Dynamics of curved fronts and pattern selection

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Dynamics of curved fronts and pattern selection

D. Bensimon, P. Pelce, B.I. Shraiman

To cite this version:

D. Bensimon, P. Pelce, B.I. Shraiman. Dynamics of curved fronts and pattern selection. Journal de

Physique, 1987, 48 (12), pp.2081-2087. �10.1051/jphys:0198700480120208100�. �jpa-00210655�

Dynamics of curved fronts and pattern ^selection

D. Bensimon, ^{P. Pelce} (1) and B. I. Shraiman

AT & T Bell Laboratories, 600 Mountain Av., Murray Hill, NJ 07974, ^U.S.A.

(Reçu ^{le 24} juillet ^1987, accepté le 31 août 1987)

Résumé.

On étudie la stabilité d’un front courbe vis-à-vis de perturbations ^{de courte} longueur d’onde dans le cadre de l’approximation ^{WKB. On trouve un} spectre discret de modes instables pour ^une famille d’interfaces à un paramètre. On propose ^un critère de sélection approché qui détermine les valeurs des

paramètres correspondant ^aux interfaces stationnaires. Dans le cas du doigt ^de Saffman-Taylor ^{et de la} dendrite, ^on prouve la conjecture que le n-ième ^état sélectionné possède n modes instables de tip-splitting.

Abstract.

The stability ^of moving curved fronts to short wavelength perturbations ^is investigated using ^the

WKB approximation. ^{A discrete} spectrum of unstable localized modes is found for a one parameter family ^of

interfaces. An approximate selection criterion for determining the values of the parameter corresponding ^{to the} steady ^{state is} proposed. ^{For the} Saffman-Taylor problem and dendritic growth ^we prove the conjecture ^that

the n-th selected state possesses n unstable tip-splitting ^modes.

Classification

Physics ^Abstracts

47.15H-47.20K-61.50K

1. Introduction.

In this paper ^we shall concern ourselves with the

problem ^of determining ^the shape ^and stability ^of moving interfaces in non-equilibrium systems. ^In

particular ^{we shall} consider the Saffman-Taylor (ST) problem [1] which addresses the motion of the interface between two viscous fluids in a Hele-Shaw cell and the dendritic growth (DG) problem [2]

which deals with the propagation ^{of a} solidification front into an undercooled melt. The major ^recent

progress in the understanding of the mathematical structure of these problems ^{was due to the} realisation of the essential and subtle role played by ^surface

tension. In its absence the ST and DG problems ^are

known to admit a continuous family ^of steady ^state

fronts propagating ^with arbitrary velocity. ^{For the}

ST problem ^{these fronts are fluid} fingers [1] occupy-

ing ^an arbitrary fraction 0 A 1 of the cell width and for DG they ^are Ivantsov’s needle crystals- parabolic ^fronts [2] ^with arbitrary tip ^{radius p.}

However, ^as proved by ^{recent numerical} [3-5] ^and

(1 ) Permanent address : Laboratoire de Recherche ^en Combustion, Universite de Provence, Centre St-Jdrome,

rue H. Poincard, 13397 Marseille Cedex 13, ^France.

analytical [6-11] studies, arbitrarily small surface tension leads to a breakdown of the family ^{to a}

denumerable discrete set of steady ^states. Inspite ^of

this dramatic effect, ^these steady ^states remain very close in shape ^{to the zero} surface tension solutions

having ^{the same} finger ^{width A or} tip ^{radius p. This}

suggests looking ^{for a} simple ^{criterion to determine}

which of the solutions of the zero surface tension

problem ^are good approximations ^{to the true} steady

states. Another interesting question ^{concerns the}

apparent connection between the existence of a

steady ^{state and its} dynamics. ^{It was} posed directly by ^recent numerical calculations of Kessler and Levine [12] and Tanveer [13]. They ^{have shown that}

of all the discrete solutions existing ^{in the} presence of surface tension only ^one, the lowest (Ào ^or Po) ^is stable. All others are unstable. The n-th solution was conjectured ^to possess n unstable localized eigenmodes.

In this paper ^we present ^a general approach ^to study ^the stability of curved interfaces to short-

wavelength perturbations. ^We will prove the afore- mentioned conjecture and propose ^a selection criterion which allows to determine the parameter values for the set of true steady ^{states. Our} starting point, following the idea of Zeldovich [14-16] ^{is a}

WKB description ^{of the} dynamics ^{of short}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198700480120208100

2082

wavelength disturbances on a steadily moving ^front.

