Solutal convection and morphological instability in directional solidification of binary alloys. - II. Effect of the density difference between the two phases

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Solutal convection and morphological instability in directional solidification of binary alloys. - II. Effect of

the density difference between the two phases

B. Caroli, C. Caroli, C. Misbah, B. Roulet

To cite this version:

B. Caroli, C. Caroli, C. Misbah, B. Roulet. Solutal convection and morphological instability in direc-

tional solidification of binary alloys. - II. Effect of the density difference between the two phases. Jour-

nal de Physique, 1985, 46 (10), pp.1657-1665. �10.1051/jphys:0198500460100165700�. �jpa-00210114�

Solutal convection and morphological instability

in directional solidification of binary alloys.

II. Effect of the density difference between the two phases

B. Caroli (+), ^C. Caroli, ^{C. Misbah} and B. Roulet

Groupe ^de Physique ^{des Solides de} l’Ecole Normale Supérieure (*), Université Paris VII, ² place Jussieu, ⁷⁵²⁵¹ Paris Cedex 05, France

(Reçu ^{le 11 mars 1985,} accepté ^le 10 juin 1985)

Résumé. 2014 Nous étendons l’analyse ^de perturbation ^du couplage ^entre déformation de front et convection solu- tale présentée ^{dans un} article récent [3] pour tenir compte de l’effet de l’« advection », c’est-à-dire de l’écoulement induit par la variation de densité qui accompagne la solidification. Nous étudions le déplacement ^{de la} bifurcation à partir ^du régime stationnaire à front plan. ^{Nous montrons} que, le long ^{de la branche} convective, ^{l’effet de l’advec-}

tion est du même ordre que celui du couplage convection-déformation, alors que, le long de la branche morpho- logique, ^{il est} largement ^dominant.

Abstract

We extend the perturbative analysis ^{of the} coupling between front deformation and solutal convec-

tion performed ^{in a recent article} [3] ^{to include} the effect of « advection _», i.e. of the flow induced by ^the density change associated with solidification. We study ^the resulting shifts of the bifurcation from the stationary planar

front regime. ^{We show} that, ^for the convective branch, the effect of advection is comparable ^to that of the convec-

tion-deformation coupling, while, ^{for the} morphological ^{branch, it is} by far dominant Classification

Physics ^Abstracts

61.50C - 47.20

64.70D

1. Introduction.

It is well known that when ^a dilute binary alloy ^is

submitted to directional solidification (i.e. pulled

at constant velocity ^{V in an} external thermal gradient),

the structure of the solid and the morphology ^{of the} growth ^{front are} primarily controlled, ^for systems with atomically rough solid-liquid interfaces, by

solute diffusion in the liquid phase. ^{At small} pulling velocities, ^the growth ^{front is} planar; beyond ^a

critical velocity Vc (which depends ^on the thermal

gradient ^{6 and on the} concentration far from the front in the liquid, C ex)’ ^{the front} develops ^a periodic

cellular structure, associated with the Mullins-Sekerka

instability [1, 2].

However, ^{such a} system ^{is never free from} hydro- dynamic motion in the liquid phase. This flow is induced by ^two mechanisms :

(i) ^the system ^{is submitted to a thermal} gradient

(*) Associe ^{au C.N.R.S.}

(+) ^{Also :} Departement ^de Physique, ^UER de Sciences Exactes et Naturelles, Universite de Picardie, 33, ^{rue Saint-} Leu, ⁸⁰⁰⁰⁰ Amiens, ^France.

Moreover, solidification of an alloy produces, ^{at the} interface, an excess or defect of solute which induces

a concentration gradient ^{in the} liquid ahead of the front In the presence of gravity, ^these gradients give

rise to buoyancy ^{forces and} may thus induce a con-

vective flow ;

(ii) the densities of the liquid and solid phases

are different The solidification of a given ^{mass of} liquid is therefore associated with a volume change,

which must be compensated ^for by ^a hydrodynamic

flow in the liquid ^{In order to make a} clear distinc- tion from buoyancy-induced effects, ^{we will conven-} tionally call this current an « advective flow ».

Hydrodynamic flows and front deformations are

coupled ^{In a recent article} [3] (hereafter ^{referred to}

as (I)), ^we studied the effect of the coupling ^between

front deformation and solutal convection on the

position ^{of the} bifurcation from the regime ^with pla-

nar front and quiescent liquid ^{in the case where the}

solid is pulled vertically, ^{the thermal} gradient ^is stabilizing, and the solid and liquid ^{densities are}

assumed to be equal.

