Fluid dynamic modelling of crystal growth from vapour

Acta Astronautica Vol. 28, pp. 197-218, 1992 Printed in Great Britain

0094-5765/92 $5.00-1-0.00 PergamonPreu Ltd

FLUID DYNAMIC MODELI.ING OF CRYSTAL G R O W T H FROM VAPOUR L.G. Napolitano, A.Viviani, R. Savino Istituto di Aerodinamica "Umberto Nobile" Facolth di Ingegneria-Universit~ di Napoli P.le V.Tecchio 80, 80125 Napoli (Italy) Abstract magnitude of these new terms during typical crystal growth experiments on Earth and under microgravity, and to the quantitative analysis of thermal stress convection in a simplified geometrical configuration.

Navier-Stokes equations along with no-slip boundary conditions have been widely used, up till now, in modelling vapour crystal growth fluidynamics. In this paper attention is focused on new types of free convection which occur in a gas or a vapour when viscous stresses, due to velocity gradients, are of the same order of magnitude as stresses due to temperature and/or concentration gradients (thermal and/or solutal-stress convection). In this case linear phenomenological relations, based on the classical principles of non-equilibrium thermodynamics, for the diffusive fluxes of momentum, energy and species concentration (conventional Stokes, Fourier, Fick laws; NavierStokes fluidynamics) become inadequate and a more general theory must be formulated to account for thermal stresses convection, included in the second-order approximation for the gas-dynamic equations (Burnett equations), and side-wall gas creep, induced by the slip boundary condition in the Knudsen layer. This work deals with the above mentioned gaskinetic phenomena, presented in the framework of non-equilibrium thermodynamics. Both phenomenological (macroscopic) and kinetic (microscopic) points of view are considered together with the question of the coexistence of these approaches. The Burnett equations are written in nondimensional form, and a-priori criteria of the non-dimensional order of magnitude analysis lead to the identification of new characteristic velocities and corresponding non-dimensional numbers; new classes of free convection (thermal stress and thermal creep convection) are discussed. The final part of the work is devoted to the evaluation of the orders of

1. Introduction Crystal growth from vapour phase has gained in recent years increasing importance in the preparation of electronic materials, insulators and metals(1); particular emphasis has been posed on new interesting developments coming from the utilization of space microgravity environment. In fact, as gravity play an important role in many transport mechanisms involved in these processes, low-gravity conditions are expected to lead to reduction of uncontrolled disturbances and, therefore, to enhancement of crystals quality (purity and homogeneity). Similarly to many other technological processes, progress in vapour crystal growth has been achieved to a large extent only through intuition and experience, but systematic analyses of growth process have been often limited by very little knowledge of the vapour fluidynamics. Vapour transport usually involves complex physicochemical mechanisms such as: phase changes at source and crystal; forced convection (Stephan-Nusselt flow) and free convection induced by non-uniform temperature and solute species concentrations; diffusion of internal energy and chemical species with coupling cross effects (Soret and Dufour); chemical reactions (bulk and surface, either in equilibrium or in non-equilibrium) (2).

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In addition, fluidynamics of vapour growth is complicated by complex geometries and boundary conditions (unsteady, mixed type, etc.). Mathematical modelling of vapour growth fluidynamics has been devoted, up till now, to the analysis of vapour transport in idealized geometrical configurations (shallow plane or cylindrical enclosures with differentially heated end walls) and with rather simple mathematical models (Navier-Stokes fluidynamics; no-slip boundary conditions on the passive boundaries; constant transport properties and Boussinesq approximations; kinetic cross effects negligible). After the pioneering work of Klosse and U l l e r s m a (3) who obtained, on the basis of the parallel flow core solution found by Gill (a) , analytical solutions for low-Reynolds natural convection in shallow plane and axialsymmetric configurations, Markham et AI. (5'6) and Jahveri and Rosemberg (7) a n a l y z e d "diffusive" and "convective-diffusive" physical vapour transport in closed ampoules by considering double-diffusive natural convection under the Boussinesq approximation. They obtained numerical results for different ampoule geometry and orientation with respect to gravity, with two kind of boundary conditions for the side walls (perfectly conductive or perfectly insulated). Other authors (8"12) proposed numerical solutions for two or three-dimensional free convection problems in closed ampoules with different boundary conditions for the temperature at the side walls (multizone physical vapour transport) or including the coupling between convection and interface transport (11) (mixedtype boundary conditions). Common denominator in all these works is that Navier-Stokes fluidynamics along with no-slip boundary conditions are considered to be adequate for a proper and complete description of the involved phenomenologies. This basic assumption is removed in this paper. We consider new types of free convection which arise in a gas or a vapour due to thermal and solutal stresses (volume driving actions) and temperature and concentration creep (surface driving actions) (t3). The former are non-linear contributions to the stress tensor related to volume differences of temperature and solute species concentration. The latter are due to tangential slip induced by surface gradients of temperature and concentration along side-walls (slip boundary condition). In many problems of interest, in particular during non-isothermal processes in low-gravity, these non-linear irreversible thermodynamic effects can become rather important sources of convection,

whereas other typical driving actions (Stephan and buoyancy-driven flows) can be n e g 1 i g i b 1 e ( 14 ) In this case, linear phenomenological relations, based on the classical principles of non-equilibrium thermodynamics, for diffusive fluxes of momentum, energy and species mass concentration (classical Stokes, Fourier, Fick laws) are no longer adequate and higher order approximations of the gas-dynamics equations (Burnett equations) must be taken into account. The first part of the paper deals with nonequilibrium thermodynamics, following both the phenomenological (macroscopic) and the kinetic (microscopic) points of view; the question of the bridge between these approaches is addressed, and open research areas along this line are indicated. Third and fourth paragraphs are focused on the non-dimensional order of magnitude analysis of the field equations: new characteristic velocities, arising from thermal stresses and creep, and the corresponding characteristic driving numbers are defined and discussed. The a-priori criteria of the non-dimensional order of magnitude analysis are used to identify and characterize the different classes of convection (thermal-stress, thermal creep, natural) and the types of flow regimes (diffusive, convectivediffusive) in terms of problem's data. The orders of magnitude of the relevant nondimensional parameters are evaluated for some vapour crystal growth experiments, both on Earth and under microgravity. A comparison between buoyancy and thermal stress convection is presented in paragraph 5, where analytical solutions for the flow between nonisothermal parallel walls are obtained and discussed. Final comments and conclusions are presented in paragraph 6, where analogous phenomena, induced in gas mixtures by concentration (diffusion) stresses and creep are indicated. 2. Chanman-Ensko~ method and non-equilibri0m Thermodynamics 2.1 General ~Qnsiderations It is well known that two different approaches can be used to describe the dynamic evolution of a continuous medium: the statistical (microscopic) approach and the phenomenological (macroscopic) approach. Each has its own advantages and shortcomings. Phenomenological theories are particularly able to cope with more complex and general situations than statistical theories, whose

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inherent complexity restricts the classes of problems that can be considered. On the other hand the statistical approach leads to quantitative results expressed in terms of physical characteristics of atoms and molecules; these results can be also used to verify and justify the postulates of the macroscopic approach. The phenomenological theories do not allow to define the kinetic coefficients of the resulting equations and to determine the area of applicability of the assumed models. These limits can be overcome either by experiments or by kinetic theory. Hence, in principle, to formulate adequate and consistent mathematical models, one has to use a suitable combination of phenomenological and statistical theories. In the next paragraphs the main features of both kinetic and phenomenological approaches are outlined, by emphasizing new research lines and work to be done in each field. 2.2 Kinetic Theory of gase~s The kinetic approach is of cognitive interest not only because gases really consist of molecules, but also because many gas-dynamics problems cannot be considered from the continuum point of view (rarefied media). The fundamental equation of the kinetic theory is the Boltzmann equation for the velocity distribution function ( F ) . Its area of applicability, in fact, covers the full range of variability of the Knudsen number Kn=l / L (where (lm) is the mean free path and (L) the characteristic flow dimension). If the distribution function (F) is known, any gas-dynamic macroscopic quantity (density (p), momentum (p_~[), temperature (T), stress tensor (~), etc.) may be easily defined as an integral of functions of molecules coordinates and velocity only (15). The corresponding field equations for these global gasdynamic variables are the mass and energy conservation equations and the momentum equation, valid for any gas and Knudsen number. These equations need additional relations to close the problem, for instance relations between stress tensor (~) and heat-flux (jq) on the one hand and velocity and temperature gradients on the other. Closed macroscopic equations can be obtained from the Boltzmann equation by finding its solution as a series in Knudsen number: F = F 0+ Kn F 1 + Kn 2F 2+ ....

