Effect of interfacial heat transfer on the onset of oscillatory convection in liquid bridge

Transcription

1 Effet of interfaial heat transfer on the onset of osillatory onvetion in liquid bridge Bo Xun, Kai Li, Paul G. Chen, Wen-Rui Hu To ite this version: Bo Xun, Kai Li, Paul G. Chen, Wen-Rui Hu. Effet of interfaial heat transfer on the onset of osillatory onvetion in liquid bridge. International Journal of Heat and Mass Transfer, Elsevier, 2009, 52 (19-20), pp < /j.ijheatmasstransfer >. <hal > HAL Id: hal Submitted on 28 Apr 2016 HAL is a multi-disiplinary open aess arhive for the deposit and dissemination of sientifi researh douments, whether they are published or not. The douments may ome from teahing and researh institutions in Frane or abroad, or from publi or private researh enters. L arhive ouverte pluridisiplinaire HAL, est destinée au dépôt et à la diffusion de douments sientifiques de niveau reherhe, publiés ou non, émanant des établissements d enseignement et de reherhe français ou étrangers, des laboratoires publis ou privés.

2 Effet of interfaial heat transfer on the onset of osillatory onvetion in liquid bridge Bo Xun 1,3, Kai Li 1, Paul Gang Chen 2 and Wen-Rui Hu 1 1 National Mirogravity Laboratory, Institute of Mehanis, Chinese Aademy of Sienes, Beijing , China 2 Laboratoire Modélisation et Simulation Numérique en Méanique et Génie des Proédés, UMR 6181, CNRS-Universités d Aix-Marseille, F Marseille Cedex 20, Frane 3 The Graduate Shool of Chinese Aademy of Sienes, Beijing , China Corresponding author Kai Li National Mirogravity Laboratory, Institute of Mehanis, Chinese Aademy of Sienes, Beijing , China Tel: Fax: likai@imeh.a.n 1

3 Nomenlature A matrix in eigenvalue problem B matrix in eigenvalue problem Re s ritial Reynolds nmber Surfae of the liquid bridge B Buoyant effet S Dimensionless stress Bi Biot number, hr k Bo stati Bond T t! z tensor, 1!! ( U + ( U ) T ) 2 dimensionless time the unit vetor tangent to the free surfae in Ca ρ0gr number, σ Capillary number, 0 2 γδt σ 0 the (r, z) plane t! ϕ the unit vetor tangent to the free surfae in the (r,ϕ ) plane D k mehanial dissipation T dimensionless temperature D th thermal dissipation T 0 mean temperature of the upper and lower ends E k Kineti energy of disturbanes E th thermal energy of T amb dimensionless temperature of the ambient air T old dimensionless temperature on the old rod g disturbanes gravitational aeleration U! dimensionless veloity vetor,( u, v, w) 2

4 g 0 normal gravitational w maximal value of w on the free surfae max aeleration, ( kg m / s ) Gr Grashof X vetor omposed of disturbane veloity, gβδtr number, 2 υ 3 pressure and temperature, ( u ', iv', w', p', T') T h heat transfer oeffiient on free surfae X the basi steady axisymmetri state,!!! { U ( r, z) = Ue + We, P( r, z), T ( r, z)} r z h(z) free surfae loal radius ( r, θ, z) ylindrial oordinates i 1 V 0 the liquid volume with ylindrial shape I Interative term in kineti energy equation Greek symbols J 1, Interative term in α thermal diffusivity oeffiient J 2 thermal energy equation deomposed in ylindrial oordinates k thermal ondutivity oeffiient β thermal expansion oeffiient L height of the liquid bridge Δ T applied temperature differene between two solid ends M, 1 M 2 Work done by γ negative temperature gradient of surfae M 3 thermoapillary fore tension 3

5 deomposed in ylindrial oordinates n! the outward-direted Γ aspet ratio, L R normal vetor of the free surfae N z number of the grid µ dynami visosity oeffiient points in axial diretion N r number of the grid points in radial diretion υ kinemati visosity oeffiient P dimensionless pressure ( ξ, η) oordinates in omputational domain P s dimensionless stati pressure ψ maximal absolute value of the stream max funtion Pr Prandtl number, α υ ρ 0 mean density Q thermal energy transport from the free σ 0 mean free surfae tension surfae R radius of the liquid bridge σ (m) the omplex growth rate of the orresponding perturbation mode Re Reynolds number, γδtr µυ Ω Volume domain oupied by the liquid bridge 4

