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Analysis of interface dynamics in the quasistationary Stefan problem
50
5.
6. 7.
On the asymptotic forms of the solution with a transition layer of a NESTERCVA V., Dokl. Akad. Nauk SSSR, 305, 6, perturbed hyperbolic system of equations, singularly 1989. 1350-1353, Singularly Pertubcd Equations in Critical Cases, VASIL’YEVA A.B. and BUTUZOVV.F., Izd. Moskovsk. Gos. Univ., Moscow, 1978. The method of the joining of asymptotic expansions IL’IK A.K. and LELIKOVA E.F., in a rectangle, Matem. Sb., 96, 4, 568-583, du-0(Z. v)u,-l(Z, v) for the equation 1975. Translated
U.S.S.8.Comp~t.Maths.Math.Phys.,Vol.29,No.5,pp.50-60,1989 Printed in Great Britain
by
E.L.S.
0041-5553189 SlO.OO+O.OO 01991 Pergamon Press plc
ANALYSIS OF INTERFACE DYNAMICS IN THE QUASISTATIONARY STEFAN PROBLEM* V.V. GAFIICHUK and L.A.
LUBASHEVSKII
The dynacics of dendritic structure formation are examined cf the quasista?ionary Stefan problem for a closed contour plane. A fie?d 0. defined on r and determining its topology introduced.
in a model F in a is
Introduction The non-equilibrium kinetics governing the formation and evolution of interfaces in systems involving a phase transiticn of the first kind led to one of the most striking - the growth of dendritic structures. Such phenomena crop up phenomena of self-organization chemistry and biology, e.g. the formation of dendritic structures in many branches of physics, in the crystallization of supercooled solutions, aggregation in diffusion-controlled reactions, mathematical terms, such effects are described by the classical probIn formai, and so on. of the interface separating the phases. lem - the propagation As an example, let us consider the problem of describing the propagation of the interas formulated in the electrostatic apprcxiior, fror: an electrolyte, face in electrodeposit In the regicr. of the electrolyte 0 (bounded by a contour F) there is a matlon (Fig. 1). q satisfying Laplace’s equation given electrostatic potentia;
with
boundary
60-0
(1)
VqI ,_,-rl&/f,
(2)
~lr=d,kc,.
(3)
conditions
8. is the characteristic Here 00 is a constant proportionai to the external potential, dr is the capillary length and k is the curvar is the radius-vector, surface potential, ture of I. The propagation of r is described by the normal translation velocity: v.=8(~$.n)lr. where
(4)
g is a constant of proportionality and n is the unit vector normalto I’. Formulae (l)-(O) in closed form determine the dynamics of motion of the interface. The distinctive feature of this motion is that a spontaneous local increase in the curvature k of I Induces a corresponding increase in the field gradient and (Vv,n) Ir thereby a local increase in the velocity of the Interface. It is the latter that determines the further evolution of such small perturbations. Thus, even in the context of this simple model the regular motion of the interface is unstable to a broad spectrum of perturbations, resulting in the formation of a complicated interface structure of the dendrltic type. In fact, even in this case, despite the apparent simplicity of the problem and its long research history 11, 21, we have practically no satisfactory non-linear theory of the formstion of even a single non-selfintersecting surface. Moreover, even the direct approach Of numerical simulation of problem (l)-(4) has not proved to be very efficient. *Zh.vychisZ.Nat.~t.Fiz.
,29,9,1331-1345,1989
51
FIG.
1
At present there are basically two approaches to the solution Of this mathematical problem. The first involves numerical simulation of the processes in a two-dimensional region, either by numerical solution of the partial differential equation itself [31, or by a Monte-Carlo technique describing the microscopic features of, say, aggregation in diffusioncontrolled reactions (see, e.g. [PI). The second approach considers the quasistationary Stefan problem (l)-14), using conformal mapping methods to analyze lnterfsce dynamics in a plane region (5, 61. In this paper, using methods of complex function theory, we define a certain effective field 1) on the interface. This field completely defines both the geometry and the velocity un of the interface. We shall thus be able to construct the effective surface dynamics of the interface and thus reduce the number of dimensions involved.
