The Structure and Stability of Propagating Redox Fronts · The Structure and Stability of...

The Structure and Stability of Propagating Redox FrontsAuthor(s): G. Auchmuty, J. Chadam, E. Merino, P. Ortoleva and E. RipleySource: SIAM Journal on Applied Mathematics, Vol. 46, No. 4 (Aug., 1986), pp. 588-604Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2101734 .Accessed: 20/11/2013 10:45

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SIAM J. APPL. MATH. ? 1986 Society for Industrial and Applied Mathematics Vol. 46, No. 4, August 1986 006

THE STRUCTURE AND STABILITY OF PROPAGATING REDOX FRONTS*

G. AUCHMUTYt, J. CHADAMt, E. MERINOt, P. ORTOLEVAt AND E. RIPLEYt

Abstract. This paper analyzes certain models for the deposition of minerals. The models describe the transport, by flow and diffusion, of an aqueous oxidant in an aquifer where it reacts with an immobile reductant to produce certain deposits. We prove the existence and the uniqueness of travelling wave solutions of the equations. In certain limits, the system can be modelled by a problem of Stefan type with planar fronts. We show these solutions are marginally stable and are linearly stable to transverse perturbations.

Key words. modelling geological redox fronts, travelling waves, moving free boundary problem, stability analysis

AMS(MOS) subject classifications. 35R35, 34C05, 86A60

1. Introduction. We have recently been studying certain processes and models of the deposition of important minerals, (see [2]). This deposition was modelled by systems involving flow, diffusion and chemical reaction and precipitation in an aquifer.

Here we shall analyze two of the simplest of these models with a view to developing methods for studying more realistic, and complex models.

The simple mathematical model will be briefly described in ? 2. For a detailed discussion of the chemical and geological basis of these models and for more complicated models see [2]. In ? 3 we shall prove the existence and uniqueness of travelling wave solutions for the reaction-diffusion-flow problems. When the reaction is very fast compared to the other time scales in the problem, the system may be modelled by a Stefan problem, identical to that for the melting of ice, except that the present problem includes flow. As a result the physically relevant base planar solution is no longer the x/21t similarity solution [4] but rather one which travels with a constant velocity. In ? 4 we study the stability of these fronts and find the constant velocity planar solution is marginally stable amongst all planar fronts and it is linearly stable with respect to transverse perturbations. In the course of this analysis we must prove the global existence of planar versions of this Stefan problem for arbitrary data. The simple maximum principle argument to prove globality [4, p. 226] does not apply for the direction of flow which is of interest in the present situation. As a result estimates must be made which involve the full nonlinear integral equations equivalent to the Stefan problem. Additionally, in proving that all planar solutions remain in an interval about the base planar one, we obtain a stronger result than the best which is available without flow [7, p. 185].

Since our original interest was in modelling uranium roll front deposits and other scalloped redox fronts, these results suggest that one needs to analyze some of the more sophisticated models described in paper [2] to observe these phenomena. In particular, we have seen the shape destabilization of planar fronts if reaction generated porosity changes are included and the imposed flow is replaced by velocities obtained from Darcy's law [2].

* Received by the editors December 13, 1983, and in revised form August 7, 1985. This research was supported by a grant from U.S. Department of Energy, Office of Basic Energy Sciences, Engineering Research Program, contract DE-AC02-82 ER12074.

t Geo-Chem Research Associates, Inc., Bloomington, Indiana 47401. 1 IHES, Bures-sur-Yvette 91440, France.

588


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PROPAGATING REDOX FRONTS 589

2. Model equations. We are interested in modelling situations where an aqueous oxidant, transported by flow and diffusion in an aquifer, reacts with an immobile reductant to produce various products. Typically the oxidant is dissolved oxygen; its concentration will be denoted by X. The reductant may be pyrite or a solid carbonaceous material and its concentration is P. These react irreversibly according to

mX + nP -products-

and will assume a rate law of the form

(2.1) Q(X, P) =kXaP

where k, a, /3 are positive constants. Assume the aquifer occupies a region Ql in R3 and portions afk1, afk2 of the

boundary are, respectively, permeable and impermeable to X. The fluid flow is pre- scribed and has velocity vi(x, t). Then the equations for the concentrations X and P are

ax (2.2) _ =div (D grad X - Xv)-mQ(X, P) at and

(2.3) d= -nQ(X, P)

in Q7 x (0, T). Here D is the diffusion coefficient for X in the aquifer and the second equation reflects the fact that P is assumed immobile.

