Morphological Instability of Similarity Solution to the Stefan Problem with Undercooling and Surface Tension

This paper concerns morphological stability of a similarity solution to the Stefan problem with surface tension and initial supercooling. The linear stability analysis shows that for a nonzero surface tension each perturbation mode with a nonzero wave number is stable. However, the solution is unstable with respect to perturbations with a zero wave number limit point in their Fourier spectrum. 0. Introduction. In this paper we address the morphological stability of a similarity solution to the Stefan problem with surface tension and initial supercooling. The study of shape instabilities in solidification with supercooling is important, in particular, because these instabilities initiate the appearance of dendrites. Following the review by Langer [1], let us recall two well-known examples of morphological instabilities. The first concerns the quasi-steady-state growth of a spherical germ into a supercooled melt [2] (Mullins and Sekerka, 1963). The respective model problem reads: 2 urr H—ur = 0 Vr R, (0.1) r 27 u\r=R = —-5-, 7-surface tension, (0.2) R dR D[ur]\r=R = ~ (0.3) (here [ ] denotes the jump across the solid/liquid interface (from solid to liquid)), tt(oo) = —8. (0.4) (Hereon equal heat-diifusivities of the liquid and solid are assumed for the sake of simplicity.) There is a simple solution to this radial problem according to which a germ with an initial radius bigger than some critical value Rq goes on growing monotonically. The Received September 13, 1995. 1991 Mathematics Subject Classification. Primary 35B35, 35K05. B. Zaltzman is a Sally Berg Foundation Fellow. ©1998 Brown University 341 342 I. RUBINSTEIN and B. ZALTZMAN linear stability analysis of this solution yields the existence of another critical value of the germ radius Rcr > Ro, uniquely determined by 7 and 6, above which the solution is unstable. The next example we would like to recall concerns the travelling wave solution to the one-dimensional Stefan problem which, upon the transformation to a coordinate system moving with the wave speed v, reads uzz + vuz = 0 Vz ^ zo, (0.5) u\z=zo = 0, (0.6) [««o = v, (0-7) ■u(oo) = Uoo, u(—00) = 0. (0.8a,b) A one-parameter family of solutions (with arbitrary v) to this problem exists for only one critical value of undercooling: = —1. For different undercoolings no planar travelling-wave solutions exist. The linear stability analysis of this solution yields a critical wave-length of perturbation, above which the solution is unstable. This critical wave-length depends on the wave speed, unspecified by undercooling. This sets the ground for the "selection" problem, resulting here from particularity of the basic solution concerned. In this sense, the aforementioned sphere growing into the undercooled melt represents a fairly exceptional example of an analytic solution to be subject to a stability analysis and completely determined by the data of the problem (undercooling). Another explicit solution of the Stefan problem, specified completely by the undercooling, is the famous planar similarity solution of the two-phase problem. An additional importance of this solution lies in the fact that it represents the long-time asymptotics of the solutions to the one-dimensional Cauchy-Stefan problems with constant temperatures at infinities, whenever supercooling is below the critical value. In this solution the speed of the free boundary varies and tends to zero as time goes on. Stability of the planar similarity solution with respect to multidimensional perturbations of finite wave-length has been proved by L. Rubinstein [3] for the normal (non-supercooled) Stefan problem without surface tension. Chadam and Ortoleva [4] addressed stability of the respective solutions for the Stefan problem with surface tension and initial supercooling. They