FIG. 2 1. Equations of motion To obtain the equations cf motion of the interface, we consider a closed COEtC'Jr r in the plane of z = z + iy, corresponding to the boundary of the regicn Q containing t.?e point at infinity of the plane (Fig. 1). The motion of this contour in the 2 plane is spccifled by the velocity u (s, t) along the normal to r (S is 2 natural parameter). Let f(o) be a conformal mappln”f of the exterior of a circle Q* (lol>R) in the plane of 09u+iv onto Q (see Fig. 1). witn a pcle of order 1 at the point 0-m such that dJ;dlLl.,.=l. Then the function In(df!do)-lnIdfidoI+irt(o). value of its real part on the boundary (1.1). satisfies the relation
Define
a field
n(6)
in the
interval
which r’= (0-R
(1.1) is regular exp(r8))
in Q:?, uniquely and, in addition,
defines the by condition
(0,2n]: q(e)-((djidolr.)-’
(1.2!
52
r under represents the dilatation of the boundary rl(0) lo identical with and the imaginary part Imh(&/do)(,-cp@) the tangent vectors to r and to r’ at corresponding points (see f0: ows immediately from the Schwarz lntcgral for the exterior of see, e.g. [71j that
The magnitude mapping /-‘. angle between addition, it (lo!>&
cf
the conformal the directed Fig. 1). In the disk
&=f(w,) and Thus the field uniquely defines the curve r given any initial point q(e) the equation of motion of the conformal mapping in fact reduces to the equation of motion of the fleid ?j(e, i) together with some distlngulshed point O-(2,(0, n(0). rl(R 1) we consider two positions To construct the equation of motion of the field where c is an infinitesimal of the contour I’ at close instants of time t and t+c, time increment (Fig. 2). and r, along the normal The distance between the contours r,+, n=n Therefore, if the conformal mapping + in3 to r, at a point 5 Is ev.(s, f). CC-&, 1s extended by continuity to a small neighbourhood of rr, then f,-’ carries into the curve r-(6-r(B)exp(ie)), r, into a circle (o'lR(t), and the contour r,+. In addition, the motion of the where, to within c, we have @)=R(t)+cV.(~(@), t)q(& 2). fixed point 0 is naturally determined by the correspondence of points Ot and O,,, along the normal to r,; hence under the conformal mapping 1-l their coordinates are &(I)-R(f) and e,(t)=e,u+t): o’=df(w) Of the bounded
by
Schwarz within
integral E
To construct a conformal mapping r,(t+r)-Ro(t)+ C&(8‘. t)q(e,, 1). onto the region of the o’ plane exterlor of the disk lUl>R(fte) and satisfying a condition analogous to (1.11, we can use the Repeating the arguments of [Bl almost word for word, we see that, to
r,,,
and
Here i.e.
and the
El=R(f-e)exp(l@J
angular
<, > denote
brackets
averaging
ever
the
ar.g:e
8.
I.r A(ejd6
Then the so that
reqcired
zonfcrmai
mapping
f;,‘,
is defined
by the
composition
I;!, (:)=b,f-‘-I,-‘(z).
(1.4)
and
0, f+e) to cvalLa:e the field corresponding to the contour r,+,,
Therefore,
to within
c
the derivatives in (1.4) are evaluated at the point namely at the point zr,,,+en(s. L)u.(s, 1). %,I+.:-
we have
I-iii-dig-’ I
11.5)
x E10rl(e’,
thtg y,eq.
