaches the tube mouth and becomes pinned there. It is also worth noting that, after almost 60 s (corresponding to L/D of 0.3, where L is the distance between the two menisci and D is the internal tube diameter), the meniscus temperature starts oscillating regularly (there are also two oscillations during the transient). These temperature oscillations (SWs) remain in place until the second receding meniscus approaches the primary meniscus pinned at the tube mouth. When the liquid left in the capillary corresponds to L/D of just over 1.5, the SWs stop. We will demonstrate with other initial filling ratios and a second tube size that there seems to be a characteristic length after which the SWs stop, which depends upon the tube size. Instead, there does not seem to be a correlation between the SWs starting and lasting lengths with the tube size. During the ending transient, the SWs disappear, and when we approach the end of the liquid bridge, the meniscus temperature increases toward ambient. It C
DOI: 10.1021/acs.langmuir.6b02970 Langmuir XXXX, XXX, XXX−XXX
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direction of the temperature gradient for low Prandtl numbers and in perpendicular direction for high Prandtl numbers. HTWs traveling in oblique directions have also been found by Pelacho and Burguete.28 Sefiane et al.29 and Sobac and Brutin30 reported on the existence of HTWs in evaporating sessile drops. HTWs can be traveling waves or SWs. In the present case, the depth of the liquid is many times the capillary length, and therefore, an unsteady toroidal flow pattern is formed18,21,23 and there should not be HTWs. However, there is another characteristic dimension, which is the tube radius along which a temperature gradient is established, and as the present results demonstrate, there is a clear SW originating in the upper part of the meniscus interface, with temperature fluctuations as big as 2.2 °C for the 2 mm tube and 1.5 °C for the 1 mm tube. In Buffone et al.,26 we argued about the possible origin of this SW being the interaction of the symmetrical and stabilizing effect of the surface tension gradient to the asymmetrical and destabilizing effect of gravity along the curved meniscus interface of a horizontal positioned capillary tube. In a classical Marangoni problem, where the liquid−vapor interface is perpendicular to the gravity vector, surface tension is a destabilizing effect and gravity is a stabilizing effect. In the present study, the meniscus is curved and the tube is positioned horizontally; therefore, gravity generates a downward force on the meniscus that competes with the symmetrical effect of surface tension along the meniscus. The outcome of this competition is an asymmetric resulting force that changes in space and time. The temperature fluctuations of the meniscus interface are very large for this kind of experiment, where evaporation is self-established. The fluid internal energy is proportional to its temperature, and if we divide the temperature fluctuations along the meniscus interface with the temperature difference between the interface and ambient, we obtain 23 and 18% for 2 and 1 mm tubes, respectively; this is a considerable change in internal energy, which must be accounted for in changes of heat from the liquid beneath the meniscus or from the surrounding of the tube. This aspect is worth further investigation. In the present study, we also capture the initial transient when the meniscus becomes pinned at the capillary mouth and are thus able to measure the temperature drop of the meniscus from ambient and compare
Figure 4. Temperature evolution of the “up” marker for the 1 mm internal diameter tube and acetone. The initial and end transients between which there is no temperature oscillations are highlighted by the vertical dashed lines.
solid lines with the number represent when the SWs stop; the number is the relative location of when the SW stops in terms of L/D. The meniscus pinned at the tube mouth is at L/D = 0, and each bar represents the location of the secondary meniscus. From this figure, it can be concluded that the SWs seem to stop at the same location for each tube size, regardless of the initial filling ratio (with a maximum measured difference of 8%, which is slightly larger than the accuracy with which we have tracked the meniscus location using an optical recording system). The distance in L/D after which the SWs start and stop seems to increase as the initial filling ratio increases, but no firm conclusion can be drawn on when they start and how long they last. The existence of hydrothermal waves (HTWs) have been predicted by Smith and Davis8 and then demonstrated experimentally by Riley and Neitzel.27 HTWs are in the same
Figure 5. Fast Fourier transform of the temperature evolution of Figure 3 for the 2 mm tube (on the left frame) and Figure 4 for the 1 mm tube (on the right frame). D
DOI: 10.1021/acs.langmuir.6b02970 Langmuir XXXX, XXX, XXX−XXX
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Figure 6. Starting and extinction of SWs for two capillary tube sizes (of 1 and 2 mm internal diameter) and different initial filling ratios.