6 Abstrat In present study, effet of interfaial heat transfer with ambient gas on the onset of osillatory onvetion in a liquid bridge of large Prandtl number on the ground is systematially investigated by the method of linear stability analyses. With both the onstant and linear ambient air temperature distributions, the numerial results show that the interfaial heat transfer modifies the free surfae temperature distribution diretly and then indues a steeper temperature gradient on the middle part of the free surfae, whih may destabilize the onvetion. On the other hand, the interfaial heat transfer restrains the temperature disturbanes on the free surfae, whih may stabilize the onvetion. The two oupling effets result in a omplex dependene of the stability property on the Biot number. Effets of melt free-surfae deformation on the ritial onditions of the osillatory onvetion were also investigated. Moreover, to better understand the mehanism of the instabilities, rates of kineti energy hange and thermal energy hange of the ritial disturbanes were investigated Keywords: liquid bridge, interfaial heat transfer, osillatory onvetion, ritial ondition 5

7 Introdution Floating-zone (FZ) tehnique is a ruible-free proess for the growth of high quality single rystal, whih the melt zone is onfined by the surfae tension of melt free surfae. However, the diameter of the grown rystal is limited under the terrestrial ondition due to easy breakage of the melt zone indued by gravity. The essential reason to go to spae for melt growth of single rystals lies in a promise that a substantial redution of gravity level ahieved in spaeraft may result in a quiesent melt pool, thereby allowing a diffusion-ontrolled growth ondition to be realized. Moreover, the mirogravity environment provides the possibility of growing large size rystal by the FZ tehnique. However, experimental fats revealed that during the FZ proess in mirogravity, thermoapillary flow driven by the surfae tension gradient of the melt free surfae, depending on suh fators like temperature, solutal onentration, eletri potential et., ours in the melt zone even when the buoyany-driven flow is greatly redued. The thermoapillary flow may be osillatory when the Reynolds number exeeds ertain ritial value and responsible for striations in rystals grown in spae [1]. Therefore, numerous experimentations and analytial studies (linear instability analyses and diret numerial simulations) have been devoted to the osillatory thermoapillary flows in liquid bridge model (see Fig.1) mimiking the half of FZ for the sake of simpliity during the last deades. It is now well established that a steady axisymmetri (2D) thermoapillary flow loses its stability first to a steady asymmetri (3D) flow and then to an osillatory flow in 6

8 liquid bridges of low Prandtl numbers (Pr 0.06) while it loses its stability diretly to an osillatory flow (3D) in those of large Pr numbers [2]. On the other hand, the orresponding ritial onditions determined through the analytial studies an not agree well with the experimental results in quantity, espeially for high Pr number fluids. It is noted that most of the analytial studies on the liquid bridge were arried out with an adiabati melt free surfae assumption, i.e., there is no heat transfer aross the melt free surfae. In pratie, interfaial heat transfer in the experimentations, espeially under high temperature onditions, may play an important role in the flow dynamis (for example, see [3-8]). Kamotani et al. [3] studied the effet of interfaial heat loss in liquid bridges of high Pr fluids onsidering the ambient air flow. They alulated the average interfaial heat transfer rate (indiated non-dimensionally as average Biot number in their notation), and found that the ritial onditions derease with the inreased heat loss when the average Biot number is less than 1.5. Melnikov and Shevtsova [7] numerially investigated the effet of interfaial heat transfer on the thermoapillary flow in a liquid bridge of Pr = 14 with aspet ratio equal 1.8 under normal gravity ondition, and the ambient air temperature was assumed to equal the old rod temperature. They found that the heat loss leads to destabilization of the flow at small Biot numbers (Bi 2) ontrary to the stabilization of the flow at large Bi numbers (Bi 5). Similarly, Kousaka and Kawamura [5] studied the thermoapillary flow in a liquid bridge of Pr = 28.1 with unitary aspet ratio under zero gravity ondition, and a linear ambient air temperature distribution was adopted. In their ase, 7

