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Experiments on the role of gas height in the Rayleigh–Marangoni instability problem
Journal of Colloid and Interface Science 289 (2005) 271–275 www.elsevier.com/locate/jcis
Experiments on the role of gas height in the Rayleigh–Marangoni instability problem O. Ozen a,∗ , E. Theisen b , D.T. Johnson c , P.C. Dauby d , R. Narayanan a a Department of Chemical Engineering, University of Florida, Gainesville, FL 32611, USA b School of Chemical Engineering, Cornell University, Ithaca, NY 14853, USA c Department of Chemical and Biological Engineering, University of Alabama, Tuscaloosa, AL 35487-0203, USA d Institut de Physique B5a, Université de Liège, Allée du 6 Août 17, B-4000 Liège 1, Belgium
Received 7 October 2004; accepted 20 March 2005 Available online 19 April 2005
Abstract Experimental evidence is provided to show the effect of gas phase dynamics on the onset of thermal convection and on the accompanying patterns in a silicone oil–air convecting bilayer. Very good agreement with three-dimensional calculations for linearized stability is obtained mostly for small and large gas heights. Reasons for this agreement as well as the results at intermediate gas heights are qualitatively explained from the perspective of well-established nonlinear analysis. 2005 Elsevier Inc. All rights reserved. Keywords: Interfacial instability; Convection; Marangoni; Bilayers
1. Introduction The purpose of this study is to examine the role of gas layer height in a liquid–gas bilayer system heated from below. The study, which involves experiments, is performed on a silicone oil–air bilayer system. It is known that convection in liquid–gas bilayers occurs due to buoyancy, i.e., Rayleigh convection, as well as surface tension gradients, i.e., Marangoni convection. For small temperature gradients the layers conduct heat, but once a critical temperature gradient is exceeded convection sets in. Typical theoretical studies [1–3] of this problem assume that the gas layer is fluid dynamically passive even though one might expect the gas layer to flow like the liquid, once convection ensues. The gas layer height has a dual effect on the heat transport in the gas phase. When the gas height is small, the upper layer is mostly quiescent offering resistance via heat conduction. Increasing the gas height is ordinarily thought to only increase the resistance via conductive heat transfer. However, * Corresponding author.
E-mail address: [email protected] (O. Ozen). 0021-9797/$ – see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2005.03.044
for deep gas layers the gas can flow more easily and enhance the heat transport. To understand the dual role of gas height, imagine bilayer convection experiments where the total temperature drop across the layers is increased in increments until a critical temperature drop is reached when convection begins. Further imagine conducting a series of such experiments with increasing gas heights. Given a liquid layer depth, an increase in gas height will initially cause a rise in the overall critical temperature drop because of an increase in the heat transfer resistance. However for deeper gas layers, buoyancy-driven convection will become important in the gas layer and will enhance the heat transport causing a decrease in the critical temperature drop. For very large gas heights the convective flow in the gas is dominant. But flow in the low viscosity gas cannot induce substantial shear driven flow in the significantly more viscous liquid phase. The gas flow does, however, generate transverse temperature gradients at the interface, which, in turn, induces weak surface tension-driven convection, i.e., Marangoni convection, in the liquid. To illustrate this, a set of calculations are presented in Fig. 1. The calculations depict the critical total temperature difference across a silicone oil–air bilayer sys-
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O. Ozen et al. / Journal of Colloid and Interface Science 289 (2005) 271–275
Fig. 1. The overall critical temperature difference for the onset of convection in silicone oil–air bilayer heated from the liquid side. The oil height is taken to be 5 mm. The thermophysical properties are given in Table 1.