(l)

199

By substituting this expansion into the Boitzmann equation and equating terms of the same order in Knudsen, we obtain a recurrent set of integral equations for the (Fi) functions (Chapman-Enskog method). The solution of the zeroth-order approximation is the local Maxwellian distribution. To obtain the functions F l ' F 2 , etc., it is necessary to solve inhomogeneous linear equations. Methods of

solving these equations, for pure gases and gas mixtures are presented in the well known monographs by Chapman and Cowling [16) and Hirchfelder, Curtiss and Bird 07). The maxwellian distribution function (F0) depends only on the gasdynamic quantities p, V and T. The functions F1, F2, etc., depend also on the derivatives of the gasdynamic properties: hence the function ( F ) , which can be represented as a series (1), is expressed in terms of gasdynamic quantities and their derivatives. In conclusion, by substituting the distribution function into the definitions of stress tensor and heat-flux, we obtain the missing relations which close the field equations. In particular, for a single-component homogeneous gas we have: $=~o + =~1+ ~z+....

(2a)

Jq = Jqo + Jql + Jq2 + ....

(2b)

_T, 0 = -pU _'r1 = 2g(VV)o g2

'

D

s

s

+ p (VpVT)o +

(VTVT) 0 + (06

s

s

}

Jqo = 0 Jql

- ~" VT

Jq2 = pg2e p { 01V'V VT + 02 (~t V T - ( V -V ) . V T ) T

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The number of terms retained in the expressions (2) correspond to the number of terms retained in the series (1). The first terms correspond to the Euler approximation, the second terms to Navier-Stokes, the third and fourth terms to Burnett and super-Burnett equations and so on. In the above expressions for X__and jq, p is the thermodynamic pressure, and Z, are coefficients of viscosity and thermal conductivity, Cp is the specific heat at constant s

pressure; ( )0 denotes the symmetric, traceless part of a tensor; co i ' 0 i are constants for molecules with a power-law interaction, which depend on the exponent. The calculation of transport coefficients (viscosity, thermal conductivity, diffusion coefficients, the latter for mixtures) for different molecular interaction laws, may be found in the above mentioned monographs. More complex problems arise in the case of more general and complex phenomenologies (plurireacting mixtures in non-equilibrium, mixtures with very different molecular masses, etc.) In these cases, velocities and temperatures of the components may be different and two or more-fluid formulations, based on macroscopic approaches, may be more appropriate C18]. Other problems of actual interest in the field of kinetic theory include: kinetics of heterogeneous reactions; phase changes; gases with internal degrees of freedom; magnetoplasma-dynamics).

which implies that Equilibrium Thermodynamics can be locally applied to describe any evolution of the system (19) (which involves interaction with its environment and/or internal productions). Considering the simple case of a singlecomponent gas, in the absence of capillarity and electromagnetic effects, the primitive fluiddynamic extensive unknowns are the mass velocity V and two thermodynamic extensive variables, e.g. specific entropy per unit mass (s) and specific volume (~). The closed set of field equations is given by: 1)A thermodynamic fundamental relation or, equivalently, two equations of state. With the energy representation the fundamental relation is: e:e(s,a~)=Ts-pv

(3)

where (e), the specific internal energy per unit mass, is a continuous, twice differentiable, convex function, monotonically increasing with (s) and monotonically decreasing with (~); (T) and (p) are the "equilibrium" temperature and pressure. The (3) is known as Euler equation. A set of state equations is given by the derivatives of (e), according to the Gibbs relation de=Tds-pdD

(4)

They are: 2.3 Phenomenological approach The phenomenological formulation includes: i) identification of "extensive" variables which characterize the (equilibrium) "local state" of the medium and its motion (primitive field unknowns); ii) formulation of an appropriate number of relationships between these state variables and the equilibrium thermodynamic variables (state equations and balance equations, the latter representing "constraints" under which the evolution takes place); iii)formulation of phenomenological relations to "close" the set of basic equations. These relations are needed because the conservation and balance equations, formulated for the primitive field variables, introduce additional unknowns (diffusive fluxes and productions of the extensive variables). The basic idea of the macroscopic formulation is the "shifting (or local) equilibrium" hypothesis,

T=

= T(s,~) ; p= -

= p(s,~)

(5)

2) The balance and/or conservation equations: DO p~- =V.V Ds

p~

(6a)

+V.Js=PS

DV p~-~ -V.

[-pU +(p+g)U+~ov

(6b)

]:pf

(6c)

0 where D = Ott + V . V denotes the material (or substantial) derivative; p is the mass density; Js and s are the entropy diffusive flux and production per unit volume; in (6c) the stress

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tensor has been split into an equilibrium ( - p U ) and a viscous part, x__v, which is further decomposed in an isotropic [+(p+~r)Ul and a traceless tensor (or deviator) x__0v(n=l/3 Tr x ); f represents external (body) forces per unit mass. 3) The expression for the entropy production and diffusive flux are easily obtained, after the shifting equilibrium hypothesis, substituting Ds the expression for p ~ obtained by the Gibbs equation (3) into the entropy equation (6b), applying the continuity (6a) and the internal energy balance equation: VT + X__0v:(VV)So} s• = ~l { (p+n)V.V-jq.-'~-" jq= T Js

(7a) (7b)

The entropy production (7a) is a bilinear form of scalar, vectorial and tensorial generalized fluxes and forces (or affinities)• The closure of the system of field equations is then obtained when the generalized fluxes are expressed in terms of generalized forces (phenomenological r e l a t i o n s ) . - T h e basic postulates of classical irreversible thermodynamics lead to linear constitutive relations: ~ v = 2g(V-~ 0 ; p+~ = ~t2 V.V

J q = - ~'VT

(8a,b) (8c)

with ~., ~t, la 2 the thermal conductivity, the shear and bulk viscosity coefficients, respectively. The (8a,b) are the well known Stokes and Fourier's laws. For multicomponent gas mixtures, when there are two or more fluxes of the same tensorial order, the Curie's postulate introduces so-called cross-coupling

effects (e.g. Soret and Dufour). Moreover, the domain of validity of classical phenomenological thermodynamics is limited to the domain of validity of linear constitutive relations. More generally, generalized fluxes may depend on the thermodynamic state of the system and on the generalized forces. However, the crucial problem is : how general this dependence may be while remaining within the limits of validity of the expression (7a) for the entropy production? (i.e. within the limits of validity of the shifting equilibrium hypothesis?). The kinetic theory shows that the expression (7a) holds as long as linear phenomenological

201

relations hold (Navier-Stokes approximation of the Chapman-Enskog method, linear functional dependence between fluxes and corresponding affinities)• Furthermore, the (7a) is based on the "local equilibrium" assumption, an hypothesis that falls down far from equilibrium conditions. Therefore, a bridge between phenomenological and statistical thermodynamics may be obtained only extending and generalizing the classical principles of macroscopic thermodynamics outside the limits of validity of linear constitutive equations and of the shifting equilibrium hypothesis. This will be addressed in future works• 3. Non-dimensional field eouations and boundary conditions 3.1 Generalities Initially Burnett equations were written to extend the range of validity of gasdynamics equations towards higher Knudsen numbers (rarefied gasdynamics) at which Navier-Stokes equations are no longer valid ~2°). However, it was shown that, in the general case, the series (1) in powers of (Kn) do not converge in the usual sense and asymptotic convergence is guaranteed only for K n ~ 0 . This is why interest in Burnett's equations has waned considerably and it has been suggested that these equations are of no practical interest• In this paragraph we shall consider some classes of continuum flows ( K n ~ 0 ) which cannot be described by N a v i e r - S t o k e s fluidynamics, because some Burnett terms are of the same order of magnitude of convective and (viscous) diffusive terms, so that they need to be allowed for to leading order. The Knudsen number (Kn) may be represented in terms of Math (M) and Reynolds (Re) numbers: Kn=M/Re. Continuum flows (Kn~0) occur in any case in two limits: at fixed Mach number, with R e~, (Euler approximation) and at fixed Reynolds with M ~ 0 (slow flows)• We are interested in the latter situation. However, specification of (M) and (Re) requires specification of the reference velocity V r. If Vr is equal to a physical velocity Vp (Vp represents, for instance, an external imposed velocity in problems of forced convection, a buoyant velocity in problems of natural convection, etc.) we focuse our attention on continuum flows characterized by: V M= ~ar ~ 0