9 in the range of small Bi numbers (Bi < 1), there is heat loss through the free surfae near the old rod and heat gain near the hot rod, and the destabilization of the thermoapillary flow by the interfaial heat transfer was also found. Reently, Wang et al. [8] studied the situation with interfaial heat gain. They found that the ritial onditions are not signifiantly affeted by the interfaial heat gain, and there is a big jump of the ritial onditions when the interfaial heat transfer is hanged from heat gain to heat loss [3]. However, the aforementioned results are fragmental due to the omputation task of 3D diret numerial simulation, and the detailed dependeny of the ritial onditions on the interfaial heat transfer is still laking. On the other hand, to the end of the manipulation of osillatory thermoapillary flow through varying the interfaial heat exhange, extensive preliminary experimental studies should be onduted on the ground due to the sare spae experiment opportunity. Therefore, the effet of melt free-surfae deformation, whih is usually ignored in previous studies suh as [7], should be taken into aount. In present study, linear stability analyses were onduted to systematially investigate the dependeny of the ritial onditions of osillatory onvetion on the interfaial heat transfer under normal gravity ondition with both the onstant and linear ambient air temperature distributions (The studies on the orresponding ases in the mirogravity environment will be reported elsewhere). Moreover, effets of melt free-surfae deformation on the ritial onditions of osillatory onvetion were investigated. 8

10 2. Governing Equations and Numerial shemes Figure 1 shows the sheme diagram of the liquid bridge onsidered in the present study. The liquid bridge is formed by 1st oil with height L, radius R and an applied temperature differene Δ T between two solid ends. The loal radius of the melt free surfae is denoted as h(z). The length, veloity, pressure and time are saled by R, γδ T γδ, µ RT and 2 R υ respetively, and the temperature measured with respet to T 0 is saled by Δ T, where T 0 is the mean temperature of the upper and lower ends, µ the dynami visosity oeffiient, υ the kinemati visosity oeffiient and γ the negative temperature gradient of surfae tension. The Reynolds number, Prandtl number, Grashof number, aspet ratio, Biot number and stati Bond number are defined as follows respetively, γδtr Re =, µυ 3 υ gβδtr Pr =, Gr =, 2 α υ L Γ =, R hr Bi =, k Bo ρ gr σ 2 0 =, 0 where α is the thermal diffusivity oeffiient, h the heat transfer oeffiient, ρ 0 the mean density, σ 0 the melt surfae tension and g the aeleration of gravity. The thermophysial properties of 1st silione oil are listed in Table 1. In the ylindrial oordinate ( r, θ, z), the non-dimensional governing equations are as follows: U! = 0! U!! Gr + Re( U ) U + P = ΔU! + Te z (2) t Re T 1 + Re( U! ) T = ΔT (3) t Pr! where U = ( u, v, w) indiates the dimensionless veloity vetor, P the pressure, T (1) 9

11 the temperature and t the time. The orresponding boundary onditions are as following: " 1 z = 0, Γ : U = 0, T = (4) 2!!!!! r = h(z) : n U = 0, tz ( S n) = tz T!!!! (5) t ( S n) = t T, n T = Bi( T T ) where ϕ ϕ T amb indiates the dimensionless ambient air temperature, 1!! S = ( U + ( U ) T ) the stress tensor in non-dimensional form. The Vetor n! is the 2 outward-direted normal vetor of the free surfae, and the vetors t! z and t! ϕ denote the unit vetor tangent to the melt free surfae in the (r, z) plane and (r,ϕ ) plane respetively. γδt Considering the asymptoti limit of apillary number, Ca = 0, the melt σ free surfae shape is symmetri to the axis of melt zone and the normal-stress balane an be approximated by the Young-Laplae equation: P s! = n + Bo amb z, (7) 0 where P s is the dimensionless stati pressure. Before alulating the flow and temperature fields, the free surfae shape h(z) is determined by Eq. (7) with the liquid bridge volume equal π R 2 L. As shown in Fig. 1, if Bo=0, the free surfae shape is ylindrial with h(z) = 1 as indiated by the dashed lines (hereafter, we use h(z) to indiate the dimensionless loal radius of the free surfae without onfusion) while Bo 0, the free surfae is urved as indiated by the solid lines. For the ase with urved free surfae shape, the body-fitted urvilinear oordinates are employed. The original physial domain in the (r, z) plane oupied by the liquid bridge is 10