tem of infinite lateral extent against the air layer height for a given oil height. In one set of the calculations, the gas layer is assumed to be fluid dynamically passive. In the other set of calculations fluid flow in the air is also taken into account. In either set of calculations the determination of the critical temperature difference is a result of a well-known procedure that is based on linearized stability and explained in [4,5]. The purpose of the calculations is twofold. The first is to show that the overall critical temperature differences are in a measurable range. The second is to show that the overall critical temperature drop at the onset of flow does not increase monotonically with the air height when the gas layer is assumed fluid dynamically active. Observe that there is a maximum value of the temperature drop. One would expect this maximum to occur at a point when the fluid mechanics goes from being liquid phase controlled to gas phase controlled. Indeed, if the air were assumed to be fluid dynamically passive, no maximum is seen in the calculations for a passive upper fluid. The goal of the paper is therefore to investigate the dual role of air height by way of experiments and compare the results with a numerical calculation and to show that it is ordinarily incorrect to ignore the fluid flow in the air for large air to liquid height ratios. As mentioned, the calculations depicted via Fig. 1 assumed a laterally unbounded geometry; however, experiments are always conducted in the presence of sidewalls. For this reason, the numerical calculations were repeated for the case of laterally bounded containers where the effects of sidewalls are included. There have only been a few experiments on free surface convection where the effects of sidewalls have been studied. In cylindrical containers the most notable studies are [6–10]. In each of these studies the
air heights were smaller than the liquid heights in order to reduce the effect of gas phase fluid flow. This is the first study where the air height is varied with the goal of investigating the dual role of the air layer on the instability. In Section 2, we will describe the experimental setup with minimum detail. Further information on the experiment, including the data acquisition system and control algorithms, is given in the thesis of Johnson [11]. 2. Experimental apparatus and procedure The experimental apparatus was designed to perform experiments for various air heights and aspect ratios. A schematic of the test section is given in Fig. 2. The lower heating element consisted of a heating plate under a hollowed cylinder made of lucite, 20 mm thick, called the lower bath. The
Fig. 2. Schematic of the experimental apparatus.
O. Ozen et al. / Journal of Colloid and Interface Science 289 (2005) 271–275
top and bottom of the cylinder were capped with thin copper plates, 6.4 mm thick, and the interior of the cylinder was filled with water. The upper copper plate named “heating block” in the figure was in contact with the oil in the test section. A magnetic stir bar was placed in the water and the entire lower heating bath sat on top of a magnetic stirrer. The stirred water helped to stabilize any temperature fluctuations, ensuring a uniform temperature up to ±0.05 ◦ C. A thermistor was placed in the lower water heating bath. The test section itself consisted of four pieces: a liquid insert, an air insert, a zinc selenide IR transparent window and a clamp for structural integrity. The oil insert, the air insert and the clamp were made of lucite. To ensure that the liquid– gas interface was flat, a pinning edge was used in the liquid insert. The liquid insert could be made of different radii and heights, to achieve the desired liquid aspect ratio. The inner radius of the air insert was always the same as the inner radius of the liquid insert. However, different air heights could be used for the same liquid insert. The lucite clamp fastened the liquid and air inserts down, preventing silicone oil from leaking and ensuring structural integrity. The outer radius of the oil inserts was 46.3 mm. The outer radius of the air inserts was 57.2 mm. The zinc selenide window had a diameter of 51 mm and was 5 mm thick. The accuracy of each piece was machined to within 0.1 mm. An Inframetrics Model 760 infrared camera was used to visualize the flow patterns. This particular model of the infrared camera is capable of measuring in the 3–5 µm range or the 8–12 µm range. Only the 8–12 µm range feature was employed. The infrared camera was placed directly above the test section and measured the infrared radiation being emitted by the silicone oil. As silicone oil readily absorbs infrared radiation, only the radiation from the first few angstroms of the silicone oil interface could be detected. The IR camera was never used to measure the bulk temperature; rather it was used to detect the variation in interfacial temperature from which the flow structure could be deduced. Zinc selenide is 60% transparent to infrared radiation in the 8 to 12 µm range. Additionally, an anti-reflective infrared polymer was coated on the zinc selenide window by II–VI, Inc. This coating was necessary to eliminate any false images generated by reflected, ambient infrared radiation. The temperature at the top of the test section as well as the temperature difference across the test section were controlled. To control the temperature at the top of the test section, an infrared transparent medium was needed. The medium of choice was air. A large lucite box enclosed the test section, the lower bath, the magnetic stirrer, the infrared camera and the heating control elements. An electric heater and a shell and tube heat exchanger were used to control the temperature of the ambient air in the lucite box. The heater would turn on to increase the ambient temperature, while cool water, pumped through the heat exchanger, continuously removed heat from the air. Proper mixing was ensured by a blowing fan. The controlled air temperature was sensed by a thermistor located at the top of the zinc selenide lens.