,"

M Re = ~ < O(1)

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where a r is the laplacian speed in the reference state. Denoting by e T the characteristic difference over the characteristic

in this case Navier-Stokes equations along with no-slip boundary conditions are no longer adequate and Burnett terms must be retained at leading order. This conclusion apply afortiori when the only relevant physical speed is the characteristic velocity associated to thermal stresses (e.g. in free convection under zerogravity conditions).

temperature temperature,

(AT)r the relative measure of Burnett eT = T r ' stresses, corresponding to temperature gradients, and diffusive (Navier-Stokes) terms, is

given

by R~ee

(see

the

3.2 Fi¢l.0 eauations

non-dimensional

To perform a correct non-dimcnsionalization of the field equations, no a-priori assumptions must be made about the orders of magnitude of the reference quantities: indeed, the appropriate choice for these quantities, among the several possible, depend on the particular problem under study (i.e. on geometry, fluid properties and boundary conditions) and must be determined via a rigourous order of magnitude analysis. In this paragraph we shall refer to generic reference quantities (denoted by a subscript r). The appropriate choice of these quatities will be obtained as a result of the order of magnitude analysis performed in the next paragraphs, where reference velocities and scale factors will be determined, for each class of convection and flow regime, depending on the orders of magnitude of the relevant characteristic numbers (21'22). The continuity, momentum and energy equations, the latter one expressed in terms of total enthalpy, are:

analysis performed in the next paragraph). When boundary conditions do not introduce a characteristic imposed temperature difference, temperature changes are defined by the conversion of kinetic into internal energy: (aT)r = O(M 2) M2 ~0 Tr (AT) In this case e T -_ LT r r <<1 and Burnett terms corresponding to temperature gradients in the momentum equation are negligible. When boundary conditions introduce a characteristic imposed temperature difference (ar)~ ; i f R e = O ( ~ T) Burnett terms in the eT = T r momentum equation and some terms in the boundary condition for the Knudsen layer (see below), associated with temperature gradients, are of the same order of magnitude of the convective and diffusive (Navier-Stokes) terms;

T (9a)

CONTINUITY

v.y_ffi-~ p~ot (9b)

MOMENTUM

9 Dt *VpffiV' { 2~(VY.fl 0.

~V'X (V~Oo÷~~

Y~O" 2 ( (VW.~VW'~'/÷ ~0/o) ~o~Cp',~VVT~ -'0~÷~--~ p (VpVT~' x -o * .. - -

(9c)

F,NF,RGY

PD (h + -~"-+ V2 ¥,-~t = V.[ XVT-p_y.• 2 p.(VY..)o.V]+V. {-~ [cp01V.VVT+ talV.Y_(VY..)O..Y.• cpO2(D VT-(V..Y.)'VT)+ T s s T ÷tai(~-~Vy..)O,V... 2 ((vyv),(vy..)o)S0,V) + Cp03p(Vy_) 0. Vp÷ ¢%Cp(VVT)o'Y_+ CpO4 V'(V~.)o+

a) c s V + ¢pO,(V.V_)0. VT + ~ T (V'IWT)o'V+ %(V~')0"(vY')0'3/] • "~(VpVT)o. s s s } where ~¢ denotes

(vwr)~ ÷

the potential e n e r g y

(g=- V ~ ) and h = e + p~ is

the thermodynamicenthalpy.

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The above set is closed by assuming for the fluid the thermodynamic model of perfect gas: h=hr+Cp(T-Tr)

; p=pRT

(10a,b)

Furthermore, for the viscosity and thermal conductivity, we assume the following dependence upon the temperature:

203

It is convenient to summarize (see Table I) the coefficients of the terms entering the balance equations. The physical meaning of the nondimensional characteristic numbers are illustrated in Table 2. Z Q

" '

o_

By denoting with a star all non-dimensional variables we assume: p = p r +(Ap)rp+ • p = p r + ~rVr p+ , Lr

(12)

T = T r + (AT)r T + ; h --" hr + Cpr(AT)r h +

(13)

>~,1

I

rY

with Pr = Pr ar2 ; hr = CprTr

The reference quantities (AP)rand (AT)rare related by the thermodynamic state equation (10b). For the transport coefficients we assume tt=ttr~t+, X=Xr~.+ whereas the dimensional time, velocity and nabla operator are expressed as: t=trt + ; V = L rI__.V+ ; V = V r _ _V+

8

(14a,b,c) a

In the (14b,c) we assumed that the flow field is isotropic, i.e. reference lengths and velocities arc the same in all field directions. In this way we do not consider, in the present formulation, anisotropic effects associated to dissipative layers, which correspond to different appropriate reference quantities for the components of vectors and tensors in direction normal and parallel to the interface. In this paragraph, in fact, we are mainly interested to obtain a correct non-dimensional form of the field equations, in order to compare the correct measure of Burnett stresses to that of classical Navier-Stokes terms. However, the limitation of isotropic reference quantities will be removed later on, when free convection induced by buoyancy and thermal stresses will bc analyzed via a rigourous order of magnitude analysis of the pertinent set of field equations. For sake of c o n c i s e n e s s the long and

cumbersome expressions for the nondimensional field equations have been reported in Appendix A. The stars have been omitted, so that all variables denote non-dimensional quantities. In Appendix A have been also reported the definitions of the various nondimensional characteristic numbers.

0

:z

xOR3~I-~

Table 2

i i St

i

.....

~rvscnoN UNSl"BADZ~.SS

.

.

.

.

.

.

.

.

i M 2 i

CONVECTION OF M O M E N T U M _ _ REVERSIBLE DIFFUSION OF M O M E N T U M ~PRESSURE~

i i Re

CONVBCTION OF M O M E N T U M LINEARIRRBVBR$IDLBDIFPU$1ON,O[ZM()MENTUM~N-S)

i Fr

CONVECTIONOF M O M E N T U M PROD,UC'rlONO F,MOMBN'I~JM DUB TO GRAVITY FIELD

'I l

I

I

! ] .

PRO~)UC'IIONO F M O M E N T U M DUE TO BUOYANCY

O

L ~m~R mRBVEISla~ ~l~.s. OE..O.~..U...O.~ ~.,..T~..K.LN..-..S2

Pc

CO~VECrlON OF ENEROY UNBARIREEVEREIBLBDIFFUSIONOF INT. ENRROYfPOUEIER)

F.