The evolution equation ^{thus derived} depends only

on the fundamental aspects of the flat interface

instability. It has the same general form for ST as

well as DG problems which allows us to treat both

problems simultaneously. ^{For the zero surface ten-}

sion solutions the spectrum of unstable localized modes is then determined by ^{a « resonance » condi-}

tion similar to the Bohr-Sommerfeld quantization

relation. (The analysis is similar to the WKB study

of scattering resonances [17-18] ⁱⁿ quantum

mechanics.) ^We will argue that the ^true steady-states corresponds ^{to the} parameter ^values (A ^or p) ^for

which the discrete set of localized modes contains a zero mode [19]. ^{From the structure of the spectrum}

it will be seen that the n-th selected state possesses n unstable localized (tip splitting) eigenmodes.

2. Evolution equation ^for shortwavelength pertur- bations.

For both the ST and DG problems (in ^the quasis- tationary approximation) ^the dynamics of the inter- face is determined by solving ^an elliptic equation ^for

some scalar field 0 (x, y) (the pressure for the ST

problem [1], ^the temperature ^{for DG} [2]) ^{the value}

of which is given ^{on the} interface, r, ^{in terms of the}

local curvature : K.

The normal velocity of the interface is then determi- ned by the field flux at the interface.

For this class of problems the flat front is unstable

[20]. ^The growth ^{rate cr of a small} perturbation ^with

wavenumber k satisfies the dispersion ^relation (see Fig. 1) :

The velocity ^{of the} moving ^{front V sets} the time scale and the effective surface tension parameter v ^deter- mines the lengthscale ^on which the interface is restabilized. For the ST problem [1, 15] ^{v2 =}

(2 7Tb /W)2 . T /12 J.L V ^{with b and W} respectively

the thickness and width of the Hele-Shaw cell, J.L ^the viscosity ^{of the more} viscous fluid (the ^other having negligible viscosity), and T the interfacial surface tension. For DG [2], v2 = lD d0 ^with lD ^{- the}

diffusion length ^and do

^the capillary length.

We proceed by writing down the WKB evolution

equation [14]. Consider the propagation ^{of a small}

disturbance y (s, t ) ^{on a} steady ^front r 0 moving ^at velocity ^{V = 1} (see Fig. 2). ^{The front can be} specified by ^the angle between the normal to the

Fig. ^1.

^The dispersion relation for the growth ^{rate a of}

a disturbance with wavenumber k on a flat front moving ^at velocity ^{V = 1.}

Fig. ^{2. - The} perturbation y (s, t ) ^{to the} steady ^state

interface To.

interface and the direction of propagation 0 (s) ^{as a}

function of arclength ^s. y (s, t ) is defined as the

displacement ^{normal to} the interface : F(s, t) = r0(s) ⁺ hy (s, t ). ^{If the} wavelength ^{of the} perturba-

tion is much smaller than the local radius of ^curva- ture, ^we may consider the front ^to be locally ^flat.

Thus the local growth ^{rate of y is} given by

equation (2). However because of the curvature, ^the

disturbance is also advected (in ^the rest-frame of the

propagating front) by ^the tangential component ^of the velocity. ^{These two effects are described} by ^an integro-differential ^evolution equation ^for y(s,t) :

For a flat interface , 0 (s ) ⁼ 7T /2, ^this equation ^is just

the Fourier transform of equation (2). The Hilbert transform on the right-hand ^{side of} equation (3)

expresses the non-locality ^{of the} problem. ^The

second term on the left incorporates the advection of the perturbation along ^the front, ^{sin 0} asy, ^{and its} stretching ^{due to the} non-uniformity ^{of the} tangential velocity ^{field, y} aS (sin 8 ). Equation (3) ^{is a} generali-

zation of the Mullins-Sekerka instability [20] ^to

curved fronts. It can be derived more rigorously (see Appendix A) by considering ^the perturbation 5 cf> (x, y) ^{to the} scalar field .0 0 (x, y ), ^{due to the}

disturbance _y (s, t), ^and showing ^that (in ^{a WKB} approximation) ^it obeys ^a Laplace equation ^{in a} locally ^Cartesian coordinate system [21]. ^{The normal}

and tangential derivatives of 5 0 ^{on the interface are}

thus Hilbert pairs ^and expressing ^{these, via the} boundary conditions equations (lb, c), ^{in terms of} l’ (s, t) and its derivatives one obtains equation (3).