Table 1. Dimensionless Numbers for the Cases Studied Experimentally diameter (mm) acetone ethanol
1 2 2
number
Ma
Ra
Bd
Bo
Ca
Pr
formula
−(∂σ/∂T)(∂T/∂D)(D2/4μα)
gβΔTD3/8vα
Ra/Ma
ΔρgD2/4σ
μV/σ
Cpμ/k
2 × 103 5.3 × 103 1.2 × 103
3.6 × 102 3.3 × 103 8.1 × 102
0.18 0.62 0.64
0.08 0.33 0.34
9.5 × 10−6 6.8 × 10−6 1.5 × 10−5
3.8 1.4
center is measured to be 6.15 and 4.56 °C for 2 and 1 mm tubes, respectively (this distance is half D); this difference is given by the differential evaporation along the curved meniscus interface and depends mainly upon the tube size and liquid used. ΔT instead is the difference between ambient temperature and the meniscus triple-line temperature where most of the evaporation takes place for this case of concave meniscus; this difference is mainly dictated by the latent heat of evaporation, the liquid partial pressure, and the tube size. From Table 1, it can be deducted that, for the tubes sizes considered, acetone has Ma and Ra higher than ethanol. In addition, Ca for acetone is more than 2 times smaller than that of ethanol. These findings are partially due to the higher volatility of acetone compared to ethanol but also quite strongly correlated to the sensibly lower dynamic viscosity of acetone. With the static and dynamic Bond numbers in Table 1 being less than 1, we can state that the SWs reported here have a thermocapillary of HTW type 1 as reported by Sobac and Brutin.30 It is also worth noting that the Rayleigh number for the natural convection outside the capillary tube is between 10 and 1 for 2 and 1 mm tubes, respectively, 2 orders of magnitude less than the one inside the tube. This is attributed to the difference in fluid properties between outside (air) and inside (acetone) the tube.
that to the evaluated one. From the temperature measurements of acetone, it is possible to evaluate the time taken to reach the lowest meniscus, with the average temperature being around t = 60 s for the 2 mm tube and t = 50 s for the 1 mm tube. We assume that, in the transient, the liquid is a perfect thermal conductor and there is no heat coming from the outside ambient through the borosilicate tube wall. Equating the sensible heat with the product of measured mass flow rate by the time by the latent heat of evaporation (mCpΔT = ṁ thfg) and assuming that the length of liquid with a temperature sensibly lower than ambient is as big as the lengths shown in Figure 6 for the “long” case times the tube diameter one obtain for 2 mm tube diameter ∼9.57 °C and for 1 mm tube diameter ∼8.25 °C. The measured temperature drop of the meniscus averaged along the interface is ∼9.65 and ∼8.49 °C for 2 and 1 mm tubes, respectively; the percentage difference of measured and calculated ΔT is 1 and 3%, respectively, which is relatively close given the approximations that we made especially in connection with no heating coming from the outside ambient, the averaged meniscus temperature that instead varies considerably along the interface, and the fluid properties taken at ambient temperature. For this analysis, the tube was not insulated from the surrounding ambient and we did not take into account the effect of the natural convection around the tube on the Marangoni convection inside the tube. It is worth introducing some dimensionless numbers (defined and evaluated in Table 1), which will help in the analysis of the present case. In the formulas, the quantities not defined yet are V as the fluid velocity along the meniscus interface (measured) and Cp as the heat capacity. There are two temperature differences governing this process. The temperature gradient (∂T) between the triple line and the meniscus
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CONCLUSION The IR temperature measurements presented in this paper for a curved meniscus interface pinned at a capillary tube mouth during evaporation of volatile liquids showed that, for the control experiment with ethanol, there is no appreciable E
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(10) Riley, R. J. An investigation of the stability and control of a combined thermocapillary-buoyancy driven flow. Ph.D. Dissertation, Georgia Institute of Technology, Atlanta, GA, 1996. (11) Potash, M. L., Jr.; Wayner, P. C., Jr. Effect of Thermocapillary on the Evaporating Meniscus, 1974; Report RPI TCTP-001, NTIS PB-235737. (12) Preiss, G.; Wayner, P. C. Evaporation from a capillary tube. J. Heat Transfer 1976, 98, 178−181. (13) Swanson, L. W.; Herdt, G. C. Model of the evaporating meniscus in a capillary tube. J. Heat Transfer 1992, 114, 434−441. (14) Reyes, R.; Wayner, P. C. A Kelvin−Clapeyron adsorption model for spreading on a heated plate. J. Heat Transfer 1996, 118, 822−830. (15) Khrustalev, D.; Faghri, A. Estimation of the maximum heat flux in the inverted meniscus type evaporator of a flat miniature heat pipe. Int. J. Heat Mass Transfer 1996, 39 (9), 1899−1909. (16) Sartre, V.; Zaghdoudi, M. C.; Lallemand, M. Effect of interfacial phenomena on evaporative heat transfer in micro heat pipes. Int. J. Therm. Sci. 2000, 39 (4), 498−504. (17) Hohmann, C.; Stephan, P. Microscale temperature measurement at an evaporating liquid meniscus. Exp. Therm. Fluid Sci. 2002, 26, 157−162. (18) Buffone, C.; Sefiane, K. Temperature measurement near the triple line during phase change using thermochromic