12 transformed into a retangular omputational domain in the ( ξ, η) plane by the transformation: r ξ = h( z) (9) η = z For the sake of brevity, the details of the transformed equations in the urvilinear oordinates ( ξ, η), whih is the same as the exellent work [9], are not shown here. X For the linear stability analysis, the basi steady axisymmetri state,!!! = { U ( r, z) = Ue + We, P( r, z), T ( r, z)}, is first determined for a given set of r z parameters (Re, Pr,Bi and Γ ), and then small three-dimensional disturbanes are added to the basi state and linearized by negleting high orders of disturbanes [10-12]. The disturbanes are assumed to be in the normal mode:! u' p' = T ' + m=! u' p' T ' m m m ( r, z) ( r, z) exp[ σ ( m) t + jmφ], (10) ( r, z) where the variables with prime denote the disturbanes, m the azimuthal wave number, σ (m) the omplex growth rate of the orresponding perturbation mode, and i = 1. The disrete form of the linearized equations an be written as a generalized eigenvalue problem: g( x, X, Re, m, Pr, Γ, Bi) Ax = Bx, (11) where T x = ( u', iv', w', p', T' ) denotes a vetor omposed of disturbane veloity, pressure and temperature. A is a real-valued non-symmetri matrix, while B is a singular real-valued diagonal matrix. The eigenvalues and related eigenfuntions of problem (8) are solved by the Arnoldi method [13]. The ritial Reynolds number 11

13 Re is obtained when the maximal real part of σ (m) for all m is zero. In order to well resolve the boundary layers at both ends, a non-uniform grid with denser points near both solid ends and free surfae is adopted in this study. The grid we used in the alulation is N r N z = , where N and r N indiate the z number of the grid points in radial and axial diretion respetively. Moreover, to validate the present ode under urved free surfae situation, we reprodued some alulations in the work [9] and [14] using their definitions of the dimensionless numbers, and the omputed results show good agreement (see Table 2). Moreover, to better understand the mehanism of the instabilities, rates of kineti energy (E k ) hange and thermal energy (E th ) hange of the ritial disturbanes were investigated in the following way: the disturbane equations for momentum and temperature [10-12] were multiplied by the veloity and temperature disturbanes respetively, and then integrated over the volume of liquid bridge and normalized by the mehanial dissipation (D k ) and thermal dissipation (D th ) respetively, thus the following equations were obtained: 1 D k de dt k = M I 1 + M 2 + M 3 + B + 1, (12) 1 D th de dt th = Q + J J (13)! u ' where = 2 T' E k dω, = Ω 2 2 S': S' T ' T ' E th dω, D k = Ω 2 dω, D th = Ω 2 dω Ω Pr are the integrations over the volume of liquid bridge ( Ω ) respetively, M 1 = D k s! u' T ' ds = M 1 + M 2 + M 3 1 = D k s T ' 1 u' ds r D k s T ' 1 v' ds ϕ D k s T ' w' ds z 12

14 the integrations over the surfae of Ω denoting the work done by the thermoapillary fore indued by the temperature disturbane per unit time, Gr B = w' T' dω Re Ω the work done by the buoyant fore, I = Re! u' S u! ' dω 2 Ω the interative term between the basi state stress tensor and the veloity disturbane, Bi Q = Pr s T' 2 ds the transport of thermal energy through the free surfae, and J Re! = ( u' T ) T' dω D th Ω Re T Re T = J1 + J2 = ( u' ) T' dω ( w' ) T' dω the D r D z th Ω th Ω interative terms between the basi state temperature and the veloity and temperature disturbanes. 3. RESULTS AND DISCUSSIONS 3.1 Constant ambient air temperature distribution ( T amb = Told ) A liquid bridge model (see Fig. 1) formed by 1st silione oil (Pr = 16) with Γ = 1.8 (L = 4.5mm and R = 2.5mm) is adopted in this subsetion. With the unitary volume ratio, V V 0 =1. 0, the free surfae is deformed under normal gravity with the maximum of h(z) = and the minimum of h(z) = The ambient air temperature is assumed to equal the temperature of the old rod. Therefore, the interfaial heat transfer is always heat loss from the melt to the ambient air. Figure 2 shows the dependeny of Re upon Bi aording to the omputed results listed in Table 3, and the orresponding neutral modes are all osillatory onvetions with the wave number (m = 1). The Re profile in the ase of under normal gravity ondition ( g = g0, Bo = ) exhibits a onvex dependeny upon the inreasing heat loss. In 13