273
Table 1 Physical properties of silicone oil and air at 20 ◦ C ρ = 968 kg/m3 µ = 96.8 × 10−3 kg/(m s) k = 1.6 × 10−1 J/(m s ◦ C) κ = 1.1 × 10−7 m2 /s α = 9.6 × 10−4 1/◦ C γ = 20.9 × 10−3 N/m
ρ ∗ = 1.2 kg/m3 µ∗ = 1.9 × 10−5 kg/(m s) k ∗ = 2.6 × 10−2 J/(m s ◦ C) κ ∗ = 2.1 × 10−5 m2 /s α ∗ = 32.4 × 10−4 1/◦ C γT = 0.5 × 10−4 N/m
Now the temperature difference between the top and bottom thermistors, i.e., the “air” thermistor on top of the zinc selenide lens and the thermistor in the hot water bath below the copper plate, that was in contact with the oil, was measured and also controlled. As the zinc selenide and copper are both high thermal conductors, the temperature drops across these materials were negligible. The overall control of the temperature difference was never more than 2.4% from the set point. When the smoothing of the temperature fluctuations due to the highly conductive plates (copper and zinc selenide) is taken into account, the actual control across the two fluid layers was better. The entire control system was made possible by using a computerized data acquisition system. The infrared images were sent to a VCR whence they were recorded and displayed on a TV. Each experiment was conducted in the following manner. First, the silicone oil was placed into the liquid insert. The corresponding air insert was placed on top of the liquid insert. Then, the lucite clamp was placed on top of the air insert and screwed into the lower bath. The level of the oil–air interface was ensured to be flat. Once the test section was secured, a temperature difference was applied across the liquid–gas bilayer. The initial temperature difference was less than the critical temperature difference necessary to initiate convection in either layer. When a temperature difference was applied, it was held constant for several time constants. The longest time constant in these experiments is the horizontal thermal diffusion time constant. Here, the thermal diffusivity of silicone oil is κ = 1.1 × 10−7 m2 /s, and the typical diameter is about 25 mm to give a horizontal time constant of d2 (25 × 10−3 m)2 = = 1.6 h. κ 1.1 × 10−7 m2 /s The temperature difference for these experiments was held constant for 4 h. However, steady state was usually reached well within the four-hour period. After the temperature difference was held constant for four hours, the temperature difference was increased to a new set point and held constant for another four hours. This procedure was repeated until the fluid began to flow. Once the fluid began to flow, the temperature difference and the flow pattern were recorded. 3. Comparison of experimental results and calculations A model was generated respecting the physics of the experiment. The calculations were performed using the phys-
274
O. Ozen et al. / Journal of Colloid and Interface Science 289 (2005) 271–275
ical properties in Table 1 to support the physical arguments outlined earlier. The properties with an asterisk as a superscript refer to air while the oil properties do not have a superscript. In the table ρ, µ, k, κ, and α are the density, dynamic viscosity, thermal conductivity, thermal diffusivity and thermal expansion coefficients respectively while γ and γT are the surface tension and surface tension gradient, respectively. The calculations were performed using a method described in [12], which assumes that a single liquid layer of silicone oil is superposed by air. The method requires the expansion of the state variables in a Chebyshev series after linearizing the equations about a conductive base state. Only neutral stability conditions were investigated, as the onset of flow is not oscillatory. The code was written for a model that assumes the air to be fluid dynamically active and that both fluids reside in a closed cylindrical container. The interface between the liquid and the gas is assumed to be flat. It turns out that the interfacial tension is high enough that this is a good approximation and results of independent calculations in infinitely wide containers support this assumption. All sides of the cylindrical container have no-
slip boundaries. The top and bottom boundaries are rigid, no-slip, conducting plates. The lateral sidewalls were taken to be perfect insulators. The calculations were performed iteratively and accounted for the viscosity change of the oil with temperature. The fluid viscosity’s variation with temperature was measured using a Cole Parmer 98936 series viscometer. The functional dependence with respect to temperature was found to be µ = −1.45T + 128.7. Here, T is the temperature in ◦ C and µ is the viscosity of silicone oil in units of cP. The air viscosity was assumed to be constant. The calculations are able to predict the critical temperature difference (equivalently the critical Rayleigh number or Marangoni number) for different azimuthal modes over a range of aspect ratios. Fig. 3 shows the calculated and experimental critical temperature differences for a range of air heights at a fixed liquid depth and radius. In Fig. 3 there are three curves from calculations for each air height. Each one of these curves stands for a different azimuthal flow pattern. The symbol ‘m’ designates the azimuthal mode. For
Fig. 3. The critical temperature difference vs air height for a bilayer system where the silicone oil depth is 8 mm and the aspect ratio is 2.5. The results of experiments are numbered. The pictures of some of these numbered experiments are given in Fig. 4.