0
where
8 and
Hence,
by (1.31,
the arguments to within e:
of complex
0 are
l
(J-o’
_*_c[u.(e)~(e)-(u.(e)?(e))l
dbf-'
Ido’ I
numbers
-‘;
0
R(r)
and
o such
that
@-6j(o),
(1.6)
53
Using C,
we
expressions find the
(?.3), required
the (1.5) and ‘1.6) and expanding equation oi mction for The field
ae
81 -rl(e,t)
i
i
PI-
[ q(e,t)Lkce,t)-G
q(B(B, e))
of
in powers
I”
aq(e.t) _-f R(f) 1
atl(e.0
function 9 (e. 0 :
2n D
c.(e’,f)~(e*,
!.lj2des-
tbtg
(1.7)
’ 3nre,,~~“.(e,,~,,e,]+ j!
+q(e,r~~..ce,f)$&
h r1(e’,thg2
e’-e
de,_
. 0 -q(e.
and the
equal:=:
In addition, if along the circle
for
the
the field r, is,
0 $k_
7 L‘. (8’. 0 q(e’. tktg A!+} . ,
radius:
see. t) by (?.3‘,
is knowr., the
equation
of mot;or? of the
fixed
point
8~
(1.9)
and SC, in the
phys;cal
z plane, d:,;d:=c.fe,.
The set of equations (1.7)-C1.101, of motion of the conformal mapFlng
2.
f)exp(ily(&)+6,)1
combined f, given
with (1.2!, the velocity
I.
il.iOi
constitutes h(e,
the
required
equations
1).
One-sided model
We first consider equations (1.7), (l.B), of and t: ri.8
a version of the Stefa:. prcblem C:)-(P), we have tc know the normal velocity of the th-ue.
Tc close contour
the system of r, as a function.
q. 2).
It should substantial localized
be noted that the formulation of the problem represented by (l)-(4) involves a simplification, in that vn is determined exclusively by the potential v of 6 to one side of F. (That is why we have called this the “one-sided model.“) must be harmonic, Since the function we can introduce a complex potential cp(r* V) and express condition (4) as Y 0)-Q@, u)+t@(+. I/) (2.1)
n-esp[i(e+$)] where is the complex representation of the normal vector the interior of Q. By the definition (1.23 of q (8. I) and the properties mapping f, Eq. (2.1) is equivalent tc
~~(8.t. q)-q(e.t)--since
p dY R(t) 38
r,
’
n pointing
of the
into conformal
54
0 are used to determine the curvawhen the boundary conditions (2), (3) for the potential k-(d/&s) (8+~+-x/2)-~(l-cd~/~8)/R ture of the ccntour the Schwerz integral for the exterior of the disk enables us to express the imaginary part of the complex potential ImY in expiicit fcrrr, as a functional cf expresde, t) and Q, and thus to reduce the required sion for the normal velocity ‘Jn to the form
c.(e,~,f)-p-
q
(8, t) R(t)
i
I”
a i t Tic;, f fl w, tktg dngoR(l)z
1
$2
de’-
(2.2)
solves the prctle? formuiated in this part of the paper, closing the equations (1.8’ cf fne field s(@. I). We *;11 analyse these eqJ8tlOns. In themselves, Eqs. (1.7) and !1.8! are not directly relareC tS tne piys::al mechanism determining the motion of the interface in real spa:e ; rat.ner , they deterr:ne the variation o f the law governing the mapping of a circle inrc t.ne contour r, given the normal veiocity V n of the contour. Thus the equation has a purely geometrical meaning, whereas the equation for v reflects the physical mechanism of
This forzla >f m-*. -r v.,_vl,
I.?),
the mo?ion cf r,. As a result, if the physical nature ofnthe system is changed, there is no need to change Eqs. (1.7) and 11.8), which retain their fcrm; what must change is expression (2.21 for vn. Nevertheiess, nctwithstanding its generality, this approach offers several substantial advantages in anaiysing interface dynamics in the two-dimensional Stefan problem. First, the governing equaticn (1.71, (1.8) and (2.2) of the field q(e. t) are one-dimensional, i.e. they concerr, surfaces, ur.?ikt the usual formulation of S?efan-type problems. Second, irrespective cf the spatial propagation and geometrical deformation of r,. the segment E)E[@. 2.11 remains fixed, SC that one can expect further development of numerical and analytic methods for sclving prctlens cf this type. Third, and finally, the method has a very attractive feature: twc ;cin:s that are close together cr. the active side cf an evolving front rl, but net necessarily close together if the distance is measured along the contcur itself, correspond to nearby points on the 6 axis. This point is of paramount importance, as will become evident later when we determine inhomogeneous distributions of interfaces in the form of dendritic structures. To investigate the properties of the solutions of Eqs. (1.7) and (1.81, we consider the linear stage of instability of a cylindrical interface. Consider a perturbation of the q==l+-by esp(ine), where n = +(1,2,3,...). Linearizing Eqs. (1.7) and field 0. 1) w we obtain the following dispersion equation: (1.8) and formula (2.2),
COB y--gy [ n-2 where
-2
(n’-l)n].