liquid crystal thermography. Exp. Fluids 2005, 39, 99−110. (19) Pratt, D.; Hallinan, K. P. Thermocapillary effects on the wetting characteristics of a heated curved meniscus. J. Thermophys. Heat Transfer 1997, 11 (4), 519−525. (20) Pratt, D.; Brown, J.; Hallinan, K. P. Thermocapillary effects on the stability of a heated, curved meniscus. J. Heat Transfer 1998, 120 (1), 220−226. (21) Buffone, C.; Sefiane, K.; Christy, J. R. Experimental investigation of self-induced thermocapillary convection for an evaporating meniscus in capillary tubes using micro-particle image velocimetry. Phys. Fluids 2005, 17, 052104. (22) Buffone, C.; Sefiane, K.; Easson, W. Marangoni-driven instabilities of an evaporating liquid-vapor interface. Phys. Rev. E 2005, 71, 056302. (23) Minetti, C.; Buffone, C. Three-dimensional Marangoni cell in self-induced evaporating cooling unveiled by digital holographic microscopy. Phys. Rev. E 2014, 89, 013007. (24) Buffone, C.; Sefiane, K. IR measurement of the temperature of an evaporating meniscus in a confined environment. Exp. Therm. Fluid Sci. 2004, 29, 65−74. (25) Brutin, D.; Sobac, B.; Rigollet, F.; Le Niliot, C. Infrared visualization of thermal motion inside a sessile drop deposited onto a heated surface. Exp. Therm. Fluid Sci. 2011, 35, 521−530. (26) Buffone, C.; Sefiane, K.; Minetti, C.; Mamalis, D. Standing wave in evaporatin meniscus detected by infrared thermography. Appl. Phys. Lett. 2015, 107, 041606. (27) Riley, R. J.; Neitzel, G. P. Instability of thermocapillary buoyancy convection in shallow layers. Part 1. Characterization of steady and oscillatory instabilities. J. Fluid Mech. 1998, 359, 143−164. (28) Pelacho, M. A.; Burguete, J. Temperature oscillations of hydrothermal waves in thermocapillary-buoyancy convection. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1999, 59 (1), 835−840. (29) Sefiane, K.; Moffat, J. R.; Matar, O. K.; Craster, R. V. Self-excited hydrothermal waves in evaporating sessile drops. Appl. Phys. Lett. 2008, 93, 074103. (30) Sobac, B.; Brutin, D. Thermocapillary instabilities in an evaporating drop deposited onto a heated substrate. Phys. Fluids 2012, 24, 032103.
temperature oscillation of the interface, whereas acetone shows an in intriguing mechanism. What we noticed is that there are three stages of the evaporation process. During the first transient stage, the meniscus temperature drops considerably below the ambient temperature, because of the evaporative cooling effect. The second stage is characterized by clear SWs, which have characteristic frequencies and oscillate between clearly defined temperature limits, which depend upon the tube size. Very interestingly is the last stage of the process, during which the SWs disappear; we notice that, regardless of the amount of liquid with which we fill the tubes, the SWs seem to stop when the second receding meniscus inside the tube is at the same location from the first meniscus pinned at the tube mouth. We believe that these SWs are generated by the interplay of the stabilizing surface tension forces and the destabilizing gravity action along the upper part of the curved meniscus interface for a horizontally positioned tube. Despite the static and dynamic Bond number (with this latter being the ratio between Marangoni and Rayleigh numbers) being similar for acetone and ethanol, acetone for a 2 mm internal diameter tube size investigated has Marangoni and Rayleigh numbers that are almost 5 times larger than those of ethanol; this is mainly due to the dynamic viscosity of acetone being more than 3.5 time smaller than that of ethanol. This fact might explain the clear SWs detected in the present study for acetone.
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AUTHOR INFORMATION
Corresponding Author
*E-mail: cosimobuff
[email protected]. ORCID
Cosimo Buffone: 0000-0003-3884-6936 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors thank Dr. Tadhg O’Donovan of Herriot-Watt University in Edinburgh for providing the IR camera. The authors are also indebted to Dr. Majid Safari, Mohammad Dehghani, and the technicians of the School of Engineering at The University of Edinburgh.
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REFERENCES
(1) Thomson, J. On certain curious motions observable at the surface of wine and other alcoholic liquors. Philos. Mag. 1855, 4 (10), 330− 333. (2) Benard, H. Rev. Gén. Sci. Pures Appl. 1900, 11, 1261−1309. (3) Pearson, J. R. A. On convection cells induced by surface tension. J. Fluid Mech. 1958, 4, 489−500. (4) Scriven, L. E.; Sternling, C. V. The Marangoni effects. Nature 1960, 187, 186−188. (5) Levich, V. G.; Krylov, V. S. Surface tension driven phenomena. Annu. Rev. Fluid Mech. 1969, 1, 293−316. (6) Zeytounian, R. K. The Bénard-Marangoni thermocapillary instability problem: On the rôle of the buoyancy. Int. J. Eng. Sci. 1997, 35, 455. (7) Smith, M. K.; Davis, S. H. Instabilities of dynamics thermocapillary liquid layers. Part I. Convective instabilities. J. Fluid Mech. 1983, 132, 119−144. (8) Smith, M. K.; Davis, S. H. Instabilities of dynamics thermocapillary liquid layers. Part II. Surface-wave instabilities. J. Fluid Mech. 1983, 132, 145−162. (9) Davis, S. H. Themocapillary instabilities. Annu. Rev. Fluid Mech. 1987, 19, 403−435. F
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