Fig. 4. Images of the temperature profile at the silicone oil–air interface recorded by the IR camera at the onset of instability. The image contrast is slightly increased for publication purposes.
O. Ozen et al. / Journal of Colloid and Interface Science 289 (2005) 271–275
example the mode m = 0 is an axisymmetric mode while m = 1 refers to a unimodal flow pattern. To better explain the convection patterns of these modes, turn to Fig. 4, which show the patterns. The fourth picture in Fig. 4 shows a clear m = 0 flow pattern where the up flow is represented by the lighter color in the middle. On the other hand, the m = 1 mode is symmetric around an imaginary diameter cutting the cylinder in the middle. In this mode, which is very clear in the second picture of Fig. 4 and somewhat less clear in the first picture, the flow rises on one side and falls down the other. The flows corresponding to the fourth and fifth points were largely controlled by the air. However, no picture is given for the fifth point. It was very faint on account of the very weak temperature gradients at the onset of surface driven flow and not of publication quality. The experiments and calculations agree very well at small and large air depths whereas, at intermediate air heights, the onset of convection is observed before the calculated critical temperature difference is reached. When the air depths are very small, the convection is due to a combination of buoyancy as well as Marangoni effects in the liquid and for large air heights primarily due to buoyancy in the gas phase. For small gas heights the convection was non axisymmetric in nature. Now it can be shown mathematically (see [13]), using a weak nonlinear analysis, that convection due to Rayleigh effects as well as convection that is non axisymmetric in flow structure are both supercritical in nature. Therefore we might expect to see the experimental result agree reasonably well with the theoretical prediction for small and large air heights. However, for intermediate air depths both air convection and liquid convection begin to play a role on the stability of the problem. Observe from the third picture of Fig. 4, that the onset of convection appears to be axisymmetric, i.e., m is equal to zero and note that the combined Rayleigh–Marangoni problem can lead to transcritical flow for m = 0 (see [3,14]). In other words, one might expect the convection to begin even before the the-
275
oretically computed critical point is reached. The extent of the transcritical region can only be assessed by a nonlinear calculation and one can expect nominal experimental imperfections such as very small tilts or slightly non-uniform heating to influence the results as well. In summary, we have experiments and calculations which show that flow in the gas layer plays an important role in determining the stability of a bilayer system. Depending on the gas layer height, excluding the fluid dynamics of gas layers could lead to errors in determining the onset conditions and flow patterns.