15 the parameter range studied, the Re dereases with the inreasing Bi up to Bi 1, and this is followed by an approximately linear inrease. Noted that the interfaial heat loss starts to stabilize the onvetion at Bi 2. 5 ompared to the orresponding adiabati ase. The isotherms and streamlines of the two-dimensional axisymmetri onvetion at the seleted Re are shown in Fig. 3a-3 respetively. The thermal boundary layer is developed in the neighbor region of the hot end while the isotherms in the old orner is ompressed due to the onentrated streamlines indued by the shift of vortex ore to the onvex part of the free surfae. Note that all disturbanes must satisfy the no-slip boundary onditions at the solid ends, the magnitude of the disturbanes inside the boundary layers should be smaller than those outside the boundary layer, and the part of the liquid bridge with largest disturbanes is the most unstable. Therefore, the stability property of the basi flow is mainly determined by the effetive temperature gradient on the middle part of the liquid bridge [15]. From this point of view, it is useful to investigate the temperature gradient at the middle part of the free surfae. It is known that the interfaial heat transfer modifies the free-surfae temperature distribution diretly and then the thermal and flow fields. In the neighbor region of the melt free surfae, the inreasing interfaial heat loss pulls up the isotherms near the old end, and the temperature gradient near the old rod is smoothed while the temperature gradient near the hot rod is enhaned. Moreover, the vortex of the flow distributes more homogeneously in the bulk region of the liquid bridge. The net effet of the above modifiations is that the temperature gradient along most part of the melt free surfae gets steeper with inreasing Bi (see Fig. 4a). 14

16 Therefore, a lower Re may be required to destabilize the stationary onvetion with inreasing interfaial heat loss. On the other hand, aording to the perturbation equation of the melt free-surfae heat transfer:! n T ' = Bi T' on r = h(z), (12) whenever a positive temperature disturbane appears somewhere on the melt free surfae, it is aompanied by an inrease of heat loss through that part of the melt free surfae, and vie versa. Therefore, the interfaial heat transfer restrains the melt free-surfae temperature disturbanes, espeially at larger Bi. To verify the remarks, Figure 2 also shows the numerial results that Bi in the perturbation equation (12) is set to be zero. With the same basi flow, the interfaial heat transfer signifiantly delays the onset of the osillatory onvetion, and the stabilization effet gets stronger with inreasing Bi. In pratie, the onvex tendeny of the Re profile shown in Fig. 2 ould be due to the ompetition of the aforementioned two mehanisms. For a further investigation of the physis of the instabilities, the kineti and thermal energy balanes normalized by the mehanial and thermal dissipation D k and D th respetively are shown in Fig. 5a. For the thermal energy of the disturbane flow, the destabilizing effet (J1) is produed by the amplifiation of the high radial gradient of the basi thermal field mainly ourred in the bulk through the radial flow disturbanes. On the other hand, in addition to the major stabilizing effet ontributed by the thermal diffusion (D th ), the stabilizing effets are produed by the amplifiation of the high axial gradient of the basi thermal field (J2) mainly ourred near the hot end and old orner through the axial flow disturbanes [14] and the 15

17 transport of thermal energy through the free surfae (Q), the only term diretly involving the effet of interfaial heat transfer. The kineti energy is insignifiant for fluids of high Pr number [14]. However, it s interesting to note that the effet of buoyant fore (B) always serves the destabilization although the liquid bridge is heated from above. The rate of hange of thermal energy with the inreasing interfaial heat loss at Re = 428, the ritial Reynolds number for the orresponding adiabati ase, is shown in Fig. 5b. It an be seen that the destabilizing effet (J1) overwhelms the major stabilizing effet of heat diffusion (D th ) at small Bi, and it is overwhelmed by the heat diffusion at large Bi. The general trend of the orresponding Re profile is determined by the relative magnitude of (J1) to the thermal diffusion (D th ). However, the details of the trend, suh as the position of the loal minimum, are also ontributed by (J2) at small Bi while by (Q) at large Bi. To evaluate the effet of melt free-surfae deformation on the stability of the stationary onvetion, an artifiial ase of under normal gravity ondition but negleting the melt free-surfae deformation ( g = g0, Bo = 0 ) is studied. The numerial results show that the orresponding flow fields are quite different from those in the ases with deformed domain (see Fig. 3) where the vortex ores shift to the hot orner with the rowded streamlines. Noted that the temperature gradient at the middle part of the melt free surfae is flattened by the onvetion with the boundary layers formed at both the old and hot ends where the temperature gradient inreases sharply (see Fig. 4b). The orresponding Re profile roughly exhibits a similar tendeny as the ase of under normal gravity, however, with muh lower 16

