We present a dynamic van der Waals theory. It is useful to study phase
separation when the temperature varies in space. We show that if heat flow is
applied to liquid suspending a gas droplet at zero gravity, a convective flow
occurs such that the temperature gradient within the droplet nearly vanishes.
As the heat flux is increased, the droplet becomes attached to the heated wall
that is wetted by liquid in equilibrium. In one case corresponding to partial
wetting by gas, an apparent contact angle can be defined. In the ther case with
larger heat flux, the droplet completely wets the heated wall expelling liquid.Comment: 6pages, 8figure

A. Onuki

Akira Onuki

L. D. Landau

N. O. Young

R. B. Bird

S. Puri

English

arXiv

Onuki, A

Kyoto University Research Information Repository

Title Dynamic van der waals theory of two-phase fluids in heat flowAuthor(s)Onuki, ACitationPHYSICAL REVIEW LETTERS (2005), 94(5)Issue Date2005-02-11URL http://hdl.handle.net/2433/49918RightCopyright 2005 American Physical SocietyType Journal ArticleTextversionpublisherKyoto UniversityDynamic van der Waals Theory of Two-Phase Fluids in Heat FlowAkira OnukiDepartment of Physics, Kyoto University, Kyoto 606-8502, Japan(Received 6 June 2004; revised manuscript received 5 October 2004; published 7 February 2005)We present a dynamic van der Waals theory. It is useful to study phase separation when the temperaturevaries in space. We show that, if heat flow is applied to liquid suspending a gas droplet at zero gravity, aconvective flow occurs such that the temperature gradient within the droplet nearly vanishes. As the heatflux is increased, the droplet becomes attached to the heated wall that is wetted by liquid in equilibrium. Inone case corresponding to partial wetting by gas, an apparent contact angle can be defined. In the othercase with larger heat flux, the droplet completely wets the heated wall expelling liquid.DOI: 10.1103/PhysRevLett.94.054501 PACS numbers: 47.20.Dr, 05.70.Ln, 44.35.+c, 64.70.FxIn the usual theories of phase transitions, the fluctuationsof the temperature T are assumed to be small and areneglected. However, there can be situations in which phasetransitions occur in inhomogeneous T. For example, wet-ting properties near the gas-liquid critical point are verysensitive to applied heat flux [1,2], and boiling processesremain largely unexplored [3,4]. To treat such problems wepropose to start with coarse-grained entropy and energy,from which the temperature is defined as a functional of theorder parameter and the energy density.Let us consider one-component fluids, where the orderparameter is the number density n. In the van der Waalstheory the energy density e and the entropy s per particleare given by e  dnkBT=2 	v0n2 and s=kB  d=2 lnT  ln1=v0n 1  const, respectively, in terms of Tand n [5]. Here v0 and 	 represent the molecular volumeand the magnitude of the attractive potential, respectively,and d is the space dimensionality. To describe situations inthe presence of interfaces, we generalize this descriptionby including gradient contributions to the total entropy andthe total internal energy asS Zdrns 12Cjrnj2; (1)E Zdre 12Kjrnj2: (2)The gradient terms represent a decrease of the entropy andan increase of the energy due to inhomogeneity of n. Herethe internal energy density is written ase^  e Kjrnj2=2; (3)including the gradient part. We assume that the coefficientsC and K are simply constants and that s in Eq. (1) dependson n and e ass  kB ln	e=n 	v0nd=21=v0n 1  const: (4)This relation follows from the van der Waals theory. Ourscheme starts with Eqs. (1), (2), and (4), where the tem-perature is not yet introduced. We then define the localtemperature T  Tn; e by1TeSn n@s@en; (5)where n is fixed in the derivatives. This definition of T isanalogous to that in a microcanonical ensemble. We alsodefine the local chemical potential ^ (per particle) includ-ing the gradient contributions by^  TSne^  Tr MTrn; (6)where e^ in Eq. (3) is fixed,   T@ns=@ne, andM  