found that a similarity solution is unstable without surface tension and asymptotically stable when surface tension, however small, is present. More precisely, they proved that, with a nonvanishing surface tension, every perturbation mode with a finite nonzero wave number asymptotically decays in time. We wish, however, to reexamine here the conclusion of the overall asymptotic stability of the planar similarity solution with supercooling in the presence of surface tension. Our ultimate claim is that a proper handling of the long-wave components of the perturbation results in instability for any given surface tension. Notation. Hereon we will denote by C with a respective subscript positive constants. We will also use the common notations O(-) and o(-), occasionally with subscripts denoting the respective infinitesimal variables. MORPHOLOGICAL INSTABILITY OF SIMILARITY SOLUTION TO STEFAN PROBLEM 343 1. Formulation of the problem. Main results. The formulation of the problem is as follows. We consider a two-dimensional Stefan problem with surface tension: ut = Au, S(x,t)/0, t>t0, (1-1) u = -7K, [VuV5] = St, for S(x,t) = 0, t > t0, (1.2) u(x, T) = Uo(x), 5(x, T) = 50(x), (1.3) tio(x) —> Uqo 6 (—1,0) as x —* oo, ito(x) —» 0 as x —> — oo. (1.4) Here 5(x, t) = 0 defines the position of the solid/liquid interface and A'(x, t) its average curvature, K = (|V5|2A5 ±V(|VS|2) • VS)/(2|VSf). We are about to analyze the perturbation of the following planar similarity solution to the aforementioned problem: S(x,t) = {x = Rs{t) = 2aVt}, (1.5a) us = f^oo +2aexp(a2)(l -erf(^=)), x>2 oty/t, \o, x < Here a is the root of the transcendental equation poo 2aexp(a2) / exp(-y2)dy = -u^. (1.6) J a Let us consider a non-planar two-dimensional perturbation of the similarity solution of the form: u£(x,y,t) = us(x,t) + eu(x,y,t) + 0(s2), (1.7a) S(x,t) = {(x,y,t): x = RE(y,t)}, Rs(y,t) = Rs(t) + eR(y,t) + 0(e2) (1.7b) with Rs(t) = 2a\Jt. The linearized initial-boundary value problem for the perturbation reads: ut = Au, x > Rs(t) = 2ay/t, t > T, (1.8) duS d 7 D „ d2 s U = + ^Rvv> u* ~q^2~R' °n x = 2aVt, t > T, (1.9a,b) u(x, y, T) = y), for x > 2aVf, (x, y) -> 0 as x -► oo, R(y, T) = r(y). (1.10a,b) Following Lev Rubinstein's idea [3] we make the change of the dependent variable QUS <"y v — u + ~~ ~^Ryy (1-11) followed by the Fourier transform in y: /OO v(x, y, t) exp(-ily)dy, (1.12) -oo W f /oo R(y,t)exp{-ily)dy. (1.13) -OO 344 I. RUBINSTEIN AND B. ZALTZMAN This yields the following one-dimensional time-dependent initial-boundary value problem for wl and /': wlt = wlxx + fuf + ~ j (/' + l2fl) exp{l2t), x > 2aVt, t > T, (1.14) wl = 0, wlx = exp(l2t), for x = 2aVt, t > T, (1.15a,b) w\x,T) = \x), f\T)=F(l). (1.16a,b) Here 4>l(x) = Jxoo(4>{x,y)+ufr(y)-lryy(y))exp(-ily)dy, F(l) = f^riy) exp(-ily)dy. It may be shown (see Refs. [3], [4]), using the respective Green's function of the heat equation, that the solution satisfies the following integro-differential equation: j'(t) = -I(t)exp(-l2t) J (fl{t) + l2fl(t)) exp(—/2(f t))(P(£,t) 4l2V(t,T))dT. (1.17) Here I(t) = Uix(2a\ft, t) and U\ is a solution of the following problem: U\t = U\xx, x > 2a\ft, t > T, (1-18) Ux{2aVt,t) =0, t>T, (1.19) Ui(x,T) = 4>'{x), x>2 aVf. (1.20) Furthermore, P{t,r) = U2xi^oi\/i, t) and U2 is a solution of the problem U2t = U2xx, x > 2aVt, t > t, (1-21) U2(2aVt,t) =0, t > r, (1.22) U2(x,t) = Ux{x,t), x>2 ol\[t. (1-23) r(£,r) is defined by replacing ux(x,t) by ^ in the definition of P(t,r). Let us recall the following results of Refs. [3, 4], summarized below in Lemmas 1-3. Lemma 1. If 1 at infinity. LEMMA 2. The following estimate holds: 0 0+. Lemma 3. The following estimate holds: 0 <^r 0, the following equality holds: fl(t) = C^{l)gl{t) exp(-62£)(l + o(l)), as l2t —> oo. (1-29) Here g\t) '£ f-vW/eIp IVt\ (1.30) is the solution of the following asymptotic equation: and I 2 Q \ ol\J12 — b2 , g' + 27F^W7t)= ui " (L31>