18 quantity (see Fig. 2). It reveals the signifiant stabilization effet on the stationary onvetion through the modifiations of the thermal and flow fields due to the free-surfae deformation. The orresponding neutral mode keeps (m = 1). On the other hand, the Re profile exhibits a loal maximum roughly at Bi = 1. Note that the temperature gradient distribution along the free surfae exhibits the similar onfiguration in the middle part of the free surfae exept that it exhibits undulation at the lower half of the free surfae in the range of small Bi (see Fig. 4b). The onfiguration transition of the temperature gradient distributions at Bi = 1 oinidently orresponds to appearane of the loal maximum of the Re profile. The details of the relationship between the undulation of temperature gradient distribution and the sudden inreasing stabilizing effet of the flow need further investigation. Figure 6 shows the orresponding results of the energy analysis. Similar to the ase with the free surfae deformation, the destabilizing effet for the thermal energy of the disturbanes flow is produed by the radial energy transfer from the basi thermal field to the disturbane flow (J1). The main stabilizing effets are ontributed by the thermal diffusion (D th ) and the axial energy transfer from the basi thermal field to the disturbane flow (J2). The rate of hange of thermal energy as a funtion of Bi at Re = 292, the ritial Reynolds number for the orresponding adiabati ase, is shown in Fig. 6b. It an be seen that the destabilizing effet (J1) generally exhibits a onvex trend while it keeps nearly onstant in the range of Bi from 0.5 to 1.0. The stabilizing effet (J2) also exhibits a onave trend with a loal minimum at Bi = 1.0. The oupling of the above ontra-effets and the thermal 17

19 diffusion (D th ) results in loal maximum suppressing of the instability at Bi = 1 in the rate of the thermal energy (E th ) hange of the disturbane, whih orresponds to the appearane of the loal maximum of the Re profile at Bi = Linear ambient air temperature distribution ( T amb = z Γ 0. 5) In this subsetion, the same liquid bridge model as that in the subsetion 3.1 is adopted exept that the ambient air temperature is assumed to be linearly distributed along the axial diretion [4, 5]. Therefore, it is heat-loss to the ambient air at the lower part of the free surfae while heat-gain from the ambient air at the upper part of the free surfae (see Fig. 7a). With the inreasing interfaial heat transfer, the isotherms and streamlines of the two-dimensional axisymmetri onvetion (not shown) at the orresponding Re behave quite similarly to the ase with ( T = T ) (see Fig. 3), amb old so does the temperature gradient distribution on the middle part of the free surfae (see Fig. 7b), whih gets steeper with the inreasing Bi. On the other hand, the temperature gradient at the lower part of the free surfae in the present ase is inreased ompared with Fig. 4a at the same Bi, the distributions of the effetive temperature gradient, therefore, behave more smoothly. Figure 8 shows the dependeny of Re upon Bi aording to the omputed results listed in Table 4. Due to the two oupling mehanisms mentioned in the subsetion 3.1 (Figure 8 also shows the numerial results that Bi in the perturbation equation (12) is set to be zero), the Re profile roughly exhibits a similar onvex tendeny as the orresponding ase with ( T amb = Told ) exept the appearane of loal maximum Re in the range of 18

20 small Bi. In the parameter range studied, the wave number of the orresponding neutral mode is also (m = 1). The kineti and thermal energy balanes normalized by the mehanial and thermal dissipation D k and D th respetively are shown in Fig. 9a. In the thermal energy of the disturbanes flow, the energy transfer from the basi thermal field through the radial flow disturbanes (J1) always serves as the destabilizing effet. Both the thermal diffusion (D th ) and the effet involving the diret effet of interfaial heat transfer (Q) serve as the stabilizing effet, while the latter is muh less important in the energy ontribution. The energy transfer from the basi thermal field through the axial flow disturbanes (J2) also behaves as the stabilizing effet until it reverses to serve as the destabilizing effet at large Bi. The rate of hange of thermal energy with the inreasing interfaial heat transfer at Re = 356, the ritial Reynolds number for the ase (Bi = 1), is shown in Fig. 9b. It an be seen that in the range of small Bi, the stability property of the stationary onvetion is determined by the ontributions from the destabilizing energy transfer (J1), the stabilizing energy transfer (J2) and the thermal diffusion (D th ). The destabilizing energy transfer (J1) overwhelms the heat diffusion (Q) and exhibits a loal minimum around Bi = 0.5. On the other hand, the destabilizing energy transfer (J2) nearly keeps onstant until Bi = 0.5. The net effet of the oupling stabilizing energy transfer and the destabilizing energy transfer results in loal maximum suppressing of the instability at Bi = 0.5 in the rate of the thermal energy (E th ) hange of the disturbane, whih may relate to the appearane of the loal maximum of the orresponding Re profile. Finally, the effet of melt free-surfae deformation on 19