The boundary conditions are that

(2.4) X(-, t) = XM (, t) on a(k1 x (0, T)

and

-(a , t) = 0 on af2 x (0, T). an X

Here XM stands for the meteoric concentration of X and is given, n' is the direction of the outward normal and a(k1 U afk2 = afd-

One assumes that, at t = 0,

(2.5) X(x, 0)=0 and P(, 0) =PO(x)

are given. Equations (2.1)-(2.5) constitute a well-posed initial boundary value problem for

X and P and one is interested in the subsequent evolution of the system. For modelling purposes, it is often convenient to assume the aquifer has uniform

and finite cross-section fQO and is of infinite extent in- the normal direction to fQO. Let the x-axis lie along this normal so that Q= (-o, oo)xflo and assume the flow is in the x direction with

(2.6) v(x, t) = v(y, z)

where i is the unit vector in the x-direction. Since diffusion will homogenize any nonuniformity in XM well before the reaction

zone, we shall assume that at the inlet, x = -00, one has

(2.7) X(-oo, y, z, t) = XM and P(-oo, y, z, t) = 0 for (y, z) E fQO



590 AUCHMUTY, CHADAM, MERINO, ORTOLEVA, RIPLEY

where the super bar denotes the cross-sectional average

XM=-I IL XM(Y, z) dydz.

Far upstream one has

(2.8) X(+o, y, z, t) = 0 and P(co, y, z, t) = POO(y, z).

The side walls of the aquifer are assumed impermeable so that

(2.9) ax (nxt)=

for all (x, t) E (-oo, o) x aflo x (0, oo). We shall only study in this paper the case where the above conditions hold and

(2.4) is replaced by (2.7)-(2.9). To facilitate the analysis of this system, we shall introduce nondimensional

variables. With XM and iv denoting the average of XM and v over the cross-section f1o, let tnew = D- I2t; xnew = D-lix; Xnew X=-X; pnew = m(nXm)-lP and Vnew = (i)-lv. Then in terms of the new variables (and omitting the superscript "new")

aX a2 dXaP (2.10) -= V2X-v-- Q(X, P), -=-Q(X, P) on (-oo, o) xf10 at ax at

where Q(X, P) is given by (2.1) with kne' replacing k where

knew= m1-ng3D( V)-2Xa+P'-1k.

Equations (2.10) are to be solved subject to an initial condition of the form (2.5), and boundary conditions of the form (2.7) and (2.8). Note that in these nondimensional variables X(-oo, y, z, t) = 1 for (y, z) E fl0 for all t. This boundary condition forces the concentration of X to become nonzero for t positive and one expects X to flow and diffuse downstream. The pyrite will be reduced and thus one expects to see concentration profiles of form in Fig. 2.1.

There are two types of special solutions which are of major interest: travelling waves and redox fronts (the latter to be studied in ? 4). Travelling wave solutions of (2.10) have the form

X(x, t) = x(x - ut, y, z),

P(x, t) = ?(x - ut, y, z).

Let = x - ut, then equations (2.10) become

(2.11) AX + (u - v) a- Q(X, .) = 0

and on f.

(2.12) u--Q(X,P ) = 0

Here A = a2/a2 + 2/ay2+a2/az2 is the Laplacian, and the boundary conditions are that

(2.13) (X,?) - (1, 0) as -> -o,




FIG. 2.1. Concentration profiles at t > 0 for fixed (y, z) in QO.

and

(2.14) (X, (D) - (O, P.O) as f o +

where PO,,(y, z) is the nondimensional asymptotic pyrite concentration (i.e. m(nXM)-fP.(y, z)). These are the nondimensional versions of (2.7) and (2.8).

Subtract (2.12) from (2.11) and average over the cross-section flO, then

d 2 - d - d

Integrating once, one has

(2.15) d-+ u(X+i) - W=A on -?o< f < ?o.

When = -00, one has A = u - v and when = +00, A = -uPO. Thus

i3 (2.16) u 1+ .

This expression shows that the speed of a travelling wave solution is determined by the asymptotic concentrations of X and P and the flow speed v. It does not depend on the specific kinetic law Q or the cross-section fkO. In the original (dimensional) variables, this wave speed is

nXMv

nXM + mPJi'

In the next section, we shall look at the problem of finding the travelling wave solution whose velocity is given by (2.16).

3. Travelling wave solutions. Assume now that v, XM and P,) are independent of position (y, z) in fQO, and look for travelling wave solutions x and 1D which depend only on 5. They obey

(3.1) X"+ (u - v)X'- Q(X, D) = O

and

(3.2) uD'- Q(X, FD) = 0.