Acknowledgments We thank the University of Florida Undergraduate Student Research Program and NSF CTS 9500393 for support.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
J.R.A. Pearson, J. Fluid Mech. 4 (1958) 489. D.A. Nield, J. Fluid Mech. 19 (1964) 341. S. Rosenblat, S.H. Davis, G.M. Homsy, J. Fluid Mech. 120 (1982) 91. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Oxford Univ. Press, Oxford, 1961. D. Johnson, R. Narayanan, Chaos 9 (1999) 124. P. Cerisier, C. Perez-Garcia, C. Jamond, J. Pantaloni, Phys. Rev. A 35 (1987) 1949. B. Echebarria, D. Krmpotic, C. Pérez-García, Physica D 99 (1997) 497. D. Johnson, R. Narayanan, Phys. Rev. E 54 (1996) R3102. A. Juel, J.M. Burgess, W.D. McCormick, J.B. Swift, H.L. Swinney, Physica D 143 (2000) 169. E.L. Koschmieder, S.A. Prahl, J. Fluid Mech. 215 (1990) 571. D. Johnson, Doctoral Thesis, University of Florida, 1997. P.C. Dauby, G. Lebon, E. Bouhy, Phys. Rev. E 56 (1997) 520. S. Rosenblat, J. Fluid Mech. 122 (1982) 395. P. Colinet, J.C. Legros, M.G. Velarde, Nonlinear Dynamics of SurfaceTension-Driven Instabilities, Wiley–VCH, Berlin, 2001.
Experiments on the role of gas height in the Rayleigh–Marangoni instability problem O. Ozen a,∗ , E. Theisen b , D.T. Johnson c , P.C. Dauby d , R. Narayanan a a Department of Chemical Engineering, University of Florida, Gainesville, FL 32611, USA b School of Chemical Engineering, Cornell University, Ithaca, NY 14853, USA c Department of Chemical and Biological Engineering, University of Alabama, Tuscaloosa, AL 35487-0203, USA d Institut de Physique B5a, Université de Liège, Allée du 6 Août 17, B-4000 Liège 1, Belgium
Received 7 October 2004; accepted 20 March 2005 Available online 19 April 2005
Abstract Experimental evidence is provided to show the effect of gas phase dynamics on the onset of thermal convection and on the accompanying patterns in a silicone oil–air convecting bilayer. Very good agreement with three-dimensional calculations for linearized stability is obtained mostly for small and large gas heights. Reasons for this agreement as well as the results at intermediate gas heights are qualitatively explained from the perspective of well-established nonlinear analysis. 2005 Elsevier Inc. All rights reserved. Keywords: Interfacial instability; Convection; Marangoni; Bilayers
1. Introduction The purpose of this study is to examine the role of gas layer height in a liquid–gas bilayer system heated from below. The study, which involves experiments, is performed on a silicone oil–air bilayer system. It is known that convection in liquid–gas bilayers occurs due to buoyancy, i.e., Rayleigh convection, as well as surface tension gradients, i.e., Marangoni convection. For small temperature gradients the layers conduct heat, but once a critical temperature gradient is exceeded convection sets in. Typical theoretical studies [1–3] of this problem assume that the gas layer is fluid dynamically passive even though one might expect the gas layer to flow like the liquid, once convection ensues. The gas layer height has a dual effect on the heat transport in the gas phase. When the gas height is small, the upper layer is mostly quiescent offering resistance via heat conduction. Increasing the gas height is ordinarily thought to only increase the resistance via conductive heat transfer. However, * Corresponding author.
E-mail address: [email protected] (O. Ozen). 0021-9797/$ – see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2005.03.044
for deep gas layers the gas can flow more easily and enhance the heat transport. To understand the dual role of gas height, imagine bilayer convection experiments where the total temperature drop across the layers is increased in increments until a critical temperature drop is reached when convection begins. Further imagine conducting a series of such experiments with increasing gas heights. Given a liquid layer depth, an increase in gas height will initially cause a rise in the overall critical temperature drop because of an increase in the heat transfer resistance. However for deeper gas layers, buoyancy-driven convection will become important in the gas layer and will enhance the heat transport causing a decrease in the critical temperature drop. For very large gas heights the convective flow in the gas is dominant. But flow in the low viscosity gas cannot induce substantial shear driven flow in the significantly more viscous liquid phase. The gas flow does, however, generate transverse temperature gradients at the interface, which, in turn, induces weak surface tension-driven convection, i.e., Marangoni convection, in the liquid. To illustrate this, a set of calculations are presented in Fig. 1. The calculations depict the critical total temperature difference across a silicone oil–air bilayer sys-
272
O. Ozen et al. / Journal of Colloid and Interface Science 289 (2005) 271–275
Fig. 1. The overall critical temperature difference for the onset of convection in silicone oil–air bilayer heated from the liquid side. The oil height is taken to be 5 mm. The thermophysical properties are given in Table 1.