(2.3)
+q,‘bq.
As is clear from Eq. (2.3:, if d,-U the main instability is due to shortwave corresponding to the prevalent physical notion of interface instability in perturbations, Stefan-type problems 12, 31. For small values of d,+O the system is unstable to perturbations over a broad spectrum of wavevectors, which explains the complex fractal structure of dendrltic formations. The surface ‘energy (capillary length) imposes an upper bound on the instability region of the wavevectors and dictates the minimum radius of curvature r of a dendrite. This damping effect of the surface energy on interface evolu1 tlon has two Important consequences: first, it determines the relatively regular structure of the Interface on scales of the order of rz, and, second, it implies that to a first approximation one can legitimately ignore the surface energy for the spatial structure of a dendrite on scales significantly exceeding r2. As will be shown below, the first o f these two features will enable us to establish a fairly simple relationship between the one-sided and two-sided Stefan problems. The second will give us a way of allowing for the effect of surface energy on dendritic growth in the quasilocal approximation.
55
Two-sided model
3.
In many physical problems, such as the propagation of the crystallization front in a supercooled liquid, the dynamics of the phase interface r
depend
regions
on the
on both
distribution
sides
of the field
of r.
In such
9
in
situations,
in the quasistationary approximation of the Stefan problem, the field q satisfies Laplace’s equaon r and at tion (1) at all points of space; the FIG.
point
at
infinity
the
described by (2) and city is defined by
3
distribution
(3),
but
of
now the
p is
normal
velo-
(3.1)
~.-B[(~O’~n)-(VO-.“)lIr,, where
q-
9’
Q-1.
(or
to
assumed keep
(or
CJ-) is
the
value
In addition,
the
be significantly
within
of the potential in the L, of the contour,
exterior
length
greater
than
rl,
which,
even
(resp.,
at the
properly
interior)
initial
speaking,
time
makes
it
of
r:
t = 0,
is
possible
to
the
bounds of the quasistationary Stefan problem. As already observed, the surface energy of the interface determines the growth of dendritic structures mainly on a characteristic minimal scale r It is thanks to this feature that we shall be able, still 1’ using the quasistationary approximation, to establish a simple relationship between the one-sided and twz-sided Stefan probiens. When the one-sided selfintersecting
we have
d,-0.
Stefan protlerr. cor~toilr
and problem
a--o (l)-,A;.
Since
a small
parameter,
Vn as a sum: for
the
one-sided
speciai
aspects
with (3),
Stefan
BP=0
in the first
enables
problen
with
(3.1)
is
exactly
mean curvat’ure
the
of
equivalent
a closed
to
non-
k(s’)d.s’=Zn.‘L:
we can
rewrite
terr
1.X n
is
the
seccnd,
d&J.
problem,
boundary
formula
corresponds
(3.1)
s:m;ly
to
the
which the
for
the
previous
essentially
set
normal
velocity
formtiia
of equations
(2.2)
re;resents
the
fli-‘31,
!3.1)
equations
Cl)-
condition.
consider us to
it
The first
of the two-sided
We will (3.1)
charge vectors of r,,
and neglecting
10IA “,==C. $.