Here -oo < e


From (2.15) and the value for A one sees that

(3.4) yv'1D(5) = X' + ( y - 1) vx + yvPOO

or

?(f() 1LX'_ -2X + P0

with Al = u-1 and /2 = (1- y)/ y = POO being positive constants. Substitute this expression for (F in (3.4) then

(3.5) X" + (y - 1) VX - Q(X,A 1X'- A2X + P,) = ?-

Let r = (1 - y)vt and u(r) = X(e) = V(r/(1 - y)v) then (3.5) becomes

d 2u du / du (3.6) d -Q4 Q1 u, -; P) =0

with P = y/ (1 - y) and

(3.7) Q1(u, w; P) = clua(w -u + vP0)P

Here cl = PAk/((l - y)2v2) is a positive constant. The asymptotic conditions (2.13)-(2.14) become

(3.8) (u(r),u'(r))-*(1,0) asr--Xoo

and

(3.9) (u(,r), u'(,r)) -*(0, 0) as r >+oo.

Equation (3.6) may be written as an autonomous first order system. Let u1= u and u2 = u', then

(3.10) :r [ ui(dr)1 [Uu2 + Q('U1, u2; v)1 The vector field defined by this equation has precisely two critical points at (0, 0)

and (1, 0). Thus any solution of (3.6)-(3.9) corresponds to finding a heteroclinic orbit for

the system (3.10) going from the critical point (1, 0) at r = -00 to (0, 0) as r - +00. The following theorems show that for any positive P and any rate-law of the form

(2.1), there are heteroclinic orbits or, equivalently, travelling wave solutions. Then we shall show these are unique.

THEOREM 1. Suppose v is positive and Q, is given by (3.7) with cl, a, A3 all positive. Then there is at least one heteroclinic orbit joining (1, 0) to (0, 0). Moreover along this orbit U2(r) is monotone decreasing.

Proof. The proof proceeds by analyzing the phase-plane for (3.10). Consider the behavior of integral curves in the open triangular region A whose vertices are (0, 0), (1,0) and (0, -1). See Fig. 3.1.

On the side S, joining (0, 0) to (1, 0), the vector field defined by (3.10) points vertically upwards and has magnitude Q(ul, 0, P) which is positive except at the end-points. Similarly on the side S3 joining (0, 0) to (-1, 0) the vector field also points out of the triangle. If one starts at any point on the hypotenuse of this triangle except at (1, 0), then the solution proceeds down the hypotenuse and out through (-1, 0).

The vector field is continuously differentiable in A, so there is a unique integral curve through each point of A. We shall denote the solution of (3.10) obeying w(0) = u^ by u(r; u^).




, 2

(0,0) Si e

FIG. 3.1

One observes that if u2(r; u^) < 0, then (dul/dr)(r; u^) < 0 so u1(r; u^) decreases as r increases so long as the solution is in the lower half-plane.

Choose u2l in (0, 1) and let Io= [ ul -1, 0]. For each A in Io, the curve {u(r; ua):

X-' 0,a ̂ =(Al, A2)} must lie in the closed triangle whose sides are {a1}xIo and the subsets of S, and S2 with ual, ul ' 1. Now no integral curves come into A and the solution curve cannot have started at r = -oo at an interior point of A, so one must have

lim u(T)= II.

When A2 is close enough to 0, then (r, d) will leave A through SI as r increases and when u2 is close enough to -1, the solution leaves A through S3.

If the curve starting at u = (AD, u2) leaves Ai through Sl, then the integral curves starting at u = Al, -2) in A with -2> A2 must leave A through S1, since the integral curves cannot intersect. Similarly if (, ua) leaves A through S3 as r increases then u(r, u*) with u* = ( *1, u*) in A and u* < ua2 leaves A through S3.

Let ui2N E Io be the largest value of u2 such that the integral curve starting at = (u1, U2N) exits A at the point (1/N, 0). Let U2N E IO be the least value of U2 such

that the integral curve starting at u*= (u^,, u2N) exits A at (0, -1/N). Define IN =U[ZN, U2N]. Each IN is nonempty, so from compactness IJ = n N=K IN

is a nonempty, closed connected subset of Io (K is chosen so that K1


Proof. Suppose there are two such orbits which we shall denote by u and u^. Both these trajectories must lie entirely in the triangular region /v and obey

du2

We shall look at the solutions of (3.11) considered as curves in the u1 - u2 plane. Suppose the two heteroclinic orbits are described by a2(u1) and -2(u1) for 0_ u1l 1. If ua2(e)> >u2(e) for some 0 < 5 < 1, then a2(ul) > -2(ul) for all 0 < ul < 1 as the integral curves could only possibly intersect at critical points.