tem of infinite lateral extent against the air layer height for a given oil height. In one set of the calculations, the gas layer is assumed to be fluid dynamically passive. In the other set of calculations fluid flow in the air is also taken into account. In either set of calculations the determination of the critical temperature difference is a result of a well-known procedure that is based on linearized stability and explained in [4,5]. The purpose of the calculations is twofold. The first is to show that the overall critical temperature differences are in a measurable range. The second is to show that the overall critical temperature drop at the onset of flow does not increase monotonically with the air height when the gas layer is assumed fluid dynamically active. Observe that there is a maximum value of the temperature drop. One would expect this maximum to occur at a point when the fluid mechanics goes from being liquid phase controlled to gas phase controlled. Indeed, if the air were assumed to be fluid dynamically passive, no maximum is seen in the calculations for a passive upper fluid. The goal of the paper is therefore to investigate the dual role of air height by way of experiments and compare the results with a numerical calculation and to show that it is ordinarily incorrect to ignore the fluid flow in the air for large air to liquid height ratios. As mentioned, the calculations depicted via Fig. 1 assumed a laterally unbounded geometry; however, experiments are always conducted in the presence of sidewalls. For this reason, the numerical calculations were repeated for the case of laterally bounded containers where the effects of sidewalls are included. There have only been a few experiments on free surface convection where the effects of sidewalls have been studied. In cylindrical containers the most notable studies are [6–10]. In each of these studies the
air heights were smaller than the liquid heights in order to reduce the effect of gas phase fluid flow. This is the first study where the air height is varied with the goal of investigating the dual role of the air layer on the instability. In Section 2, we will describe the experimental setup with minimum detail. Further information on the experiment, including the data acquisition system and control algorithms, is given in the thesis of Johnson [11]. 2. Experimental apparatus and procedure The experimental apparatus was designed to perform experiments for various air heights and aspect ratios. A schematic of the test section is given in Fig. 2. The lower heating element consisted of a heating plate under a hollowed cylinder made of lucite, 20 mm thick, called the lower bath. The
Fig. 2. Schematic of the experimental apparatus.
O. Ozen et al. / Journal of Colloid and Interface Science 289 (2005) 271–275
top and bottom of the cylinder were capped with thin copper plates, 6.4 mm thick, and the interior of the cylinder was filled with water. The upper copper plate named “heating block” in the figure was in contact with the oil in the test section. A magnetic stir bar was placed in the water and the entire lower heating bath sat on top of a magnetic stirrer. The stirred water helped to stabilize any temperature fluctuations, ensuring a uniform temperature up to ±0.05 ◦ C. A thermistor was placed in the lower water heating bath. The test section itself consisted of four pieces: a liquid insert, an air insert, a zinc selenide IR transparent window and a clamp for structural integrity. The oil insert, the air insert and the clamp were made of lucite. To ensure that the liquid– gas interface was flat, a pinning edge was used in the liquid insert. The liquid insert could be made of different radii and heights, to achieve the desired liquid aspect ratio. The inner radius of the air insert was always the same as the inner radius of the liquid insert. However, different air heights could be used for the same liquid insert. The lucite clamp fastened the liquid and air inserts down, preventing silicone oil from leaking and ensuring structural integrity. The outer radius of the oil inserts was 46.3 mm. The outer radius of the air inserts was 57.2 mm. The zinc selenide window had a diameter of 51 mm and was 5 mm thick. The accuracy of each piece was machined to within 0.1 mm. An Inframetrics Model 760 infrared camera was used to visualize the flow patterns. This particular model of the infrared camera is capable of measuring in the 3–5 µm range or the 8–12 µm range. Only the 8–12 µm range feature was employed. The infrared camera was placed directly above the test section and measured the infrared radiation being emitted by the silicone oil. As silicone oil readily absorbs infrared radiation, only the radiation from the first few angstroms of the silicone oil interface could be detected. The IR camera was never used to measure the bulk temperature; rather it was used to detect the variation in interfacial temperature from which the flow structure could be deduced. Zinc selenide is 60% transparent to infrared radiation in the 8 to 12 µm range. Additionally, an anti-reflective infrared polymer was coated on the zinc selenide window by II–VI, Inc. This coating was necessary to eliminate any false images generated by reflected, ambient infrared radiation. The temperature at the top of the test section as well as the temperature difference across the test section were controlled. To control the temperature at the top of the test section, an infrared transparent medium was needed. The medium of choice was air. A large lucite box enclosed the test section, the lower bath, the magnetic stirrer, the infrared camera and the heating control elements. An electric heater and a shell and tube heat exchanger were used to control the temperature of the ambient air in the lucite box. The heater would turn on to increase the ambient temperature, while cool water, pumped through the heat exchanger, continuously removed heat from the air. Proper mixing was ensured by a blowing fan. The controlled air temperature was sensed by a thermistor located at the top of the zinc selenide lens.