~ O Y D[,~[FAT1ON L],I~AR IREBVEESIELEDIFFUSIONOF INT. ENERGYIFOURIER)

CONVECTION OF VOLUME PRODUCTION OF VOLUME (DIVRROENCE OF V ELOCIT=Y) F'T R(:

NON,UNEARDIFFUSION0P M O M E N T U M (THEJtMALEURN]B'i~frfRBSSRS) LINEARIRREVERSIBLEDIFFUSIONOF MOMENTUMIN4)

M_~2

NON-LINEAg DIFPU$1ON OF M O M E N T U ~ [VBLocrrY B U l t l ~ STatUSES)

Re L NEAR IRREVBR$IBLSDIFFUSION OF M O M E N T U M (N4) !.................................................................................................................................................... ii -M- 2 Rc2

NON-UNBAR DIPPU$|ONOF MOMENTUM(~tBHUR~BUItNETT r/1J~$U) LINEARIRRI~VR]t$1BLZDIPFUEIONOF MOMENTUMIN-S)

M 2Pr i-'~e ! I M4Pr !~

NON.LINEARDI]~U$1ONOF INT. ]Lq~OY ~'H]RRMALBURN]rI'TglltESERSI LINBAItIIItEVRItSIBLBDIFFUSIONOF INT. BNBROY(POUE/I~) NON.I~INBARDIFFUSIONOF IUNBTICBNBIOYfVRLOClTYEURNETTSTR~qSBE) L|NEAIt IRRBVRitSO~[J~DIFFU$1ONOF INT. BNERGY (FOURIER)

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of (15a,b), we have:

3.3 Slio boundary conditions. In conventional Navier-Stokes fluidynamics, no-slip conditions are accounted for, i.e. gas v e l o c i t y and temperature, on the solid boundaries, are assumed to be continuous. M a x w e l l (13) was the first to show, from the kinetic point of view, that these conditions are approximate and in some cases "slip" and "temperature jump" may be relevant. In fact, the fluidynamic equations, in their different orders of approximation, hold only at distances considerably greater that the free-path length (i.e. outside the Knudsen layer, where the full Boltzmann equations should be considered), More accurate boundary conditions for temperature and velocity have been formulated by introducing fictitious slip velocity and temperature jump on solid walls : V

3Lt (n=0) = ~rr cl n. (Vy_.t) + c 2 v r ~TslnT

( 1 5a)

5T (n=0) =

(15b)

c~Z'r PrCprar

n.VT -

Here 5T denotes the temperature jump between solid wall a n d gas; V s is the surface nabla operator, v the kinematic viscosity The nondimensional coefficients cl, c 2, c 3 depend on thermodynamic state and interaction laws of molecules with the wall. These boundary conditions ensure that the solution of Navier-Stokes equations, outside the Knudsen layer, differs from that of the Boltzmann equation by orders of Kn 2, and inside the layer by O(Kn). Since Navier-Stokes equations require K n - ~ 0 , when dealing with continuum flows the above slip boundary conditions have been never accounted for. Nevertheless, there are some continuum flows for which gas temperature and velocity on solid walls are not continuous, and the corresponding phenomenologies may be completely described only considering appropriate "slip" boundary conditions (radiometric effect, thermophoresis, diffusiophoresis). In the above mentioned case of slow nonisothermal

flows

Re T r = 0 ( 1 )

surface

temperature gradients may induce a slip velocity of the order of the "viscous" velocity, which can become an important source of convection (thermal creep convection). In fact, considering the non-dimensional form

a 1, VIT V_.t (n=0) = c I Kn n. (VV.t)+ c2 Re I+eTT Kn

8T (n=0) = c 3 P r II'VT The first and (17) term in eT=O(Re)

(16)

(17)

terms in the right hand sides of (16) are negligible for Kn-o0. The second (16) becomes relevant in the case (thermal creep driving action).

4. Order of magnitude analysis 4.1 Problem

formulation

Starting from the non-dimensional equations formulated in the previous, paragraph, we shall perform an order of magnitude analysis for a problem of free convection induced in nonisothermal gases either by buoyant forces or by thermal stresses and thermal creep. Buoyant forces and thermal stresses are "volume" driving actions, whereas thermal creep may be induced by slip effects on the boundaries, caused by surface gradients of temperature, and can be interpreied as a "surface" driving action. Generally speaking, the characteristic temperature differences associated with volume and surface actions may be different, so that we shall use in this formulation a temperature difference AT in the pertinent expression for the buoyancy and thermal stress measures, and a temperature difference (AT) s related to surface gradients of temperature on the boundary, in the expression of the thermal creep driving action. In the case of isothermal surfaces (AT)s=0, i.e. thermal creep effects are negligible; in other problems where non-uniformly heated surfaces are present ATs~0 and in particular one may have ATs=AT. Considering for instance a vertical plate heated at temperature T w(x) greater then the surrounding gas temperature T a (see Fig.la), there are two characteristic temperature differences: A T = T w - T a which induces buoyant forces and thermal stresses, and (AT) s , related to the surface gradient of Tw(X) which drives slip velocities on the plate. In the case of plane or axially-symmetric enclosures with uniformly heated end wall (see Fig. l b), the only characteristic temperature

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difference is AT= (AT)s = T2-T l where T t and T 2 > Tt are the temperature values on the vertical side walls.

IV T w

I£

Tw(x)

L

TA

(a)

Congress

205

We emphasize that the difference between equations (18a,b,c) and the corresponding equations in the Navier-Stokes approximation consists in the presence of the last two terms on the right hand side of momentum equation (1 8b) (thermal stresses). To. evaluate the order of magnitude of the driving term associated to thermal stresses, it is convenient to rearrange the momentum equation (18b) combining the gradient part of Navier-Stokes and Burnett terms (see Appendix B). Accounting for eqns. (I1), the (18a,b,c) become:

GAS

~p

V.V = V__.Vp -1+pep Y

Re (l+pep) V.VV+ VII + K 1 ~:~Re(I+TeT)Z'I V.(LVT) VT =

AT=T-T A : (AT)~Tw y

(19a)

= V. [2~t(V_~'] + Gp i 8 - Srs (l+T~r) 2z'2 [(VT) 2 VT] (19b)

~-g

(b)

(19c)

Pe (l+pep) V . V T = V. [XVT]

where: TI

GAS AT - (AT),--

i

Tz > Tl

- I"i

We assume: the motion is slow (M2<>l); hydrostatic gravitational effects and viscous dissipation are negligible (Fr>>l; E < < I ) . The gravity vector is constant in magnitude and direction. Under these hypotheses, the non-dimensional balance equations (Appendix A) are given by: E:p l+pep V.Vp

I.t2 V2T +

(20)

+ K 3 el'2 (l+TeT)2Z-I (VT) 2 Re Vv STS= K 2 ~T3 ~-- ; f

Fig. 1 Geometrical configurations and corresponding volume and surface temperature differences

V.~_= -

2 2 ar FI = p + ~ t.tV.V_+~ co3 ~

(18a,b,c)

Re (l+pep) V_.VV + Vp = V. [2p.(VV)o ] + Gp ig +

The non-dimensional constants K i (i=1,3), order of magnitude one, are defined by:

of

K 1 = co5 -2co3z; K 2 = o)3z - to5/2; K 3 = co3z + 0)5/2. The quantity 1-I can be considered as a generalized pressure, whose structure is known only when, a part from the thermodynamic pressure, the contributions associated to divergence of velocity and thermal stresses are calculated. Last term in equation (19b) can be interpreted as an external force proportional to [VT[3, direct opposite to the temperature gradient. In the case of non-uniform temperature distributions on the wall, there is a further driving action due to thermal creep. The boundary condition (16), by dropping the terms in Kn, reduces to: V T

Pe (l+pep) V.VT : V. [~.VT]

V---t(n=0)= STC I+ETT

(21)

206

42rid IAF Congress

with STC =

c2(AT) ~ Vv (AT) e'r ~rr ;

4.2 ~haracteristic numbers

soeeds

4.3 Determination of reference ouantities and flow re~,imes and

non-dimensional

In the eqns (19), (21) there appear following non-dimensional numbers: Re= Vr V~

G=

Pe = V_~ Va

the

(22a,b)

_

Vr

~ T S - Vr

(23a,b,c) STC

VT~ = Vr

defined in terms of the characteristic speeds: Vv= gr PrLr

Va= a_L

Vg = g(Ap)rLr2 ~r

K2(AT)3 lar VTS= Tr----T~ P rL r = K2 ~T3 Vv

Lr

(24a,b)