To study equation (3) ^it is convenient to introduce the complex ^function 1/1’ (z, t ) analytic in the upper half plane :

which is the analytic continuation of equation (3). ^It

is valid within the WKB approximation : ^as long ^as 41 (s) ^varies rapidly ^and ø (s) slowly ^{the Hilbert}

transform H { ø 1Jt} ^is ø H { 1]1’ } up ^{to a} correction

exponentially ^{small in v -1. That assures} the consis-

tency ^of equation (4) ^and equation (5).

3. The eigenmodes ^{of the evolution} equation.

Let us now study ^the spectrum of unstable short

wavelength modes. These fall in ^two classes : localized modes which decay ^as s --i. :t oo and ex-

tended modes which do not. The latter are associated with a convective instability [14, 23] and describe disturbances advected by the flow away from the tip along ^the propagating front. The localized modes

are associated with absolute instabilities [22] ^which

are stationary in the frame of the front. These are

the modes we will be concerned with.

Let ^us look for the unstable localized modes of

equation (5). ^{We set 41} (z, t ) ^{= eut 1]1’} (z ) ^{and assume}

1 « a 1 / v . Within the WKB approximation equation (5) ^{has two} solutions :

a slowly varying ^one

and a rapidly varying ^one

Equation (5) ^{has a} third solution which diverges ^as

Im z - oo and thus cannot be considered. Since 0 (s) - ⁺ w /2 ^{as s} -> ± oo the slow mode, ’/I’s, is

localized while the fast mode, Vlf is extended. These two modes are related to the two degenerate ^inde- pendent ^{modes of the} planar interface : the Fourier modes with wave numbers ks, k f satisfying equation (2) ^{for a fixed} positive value of a (see Fig. 1). By analogy ^one may think that ’/I’s ^and Wf ^{f are also} independent and conclude that there exists a localized unstable eigenmode for every

positive ^a. However, ⁱⁿ general ’/I’s ^{is not an} eigenmode ^{of the} problem. ^{If we let lim ’/I’} (z) =

Z - - 00

’/I’ S (z ), ^{then in} general ^{as z -. oo the two modes will}

mix : lim 1J’(z) = a1Jl’s(z) ⁺ b’/l’f(Z). ^The subtlety

z->oo

arises from the exponential disparity ^{in the} amplitude

of the two modes. While either of the modes is

locally ^an approximate solution of equation (4) ^a

careful analysis ^is needed in order to determine the solution globally. ^It has been described by ^Landau

and Lifchitz [17] ^{in the} analogous quantum ^mechani-

cal context of reflection of a particle moving ^{above a} potential ^{well and} requires ^the analytical ^continua-

tion of the WKB wavefunctions in the complex plane. ^The mixing of the wavefunctions near the

singularities ^{of the} potential ^involves solving ^an

inner problem ^and matching ^{with the outer WKB}

solutions. However we are not interested in calculat-

ing ^the mixing coefficients (a, b), ^{but want to know if}

the resonance condition, ^{b =} 0, ^{is satisfied so that}

the localized mode 1JI’s (z) is ^an eigenmode ^{of the} problem.

4. Selection criterion.

To derive the resonance condition we follow Pok-

rovsky and Khalatnikov [18] and consider the solu-

-z

tion on the line Re I f d (ks - kf) ^{= Const.,}

where both modes are of comparable magnitude.

This line must pass through ^the singularities ^of

ks - kf [17 , 18] in the lower half plane ^{which we will}

2084

I I I

Fig. 3. - The Stokes line and the complex plane sing-

ularities for the Saffman-Taylor problem.

assume to be at the points ^z_ and z+ (see Fig. 3). ^At

these points ^we define the 2 x 2 monodromy ^mat-

rices [24] ^{A ± which} provide ^{a linear} transformation

connecting ^{the outer solution on the left of the} turning point ^{to the outer solution on its} right : lp R = A± WL with W R, L (,p s R, L, 1/’ f’ L). ^{The ele-}

ments of these matrices {A±ij} ^are determined by ^the

solution of the inner problem (2) ^{and are thus out of}

reach of our WKB approach. ^{Far to} the left of

z_ ^we set 1/’ (1)(z) ⁼ 91, (z). ^{Then the outer solution}

to the right ^{of z_ is :}

Which to the left of z+ ^we rewrite as :