(1:1-i3i,
Lt”rZ,
the
restate
latter the
is distributed on a contour illustrating the identity measured, e.g. from a point
situation.
two-sided
In this
problerr
at
the
case
se?
of
follows: a certain I‘, (the circie in Fig. 3 shows the distribution of (3.5)) with surface density q(s\ !S is a natural parameter 4) satisfying the integral eqaatior.
q-0
as
(3.2;
and,
since
r2/Lt
is small.
the
totai
charge
is,
to
a first
approximation,
4 I 0 The normal
velocity
Ye introduce
u (”
of
r,
in this
q ($1ds=O.
case
by
expression
(3.3)
is
for
the
defined
by
u,=4x;q(s).
annew variable j q(s’)ds’.
p(s)=
which,
is
(3.3)
a well-defined potentials
function q-* cc-
q(r) --
0
of s
2
on
1‘,.
be written
can
In terms as
of
this
new variable
jdsl-dlz.-zl 127-11
the
(3.4)
r,
follows
For further manipulations of the integral (3.4), directly from the geometry of the problem (see
we need Fig. 3):
a system
of
identities
which
56
d(z,-zJ
1
-li-
(s,-r)ds
IL--iI’
Il.-2)
(3.5a)
(z,-z)ds-Rel(z,-2)~~-i(2.-z)Xdsl-Re[(z~)ds],
where zs-z is the radius-vector corresponding complex number, along
ar,d
r,
the
between
ccrresponding
complex
functions
, we can associate’ Y-: Re V--othe in O-; thanks tc ‘t.41, (2.5’ and
the
ds and
and with
the
the
potentials
Z;-Z is the conjugate of the an infinitesimal displacement Since 0’ and v- are harmonic
Yy-: Re Y*-q+
in
to
q(S)
potentials
of
Identity
of
vector
respectively.
complex
relaticnship the
are
number,
them
S and zs,
points
ds
(3.5b)
the
by a Cauchy
Is,-tl’-(L.-Z)(~).
Y(z)--
2
variable
Q* and is
defined,
integral
jP(S)A, .
(3.6)
r, in
the
sense
va:ue
The except of
that,
cf the
pci.nts
the
Y-(z)
function
rcle
f ST the
of
pfs)
contour
2
the point Y-(:) or
potential
hclocorphic
tf,ar
as
according
complex
wii: (see
then
lies
in
in Q-
can
be played
2 c in
Q-.
is a real ccnstant. In order to est;mate 6s
reiaticnsh:; the
will
be
between
tangent
the 7
vectcr
d:,=dsexp(lQ).
:ke
evident
real frc~
Se(s)
to
the cne
constant the
c.ur;‘atore
On the
also a
by
the
be
integral
(3.6)
gives
by a Cauchy
represented
Certain
d:,+e 2.-z
Tt c
Q-,
the
purely
imaginary
integral,
function
g(S)
IS;): V’_(z)=2i
where
Q’ or
Y-(z).
of
kis!
contour hand,
c,
sequel, and
the
we consider
a charazteristlc whence
g(s)-d,k(s)q,. ccntcur ar,d
the
tke
(3.7)
’
axis,
03
integrai
tne
that
(3.7)
mean it
angle
point
fellows,
using
8-e-Ij-X/2 ds.
k(s) s-d!!
the
betweer.
and
mcreover
yields
Re Y-(z.)-d,q,!R:-c. vhere
Rc is
potential, +he c
Thus
a ckaracteristic of
real
the
same
constant
linear crder
cf
unit
a small
c is
assumed
here we car; put C = G. Comparing the integrals
points
of C-:
on
magnitude
parameter
(3.6)
On the
r:.