(25a,b,c) VTC =

c~(AT)~ i.tr c2(AT)s Tr PrLr = (AT) e'l" Vv

and the still unknown reference velocity V r. The velocities (24a,b) are the "resistivity", momentum and energy, speeds; the (25a,b,c) are "driving" speeds. The velocity (25a) is the well known buoyant speed (volume driving). The velocities (25b,c) can be interpreted respectively as volume driving (thermal stress speed) and surface driving (thermal creep speed). In the same way the characteristic numbers (22a,b), Reynolds and Peclet, are rensponse numbers, whereas the (23a,b,c) are the driving numbers of the problem: the (23a,b) are "volume driving" numbers, the (23c) is a "surface driving" number. By differentiating the state equation (10b) and (Ap) r M 2 considering that Pr - Re <<1, it follows that ep = O(eT). In particular we assume APt = Pr 13T AT with J3T the coefficient of thermal expansion, so that, for the buoyant speed (25a) we obtain: Vg=

glBT~TLr2 vr

To perform a rigourous and correct order of magnitude analysis, the following facts must be reckoned. a) The set of reference quantities cannot be chosen arbitrarely a-priori but must be determined, in terms of problem's data only, from a rigourous analysis of the field equations, by imposing general physical constraints in order to include all possible alternatives and to consider, consequently, all possible classes of flow regimes. b) When dissipative layers are present, the flow field is not isotropic. A physically correct nondimensional form of the field equations must exhibit this anisotropy so that, in particular, reference quantities for velocity and length in directions normal and parallel to these layers must be allowed to be of different orders of magnitude (anisotropic reference quantities for vectors and tensors). Therefore, we remove the limitation reference velocities and lengths, by (Lrn, L) and (Vrn, Vr) as reference velocities in directions normal and the main flow direction. Due to this

of isotropic introducing lengths and parallel to assumption:

1) the ratios (Lrn/L) and (Vrn/Vr) are of the same order of magnitude, upon the continuity equation. This common order of magnitude will be referred to as "normal scale factor" and denoted by (I). The scale factor (I), and the reference velocity V r are unknown. 2) the introduction of different reference lengths and velocities leads to different orders of magnitude of the various terms of the field equations corresponding to normal and tangential directions. In particular, the leading order of magnitude of convective terms, in the momentum equation, is Rel 2, instead of Re, and in the energy equation is Pel 2 , instead of Pe; the correct non-dimensional measure of buoyant forces, in the momentum equation, is G In, instead of G, where n=2,3 according to whether the gravity vector is parallel or normal to the main flow direction C23). According to N a p o l i t a n o (22) the relevant diffusive speed must be determined as: V D = min { V v , V t, }. The i'elative importance of

42rid lAP"Cotlgrc~a

V v, V a depends on the order of magnitude of the Prandtl number. Since for gases Pr=O(l) they are of the same order of magnitude and we can assume, for instance, V D = Vv . The reference velocity and the normal scale factor are obtained from the following constrained maximum problem:

v

ax{V ,n

V c}

( 6a>

Vr 12< O(1) VV

(26b)

I ~ 0(1)

(26c)

207

determines the prevailing flow regime. For thermal stress or creep convection we have diffusive or c o n v e c t i v e - d i f f u s i v e regime according to eT < O(I) or eT = O(1). In the case of natural convection the scale factor must be determined according to the value of the number Reg. It is equal to 1 for Res O(1) it follows the well known l/(n+2) power law (natural boundary layers).

4.3 In the particular case AT=(AT)s we have:

Combining the relevant driving velocities, we obtain two independent non-dimensional numbers, function of the problem's data only: _ Vs ffi ~TTrgL3 (AT) G'rc - V.rc c2vr (AT), (27a,b) S = VT~ K~(AT) ~2 VTc = C2(AT)s The first is a conditional Grashof number based on the thermal creep velocity; the second measures the relative importance of thermal stress and creep effects. The possible alternatives for the orders of magnitude of G T ¢ , Sand G T s = G T C / S determine the type of prevailing convection and flow regime. These alternatives are shown in Tab 3, together with the corresponding values of the reference velocity Vr. For each class of convection (natural: Vr = Vgl n thermal stress: Vr = VTS; thermal creep: Vr=VTc ) V the relevant transport number [ R e g -- - - g - . Vv VTS ReTS = Vv = K2 8T3 " ReTC = VT¢ = c~(AT)v eT] ' Vv (AT) - - -

GTC ffi

[~TTrgL3 C2Vr 2

S = ~ &r2 C2

The number G Tc (and then the relative importance of buoyancy and creep effects) does not depend on the imposed temperature difference. For ideal gases ~)TTr= O(I), so that GTC= O I..~2~I

~,v, )

and it is possible to define a characteristic length L*=

for which GTC ffi O(1) (i.e.

buoyancy and thermal creep effects are of the same order of magnitude). For L/L* < O(1) thermal creep convection prevails; for L/L* > O(1) buoyancy effects are dominant (natural convection). The characteristic length (L*) depends only on the gravity level (g) and the transport properties (v). In particular it increases for decreasing gravity level (microgravity conditions) and for increasing kinematic viscosity (e.g. at low partial pressures typical of vapour crystal growth processes). For a given substance the nirio between the length L*, at a given gravity level (g) and the

Table 3 Grc ~0(1)

S SO(1)

>0(1) < O(1) >O(I)

M 28-0

> 0(1)

GTS = Grc/S ~0(1)

Vr

TYPEOFFREECONVECTION

Vzc

THERMALCREEPCONVECTION

sO(l)

VTs

THERMALSTRY~SCONVECTION

>0(1)

Vs I n

NATURALCONVECTION

~0(1)

VTs

THERMALS'[RF~SCONVECTION

>0(1)

208

42rid IAF Congress

corresponding length L 0 in conditions decreases as:

normal

gravity

L._=i* (.g_~-113 LO* =

The possible flow regimes, in this case, are classified according to the values of G TS and Reg, ReTs . They are shown in Table 4, together with the corresponding values of normal scale factor and reference velocity. In the same table are also shown the controlling inequalities, in terms of the relevant driving and diffusion speeds, that determine the class of convection (natural, thermal creep, combined) and the flow regime (convective, diffusive, c o n v e c t i v e diffusive). stress effects are For Vg _> V TS thermal negligible: we fall within the case of natural convection, described by conventional NavierStokes fluidynamics. The controlling transport parameter is the conditional Reynolds number V ~ Diffusive (so called Stokes) regime, Reg = - V

(28)

Lgo)

The other relevant non-dimensional

number is

S = 0(%2). Since ST< O(1), in the case A T = ( A T ) s the thermal stress speed (VTs) can be never greater than the thermal creep speed ( V T c ) a n d for % < O ( I ) thermal stress effects are negligible. 4.4 Thermal stress and buovancv free convection Let consider now the case of isothermal surfaces [ S > O(1); GTC> O(1)]. Thermal creep effects are negligible, so that the only driving actions are due to buoyant forces and thermal stresses (volume driving actions). The relative importance of these forces is measured by the V non-dimensional parameter GTS = G T C / S = VTS = 13TTrgL 3 K2Vr2 ET_-2 which can be interpreted as a

convective-diffusive (Navier-Stokes) or natural boundary layer regime occur according to whether the Reynolds number is of order of magnitude less, equal or greater than one. In these cases, the normal scale factor is equal to 1 (for Stokes and Navier-Stokes regimes), or follows the well known l/(n+2) power law (natural boundary layes). The reference velocity is equal to the buoyant velocity (Vg) for Stokes and Navier-Stokeg regimes, whereas in the case of natural boundary layer regime, it When the is equal to Vg 2 / ( n + 2 ) V vn / ( n + 2 )

conditional Grashof number based on the thermal stress velocity VTS. Similarly to the previous case, we can introduce

thermal stress velocity VTS is greater than or equal to the buoyant velocity, a new type of free convection arise. In this case the reference velocity is equal to VTS = K 2 V vST 3, and the controlling transport number is equal to

(V 2M/3

a characteristic length

t.= I'-~-r \ g ) | ~T2/3 for which GTS = O(I) (buoyancy and thermal stress effects are of the same order of magnitude). For

L/L

<

O(I)

thermal

stress

convection ReTS = VTS=vv K 2 ET3 so that convective terms in

prevails; for L/L > O(1) buoyancy effects are dominant.