Then to the right of z+ , the coefficient of 1JI’ f ^{is :}

In general ^{b =A} 0. However for the symmetric ^Saff- man-Taylor fingers ^the magnitude ^{of the two terms}

(2) To find that solution, ^{one must} go back ^to the time

dependent McLean-Saffman equations ^{in the ST case, or}

to the Nash-Glicksman equations ^{in the DG case. This is}

done in references [7, 10] ^{for the} stationary (time indepen- dent) ^solutions.

on the right hand side is the same : the symmetry under reflection s -> - s requires ^{A + =} (A - )- 1, ^so

that Ai A11 ^{= -} A+22 A21. ^{The bar} denoting complex

11 21

quiring ^{the fast} extended mode to vanish as

s --> oo implies :

This resonance condition is similar to the Bohr- Sommerfeld quantization ^{relation and like it, it is}

valid for large ^n. (The ^{constant in} Eq. (7) ^{is deter-}

mined by the solution of the inner problem.)

Let us now determine the resonance condition

(and ^{hence the} spectrum) ^{for the zero surface}

tension family ^of solutions. For the ST problem ^this family ^is parametrized by the relative width of the

finger ^{A and is} given implicitely by :

singularities ^{of eitl(Z)} in the lower half-plane ^{are the}

two branch points ^at Z;j: = 7r /2 (- i :!: J a) ^(corre-

sponding ^{to q =} i ). Substituting ^the explicit ^{form of} ks (z ) ^and kf(z) (Eq. (6b, c)) ^{into the resonance} condition, equation (7), changing variables and per-

forming ^the integration yields :

This relation defines a set of lines in the a, A plane.

At the selected values of the width, {Am} , ^{the zero}

surface tension solutions are a good approximation

to the true steady ^states. Therefore the spectrum ^of localized unstable modes of the full problem ^consists

of the values of o, satisfying equation (8) ^{for a} given À m. ^In general, because of the translational invar- iance,. ^{for a} steady ^state front there should be a

marginal (u ⁼ 0) ^localized mode. On the other hand as seen from the resonance condition the existence of such a mode implies ^a non-trivial constraint on the shape ^{of the front. For the « true »} steady ^{states this} constraint would be satisfied identi-

cally by virtue of the underlying symmetry. ^{For a} family ^of approximate ^{solutions it} is satisfied at the values of the parameter for which the real solution is

nearby. ^{Hence the} existence of the localized zero

mode can be used as a selection criterion. [19]. ^The

values of À thus obtained (by setting a ^{= 0 in} equation (8)) ^are precisely ^{the ones obtained} by ^a

more rigorous analysis [7]. ^An immediate conclusion is that the n-th selected state has n unstable localized

eigenmodes corresponding ^to tip splitting (see Fig. 4) which is in agreement with the numerical observations [12, 13, 24].

Fig. ^4.

^{The resonant} lines for the Saffman-Taylor prob-

lem from equation (8) (the ^{constant was} arbitrarily ^{set to} 0). For each selected state (open circles) ^the spectrum ^of unstable localised eigenmodes ^{is shown} (black circles).

Clearly ^{the n-th state} (n ⁼ 0, 1, 2, ...) has n unstable localized eigenmodes.

.

Similar conclusions can be drawn for DG [21] (see Appendix B). ^{In that case the zero} surface tension solutions are parabolas :

(The lengths ^are in units of the tip ^{radius, p and}

therefore v 2 = 2 Ddol V p 2). ^The anisotropy ^{of the}

surface tension is accounted by f ( (J) = 1 - a

cos 4 0 multiplying ^{the first term on the} right ^hand

side of equation (5). ^{For a 1 there are three}

relevant singularities : ^at

The resonance condition turns out to be similar to

equation (8) ^{with the} integral computed ^between

zo and z+. Expanding ^the wavevectors kf(z) ^and ks (z ) ^to second order in ^w and computing ^the

resonance condition yields (Appendix B) :

Here again ^{as we set a =} 0, ^we regain ^{the selected}

states [10, 11].

In conclusion the general approach proposed

above provides ^a link between the dynamics ^and stability ^of moving fronts and the existence of

steady-states. ^{In the two} examples ^studied here, ^we

were able to derive the spectrum of unstable local- ised modes for the selected states and to show that the n-th solution possesses n unstable modes. This

implies that the n ⁼ 0 solution is the only ^stable steady ^{state, the one} actually observed in real and numerical experiments.

We thank P. Hohenberg, ^{A. Libchaber and A.}

Pumir for helpful discussions. D. B. acknowledges

the partial support ^{of a Weizmann} Fellowship.