Assume a2(u1) '- -2(u1) on (0, 1). Then

-(U^2d U2)= C2 u[U21(U2-u 1 + 1 - 1(2- u1 + 1)] du,

= C1ul(U^2- u27 -

( - u1 + 1) 1[1-u1 + (1-/)]

where U2(u1) ' l a2(u1) from the mean value theorem. For small ul, the right-hand side is negative and it is zero if and only if u2 = u2. Now U2 = U2 = 0 when u1 = 0, so for small u1, one has

U2(u1)= U2(u)+ f (U2 - U2) d4a U2(U1)- JO de

This can only happen if u2(u1) =2(u1) for all u1 or, equivalently, there is a unique heteroclinic orbit.

4. Shape stability of nearly planar redox fronts. The morphological stability of nearly planar solutions cannot be treated in the context of the full problem (2.10) because the base planar solution, even with nondimensional v = 1, XM = 1, POO being constant (i.e. solution of problem (3.1-3.2)), is not available in closed form. For this reason we shall restrict the study in this section to the fast reaction limit of equations (2.10). For specificity and completeness we shall give, in ? A, this limiting version-of the equations (2.10). Motivated by the results of ?? 2 and 3, an explicit special planar travelling front solution to these equations is found in ? B and it is shown to be asymptotically marginally stable within the class of planar solutions. One cannot do better than this because of the translational invariance of the problem. In ? C a complete linearized stability analysis about this special planar solution shows that all transverse perturbations will decay.

A. Redox fronts. When the reaction is much faster than the transport (as in aquifers of low porosity), the redox zone becomes very narrow and may be idealized as a front. If Q(X, P) = 8-VG(X, P), then a formal singular perturbation analysis (see [2]) yields the following free boundary problem in the limit as ? -* 0+ (with v = 1, XM = 1, POO constant).

Let the unknown redox front be given by

(4.1) S(x,* t) =0.

Then, behind the front, one has

(4.2) dt=div (grad X-Xi) and P(, t) =0 in S(, t) 0.




At the front, S( , t) =0, one has

(4.4) X(x, t)= 0

and

(4.5) as = P-1 (grad X) - (grad S). at 0

When x - -oo, one has

(4.6) X(x t) ->I

and one must specify X(x, 0) = X0(x+) for all xv obeying

(4.7) S(x+ 0)< 0.

In this section we shall consider doubly-infinite cross-sections but the analysis is similar for finite cross-sections with Neumann boundary conditions. The system (4.1)- (4.7) is a one-phase (the "phase" ahead of the front is trivial-see (4.3)) multi- dimensional version of the classical Stefan problem [4], [7] and also includes flow.

B. Planar redox fronts. With S = x - s(t), the planar version of (4.2)-(4.7) is

aX a2X aX (4.8) ax= ax2 ax X < s(t),

(4.9) X=0

ax onx=s(t), (4.10) ax_-POOs

(4.11) X -1

(4.12) ax0 asx-o-00a and

(4.13) X(x, 0) = X0(x) for x < s(0),

where the limits in (4.11)-(4.12) are uniform in bounded t sets (the last being added for technical reasons). A straightforward calculation shows that

(4.14) Xp(x, t) = (1 - exp [(1 - V)(x - Vt)])

and

(4.15) s*(t) = Vt with

(4.16) V= 1 1+ P0

is a solution of equations (4.8)-(4.13) with special initial datum

(4.17) Xp(x, 0) = (1 -exp [(1- V)x]),

(4.18) Sp (O) = O. This is the unique travelling front solution of (4.8)-(4.12) and its velocity, not surpris- ingly, is precisely that obtained in ? 3. We claim it is marginally asymptotically stable among all planar solutions (i.e. with arbitrary Xoo(x), s(0)). On the other hand it is not a priori obvious that solutions of (4.8)-(4.13) with arbitrary data even exist globally in time. We begin by addressing this question.




DEFINITION 1. We say X(x, t), s(t) is a solution of problem (4.8)-(4.13) over the interval 0 < t < T if (i) a2X/ax2 and aX/at are continuous in -oo < x < s(t), 0 < t < T; (ii) X and aX/ax are continuous in -oo < x _ s( t), 0 < t < T; (iii) X is also continuous in -oo < x ' s( t), 0 _ t < T; (iv) s(t) is continuously differentiable in 0 ' t < T and (v) the equations (4.8)-(4.13) are satisfied.

In order to make closer contact with existing literature [4], [7], [1], we rephrase problem (4.8)-(4.13) as follows. Make the change of variables x'= x +t, t'= t and denote R(t) = -s(t) + t and u = -X. Dropping the primes, one obtains

aU a2U (4.19) - 2,X > R(t),

(4.20) u = 0,

(4.21) au p(P . x=R(t)- ax

(4.22) ~ ~ ~ u X->+o19 (4.23)

au 0,

(4.24) u(x, 0) = uo(x) for x > R(0).