21 the stability of the stationary onvetion was briefly studied as shown in Fig. 9. For the artifiial ase of under normal gravity ondition but negleting the melt free-surfae deformation ( g = g0, Bo = 0 ), the orresponding Re profile exhibits a similar onvex tendeny as the ase of under normal gravity. Noted that the profile does not exhibit any loal maximum in the parameter range studied. The omputed results also reveal the signifiant stabilization effet on the stationary onvetion through the modifiations of the thermal and flow fields due to the free-surfae deformation. 4. Conlusions In the present paper, effet of interfaial heat transfer on the onset of osillatory onvetion in liquid bridges formed by 1st silione oil on the ground is systematially studied in an extended range of Bi. With both the onstant and linear ambient air temperature distributions, the numerial results show that the interfaial heat transfer modifies the free surfae temperature distribution diretly and then indues a steeper temperature gradient on the middle part of the free surfae, whih may destabilize the onvetion. On the other hand, the interfaial heat transfer restrains the temperature disturbanes on the free surfae, whih may stabilize the onvetion. The two oupling effets result in a omplex dependene of the stability property on the Biot number, and the appearane of the loal maximum of the Re profile losely relates to the ambient temperature distribution. Moreover, the omputed results reveal that the effet of free-surfae deformation serves a signifiant 20

22 stabilization on the two-dimensional axisymmetri stationary onvetion. To better understand the mehanism of the instabilities, rates of kineti energy hange and thermal energy hange of the ritial disturbanes were also investigated. The thermal energy of the disturbanes, whih is important in liquid bridges of large Prandtl number, is mainly produed by the interation between the basi thermal field and the flow disturbanes and the thermal diffusion (D th ). The net work of the oupling effets above determines the stability property of the osillatory onvetion in the liquid bridge. 21

23 Referenes [1] D, Shwabe, in: H. C. Freyhardt (ED.), Crystals 11, , Springer, Berlin, [2] M. Levenstam, G. Amberg, and C. Winkler, Instability of thermoapillary onvetion in a half-zone at intermediate Prandtl numbers, Phys. Fluids 13 (2001) [3] Y. Kamotani, L. Wang, S. Hatta, A. Wang, and S. Yoda, Free surfae heat loss effet on osillatory thermoapillary flow in liquid bridges of high Prandtl number fluids, Int. J. Heat and Mass Transfer 46 (2003) [4] M. Irikura, Y. Arakawa, I. Ueno, and H. Kawamura, Effet of ambient fluid flow upon onset of osillatory thermoapillary onvetion in half-zone liquid bridge, Mirogravity Si. Tehnol. XVI-I (2005) [5] Y. Kousaka, and H. Kawamura, Numerial study on the effet of heat loss upon the ritial Marangoni number in a half-zone liquid bridge, Mirogravity Si. Tehnol. XVIII-3(4) (2006) [6] A. Mialdum and V. Shevtsova, Mirogravity Si. Tehnol. XVI, (2006) [7] D.E. Melnikov and V.M. Shevtsova, Thermoapillary onvetion in a liquid bridge subjeted to interfaial ooling, Mirogravity Si. Tehnol. XVIII-3(4) (2006) [8] A. Wang, Y. Kamotani, S. Yoda, Osillatory thermoapillary flow in liquid bridges of high Prandtl number fluid with free surfae heat gain, Int. J. Heat and Mass Transfer 50 (2007)