We showed that, ^{close to} the convective and mor-

phological bifurcation, the effective coupling ^between

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198500460100165700

1658

convection and front deformation is small at usual values of the thermal gradient, ^{and that a} first-order

perturbation ^{treatment is well} justified ^We found,

in particular, that the convection-induced shift of the Mullins-Sekerka (MS) bifurcation is, ^{under usual} experimental conditions, extremely ^small.

In this article, ^{we want to} study ^the ways these results are modified when advection effects ^are taken into account It is clear that advection couples ^to

front deformation : indeed, ^the corresponding ^flow

must be normal to the solid surface. Thus, ^{a defor-}

mation of the front distorts the flow, this modifies the transport ^of solute from (or towards) ^{the inter-} face, and that in turn reacts on the front deforma-

bility.

Of course, the relative density changes ⁸

(ps - PL)/PL ^{are small} (typically, of the order 10-2 _to, at

most, 10-’) ^{and so are} usually considered ^as negli- gible. However, advection effects are unavoidable

(even ⁱⁿ microgravity experiments). ^{Moreover, since}

it is found that the bifurcation shifts due to the convec-

tion-deformation coupling ^{are small, the} question

arises of comparing ^{them to} those induced by ^advec-

tion.

In § ^{2 we} write down the equations describing ^the

solidification problem in the presence of both advec- tion and solutal convection and calculate the solution with a planar stationary solid front We set up the linear stability analysis ^{of this solution and write}

down the condition of existence of marginal ^modes.

In § ^{3 we} study the effect of advection on the decou-

pled MS and convective bifurcations.

In § 4, ^we include the convection-deformation

coupling and calculate the position of the bifurca- tions in a first-order perturbation approximation.

2. Dynamical equations ^and planar stationary ^solu-

tion

We assume that the dilute binary mixture is pulled vertically (in ^{the (- 2)} direction) ^at velocity ^{V, and}

that the system ^is quasi-infinite ^{in the} (x, y) directions.

Following (I), ^{we also assume} that the thermal _gra- dient is stabilizing ^and neglect ^thermal expansion (i.e.

thermally-induced convection).

We define the following dimensionless variables :

where the tilded variables are the physical ^ones.

f, ti, dr ,, ^{C are the} temperature, liquid velocity,

front velocity, and solute concentration in the liquid

is the relative density change ^at solidification (for

most materials, s > 0). ^D is the solute diffusion coef-

ficient in the liquid, TM ^the melting temperature ^of the _pure solvent, Coo the solute concentration in the

liquid far from the front, ^{and K the} equilibrium ^solute

distribution coefficient

Following (I), ^we neglect, in the non-dimensionaliz- ed equations ^{of the} problem, ^{all terms} proportional

to D/Dth ^{and to} D/v (where Dth ^{is a heat} diffusion coef- ficient and ^v the kinematic viscosity ^{of the} liquid).

We also neglect diffusion in the solid phase, ^{treat the} liquid ^as incompressible, ^{and assume} quasi-instan-

taneous local thermodynamic equilibrium ^{on the}

front

The system is then described by ^the following equa- tions :

(i) ^{In the solid} phase :

Heat diffusion :

(ii) ^{In the} liquid phase:

Heat diffusion :

Solute diffusion, :

(z is the unit vector along Oz)

Mass conservation :

Momentum conservation :

a PL ^TC- is the solutal expansion coefficient;

PL

the gravity g ^is positive.

(iii) At the interface (z

zs(x, ^y, t)) : No-slip condition :

where n is the unit vector along the normal to the front pointing ^{into the} liquid.

Mass conservation :

Continuity ^of temperature :

Heat balance :

where n

ks/kL is the ratio of the thermal conducti-

vities.

Concentration balance :

Curvature-induced local interface temperature ^{shift :}

M

ML CwIKTm, ^where mL is the slope ^{of the} liquid-

us curve of the mixture at (T

T M, ^C

0).

r

y VIED, where y ^{is the} solid-liquid ^surface tension, C ^the specific latent heat of fusion, ^{and X}

the curvature of the front, ^{defined as} positive ^{for a}

convex solid

Moreover, ^{we assume that the} hydrodynamic ^flow

is free at infinity ^{in the} liquid (z ^-+ oo), ^{where the}

pressure is fixed.