Proceeding formally, if u, R is a solution of problem (4.19)-(4.21) over 0< t < T in the sense of Definition 1 (modulo the above change of variables), then one has

00 ~ ~ ~ (t au (4.25) u(x, t) = E(x - z, t)uo(z) dz- E(x - R(s), t - s) - (R(s), s) ds

R(O) t ax where E(x - z, t - s) is the fundamental solution of equation (4.19). Now, differentiating with respect to x, taking the limit x -> R( t) + 0, noting the jump discontinuity in the second integral, one obtains for U(t) = (au/ax)(R(t), t)

U(t)= 2[ J -(R(t)-z, t)uo(z) dz

(4.26) R(O) ax -t aE -(R(t)-R(s), t-s) U(s) ds J0 ax

(4.27)=- 2[ -f E(R(t) - z, t)u'(z) dz

(4.27) R(O)

- dx (R(t)-R(s), t-s)U(s) ds

if uo is differentiable and, as required by part (iii) of Definition 1, uo(R(0)) =0. Integrating equation (4.21) one obtains

(4.28) R(t)=R(O)+t+P P |1 U(s) ds.

By a straightforward modification of the classical (v = 0) proof [4, p. 216] one obtains the basic result of this integral equation approach-that the differential equations (4.19)-(4.24) and the integral equations (4.27), (4.28) are equivalent.

THEOREM 3 [4, p. 221]. u(x, t), R(t) is a solution of problem (4.19)-(4.24) over the interval 0 < t < T if and only if U(t) is a continuous solution of the integral equation (4.27) over 0-< t< T, where R(t) is given by (4.28).




The integral equation can then be solved by a contraction mapping argument [4, p. 222] in the space C(O, T) with T sufficiently small provided the integral involving the initial data in (4.27) is continuous in 0 _ t < T This local solution can be extended globally (i.e. T = Xo) if I U(t)I remains finite [4, p. 223]. This is the only thing which requires checking in the present case.

THEOREM 4. If O 'uE C1(R(0), oo), u(R(O)) = O and uO(x) + 1 EL(R(O), oo), the problem (4.19)-(4.24) has a unique global solution.

Proof In the interval of existence U(t) is bounded from above by the maximum principle. Specifically, uo c 0 on R(0) < x < oo and u = 0 on x = R( t), 0 < t < T so that u ' O in R(t) < x < oo, 0 < t < T. Thus U(t) = (au/&x)(R(t), t)

- O. The lower bound can be obtained via the integral equations (4.27), (4.28). Indeed,

00 U(t) = -2 E (R(t)-z, t) u'(z) dz

J2R(O)

(4.29) rt 2(R(t)-R(s)) [ (R(t)-R(s))21 -2 J i 88r( t-s)3/2 U(s) exp - 4(t-s) Jds

Because U(t)O0, R(t)-R(s)'(t-s). If the (scaled) flow velocity were -1, then the last integral is positive so that

00 U(t)'-2 E(R(t)-z,t) dz maxlu'(z)l

(4.30) R(O) ? -2 max Iu (z)I.

Unfortunately in the case of interest v = 1 > 0, so that the last integral must be retained. Using (4.26) one has

1 1irt (4.31) U(t) ' -2 max l uo(z)I 2(t)1/2 +y-J U(s)(t -s)112 ds. This can be reduced to a Gronwall estimate as follows. Let

't

(4.32) V(t) = U(w)(t - W)-112 dw. 0

Then one has

V(t) - max l uo(z)I [ t 2Gr 0

(4.)+2 : (t -W)-l/2[ U(s)(w - S)-1/2 ds]

Upon changing the order of integration in the last integral it becomes

f U(s)[f (t - W)-1/2(W - s)-1/2 dw] ds

(4.34) t

= U(s)[ (t-s-w)112w-112 dw] ds.

The interior integral and the first one in (4.33) are equal to V. Thus

(4.35) V(t)'--maxu(z)j+ U(s) ds.




Combining this with (4.29) one has

(4.36) U(t) -(2+1r) max(Iuo(z)I+Iu (z)I)+2{ U(s) ds,

which gives a finite lower bound for U(t) for all finite t, completing the proof. We shall now show that these solutions remain in an interval about the special

solution (4.14) (modulo the change of variables above (4.19) with the same asymptotic value of XM = 1 (or equivalently UM = 1)). The proof we give is in the spirit of Friedman's (zero velocity, small data) proof [4, p. 226] but does not require the smallness condition. Moreover it is stronger than the best result available for zero velocity [7, p. 185] in that it does not require arbitrarily small adjustment factors in the velocity. For the proof it is convenient to look at a third, equivalent, version of problem (4.8)-(4.13) or (4.19)-(4.24). In the former, let x'= -x, t'= t and denote r(t) = -s(t) and u =-X. Dropping primes one obtains

(4-37) -~ a2 ,

a9 x > r(t),

(4.38) u =09

au x=r(t),

(4.39) axP. il (4.40) u- 1

(4.41) 4x->0 (4.42) u(x,, 0) = uo(x) for x > r(0).