24 [9] V. Shevtsova, Thermal onvetion in liquid bridges with urved free surfaes: Benhmark of numerial solutions, J. Crystal Growth 280, (2005) [10] G. Chen, A. Lizée, and B. Roux, Bifuration analysis of thermoapillary onvetion in ylindrial liquid bridges, J. Crystal Growth 180 (1997) [11] H.C. Kuhlmann and H.J. Rath, Hydrodynami instabilities in ylindrial thermoapillary liquid bridges, J. Fluid Meh. 247 (1993) [12] G.P. Neitzel, K.T. Chang, D. F. Jankowski and H.D. Mittelmann, Linear stability theory of thermoapillary onvetion in a model of the float-zone rystal-growth proess, Phys. Fluids A5 (1993) [13] G.H. Golub, and C.F. Van Loan, Matrix Computations, Johns Hopkins University Press, Baltimore, [14] CH. Nienh ü ser, H. C. Kuhlmann, Stability of thermoapillary flows in non-ylindrial liquid bridges, J. Fluid Meh. 458,(2002) [15] D. Shwabe, Hydrothermal waves in a liquid bridge with aspet ratio near the Rayleigh limit under mirogravity. Phys. Fluids 17, (2005)

25 Table 1: Thermophysial properties of 1st silione oil 3 ρ 818( kg / m ) β (K -1 ) 0 υ ( m / s) γ ( kg / s ) K ( al / m s k) 0 σ ( N / m) 24

26 Table2: ode validation (a): Pr = 0.02, Re = 2000, g = 0 Aspet ontat angle at Present ode Ref. [9] ratio hot rod: α h N r N z = ψ Γ = max Γ = w on free max surfae Γ = (b): Pr = 4, Γ = 1, g = 0, Bi = 0 Γ = ontat angle at hot m Present ode Ref. [14] rod: α h N r N z =

27 Table 3: Re versus Bi when T = 0. 5 amb Bi g=g 0, Bo=0 g=g 0, Bo=

28 Table 4: Re versus Bi when = z Γ 0. 5 T amb Bi g=g 0, Bo=0 g=g 0, Bo=

29 Figure 1. Sheme diagram of a liquid bridge. 28

30 Figure 2. Critial Reynolds number versus Bi for ases of T amb = -0.5: (g = g 0, Bo = 0)-diamond points on solid line; (g = g 0, Bo = 2.965)-triangle points on solid line and (g = g 0, Bo = 2.965) when Biot number is set to zero in the perturbation equation (12)- triangle points on the dotted line respetively. 29

31 (a) (d) (b) (e) () (f) Figure 3. Isotherms and streamlines at the orresponding Re for different Bi under normal gravity ondition for T amb = -0.5 (left: Bo = 2.965, right: Bo = 0): (a) Bi = 0, Re = 428; (b) Bi = 1, Re = 356; () Bi = 4, Re = 373; (d) Bi = 0, Re = 292; (e) Bi = 1, Re = 240; (f) Bi = 4, Re =

32 (a) (b) Figure 4. Temperature gradient distribution along the melt free surfae at the orresponding Re for T amb = -0.5: (a) g = g0, Bo = ; (b) g = g0, Bo = 0. 31

33 (a) (b) Figure 5. Kineti and thermal Energy balane for ases of T amb = -0.5 when g = g0, Bo = at the orresponding Re (a) and at Re equals 428(b). 32

34 (a) (b) Figure 6. Kineti and thermal Energy balane for ases of T amb = -0.5 when g = g0, Bo = 0 at the orresponding Re (a) and at Re equals 292(b). 33

35 (a) (b) Figure 7. Temperature (a) and Temperature gradient (b) distribution on free surfae at the orresponding Re for linear T amb under normal gravity ondition. 34

36 Figure 8. Critial Reynolds number versus Bi for ases of linear ambient temperature (g = g 0, Bo = 0)-diamond points on solid line; (g = g 0, Bo = 2.965)-triangle points on solid line and (g = g 0, Bo = 2.965) when Biot number is set to zero in the perturbation equation (12)- triangle points on the dotted lines respetively. 35

37 (a) (b) Figure 9. Kineti and thermal Energy balane for ases of T amb is linear when g = g0, Bo = at the orresponding Re (a) and at Re equals 356(b). 36

38 (a) (b) Figure 10. Kineti and thermal Energy balane for ases of T amb is linear when g = g 0, Bo = at the orresponding Re (a) and at Re equals 240(b). 37