Table 4 BASIC INEQUALITIES V

Scale factor I

Vr

FLOW REGIME

C~ > O(1)

Sel= ~v < O(1)

1

Vs

DIFFUSIVE (STOKES)

(Vs> VTS)

ReI= O(1)

1

V|

CONVECTIVE-DIFFUSIVE (NAVIER-STOKES)

~ g~ l/(n+2)

ReI > 0(1) V O.r$< O(1'~ Rtrs=L~v
(V, s V.m)

R~rs= O(I)

I

V BI"

VTs V.m

NATURAL BOUNDARYLAYER

THERMAL STRESS DIFFUSIVE THERMAL STRESSCONYECIIYE-DIFFUSIVE

42nd lAF Consre~r

209

reported in Tab.5. In Fig. 3 the corresponding points have been reported in the plane (Reg, GTC): we see that experimental points fall in different flow regimes (natural boundary layers, Navier-Stokes, Stokes; diffusive and convective-diffusive thermal stress regimes), depending on the appropriate combination of characteristic dimension (L), kinematic viscosity (v) and gravity level (g/g0)" As analyzed before, thermal stress convection prevails at low characteristic flow dimension (small ampoules) for fixed gas properties and gravity level; at high kinematic viscosities (dilute vapours) for fixed characteristic length and gravity level; at low-gravity level (microgravity conditions) for fixed (v) and (L). Space experiments (Cadoret, SLI; Van den Berg, SL3) are characterized by very low values of the non-dimensional numbers Grc (10"2+10 "3) and Reg(10"3÷10 "5) so that thermal stress and slip effects may be relevant under microgravity conditions.

the momentum and energy equations are negligible for eT < O(1) (diffusive regime) or they are of the same order of diffusive terms for e T = O(1) (convective-diffusive regime). The map of flow regimes in the plane (Reg, GTs) is represented in Fig. 2. 4.5 AnDlication to vanour crystal 2rowth

exoeriments Typical set of vapour transport properties and experimental conditions can be found in Kaldis, Cadoret and Shoner (]).

The orders of magnitude of the characteristic lengths (L*, L ) and of the non-dimensional numbers (GTC, S, GTS, Re8) defined in the previous paragraphs, have been evaluated for several experiments of crystal growth from vapour, e.g. physical vapour and chemical vapour transport, either during Earth-gravity or under microgravity conditions. They are

GTSl

DIFFUSIVE J ~ 0

4"~BOUNDARYLAYERS

NATURAL BOUNDARY LAYERS 2 STOKES

REGIME

V=V rg

n

I

Vr = Vn'2g V: "? Io

[=1

[ " Regn-2

I

i Q w

n° 2

G ~ ~.e~ THERHAL STRESS AND IO -4

IO-3

10-2

iO-)

16

16 2

10 3

THERHAL 5TRESS DIFFUSIVE REGIME I Vr= VTS

\

\

VT5 V

/=l

v

.~.1i0 -2 I I t I 1.,,~i 0 "3

Fig. 2

Regimes plane of buoyancy and thermal stress convection

10 4

l0 s

_

RUg

42rid lAY Consreaa

210

Table 5

SUBSTANCE

g/g0

Tr(K)

AT L(cm)L°(cm)[,(mm)fL'~C~-';J /

S(~';j2

Re g

[

H$I;

1

393

10

10

1.5

1.5

O(102)

O(10 "2)

0(I)

Omaly et AL (1983)

HS11

1

375

1

10

1.5

0.2

O(102)

O(10 -4)

O(10 "t)

Kobayuehl (1983)

HgI I

1

375

6

20

13

0.9

O(103 )

O(10 d )

0(10)

Ctdoret (1985)

H811

10 -4

385

1

20

30

6

O(10 "1)

O(10 -4)

O(10-4)

Vm den I~ 8 (1985)

Hglg

10 .4

380

45

8.5

30

70

O(10 .2)

O(10 "2)

0(i0 "3)

Sl~ab~ It

AI. (1974)

Ge

1

460

80

22

0.5

1.5

O{105)

O(10 "t)

O~ 104 )

MilieU, ZappoU 0983)

Ge

10 .4

760

44

lO

I0

15

0(I)

0(10 .2)

0(10 -2)

~,

CuPc

1 1

600

330

7.5

1.8 ~.3

12 5

O(10) 0(104)

O(1) 0(]01)

oct)

** ~ . t l . 7 I

I~C12

Siall]a It AL (19119)

250

50 20

0(103 )

Grc 1o

4

1o31 e

IOmtly 1983,11112 i/10. 1 • I

Koblylaclatl~3.11|ll

s/so-I [

L0 eScoeber1974, H|lz|/|O. 1

II LaunayLqS2, "Ge

|/1%'1

I

@l~be et al. 1987.¢uPcl I/i 0- II lO i Miguola.~ppoli [983, DI Ge I @ I 10 -4

I 10-3

I 10 -z

I 10 -]

1

I

I

I

10

10 z

103

10"1

I

"

Cadoret 1911~.SLI 111112

I 10 "z

J

vtn den Beril@19$6.SL3 Kil z I

.10-3

Fig. 3

Experimental points in the regimes plane

I

10 (

I

105

v

Re

42rid IAF Congreu

211

5. Thermal stress and buoyancy convection between non-isothermai oarallel walls

y~

In this section we shall consider a simplified geometrical configuration to give a first quantitative analysis of thermal stress and buoyancy effects.

T-x GAS

5.1 Mathematical formulation T-0 The geometrical configuration is shown in Fig. 4. We consider two non-uniformly heated parallel plates, with the gravity vector acting normally to the plates. The bottom wall has uniform temperature (T=0) whereas the temperature of the top wall increases linearly with the abscissa (T=x). For the density, viscosity and thermal conductivity we assume a linear dependence upon the temperature (p = - T; ~ = I~ =1 + eT T). The non-dimensional field equations are obtained from the (19a,b,c), in cartesian coordinates, under the shallow geometry approximation: Ux + Vy = 0

Fig. 4 Geometry of the problem

After the position ¢~= (1 + eT T) Uy, combining the y-derivative of (29b) and the x-derivative of (29c), in order to eliminate I'l, we obtain:

~yY=- 4

e~y ey 2 G £Tex 6T3(I+eT20)3/2 + 2 (l+eT20) 1/2 (32)

The solution of (32) is:

(29a) , = STS_~S (1 +6T20)1/2 + 2 G eT30y2 O~ (l+eT20) 3/2

I'lx = (1 + vTT) Uyy + ~rTy Uy - STS Ty2 Tx (29b) Fly = G T - STS Ty3

(29c)

(1 + ~r T) Tyy + f.TTy2 =0

(29d)

Here, u and v denote velocity components along x and y, respectively. The energy (29d) and momentum (29b,c) equations are uncoupled. For the temperature we find the solution:

T=

-1+'~ 1+~.T2O(x,y,e.T)

where O(x, y, eT) = xy (x

4 0~ U= STS'-~ " y + ~" G £T5Oy3 (I+~T2O)3/2 + (34)

2 (I+~T20)3/2"~ 2 +

FI(x' eT) y'~/l +ET20 - -+ eT2ey 3

temperature

/

Fi(x, eT ) must be determined by

1) the boundary conditions at y=0 and y=l: u(x,0)=u(x,l)=O

T(x, y, e t a 0 ) = xy

~T 0y

2F2(x, £T) ~/I+F1"20 + F3(x' ~T) + ET20y The functions imposing:

the

(33)

The x-component of velocity is given by:

(30)

+ ~)

In the limit case 6T---~O distribution is given by:

+ FI(X, %) y + F2(x, eT)

for no-slip bot~ndary conditions

(31 ) 2) the integral condition

The temperature profiles, at given abscissa x, for different e T , and viceversa, have been reported in Figs. 5, 6 and 7. We see that these profiles are linear for e T a 0 , whereas typical non-linear behaviours appear for eTa0 an d increase versus eT.