When comparing equations (1-12) ^{with the corres-} ponding equations ^of (I), ^{it is seen that} taking ^advec-

tion into account results in the following :

-

a redefinition of the non-dimensional variables that is, ^a rescaling of the diffusion length and time of the system ^which become, respectively, D/(1 ⁺ 8) V and D/(1 ⁺ 8)2 V2 ;

-

a factor (1 ⁺ s)- 3 ^{in the} coefficient of the second term of the hydrodynamic equation (7). As shown in

(I), this coefficient is directly ^{related to the} Rayleigh

number for our problem

where

and the (1 ⁺ E)-3 ^factor naturally ^{accounts for the} rescaling of the diffusion length;

-

the presence of ^{a term} proportional ^{to 8 in} equation (9), which describes how the volume defect

resulting ^from solidification must be fed by ^advection.

We first look for a stationary ^solution of the above

equations corresponding ^{to a} planar ^front Choosing

the origin ^{of the} z-coordinate at the front position,

one finds that this solution is described by

GL ^{is the} positive dimensionless temperature gradient

in the liquid :

with 6 the physical ^thermal gradient

We are interested in the thresholds of instability

of this solution, ^{so we want to} study its linear stability.

That is, ^{we assume} that the planar ^front undergoes

a small deformation of wavevector a (chosen along ^the arbitrary x-direction)

This induces variations of wavevector a and growth

rate a of the various fields. One then standardly expands equations (3.12) ^{to first order in} these varia- tions. The corresponding equations, ^{which are written}

and solved in the Appendix, provide ^{the «} dispersion

relation » a

f (a). ^{We assume} that the bifurcations

we study ^{are not} oscillating. As shown in (I) ^and

confirmed by numerical results [4], this is indeed true in the absence of advection, _except ^at unrealistically

small thermal gradients ^{or for} very small values of the distribution coefficient K. Since the parameter e, which measures the strength ^{of advection} effects, ^is

small (s 10-1), ^one may reasonably expect ^that the introduction of these effects does not alter the character of the bifurcations. This can be proved explicitely [5], ^to first order in _8, with the help ^{of a} perturbation expansion.

The equation determining the neutral modes of the system then reads (see Appendix) :

where

with

The functions A and B which _appear in equation (19)

are the same as those appearing ⁱⁿ (I) ; ^the presence of advection changes ^their argument from R,, ^into ,k.

The new quantity C(.,k, a) is defined in the Appendix.

3. The uncoupled bifurcations.

Following ^the analysis performed ⁱⁿ (I), ^{we first}

define the ^« _{pure »} ^or uncoupled bifurcations from the planar solution, namely :

- the _pure morphological one, i.e. the MS bifur- cation at zero gravity, ^and

- the _pure convective one, Le. the onset of convec-

tion in the liquid ^{ahead of a} planar non-deformable solid front.

In this section, ^we study ^{how these} uncoupled

bifurcations are shifted by ^advection.

1660

3.1 ADVECTION-INDUCED SHIFT OF THE MS BIFUR- CATION.

The equation ^{for the} corresponding

neutral modes is obtained by taking ^the l§’ ^{-+ 0}

limit of equation (19). ^In this limit (see Appendix)

where

The neutral mode equation (19) then becomes

The determination of the bifurcation proceeds

in the standard _{way :} one looks, ^{at fixed} p, ^{for the} position CbMS, aMS) of the minimum of the neutral

curve 1Jc(a).

The critical wavevector is the solution of

Equations (24) ^and (25) provide ^a parametrization

of the bifurcation curve -Gms(P).

Since E 1, they ^{can be, to a} good approximation,

solved to first order in £. One obtains :

where the critical wavevector is, ^to the first order, unshifted It is given by

With the help ^of equations (20), (21), ^and (27),

we obtain from equation (26) the relative shift of the bifurcation at constant G, ^{V :}

As discussed in (I), ^for not-too-small thermal _gra- dients (typically 6 > 1 K/cm) ^and not-too-large pulling velocities (V ¹ cm/s)

and

That is, ^{on the small} velocity side of the bifurcation

curve CII(V), ^which corresponds ^to usual directional solidification conditions,

-

if 8 > 0 (ps > PL) the effect of advection is

destabilizing : ^{the MS} bifurcation for given Coo ^occurs

at smaller velocities,

-

if 8 0, ^{it is} stabilizing.

At _very large velocities (ams ^{1 ; V in the} cm/s

range ^{or more} (’)), ^{it can} be checked from equations (26), (27), ^and (28) ^that

i.e., advection becomes stabilizing (resp. destabilizing)

for 6 > 0 (resp. ^E 0).