as
and
is
of type
P /L,
(3.“;,
that
the
enables
us
an explicit sion
exFresrion in
relation
explicitly
to
expression
immediately By (3.6!,
yields the
is
p(s)-q(r) to
cur
determine
for
g as
tcundary
lim:t
the
to of
expression
condition
of
ar. identity
the
value
of
valid
at
all
in
c-+
interior
for
p(s)
geometrical the
the
and
of
r,
g!s);
velocity
function
of u
r,,
of
Rr Y’--qn
(see if
this
where
we can
(3.5t!
and
the
is the angle between express ,3.6‘ as
aI
geome trical
?he
situation
radius-vector
#(s’)dazI
-5
r,
l.~-.,Wr,-
S
- s and
4q.k
(~1.
expres-
can
be
written
(3.6)
illustrated
2
we have
r,.
-doq.k(s).
identity
i21).
result
hence,
parameters
normal
this
r-c-4-r,
Using
the
accuracy
the
hclomcr~h~r
geometry
between
the for
(3)
mear.
tc w:ttir
a fnnrticn
describe
relationship
a function
des ired
the
the
namely
goal,
SC
2
the
Rt ~((;,)-dc~,(k)-do~,!L,.
d---IL-o, :,--:
I Ip(s)-igb)]
so
hand,
q(&)-
we ottain
T,
Considered
other
T‘(” ),
:r.
the
Fig.
0~ axls.
3,
we have
Using
this
formuia
(3.9)
as
57
a+ n
b
a FIG. Isolating
le,-q* of
on the
term
the
integrating over a part the equivalent equation
of
(3.5~
due to
contained
r,
4
in
the
an
singularity
in the
2ng(s)-2! f(s’)da, r,
behaviour
nelghbourhood
infinitesimal
of
of s,
Qz and
we obtain
(3.10)
-d,q,k(s),
It should be noted that in this which the integral tern EC longer has a singularity. Eq. (3.10) is considerably simpler thar. the equivalent equation (3.2), since it is essentialiy a Fredholm integral equaticr cf the second kind, whereas (3.2) is a Fredholm eqilaticr. cf the first kind. If the interface is ir fact a circle, i.e. the characteristic curvature of the contour kar-‘, then, since the cara-eter r .:5 :s small, the first term in (3.10) is iS SUCh that 1 g(s)=(2n)-‘d&(s). dominant and in
case
New ccnsider
the
quar.ri:v
A,-[;rk(s)l-‘Sk(s’)da.
(3.11)
r,
for a developed $eomeTric strdct’we r2 linked by straight lines cf lengtt the
integral
For
“flatter”
(3.11)
character;st::
interfaces,
developed
strongly the
interface
ing
larger
is
for
the
denditic
geometry of
values
In view a solution
FCirtS
absclute
illustrated A
valce
structure
(see
Fig.
aforesaid, of the cquatlor.
of
IF:&. La‘, :, w5ere
the
e.g. a ccl:ectior. cf se?lcir::es Cirert by ass;mFtion 1 >> r 1’ .L=O.i6 8_ and F shows that
of radius evaiuat:or.
Than
fcr
cI
one has in Fig
4c),
but
.i
is
/.i_iai,
zess
;,
and
moreover,
as may be verified
4b.
Naturally,
s;ich
formations
a first apprcximation in the form (3.10):
of
there are the
exist rare
in
solution
a
by analys:ng.\, topologies denditic under
for
cfr,,yieldgrowth.
consideration
g(si=~(2rr)-‘d,~,k(s), where
the
parameter
form-factor value
A,
constant of F-((l-.1.)-‘).