I

I udy = 0 0 By considering the limit for e T ~ 0 (constant transport properties), the (34) become:

212

42nd IAF Congress

U(X, y, £T.-)0) = - STS x2y

- y- + ~ +

(35)

5.2 Discussion ,Of results The velocity profiles corresponding to thermal stress convection (S.rs=l, G=O) have been reported at x=l, for different values of the parameter s.r (see Fig. 8). For ST=0 (formula 35) the velocity profile is symmetric, and the motion is directed opposite to the thermal gradient. Increasing ca.(see the corresponding temperature : profiles of Fig.5) the velocity decreases (due to corresponding increase of viscosity) and the symmetry is destroyed. Figs. 9 and 10 show the profiles of (u) for different values of the abscissa (x), for eT=0 and eT= 1, respectively. The velocity increases versus x, but, as seen before, in the case eT=0 the profile is symmetric, whereas for eT=l higher velocities occur near the bottom wall. In Fig. 11 are reported velocity profiles corresponding to the case of combined thermal stress and buoyancy convection, for different values of the ratio GTS = G/STs. The limiting cases G T s = 0 (thermal stress convection) and GTS ~ * * (natural convection) correspond to opposite directions of driving actions. In the case of natural convection the flow near the bottom wall is directed along the positive direction of x, whereas for thermal stress convection it has opposite direction. For 0 < GTS < .o we have velocity profiles which share some features typical of purely natural or thermal stress convection. For small values of G,rs the profiles are similar to the case G T S = 0 and viceversa, for GTS >>1. For GTS -= 3 the velocity is very low, due to the opposing effects corresponding to thermal stress and buoyant forces.

6 . ~ A fluid-dynamic model of vapour crystal growth has been formulated which includes non-linear irreversible thermodynamic effects. Attention has been focused on the correct formulation of boundary conditions on nonisothermal side-walls (thermal creep effects) and on the role played by thermal stresses in the vapour fluid-dynamics. Both these effects can become important sources of convection

under certain conditions that this work has pointed out. An important contribution of this paper is represented by the non-dimensional order of magnitude analysis: the 3 characteristic velocities, corresponding to buoyant, thermal stresses and creep effects lead to the identification of 3 characteristic nondimensional numbers, functions of problem's data only, which give the measure of the relative importance of two driving actions (GTc, S, GTS). The possible alternatives for the orders of magnitude of these numbers have been considered, and the corresponding classes of convection and flow regimes have been identified and discussed. Characteristic lengths (L*, L ) at which buoyant and thermal creep, and buoyant and thermal stress effects are of the same order of magnitude, have been defined, and their dependence upon fluid properties and gravity level have been discussed. Their orders of magnitude, together with the numbers ( G - r e , S, GTS) and the transport number Re s, have been evaluated for several experiments of vapour crystal growth, on Earh and under low-gravity. In particular this analysis pointed out that thermal stress and creep effects may exert an important role under microgravity conditions. A first comparison between thermal stress and natural convection has been presented considering the flow between non-uniformly heated parallel plates. Due to the originality of the phenomena considered, the work has been focused on free convection of a monocomponent gas or vapour. As already remarked in the introduction, the fluid-dynamics of vapour growth usually involves many other complex effects. Generally one has gas mixtures with non-uniform distributions of temperature and mass species concentrations and imposed longitudinal mass flow (so called Stephan-Nusselt flow) due to evaporation at source and condensation on the crystal. In the case of pluricomponent gas mixtures, analogous phenomena which arise from concentration stresses of the Burnett tensor, and from diffusion slip on solid walls should be considered. Other problems to be addressed include cross coupling effects between diffusive fluxes of internal energy and mass concentrations (Soret and Dufour), bulk and surface chemical reactions, either in equilibrium or in nonequilibrium, impurities models and a rigourous modelling of phase changes at source and crystal.

42nd IAF Congress

213

TEMPERATURE PROFILES

TEMPERATURE

x=l 1.00

-

. . . . . . . . . . . . ., . . . . . . .I . . . .~ . . . . .

T

I I

I I

I

I

I

o.so

.

.

.

.

.

.

.

.

.

.

.

/

2.00

. . . . .

I

II I

T

1

,

. . . . . . . . .

, 0.40

.....

L

- ~ :

~7/J

y,

L/ . . . . . . .

I

. . . . . . .

t_

. . . . . . .

....

l

-

:

:

~

,

,

I

I

I

U .....

L .....

I

I

I I

I I

I

0.20

.

.

.

0.00 ~ 0.00

.

. I

.

i

L 1

. . . . .

I-I

ji2nx__ 1

/ i

I

II I

. . . . .

i

L I

I

r

. . . . .

V

. . . . .

. . . . .

I

I

I

I

{- . . . .

-

I

I

'

'

I

I

I

I

I

I

I

I I

I I

I I

I I

L

. . . .

/'

L

.

.

.

'~ =

1.5' --

.

.

.

.

"

I I

I I

L

. . . . .

I

I

I

I

I

I

I

1

I

I

I

'

.

L I

1.00

. . . . .

I

I

I-

I I

(0,1,2,3,4) i

x

_ - - - - - ~ '

II I

I

I

I

.....

II I

I

I

, L

VARIOUS

I

....

L

. . . . .

ooo

AT

I

I

L

.

. . . . .

PROFILES

el = 1 . . . . . . . . . . . . . . . . . . . . . . . . . I I I

. . . . .

I

0.50

.

I I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

.

.

.

.

:

I

I

I I

I I

I I I

I

0,511 I

I

I

I

I

0.00 0.20

0.40

0.60

0.80

1.00

LO0

0.20

0.40

0.60

Y

Fig. 7

TEMPERATURE PROFILES AT VARIOUS x =

1.00

5/

Fig. 5

~T

0.80

THERMAL STRESS CONVECTION

0

VELOCITY

PROFILES x=l

2.00

.....

T

i .....

i .....

i .....

i- . . . . .

,

I I I I I

I I I I I

I I I I I

i I I

, I I I

I

1.5o

. . . . .

I

F

0.50

0.00

. . . . .

. . . . . .

~

0.00

P

.

.

.

.

.

I

I

I

I

I

I

I

,

,

I

I

I

I

/:

,

L

. . . . .

L

I

I

I

I

r

--

-r

. . . .

I I

I

I I

__

- -

', -L

. . . . .

I

I

I

r

. . . .

I I I I I

X=0.5

0.40

0.60

Y

Fig6.

0.80

/ .....

/ .....

I I I I I P

I I I I I

I I I I I

. . . . .

F

.

I I

I I

,

,

.

.

i .....

, I I I I I

.

.

.

.

I

I I

i

, ,

0.00

I

I

i .....

i

. . . . .

L-----

I

I

I

-0.01

. . . . .

r

. . . . .

i

I I I I I

-0.02

0.20

. . . . .

I I I

[- . . . . .

I t I

......

I 0.01

J; L _

I

. . . .

I

. ~ .

I

,

1.00

. . . . .

J/ I

_ / ~

I

F

. . . . .

0.02 ,,

1.00

.........

LO0

I .........

0.20

I .........

0.40

t ..................

0.60

y

Fig8.