This is illustrated in figure 1, ^{where we} plot ^the

pure MS bifurcation curve in the (Coo, V) plane ^{for a} PbSn alloy (s

^0.04) pulled ^{in a thermal} gradient i7

200 K/cm.

3.2 ADVECTION-INDUCED SHIFT OF THE UNCOUPLED CONVECTIVE BIFURCATION.

This corresponds ^{to the}

limit of a non-deformable front or, equivalently,

of infinite interface tension r, ^i.e. to ^oo. Equa-

tion (19) ^{then reduces to}

The relative density change ^s only appears in equa- tion (32) ^via the definition of the Rayleigh ^number Rs. ^{That is, the} corresponding marginal ^{curve is}

(1) ^Note that, ^{at such} large velocities, ^the present ^model should be improved ^Thermal transport, ^{latent heat} pro- duction and interface kinetics in particular, ^{should be taken}

into account

Fig. ^1.

Bifurcation diagram ^{for the} PbSn ^{system with}

G

200 K/cm (the values of the parameters ^{are taken} from reference [4]). Full line : Bifurcation curves in the presence of advection and of the convection-deformation

coupling; ^{the dots} correspond ^{to the} perturbative result, ^the

crosses to numerically computed results. Dashed line : Pure Mullins-Sekerka and pure convective bifurcation curves

without advection. Dotted line : Pure convective bifurcation

curve with advection. (Note ^{that the} corresponding ^dotted

line for the Mullins-Sekerka branch is not distinguishable

from the full line on the scale of the figure.)

-given by

where the function & ^{is the same as} for the advec- tion-free system. ^Let (Rs*, ^a*) be the coordinates of the minimum of R.,,(a); they depend only ^{on the dis-}

tribution coefficient K (see (I)). ^{The position of the}

convective bifurcation is then given, ^{with the} help

of equations (14) ^and (15), by

So, ^to first order in _s, the relative shift of the convec-

tive bifurcation

is constant along the bifurcation curve.

4. Effect of the convection-deformation coupling.

It has been shown in (I) that for advection-free systems under usual experimental conditions, ^the effective

coupling between the two above ^« _{pure »} bifurcations is small enough ^{for a} first-order perturbation approxi-

mation to be justihed

This results primarily ^{from the} large ^mismatch

between the critical wavevectors a* and a’s which characterize convection and deformation. In typical

situations a* N 0.3, while, ^{close to the} point ^of

parameter space where the uncoupled bifurcation

curves cross (C.,O, Yo), aMS > 1.

We have seen in the preceding ^section that, ^{in the}

presence of advection a* is independent ^{of 8 and}

the shift of aMS is at most of order s’. Thus, ^{the effec-}

tive strength of the convection-deformation coupling obviously ^{remains small, and its effects can} be calculat- ed within the perturbation approximation developed

in (I). Since, moreover, due to the smallness of _s, advection effects are small and can be treated to first

order, ^{we choose to rewrite} equation (19) ^{under the} equivalent ^form

where A - A(,k, ^{a), etc.}

Following (I), ^we then solve equation (35) ^{for the}

shift of each bifurcation by first-order iteration around the neutral modes associated with each of the l.h.s. factors.

4.1 THE MORPHOLOGICAL BIFURCATION.

It has been shown in (I) that, along the MS bifurcation curve, A, ^{B and C can be} calculated to first order in

Rg ^{to a} good approximation. ^{This is} obviously ^true

-far from the crossing point (C 00,0’ ^{Yo) of the two}

bifurcations, since, then, R; I along ^{the MS}

curve. Close to the crossing point, R; -- ]?;’ =- ^10.

But in this region, aMS > ¹ and the relevant Rayleigh

number for a perturbation of wavevector aMS

which is the true expansion parameter

^{is not} Rs’, ^but R s 11(aM3 ^1.

We therefore calculate the shift

(where -6’ corresponds ^{to the} pure MS bifurcation of the advection-free system) ^to first order in s and in R;, neglecting ^{terms of order} E2, Rs’2 ^and ERs’.

We thus find (with ^{the help of-} equations (A. 21

and A. 22) ^of (I))

The shift of the bare MS bifurcation is given ^thus, naturally

^{since we} keep ^to first-order expansions

by simply adding ^the convection-induced shift studied in (I), corrected for advection by ^the rescaling

and the _pure advective correction (Eq. 26).