F la certain by the
formula
the
of
A,=-1.
and
(3.12) order of unity) We also note
is related to the that the correction
mean terms
to
formula (3.12) for the solution of Eq. (3.10), due to its non-local nature, are oscillato the value of the function ptr, need not have a cardi; hence their contribution nal effect on the expression obtained using (3.12). Summing up, we see that (3.12) provides an approximation for the normal velocity of p(s)+i2b‘gr*(s) where the contour function y(d) (t)+(dlds)P(s). is the limit the contour
tory
of a certain cimplex function holomorphic analogous to the equivalent assertion for
length
region
Q*.
one-sided
Stefan
This
assertion
problem
with
is
completely
capiliary
2Fd,. In other
problems
in the the
are
words,
equivalent,
in the
approximation
subject
to
of
a suitable
(3.12)
the
one-sided
renormalization
and two-sided
of the
capillary
Stefan length:
A--2Fk.
the
Quasilocal approximation 4. the We now formulate an approximation that establishes normal velocity un and the field n for the quasistationary
local
relationship
Stefan
problem.
between This
approxi-
simulation of dendritic structures, considerably enhances the prospe,. -7s for numerical since the computation of the double integrals in (2.2) is a lengthy affair. In investigating the effect of surface energy on dendritic growth dynamics, we limit ourselves as before to studying the least characteristic scale r2 on which the interface we consider the one-sided geometry is relatively regular geometry. To simplify matters, that, as shown in the previous section, the results can Stefan problem, noting, however,
mation
USSR
29-5-E
58
be immediately generalized cut Ijefore serting the physical background.
to the two-sided problem. the mathematical theory of quasilocal Consider a relatively regular interface
where
integer
Ri
-2’r
say that
P
of
the
with
1’ is
interface
radius
fi.
(Fig.
tionai
tt
the
3!,
and,
radius
We no*
whose absolute
regular
containing R.
on these
surface
energy,
to
inhomogenelry
of
I-,
w:tr,;n of
s scaled
the tc
value
scales,
a characteristic second,
that
the
length
(see
Fig.
3).
-/?.
gradient
the
tounds
s-ale
Rir
Ri
is
is
of the order
we mean,
inhomogeneity
s.
es:imatez~hetnorkal
to
at
i an
reiatively
si
of of
‘?*p
of
the
present
approximately
approximation, on several
first,
is
that
part
entire
in a disk
the
of
When we
unity. the
localized
potential
the
of
that
we describe scales Ri,
contour
p on the
model.
The limiting
d&k,.
where ki
part of
is
propor-
interface
potential
is
the
due due
ql
mean curvature
i.e., ,+a,
5 k(s’)ds’
k,-R,-’
Therefore, value
the
of
the
contribution
rorma;
ccrrespcnding
gradient
is
to
each
all
and the
is
(V,q),--d,q,k,/R,
these
contributions:
scale
by summing
cttained
total
(4.1) ,-I,
Since
“rL
by its
15
functlcn
ex~oneztla;
ar.
continuo,us
of
i,
we can replace
the
estimating
expression
(4.1)
limit:
lA.2!
Corpar;ng identical :f
are
which
is
‘4.3; with we identify
leg;timate,
pra:tically
the
since
same for
or
v
of
mation: gral
is
.
the
the
sign,
quantity assigning
analogous
we obtain
contcur
to
the
the
it
undertaking
r
direct?
y tc
derive energy
rl(8.i the
in the
integral
value
Pcincare-Bertrand
required
This
surface
fixed
relation
of
from
‘i.2’,
t.he contc,‘~r
Is’-si:r;,
due tc
5. Simulation Equations (1.71, this
scaies
pclnts
quasilocal approximation. We will now proceed city
;ts rigcrc~s ana;og,;e tne integrals
I-, is
the (Z.2:
that
1s in fart
in far:
required
and its
the
formula
:wt
ds,d6
ass;mpt:on
between
the
expressicns
and
reg;:ar
the
cf
ve:o-
apprcxi-
thrcugh
forwart
fcllcw;;.g
is
the
norma;
quasiloca:
i .7 tr.e
car: be crc:gnt using
the
basic
re;atio-
gecmetr::.