Notice that between these equations and (4.19)-(4.24) the u's are related by a translation of - t in the x variable which is compensated for by R (t) = r( t) + t but that the initial data are identical since R(0) = r(0).

THEOREm 5. Suppose -1 uo(x):50,, u0(r(0)) =0,, UE C((r(0),ao0)) andUo u0+E L'(r(0), oo). Then there exist positive a, A such that for all 0 < t < 0,,

_ _ 1 (4.43) -1 t-pw a r(t) -r(0)? I wt+A.

Proof. The fundamental quantity in many of these calculations [4,, p. 226], [1, Thin. 2.4] appears to be q(t) = %t (u(x,, t) + 1) dx. Differentiating, one obtains

Integration gives

_ _ _ 1~~~ 1 (4.45) r(t)= 1 + wt +r(0) + . q (0)

- + q(t).

Because -I-- uo ?0 and u = 0 on x = r( t), one has, by the maximum principle, that q (t)?:i-0 so that

(4.46) r(t) -r(0)- t+A 1 + P00

with A = q(0) (I + P,,)-', giving the second part of (4.43). The trick now is to use this estimate to obtain the reverse one.




Letting w= u + 1, one has aW a32W aW

(4.47) at ---= +-, x> r(t),

(4.48) w= 1,

(4.49) ___aw x =r(t),

ax J (4.50) w > 0,

aw x -0 00, (4.51) ax J (4.52) w(x, 0) = wo(x) = uo(x) +1, x > r(O).

We shall bound this solution with that of

az_a2z4 az X> - Vt+ B, at ax ax

(4.54) z=1, x=-Vt+B,

(4.55) z(x, 0) = wo(x), x > B,

where V=(1+Po,) - and B= r(O)+A= r(O)+q(0)(1+Pcf) -. Now by (4.46)

r(t) ---- Vt +B,

so that by the maximum principle w(x, t) c z(x, t) for x > - Vt + B. Thus oo -Vt+B ao

q(t)= w(x, t) dx= w(x, t) dx+ w(x, t) dz

(4.56) r(t) r(t) -Vt+B Vt+B cc

w(x, t) dx+ z(x, t) dx. r(t) -Vt+b

Because 0 _ wo(x) - 1 and w(r(t), t) = 1, by the maximum principle w(x, t) 1 for r(t)


and E is the fundamental solution of (4.53),

(4.62) E(x -y; t -s) = 1

e ( x - y +

P[ (t-s SW

Now, straightforward calculation gives

aG (x - b(s); t - s) = E(x- b(s), t s)[ (t)

(4.63)

=-2- E(x- b(s); t-s)-(1 - V)E(x- b(s); t-s). ax

Thus

00

e(t)= z(x, t) dx b( t)

(4.64) = 5 G(x -y; t) wo(y) dy dx b( t) B

-|Lt [2 5 E(x-b(s); t-s)+(l- V)E(x-b(s); t-s)} ds] dx.

The first term is clearly dominated by |B wO(y) dy which is assumed finite, and the second, after interchange of order of integration, is

IT lo 4 ]

(4.65) -( ) | [J)(1 V)jf [ (x-b(s) +(-S))] dx] ds

= p exp[- 4 ]dp -r[J(_ V)(t_S)112/2 y] with the changes p = t - s, y = [x - b(s) + (t - s)]/2(t - S)1/2 respectively. The first is obviously uniformly bounded by the same integral out to +oo, and the second, by interchanging the order of integration again, is

i at fOO 1 y2 e-Y2 dy+ t e-Y2 dy, a 0 Jat

where a = (1- V)/2, and these last pair of integrals are uniformly bound.

C. Linearized shape stability analysis. The planar solution (4.14)-(4.16) of problem (4.2)-(4.7) will lose stability to a more complicated developing shape only at those modes which are shown to be unstable by the linearized analysis. In this section we shall prove that all transverse modes are stable at the linearized level. Again, in problem (4.2)-(4.7) we write S(x, y, t) = x - s(y, t), let x' = -x, y' = y and t' = t, denote r(y, t) = -s(y, t) and u(x, y, t) = -X(x, y, t) to obtain (dropping primes)

(4.66) a- =Au+-a-, x>r(y,t), at ax'




(4.67) u = 0,

(4.68) au auyar ar x=r(y,t), (4-68) ~ax ay ay 00 t'

(4.69) u > -1,

au X -> 0, (4.70) Ju 0

(4.71) u(x, y, 0) = uO(x, y), x > r(y, 0)= ro(y).