Appendix ^A.

In the following ^we shall derive equation (3). ^Let y (s, t ) ^{be a} shortwavelength perturbation ^{on the} steady moving ^interface r 0 : ’Y (s, t) =

exp [S (s, t )/v ]. ^This perturbation ^{causes a disturb-}

ance 5 cP (x, y, t) ^{of the} steady ^state pressure (or temperature) ^field 00(x, y ). ^Both y (s, t ) ^and 5 cp (x, y, t ) vary ^{on a} lengthscale ^of 0 ( v ) ^{which is}

much smaller than the local radius of curvature of

To. Consider the analytic ^function f (z, t) = 5 P (x, y, t) + i 5 t/J (x, y, t). ^{For a} point z (s ) ^on ro, f (z (s ) ) satisfies :

It is straightforward ^to show that :

Since 5 0 (s, t ) ^and 6 Q (s, t ) vary ^{on a} lengthscale ^of

0 (v ), equation (A2) yields :

2086

From the Cauchy ^relations ðn8 t/J ⁼ ðs8 cp ^{one ob-}

tains :

We may ^{now use} the boundary conditions :

to relate anS cP ^and ass cP to y (s, t ) and its derivatives

on To. (Notice ^that Eq. (A6) ^is identical with

Eq. (lb, c) written in ^a frame of reference moving

with the front.) ^{Let r =} ro + e01’(s,t) ^{and 0 =} 00 ^{+ E50. A} perturbation expansion ^to O(E) yields :

In a coordinate system ^{attached to} the interface (see Fig. 2) ^{one has on} To :

Since by ^the boundary condition, equation (A6), ðncPo ^--- 0, ^{one has} a2oo I r = - ^a2oo I F . ^{0 And since}

ðscPo = v2ðsKO - sin 0, ^one readily derives equa- tion (3) :

Appendix ^B.

In this Appendix ^we shall derive the Bohr-Sommer- feld selection rule for dendritic growth (Eq. (9)).

The WKB equation for the evolution of perturba-

tions on a steady ^{state needle} crystal ^{is :}

v 2 f ( (} ) a; 1]1’ + a z [ ei 0 1]1’] - i a,W ^{= 0 .} (Bl)

Where f ( (}) = 1 - a ^{cos 4 0 is a} surface tension

anisotropy factor. This equation has WKB solutions of the form :

The singularities of kf - ks ^{are at the zeros of} f (0 ) ^{which for a 1 are located} at : z, =

-

i 7T /4::!: (8 a )3/4/3 ^{and zo} i 7r/4 ^± i (8 ^a )3/4/3. ^{The most} dangerous singularities (clos-

est to the real axis) ^{are at :} z± and at zo (hereafter

denoted zo). Introducing ^the monodromy ^matrices A+ , Ao, ^{which connect the outer solution on the} right ^{of the} singular points, z, , zo, ^to the outer solution ^on their left, yield ^the following result for the coupling between the localised (Alr,) ^{and ex-}

tended modes (AFf) :

The reflection symmetry (s --> S ) imposes :

k(-s)=-k(s), ^{A+ -} (A- )-1, A ° A ° = I . Thus

A°1 ⁼ A?2 ^{= 0 and} Aii Ail ^{= -} A22 A2i . ^{The reso-}

nance condition b ⁼ 0 yields ^the Bohr-Sommerfeld selection rule :

Using equations (B3), (B4) ⁱⁿ equation (B6), ^and performing ^the integration yields equation (9) :

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[19] ^{Such an} argument ^{was used} by Kessler and Levine

(see ^Ref. [12]) ^{to determine} numerically ^the

selected fingers by searching ^{for zero modes of}

the stability operator linearised around a zero

surface tension solution.

[20] MULLINS, ^{W. W. and} SEKERKA, ^R. F., J. Appl. Phys.

34 (1963) 323 ; ³⁵ (1964) ^444.

[21] ^{For dendritic} growth ^{the current} approach is valid for small Peclet numbers : p ~ lD.

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[23] ^{ARNOLD, V. I.,} Geometrical Methods in the Theory of Ordinary Differential Equations (Springer- Verlag) ^1983.

[24] TANVEER, S., preprint ^Caltech (1987) ^and private communication, ^has independently ^{obtained our}

result for the Saffman-Taylor problem (Eq. (8)) using ^a different method (submitted ^to Phys.

Fluids).