CT  K: (7)Maximization of S under a fixed total particle numberRdrn and a fixed total energy E leads to the equilibriumconditions T  const and ^  const. As first derived byvan der Waals [5,6], the equilibrium interface densityprofile n  nx is determined by n; T Md2n=dx2 cx with the surface tension   MR11 dxdn=dx2,where cx is the chemical potential on the coexistencecurve [7].Next we set up the dynamic equations [8]. Hereaftergravity is neglected. The mass density   mn (m beingthe particle mass) obeys @=@t  r  v, where v isthe velocity field. The momentum density v obeys@@tv  r  vv  r  $  $: (8)The $  fijg is the reversible stress tensor invariant withrespect to time reversal v! v. The $  fijg is the dis-sipative stress tensor of the form ij  rivj rjvi   2=dijr  v with ri  @=@xi in terms of theviscosities  and  . The (total) energy density eT  e^v2=2 including the kinetic part is governed by@@teT  r  	eTv $  $  v  r  rT; (9)PRL 94, 054501 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending11 FEBRUARY 20050031-9007=05=94(5)=054501(4)$23.00 054501-1  2005 The American Physical Societywhere  is the thermal conductivity. We construct the stresstensor ij such that the entropy production rate within thefluid assumes the usual form [8],@@tS Zdr1Tr  rT Xijijrivj: (10)If there is no heat flow from outside, the above quantitybecomes non-negative-definite. To obtain Eq. (10) we needto require the general relation,r 1T$  nrnSe^ e^re^Sn: (11)Using the van der Waals pressure p  nkBT=1 v0n 	v0n2 we then findij  p p1ij Mrinrjn; (12)where p1  Mnr2n jrnj2=2  Tnrn  rM=T.As an example, we consider a bubble with radius R inheat flow. Thermocapillary bubble migration has longbeen observed, especially in space [9–11]. Theoretically,however, the bubble motion has been examined in the ab-sence of first-order phase transition at the interface such asin the case of air bubbles in silicone oil [9]. In such casesthe temperature dependence of the surface tension  causesa bubble velocity of order vY  R1dT=dz1= in agiven temperature gradient dT=dz1 (Marangoni effect),where 1  d=dT > 0 and  is the viscosity in liquid.In one-component fluids the effect is drastically altered dueto latent heat generation or absorption at the interface [12].We point out that a convective velocity of order vO dT=dz1=0Ts inside the bubble is sufficient to carrythe applied heat flux, where 0 is the mass density in gas,s is the entropy difference between gas and liquid perunit mass, and  is the thermal conductivity in liquid. Notethat the ratioM1  vY=vO  R10Ts= (13)is a very large number of order R=a with a being a micro-scopic length. With phase change the bubble velocity is oforder vO and is very slow [12]. Furthermore, the tempera-ture inside the bubble should be nearly homogeneous dueto the latent heat convection. This is a natural consequencein one-component fluids without contamination [1,12],because the pressure should become nearly homogeneousoutside the bubble and the fluid near the interface should beclose to the coexistence curve in the p-T phase diagram.In our simulation we set K  0, and then the interfacethickness is ‘  Cv0=2kB1=2 far from the critical point[since cx=kBT is of order 1 for n v10 ]. Space ismeasured in units of ‘, and the system length is L  202‘.Time is measured in units of t0  m‘2=v0. Here weassume constant viscosities with    . The strength ofdissipation is represented by the normalized viscosity  v0=‘m	p. We set   0:11=2. On all the bound-ary walls we assume [13,14]n  nw  ‘w  rn; (14)where  is the inward normal unit vector at the surface. Weset nw  5nc=2 and ‘w  22p‘, for which the wall iswetted by liquid in equilibrium in our simulation. Weprepared an initial equilibrium state with a circular gasbubble with radius 50 placed at the center of the cell. Thetemperature was at T  0:875Tc, where the density is0:37nc in gas and 1:74nc in liquid. The bottom boundary(y  0) was then increased by a constant T, while the topboundary (y  L) was held at the initial temperature Tt 0:875Tc. (Here we use ‘‘bottom’’ and ‘‘top’’ though nogravity is assumed.) The side boundaries (x  0 and L) areinsulating (  rT  0). Furthermore, the density depen-dence of the thermal conductivity is crucial away fromcriticality [15], so we set   v0=kBmn with the ther-= 0.00675∆T/TcFIG. 