The planar solution (4.14)-(4.16) in these variables is

(4.72) up (x, t) = - (1 - e-(1 - V)(X+ Vt)

(4.73) rp(t) =-Vt

where

(4.74) V= 1 1+ Poo

Writing the solution of problem (4.66)-(4.71) as

(4.75) U. (x, y, t) = up (x, t) + --U(x, y, t) + 0 (- 2),

(4.76) r. (y, t) = rp(t) + -r(y, t) + 0(2),

the linearized equations in terms of u, r are

(4.77) at = au+xau x>-Vt,

(4.78) u = POOVr,

au ar t

(4.79) aa" =Pocg-PoV(1-V)r, (4.79) a~~x at

(4.80) u (x, y, t) -- 0,

(4.81) au (x, Y, t) J 0,

(4.82) u (x, y, 0) = (x, Ny), x > 0.

Because equations (4.77)-(4.82) are linear, there is no loss of generality in considering solutions of the form

(4.83) u(x, y, t) = Um(X, t) cos my,

(4.84) r(y, t) = rm(t) cos my,

with m discrete if the region has finite extent in the transverse direction (e.g. -L ' y ' L) with zero flux boundary conditions and m continuous otherwise. Equivalently we have expanded the solution in a Fourier cosine series (taken the Fourier cosine transform of the solution respectively). One has, dropping the subscript m,

(4.85) U-au a2u n2u+aU x>-Vt, a t ax 2ax




(4.86) U = POO Vr

(4.87) - = Pooi - Pc V(1 - V) r, (4.88) U(x, t) -> 0,

(4.89) aJ(X,t) -> 0,

(4.90) U(x, 0) = p(x), x > 0.

The existence of unique global classical solutions to problem (4.85)-(4.90) can be proven using functional analytical/integral equation techniques outlined in [3] and to be given in detail elsewhere. They can also be solved, in principle, using Laplace transforms. Here, however, we are primarily interested in the behavior of r.m (t) (the amplitude of the bump) as an indicator of whether the planar solution loses stability to the m-mode.

Before carrying out this analysis we scale equations (4.85)-(4.90). Letting x' = (1 - V)(x+ Vt), t'= (1 - V)2t and m'= mr(I - V) 1, r'= POOVr and recalling that 1 - V> 0, these equations can be written (dropping the primes) as

(4.91) at x>0

(4.92) U=r

(4.93) -=p.rP x=o,

(4.94) U(x, t) -> 0, t) u x -> 0

(4.95) ~~- (x, t) ->0, (4.95) ~~~azJ (4.96) U(x, 0) = p(x), x> 0.

Letting

(4.97) w(x, t) = (U(x, t) - r(t)) em 2t

in equations (4.91)-(4.96), one obtains

(4.98) aiia ? a+(+r+2r)em2t -0 x>0,

(4.99) w =0,

(4.100) - = (P, r-r) em'Xx= aw

(4.101) w -> r(t) e

(4.102) aw x - x 0, (4.103) w(x, 0) = p(x) - r(0), x > 0.

Now the Green function for (4.98) in the half-space x > 0 is

(4 104) G(x, z; t-s)= 1 ex _(x-z+(t-s))2 [-exp [ xz 2,rr( -S)12

P 4(t- S)ts




Thus the solution of problem (4.98)-(4.103) can be given explicitly as

co W(X, t) = f ((z) - r(O)) G(x, z; t) dz

(4.105) 0

0 ds [f G(x, z; t -s)(i(s)+m2r(s)) em2sdz]

From this one can obtain an integro-differential equation for the amplitude by differentiating with respect to x, setting x =0, using (4.100) and multiplying by em2t. Specifically, one gets

00

Poj1(t) - r(t) = f(p(z) - r(O)) em2tGX(0, z; t) dz (4.106) ?

- ds [(i(s) + m2r(s)) em2(ts) I GJ(0, z; t- s) dz]. Direct calculation gives

I' 1 1 0 (4.107) G,(0, z; t) dz = - e-t/4+ e-Ydy

whose Laplace transform is

(L40 Gx(0, z; t) dz) = (S+)1/2+-F 4 _ 1+ 1/2

(4. 108) 2 4

(S 2s

4s)S124 ~+ (s + )1/2 4

Thus, taking Laplace transforms of (4.107) one obtains, with F(s)= L(r),

P.(sF(s) - r(O)) - F(s) = L( v(z)Gx(0, z; t) dz e-m) - r(0)[2 s(+M2t4)]

(4.109) -(sr(s) - r(O) +m2r(s)) [2(S + M

41

=L(initial term) - F(s)E+( +Sm2+1) 1/2]

Solving,

(4.110) [P 2- + (s+m2+)1/2]r(S) =L(initial) + Pr(0).