1662

Equation (37) ^{can be rewritten as :}

In order to estimate the relative orders of magnitude

of the two contributions to the shift, ^{we rewrite} equation (38) ^{in the} vicinity ^{of the} crossing point (Coo ,°, Vo), ^{where aMS » 1} (for not-too-small 6’s).

Using equation (30), ^and equation (57) ^of (I), ^one

gets :

For PbSn alloys, using the material parameters given in reference [4], ^this gives

at the crossing point and, ^{for all} reasonable values of 6(6 > ¹ K/cm), ^the convective (first-term) ^cor-

rection is usually negligible compared ^{with the}

advective (second term) ^one.

As one moves away from the crossing point along

the MS _curve, the convective shift decreases while the advective one remains quasi-constant (for ^usual pulling velocities), and thus dominant

4.2 THE CONVECTIVE BIFURCATION.

Its shift is obtained by iterating equation (35) ^to first order around the solution of

i.e., around the convective bifurcation (R,,*, a*) ^of

the non-deformable system ^{in the} presence of advec- tion.

Using equations (58) ^and (59) ^of (I), ^{one obtains}

where j = (K/RS*) (dRS*/dK) has been tabulated in

(I) ^{as a} function of K. Typically, ^{for most} materials, 2 x 10-1.

The starred quantities A *, ^{C* and m* are to be}

calculated for Rs = ^{and a}

^a*.

Developing equation (42) ^{to the first order in} E, ^one obtains for the relative velocity-shift of the convective bifurcation

where

As shown in (I), along the convective curve A is maximum at the crossing point, where, ^for aoms >> ^1,

it is of order K(m ^{+ K -} 1 ) -1. ^We have calculated the quantity C*IKA* R.* numerically. ^{Its variations}

with the distribution coefficient K are tabulated in table I. One _may then check that, since ^s 10- 1 and (/3 10-1, the cross-correction proportional

to BÇ appearing ⁱⁿ equation (43) ^is negligible, except for mixtures with _very small values of K (2).

Table I.

(2 ) ^This is, ^for example, ^{the case} of succinonitrile-etha- nol [6]. ^{Note that, for such mixtures, the effective convec-}

tion-deformation coupling ^{is not} small, ^{and our} pertur-

bative approximation ^{is not valid}

For 8 > 0 (PS > PL), which is the case for most

materials, ^{both the advection} effect and the coupling

to deformation are stabilizing.

5. Conclusion.

Thus, ^it appears that the shifts due to convection and advection _{are, to} a good approximation ^additive.

This results from the fact that each of them is small and _may thus be calculated in a first-order pertur- bation approximation.

The smallness of the advection-induced shifts

simply ^results from the size of the relative density change during solidification.

The physical origin of the smallness of the con-

vection-induced shifts has been discussed at length

in (I). ^{Let us} briefly recall that it is due to the fact that, except ^for very small thermal gradients ^or segre-

gation coefficients, ^the wavevectors of the uncoupled

convective and morphological instabilities are widely separated The mode that is becoming marginal

e.g., ^to fix ideas, the surface deformation mode, only couples ^to the convective mode with the same wave-

vector, ^{which is} always rapidly relaxing, ^{and thus}

follows almost adiabatically the slow mode. Therefore,

the marginal deformation (resp. convection) ^mode only ^« drags ^{a small amount of} convection » (resp.

deformation). Moreover, ^{one finds} that, ^{due to the} larger ^curvature of the convective spectrum, ^the convection-deformation coupling ^{affects the convec-}

tive instability ^more strongly than the MS _one, the shift of which is, under usual conditions, negligible

in practice.

We have shown here that the shifts of the two instabilities induced by ^{advection are of} comparable

orders of magnitude.

For the convective branch, ^{in the} vicinity ^{of the} crossing point, this shift is comparable ^{to that due to}

the coupling ^to surface deformation. For example,

for PbSn ^{at a thermal gradient of 2000/cm} (see (I))

the deformation-induced shift is of the order of 9 %,

while the advection-induced ^{one is} 4 %. ^{When one}

moves away from the crossing point, ^the deformation- induced shift decreases as ( v/ vo)213, ^{while the advec-}

tive one is constant and thus becomes dominant at small velocities.