273
Table 1 Physical properties of silicone oil and air at 20 ◦ C ρ = 968 kg/m3 µ = 96.8 × 10−3 kg/(m s) k = 1.6 × 10−1 J/(m s ◦ C) κ = 1.1 × 10−7 m2 /s α = 9.6 × 10−4 1/◦ C γ = 20.9 × 10−3 N/m
ρ ∗ = 1.2 kg/m3 µ∗ = 1.9 × 10−5 kg/(m s) k ∗ = 2.6 × 10−2 J/(m s ◦ C) κ ∗ = 2.1 × 10−5 m2 /s α ∗ = 32.4 × 10−4 1/◦ C γT = 0.5 × 10−4 N/m
Now the temperature difference between the top and bottom thermistors, i.e., the “air” thermistor on top of the zinc selenide lens and the thermistor in the hot water bath below the copper plate, that was in contact with the oil, was measured and also controlled. As the zinc selenide and copper are both high thermal conductors, the temperature drops across these materials were negligible. The overall control of the temperature difference was never more than 2.4% from the set point. When the smoothing of the temperature fluctuations due to the highly conductive plates (copper and zinc selenide) is taken into account, the actual control across the two fluid layers was better. The entire control system was made possible by using a computerized data acquisition system. The infrared images were sent to a VCR whence they were recorded and displayed on a TV. Each experiment was conducted in the following manner. First, the silicone oil was placed into the liquid insert. The corresponding air insert was placed on top of the liquid insert. Then, the lucite clamp was placed on top of the air insert and screwed into the lower bath. The level of the oil–air interface was ensured to be flat. Once the test section was secured, a temperature difference was applied across the liquid–gas bilayer. The initial temperature difference was less than the critical temperature difference necessary to initiate convection in either layer. When a temperature difference was applied, it was held constant for several time constants. The longest time constant in these experiments is the horizontal thermal diffusion time constant. Here, the thermal diffusivity of silicone oil is κ = 1.1 × 10−7 m2 /s, and the typical diameter is about 25 mm to give a horizontal time constant of d2 (25 × 10−3 m)2 = = 1.6 h. κ 1.1 × 10−7 m2 /s The temperature difference for these experiments was held constant for 4 h. However, steady state was usually reached well within the four-hour period. After the temperature difference was held constant for four hours, the temperature difference was increased to a new set point and held constant for another four hours. This procedure was repeated until the fluid began to flow. Once the fluid began to flow, the temperature difference and the flow pattern were recorded. 3. Comparison of experimental results and calculations A model was generated respecting the physics of the experiment. The calculations were performed using the phys-
274
O. Ozen et al. / Journal of Colloid and Interface Science 289 (2005) 271–275
ical properties in Table 1 to support the physical arguments outlined earlier. The properties with an asterisk as a superscript refer to air while the oil properties do not have a superscript. In the table ρ, µ, k, κ, and α are the density, dynamic viscosity, thermal conductivity, thermal diffusivity and thermal expansion coefficients respectively while γ and γT are the surface tension and surface tension gradient, respectively. The calculations were performed using a method described in [12], which assumes that a single liquid layer of silicone oil is superposed by air. The method requires the expansion of the state variables in a Chebyshev series after linearizing the equations about a conductive base state. Only neutral stability conditions were investigated, as the onset of flow is not oscillatory. The code was written for a model that assumes the air to be fluid dynamically active and that both fluids reside in a closed cylindrical container. The interface between the liquid and the gas is assumed to be flat. It turns out that the interfacial tension is high enough that this is a good approximation and results of independent calculations in infinitely wide containers support this assumption. All sides of the cylindrical container have no-