The asymptotic behavior of r(t) is determined by the singularities of F(s), which are the solutions of

(4.111) (s+m 4 2 _

Since P. >0, one can easily see that (4.111) has negative, simple zeros if I mI>0 and a simple zero at s = 0 when m = 0. In summary, we have

THEOREM 6. For general data v, r(O), the amplitude rm(t) in equations (4.91)- (4.96) (or equivalently (4.106)) is asymptotically uniformly bounded for m = 0 and exponentially decreasing from ImI >0.



604 AUCHMU>TY, CHADAM, MERINO, ORTOLEVA, RIPLEY

The above analysis of planar redox fronts developing from fast reactions in the simple, irreversible model (? 2) suggests that these fronts are stable with respect to both planar and transverse perturbations. If one thinks of scalloped or other mor- phologically complex fronts as nonlinear restabilizations of linearly unstable modes [8], [6] then one must look to models with more structure [2] for the occurrence of this phenomenon. In particular, if one includes reaction-induced porosity changes behind the front and prescribes the velocity through Darcy's law then numerical simulations indicate that planar fronts lose stability to perturbations with long wavelengths.

Acknowledgments. The authors would like to thank the late Charles Conley for suggesting the method of proof of the original version of Theorem 2, and one of the referees who proposed the present proof of a stronger result.

REFERENCES

[1] J. B. BAILLON, M. BERTSCH, J. CHADAM, P. ORTOLEVA AND L. PELETIER, Existence, uniqueness and asymptotics for planar, supersaturated solidifying Stefan-Cauchy problems, Proc. Brezis-Lions Seminar, College de France, 1984, to appear.

[2] P. ORTOLEVA, J. CHADAM, J. HETTMER, E. MERINO, C. MOORE AND A. SEN, Self-organization in water-rock interaction systems, II: The reactive-infiltration instability, Am. J. Science, submitted, 1986.

[3] J. CHADAM AND P. ORTOLEVA, The stabilizing effect of surface tension on the development of the free boundary in a planar, Cauchy-Stefan problem, IMA J. Appl, Math., 30 (1983), pp. 57-66.

[4] A. FRIEDMAN, PartialDifferentialEquations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ, 1964. [5] P. HARTMAN, Ordinary Differential Equations, John Wiley, New York, 1964. [6] J. S. LANGER, Instabilities and pattern fornation in crystal growth, Rev. Mod. Phys., 52 (1980), pp. 1-28. [7] L. I. RUBENSTEIN, The Stefan problem, Trans. Math. Monographs 27, American Mathematical Society,

Providence, RI, 1971. [8] D. J. WOLLKIND AND L. A. SEGAL, A nonlinear stability analysis of thefreezing of a dilute binary alloy,

Phil. Trans. Roy. Soc. London, 268, (1970), pp. 351-380.



Article Contentsp. 588p. 589p. 590p. 591p. 592p. 593p. 594p. 595p. 596p. 597p. 598p. 599p. 600p. 601p. 602p. 603p. 604

Issue Table of ContentsSIAM Journal on Applied Mathematics, Vol. 46, No. 4 (Aug., 1986), pp. 525-720Front MatterVariational Methods for Approximating Solutions of ∇ u(x) = f(x) + k(x)u(x) and Generalizations [pp. 525-544]A Note on the Three-Particle Lattice and the Henon-Heiles Problem [pp. 545-551]Stability Criteria for a System Involving Two Time Delays [pp. 552-560]Radiative Transfer as a Propagation Mechanism for Rich Flames of Reactive Suspensions [pp. 561-581]The Stability of Liñán's "Premixed Flame Regime" Revisited [pp. 582-587]The Structure and Stability of Propagating Redox Fronts [pp. 588-604]Uptake Curves for Fickian Diffusion [pp. 605-613]A Cell Kinetics Justification for Gompertz' Equation [pp. 614-629]Global Analysis of a System of Predator-Prey Equations [pp. 630-642]First Passage Times for Combinations of Random Loads [pp. 643-656]On the Performance of State-Dependent Single Server Queues [pp. 657-697]An Automatic Method for Generating Random Variates with a Given Characteristic Function [pp. 698-719]Back Matter [pp. 720-720]

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