1. Suspended gas droplet in a steady state in heat flowwith T  0:006 75Tc. The arrows indicate the velocity.00.10.20.30.40.50.60.70.80.910 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.8750.8760.8770.8780.8790.880.8810.882    cT/T=0.00675∆xycT/TFIG. 2. The temperature in the steady state in Fig. 1. It is flatwithin the gas droplet due to latent heat transport.PRL 94, 054501 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending11 FEBRUARY 2005054501-2mal diffusivity DT being of order 1 (in units of ‘2=t0 v0=m). Then  in liquid is larger than that in gas by5 times.(i) First, for very small T, the bubble position is shiftedtoward the bottom without shape changes. Figure 1 showssuch a steady state at T  0:006 75Tc taken at t  3:4104 after the application of heat flux. Here the migration isstopped by the repulsion from the walls due to the wettingcondition in Eq. (14). The velocity field v is almost parallelto the interface, but there is a small velocity componentthrough the interface. Thus, at the interface, first-orderphase transition takes place, and latent heat is generatedor absorbed. As demonstrated in Fig. 2, latent heat trans-port is so efficient that the temperature gradient almostvanishes within the bubble.(ii) Second, for the case T  0:054Tc, Fig. 3 showsdroplet migration toward the bottom. Figure 4 displays thevelocity field in the steady state taken at t  8 104. Thevelocity component through the interface is more increasedthan in Fig. 1, again resulting in a flat temperature insidethe droplet as in Fig. 5. As can be seen in Figs. 4 and 5, athin liquid layer is sandwiched between the bottom bound-ary and the droplet. The thickness of this layer is so thinthat we can define an apparent contact angle 'eff , which is adecreasing function of T. In accord with these results,Garrabos et al. observed that gas spreads on a heated wallinitially wetted by liquid and exhibits an apparent contactangle [1].(iii) Third, we increased T to 0:101 25Tc, where thebottom temperature is 0:97Tc. As shown in Fig. 6, phaseseparation occurred to produce a gas domain at the bottom,into which the suspended droplet was finally absorbed. Inthe steady state in Fig. 7, the bottom is covered by a gaslayer or is completely wetted by gas [16], but the interfacewith the bulk liquid is still curved and a small velocity fieldis induced. As can be seen in Fig. 8 taken at t  3 104,the temperature gradient within the gas layer becomessteeper than that in the liquid. It is well known that a gasfilm on a heated wall suppresses heat transfer to the bulkliquid and, as it thickens, film boiling is induced in grav-ity [3].In the steady states the velocity is of order vO DTT=TL in terms of DT  =Cp from Cp s. Asa result, the Reynolds number (  Pr1RT=LT) is verysmall, where the Prandtl number Pr  v0=mDT is heretaken to be of order 1. In fact, in Figs. 1, 4, and 7, thevariancehv2ip is 0:12, 1:9, and 0:63, respectively, in units    = 0.054∆ cT/T1000t= 7000t= 80000t=FIG. 3. Migration of a gas droplet for T  0:054Tc. It ap-parently wets the bottom partially in the steady state (left), wherea thin liquid layer is still sandwiched.= 0.054∆T/TcFIG. 4. The velocity field around the gas droplet at t  8104 in Fig. 3. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.870.880.89 0.90.910.920.93    = 0.054∆ cT/TcT/Ty xFIG. 5. The temperature in the nearly steady state in Fig. 4.300 800 30000t= t= t=    = 0.10125∆ cT/TFIG. 6. Time evolution for T  0:101 25Tc. Phase separationoccurs at the bottom. The bottom is completely wetted by gas inthe steady state.PRL 94, 054501 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending11 FEBRUARY 2005054501-3of 103‘=t0, where ‘=t0  DT=Pr‘. In our simulation theviscous dissipation is very strong even in transient states,while in the theory in Ref. [11] the inertia effect (vaporrecoil) is important in bubble motion on a heated wall. Thisdifference stems from the fact that our droplet size is verysmall if ‘ is microscopic.In the diffuse-interface methods for crystal growth [17]and thermocapillary flow [18,19], the interface width ‘may be conveniently treated as a free parameter. This isbecause the phase field equations yield the moving-interface problems with appropriate interface boundaryconditions as long as ‘ R ( domain size). Also influids we may take ‘ as a free parameter in the range‘ R as far as we treat the steady state profiles (wherethe Marangoni effect is suppressed and the Stokes approxi-mation is valid). However, to describe rapid fluid motions,the normalized viscosity / =‘ should be made verysmall if ‘ is enlarged.In summary, we have developed a scheme to study phasetransition dynamics with inhomogeneous temperatureunder mass and heat fluxes. It can serve as a phase fieldmodel in fluids, though we should examine its sharp-interface limit [19]. For one-component fluids we mentionwetting dynamics with evaporation and condensation[2,13], as exemplified here, and boiling or convection ingravity [4,5]. These effects can in principle be studied inthe scheme of thermocapillary hydrodynamics supple-mented with interface conditions. However, hydrodynamiccalculations accounting for latent heat transport have rarelybeen performed.This work is supported by Grant for the 21st CenturyCOE project (Center for Diversity and Universality inPhysics) from the Ministry of Education, Culture, Sports,Science and Technology of Japan.[1] Y. Garrabos, C. Lecoutre-Chabot, J. Hegseth, V. S.Nikolayev, D. Beysens, and J.-P. Delville, Phys. Rev. E64, 051602 (2001).[2] L. M. Pismen and Y. Pomeau, Phys. Rev. E 62, 2480(2000).[3] V. S. Nikolayev and D. A. Beysens, Europhys. Lett. 47,345 (1999).[4] A. Onuki, Physica (Amsterdam) 314A, 419 (2002).[5] A. Onuki, Phase Transition Dynamics (CambridgeUniversity Press, Cambridge, 2002).[6] J. S. Rowlinson, J. Stat. Phys. 20, 197 (1979).[7] For the van der Waals model  / M1=21 T=Tc3=2holds well for T > 0:8Tc within a few percent.[8] L. D. Landau and E. M. Lifshitz, Fluid Mechanics(Pergamon, New York, 1959).[9] N. O. Young, J. S. Goldstein, and M. J. Block, J. FluidMech. 6, 350 (1959).[10] B. Braun, Ch. Ikier, H. Klein, and D. Woermann, Chem.Phys. Lett. 233, 565 (1995).[11] D. Beysens, Y. Garrabos, V. S. Nikolayev, C. Lecoutre-Chabot, J.-P. Delville, and J. Hegseth, Europhys. Lett. 59,245 (2002).[12] A. Onuki (to be published). The temperature gradient in-side a droplet is smaller than that far from it by 1=M1M2,where M2  R0s=1 is also of order R=a 1.[13] P. G. de Gennes, Rev. Mod. Phys. 57, 827 (1985).[14] S. Puri and K. Binder, Z. Phys. B 86, 263 (1992).[15] R. B. Bird, W. E. Stewart, and E. N. Lightfoot, TransportPhenomena (Wiley, New York, 2002), p. 272.[16] At large heat flux the density steeply changes within a fewlattice points at the heated wall. Here the boundary con-dition Eq. (14) is questionable.[17] G. Caginalp, Phys. Rev. A 39, 5887 (1989); R. Kobayashi,Physica (Amsterdam) 63D, 410 (1993); A. Karma andW. J. Rappel, Phys. Rev. E 57, 4323 (1998).[18] D. Jasnow and J. Vin˜als, Phys. Fluids 8, 660 (1996);R. Borcia and M. Bestehorn, Phys. Rev. E 67, 066307(2003).[19] D. M. Anderson, G. B. McFadden, and A. A. Wheeler,Annu. Rev. Fluid Mech. 30, 139 (1998). Here an attemptof taking the sharp-interface limit for fluids is given.= 0.10125∆T/TcFIG. 7. The velocity field around the gas region at t  3 104in Fig. 6. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.88 0.90.920.940.96    = 0.10125∆ cT/TcT/TyFIG. 8. The temperature in the nearly steady state in Fig. 7.PRL 94, 054501 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending11 FEBRUARY 2005054501-4

Dynamic van der Waals Theory of two-phase fluids in heat flow

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