0.80

1.00

214

42nd 1 0 Congress

THERMAL STRESS CONVECTION VELOCITY PROFILES AT VARIOUS X er = 0

THERMAL

STRESS ANO BUOYANCY FREE CONVECTION VELOCITY PROFILES FOR VARIOUS Gm tr=0

x=l

0.08

I

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Fig. 9 7. References O)Kaldis E., Cadoret R., Schonherr E. in "Fluid Science and Material Science in space" ed. THERMAL STRESS CONVECTION VELOCITY PROFILES Walter (1988) AT VARIOUS x cr = I (2)Napolitano L.G., Viviani A., Savino R. 2 n d Scientific Meeting of Low Temperature 0,04 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I I I Vapour Growth Facilities Users, Zurich I I I I I (1990) U I I I I (3)Klosse K., Ullersma P. J. Crystal Growth . . . . . . . . . . . . L . . . . . I 0.02 111,167 (1973) I I I I (4)Gill A.E. Journal Fluid Mechanics 26, 315 I I I I (1966) I I 0.00 (S)Markham B,L., Greenwell D.W., Rosenberger F. . . . . . . . . . . . L I i i J.Crystal Growth 51, 426 (1981) I I I I I I (6)Markham B.L., Rosenberger F. J. Crystal t I Growth 67, 241 (1984) _L . . . . . . . . . . . . . . . . . I -0.02 I I (7)Jhaveri B.S., Rosenberger F. J. Crystal Growth I I I I 57, 53 (1982) I I I I (S)Bontoux P., Roux B., Schiroky G.H., Markham L . . . . . . . . . . . . . . . . I -0.04 . . . . . . I I B.L., Rosenberger F. Int. J. Heat and Mass I I Transfer 29, 227 (1986) I I I I I (9)Extremet G.P., Roux B., Bontoux P., Elie F.J. I Crystal Growth 82, "761 (1987) -0.06 ........ I ......... I ......... I ......... I ......... I 0.00 0.20 0.40 0.60 0.80 1.00 (l°)Launay J.C.J.Crystal Growth 60,185 (1982) Y Ol)Mignon C., Zappoli B. Acta Astonautica 17, 1235 (1988) Fig. 10 O2)Mignon C., Zappoli B. J. Crystal Growth 94, 783 (1989)

42rid IAF Congress

(13)Maxwell J.C. in "The Scientific papers of James Clerk Maxwell" (Dover, New York, Vol. 2 pag. 681 (1962) (t4)Rosner D.E. Phys. Fluids A 1, 1761 (1989) (XS)De Groot S.R., Mazur P. "Non-equilibrium Thermodynamics", North-Holland, Amsterdam (1962) (16)Chapman S., Cowling T.G. "The Mathematical theory of non-uniform gases". Cambridge University Press (1970) (17)Hirchfelder I.O., Curtiss C.F., Bird R.B. "Molecular theory of gases and liquids". John Wiley, New York (1954)

215

(18)NapolitanoL.G.

Int. Syrup. on Fluid Dyn. Heterog. Multiph. Continuous Media, Naples

(1966) (19)Burnett D., Proc. Lond. Math. Soc. 39, 385 (1935) (2°)Napolitano L.G. "Thermodynamique des sistSmes composites" M6moire de Sciences physiques (Gauthier Villars) (1971) (21)Napolitano L.G. 2nd Levitch Conference, Washington (1978) (22)Napolitano L.G. Acta Astronauticag, 199 (1982) (23)Napolitano L.G., Viviani A., Savino R. 41st IAF Congress, Dresden (1990)

APPENDIX A NON-DIMENSIONALBALANCEEQUATIONS CONTINUITY

"

l+p~p

~'t +

V.V

p

MOMENTUM

~''fl k ~V~

• Re ( l + p ~ , ) I ~ .

)

+ V _ . v V + Vp : V.

[2.(vvt]-o u~'~

l+~e P

-~v. Re

e'r2 I Re V,

; M2 co3 (VVT)o +

l+~p la2 M2

l+~p

V. - -M2

~p

IM2

o, 4 ( VpVT)

+~-~e p

+

1

i .s

216

42rid IAF Congress

ENERGY

[1 ~ ( M2V2 M2 ) ( M2V2 F---~r/] M E~p • Pe(l+pep) ~ ~ h + --aT --2 + ~F-r ~/ + v . v h + --aT --2 + - St0t M2pr V'[~,VT]+EV'[-pV+2B(VV-.)o'V]__ -- + - - ~ e V. {

M 2l't2

M2

[01V.VVT_ +-~COlV'V(VV)So'V+_ _ _

l+~eP +gi g

0 2 V T + - - " ~ 2 ( V ~ S o "v + v . v r-.r -

02VT+ --,,2(VV);.V aT -- -

-2

I+eTTM 2 (vV)So'VP+ m3 (VVT)So"V+ 04(1+~T) V'(VV---)So +

~e03

l+R--~-pe

(

Re=

P'rVr ep APt p,

PrVrLr I.tr

M2= V 1"2 a r2

G= gr(AP)rLr2

:

o)4/ReM 2

"

.V+

+

1+ Ree p

+ 1+ ~---~

where: tr St= Lr/V r

- - ,,2 ((VV).(VV_) ~r

-o

- o,o-

Pe=

VrL r = Re Pr otr

V2 r Fr= grLr

Pr=

C prl'l'r

~kr

V2 I E=Cpr(AT)r Pr

AT r

~

:

"r

The reference thermal diffusivity (Ctr)is defined as Ctr=

~r

PrCpr

APPENDIX B

RELEVANT TENSORIAL FORMULAS THAT LEAD TO (19b) We decompose the following symmetric traceless tensors as: v. [2~,(vv__oo] : 2 v. 6 ( v v ) q

2 -~- v ( , v . v ) 1

(B.la) (B.lb) (B.lc)

217

42nd lA! r Consre:$

Consider the first term in the right hand side of (B.lb)

V" [II2(VVT) s] = I.t2V'(VVT)S+ VI.t2" (VVT) s

(B.2)

Since It= (l+TeT)Z, VI.t2= 2z (l+TeT)2Z-I eT VT

(B.3)

V.(VVT) s

= VV2T ; VT. (VVT) s = 2 V(VT)2

(B.4)

the (B.I) becomes: V" [I.ti(VVT) s] = I.t2VV2T +z
(B.5)

In addition we have:

la2 Vv2'r = V(giV2T) - 2 Vlx (ttV2T) =V(tt2V2T) - 2V~t [V.(IaVT)-V~t.VT]

(B.6)

(1 +T~) 2z'l eTV(VT)2 =V([1 +TeT]2Z"1(VT) 2) - (VT) 2 V( 1+Ter )2z'l

(B.7)

= V([I+Tar]2Z'I(VT) 2) - (2z-l) (l+TET) 2z'2 eT(VT) 2 VT V~t = z (l+Ter)z'l e.rVT

(B.8)

Accounting for (B.6), (B.7), (B.8) the (B.5) becomes:

(B.9) -2z (l+Te.T)Z'l ~rVl V.(I.tVT) + z (l+Ter)2z'2 eT2(VT)2 VT Considering now the first term in the right hand side of (B.lc):

V • [l+~

( VTVT)S] = (l+T~)2z'l V" [(VTVT)'] + V(I+TeT)2Z'I " (VTVT) s

(B.IO)

Considering that:

V" [(VTVT) s] : VT V2T + ~ V(VT)2 ; V(I+TeT )2z'l : (2z-l)(I+TeT )2z'2 eT VT VT.

(B.ll,12) (B.13)

(VTVT)': VT (VT) 2

the first term in the rigth hand side of (B,10) can be written as: 1

(l+Tar)2Z'l V. [(VTVT) s] = (I+TeT)2z'1 VT V2T + 7 (l+TeT)2Z'l V(VT)2 =

(B.14)

218

42nd IAF Congress

= (l+T~..r)Z'l VT [V.(I.tVT) - VI..t.VT] + ½ 7([I+TET]2Z-I(VT) 2) 1 ~(VT) 2 V(I+T~T)2Z'i

Accounting for (B.8) and (B.12) the (B.14) becomes: (l+Te.r)2z'l V. [(VTVT) $] = (l+TeT)Zl VT V-(gVT)

(B.15)

1 1 +7 V([I+TeT]2z'I(VT)2) + ( 2- 2z) eT (I+TeT)2z'2 VT (VT) 2

Moreover, by considering the (B.12) and (B.13) : V(l+T~r)2Z'l .(VTVT) s = (2z-l) (I+TeT)2z'2 ET VT(VT) 2

(B.16)

From (B.15) and (B.16) the (B.10) can be written as:

V • [ 1+~-~ ( VTVT)S] = (I+TET)Z'I VT V'(gVT) +~-1 V([I+TeT]2Z.I(VT)2 )

1 ~_~T(I+TeT)2Z_ 2 VT(VT) 2

From (lgb), (B.la,b,c), (B.9) and (B.17) it follows the (19b)

(B.17)