slip boundaries. The top and bottom boundaries are rigid, no-slip, conducting plates. The lateral sidewalls were taken to be perfect insulators. The calculations were performed iteratively and accounted for the viscosity change of the oil with temperature. The fluid viscosity’s variation with temperature was measured using a Cole Parmer 98936 series viscometer. The functional dependence with respect to temperature was found to be µ = −1.45T + 128.7. Here, T is the temperature in ◦ C and µ is the viscosity of silicone oil in units of cP. The air viscosity was assumed to be constant. The calculations are able to predict the critical temperature difference (equivalently the critical Rayleigh number or Marangoni number) for different azimuthal modes over a range of aspect ratios. Fig. 3 shows the calculated and experimental critical temperature differences for a range of air heights at a fixed liquid depth and radius. In Fig. 3 there are three curves from calculations for each air height. Each one of these curves stands for a different azimuthal flow pattern. The symbol ‘m’ designates the azimuthal mode. For
Fig. 3. The critical temperature difference vs air height for a bilayer system where the silicone oil depth is 8 mm and the aspect ratio is 2.5. The results of experiments are numbered. The pictures of some of these numbered experiments are given in Fig. 4.
Fig. 4. Images of the temperature profile at the silicone oil–air interface recorded by the IR camera at the onset of instability. The image contrast is slightly increased for publication purposes.
O. Ozen et al. / Journal of Colloid and Interface Science 289 (2005) 271–275
example the mode m = 0 is an axisymmetric mode while m = 1 refers to a unimodal flow pattern. To better explain the convection patterns of these modes, turn to Fig. 4, which show the patterns. The fourth picture in Fig. 4 shows a clear m = 0 flow pattern where the up flow is represented by the lighter color in the middle. On the other hand, the m = 1 mode is symmetric around an imaginary diameter cutting the cylinder in the middle. In this mode, which is very clear in the second picture of Fig. 4 and somewhat less clear in the first picture, the flow rises on one side and falls down the other. The flows corresponding to the fourth and fifth points were largely controlled by the air. However, no picture is given for the fifth point. It was very faint on account of the very weak temperature gradients at the onset of surface driven flow and not of publication quality. The experiments and calculations agree very well at small and large air depths whereas, at intermediate air heights, the onset of convection is observed before the calculated critical temperature difference is reached. When the air depths are very small, the convection is due to a combination of buoyancy as well as Marangoni effects in the liquid and for large air heights primarily due to buoyancy in the gas phase. For small gas heights the convection was non axisymmetric in nature. Now it can be shown mathematically (see [13]), using a weak nonlinear analysis, that convection due to Rayleigh effects as well as convection that is non axisymmetric in flow structure are both supercritical in nature. Therefore we might expect to see the experimental result agree reasonably well with the theoretical prediction for small and large air heights. However, for intermediate air depths both air convection and liquid convection begin to play a role on the stability of the problem. Observe from the third picture of Fig. 4, that the onset of convection appears to be axisymmetric, i.e., m is equal to zero and note that the combined Rayleigh–Marangoni problem can lead to transcritical flow for m = 0 (see [3,14]). In other words, one might expect the convection to begin even before the the-
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oretically computed critical point is reached. The extent of the transcritical region can only be assessed by a nonlinear calculation and one can expect nominal experimental imperfections such as very small tilts or slightly non-uniform heating to influence the results as well. In summary, we have experiments and calculations which show that flow in the gas layer plays an important role in determining the stability of a bilayer system. Depending on the gas layer height, excluding the fluid dynamics of gas layers could lead to errors in determining the onset conditions and flow patterns.
Acknowledgments We thank the University of Florida Undergraduate Student Research Program and NSF CTS 9500393 for support.
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