The reason why advection stabilizes the system against convection, ^{for 8} > 0, ^is apparent ^from equation (14). ^{For a} non-deformable interface, ^the only effect of advection is to rescale the thickness of the ^« convective box » ahead of the front and the concentration gradient by ^factors (1 ⁺ E)-1 ^and (1 ⁺ ~); ^this rescales the Rayleigh ^number by (1 ⁺ S)3,

thus reducing the critical velocity ^at given C..

For the Mullins-Sekerka branch, ^as explained above, ^the shift due to convection is, ⁱⁿ general, negligible compared ^{to that due to} advection. For

positive ^{8, in a wide} range of values of V above the

crossing point (corresponding ^{to usual} experimental

conditions in directional solidification), ^advection

is destabilizing. ^This may be analysed qualitatively

in the following way.

a) ^{On the one} hand, ^as already mentioned, ^advec-

tion increases the value of the concentration gradient

at the interface. This increases the amplitude ^{of the} destabilizing effect of diffusion.

b) On the other hand, when the interface deforms,

the streamlines, being constrained to remain _per-

pendicular ^to it, ^{must curve. When E} > 0, ^{this amounts}

to the creation of a current along ^the average direction of the surface which carries concentration towards the regions where the solid bumps ^{into the} liquid

This is equivalent ^to increasing ^{the local} production

of concentration excess (or defect, depending ^on

the sign ^{of K -} 1) ^{at the interface on the} bumps.

For a given concentration gradient, ^{this must be} compensated ^for by ^a decrease in the local growth velocity. ^{So, for E} > 0, ^{this effect is} stabilizing, ^It

increases with the amplitude of the advective current, which is proportional ^{to V, but} it decreases when the

wavelength ^{of the} instability increases, ^{which occurs}

when V increases.

Thus, ^{it is seen} that the overall stabilizing ^or destabilizing character of the advection effect results from the rather subtle interplay between these effects.

At not-too-large V’s, ^{the MS} wavelength is of order

V-’I’, ^{and one} finds that the overall effect is destabi-

lizing. ^{A numerical} inspection ^of equation (38)

shows that it changes sign ^at large velocities (typically

V > 1-10 cm/s).

Finally, ^{in order to} check the accuracy of ^our

perturbative results, ^we have calculated numerically. ’

the bifurcation curves from the exact equation (19)

for the case of PbSn ^{alloys in a thermal gradient}

G

200 K/cm ^and compared them with the approxi-

mate curves obtained from our perturbation approxi-

mation. The results are plotted ⁱⁿ figure ^{1. It is seen} that, ^{at such a value} of G, the accuracy of the pertur- bative prediction is excellent. Of course, the agreement would become _poorer for _very small distribution coefficients and, ^as discussed in (I), ^for very small thermal gradients.

Appendix.

We assume that the planar ^front undergoes ^{a small}

deformation :

To first order in (, ^this deformation induces _responses of the concentration, temperature ^and velocity ^fields

of the form

(with ^f

(C, T L, T g, u)).

Equations (3-12) ^{can then be} written, ^{to first}

order in C.

1664

(i) ^{In the solid} phase :

(ii) ^{In the} liquid phase :

(iii) ^At the interface :

with the boundary ^conditions

Equations (A. 3), (A. 4), (A. 8) ^and (A. 9) give

Elimination of Cl(z) ^between equations (A. 5) ^and (A. 7), ^and of C ^between (A. 10) ^and (A. 11) finally

reduces the problem ^to solving

with the boundary ^conditions

and

where

and band p ^are defined in equation (20).

The general solution of equation (A. 17) ^{can be}

written

where the three functions u(i)(s) have been defined in the Appendix ^of (I). Plugging this solution into the interface conditions (A. 19, 20, ^and 21) ^and writing ^the compatibility condition of the resulting equations, ^we obtain, ^{for a}

0, ^the equation defining

the neutral modes :

The functions A, ^Bare those defined in equations (25)

of (I). ^{C can} be written as

the coefficients of which are defined in equation (26)

of (I). Then, using ^the small-R" expansions given ⁱⁿ equations (A. ^{8-20) of} (I), ^we easily ^{obtain, for the}

limit R.’ -+ ⁰

References

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[2] WOLLKIND, ^{D. J. and} SEGEL, ^L. A., Philos. Trans. R. Soc.

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[3] CAROLI, B., CAROLI, C., MISBAH, C., ROULET, B., ^J.

Physique ⁴⁶ (1985) ^401.

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CORIELL, ^S. R., private communication.

[5] MISBAH, C., Thèse de Doctorat de 3e Cycle, ^Université

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