Then,
n(@). comm’utatioa
we see
the
ider.tity,
inte-
wr.:ch
:G:,
(2.2):
dendritic
structures
(1.8) and (4.3) was the approximation
been investigated numericaliy. Underlying integrands f by trigonometric pc?ynomia?s [lo]:
have
of zr-,
1.W;~
!(edsin[n(e-8,)
e-e, (5
]ctg-_,
._I
where 8,=nk n. k-0. 1,. . ..2n-1. Expansions of this type make it possible quadrature formulae to evaluate singular integrals [lo]. As a result, tions (1.7), (1.8) and (4.3) reduces to a system of ordinary non-linear ~,-o,(n~
.Q.._,. R)
with
full
Jacobian;
this
system
was
integrated
to
use
wei;-known
the system equations using
of
a program
equa-
59
FIG.5 do=lO-‘. r+,-~-1. I,-O. fr-750. f,-z 1 lo’. f,-3.4.10’, ~(Io!-1-0.1 COSi38), R(f,)_lO
FIG. 6. Initial perturbations ~(9)~,~~1-10-~~~~(39): &-9.i. opp~.1; lo-O. R(t,)=R,=10. 11-667. RV,)-R,-38, r,=z 10’. R(I*)=&= 69. r,-3 lo’, R(W-&-94; graph c - field distribution at time f0 and 13 from [ill. The required accuracy was achieved by suitable choice of the number N = 2n and the local error e. In the actual numerical experiments we took A’ = 61 and c=IO-“, the required computing time on the EC-1060 computer was generally at most one hour. Knowing the values of and and the appropriate radii R(t), and applying the tl(6, t) ct(6, f), transformations dy--
R(tld6 sinlB+rC(B,t)+n/2], l-Ice*t)
was possible to reproduce geometrically defined interfaces in parametric form. Figure 5 and 6 illustrate the characrerlstic form of non-uniform distributions of the field (graphs a) and the COrrespOnding interfaces (graphs b). As examples, l(e, t) UC determined different snow-flake shapes, depending on the form of the initial perturbatlon. The reader should observe that relatively smooth variations of the interface correspond to quite marked variations in the field s(e, i) and hence also in the angle O(8, t). Thus, relatively small errors in determining do not cause marked changes in the n(g. 1) intsrface geometry. In addition, as is evident from the figures, relatively even long sections of the interface contract when mapped into the interval 10, 2x1, whereas pronounced changes in the contour f. conversely, are etretched out. In this sense such mapit
60
with respect to information about the detailed are “adaptive” pings cf r onto [O. 2x1 peometlc struct’ure of the interface. as these solutions Correspond to existing physical ideas, In conclusicc, we note that, there seem tc be good prospects for further success in the purposeful numerical modelling of when the system involves the effect of uncorrelated sources of noise non-uniform structures, on the phase interface; it should aiso be possible to analyze the behaviour of the solutions This will make it possas a function of the surface energy and anisotropies in the system. to reconstruct the entire picture of self-OrganizatIonal phenoible, in the finai analys:s, mena in the interface. Incidentally, the universal nature of the governing equations as derived in this paper indicates that they should be applicable in many problems of physics, chemistry and biology. the ast5ors wccld ;ike to thank 1.1. Lazurchak and I.V. Tyslyuk for their Finally, and also A.A. Dorodnltsyn for his constant help ln carrying out the numericai experiments, interest. REFERENCES
1.
2. 3. A.
6. 7.
8. 9. 1C
li
Fror,t,
Interfaces and Patterns, Fhyslca. D12, NOS. 1-3, 1984. LANGEF J.S., Instabiiitiesand pattern formation in crystal growth. Revs. Mod. Phys., 52, 1, l-26, 198f. lJMAN”=EV A .,,F. VIN3GKAXC V.V. and B%!SW V.T., Simulation of the evolution of .Krisrallographiya, 3i, 5, 1002, 1986. dendritic s*ructure. c Monte-Carlo approach to dendritic growth. J. Phys. SZEP J., CSEFTI 2. and KEIESi J., 1965. A, i8, 8, Liz3-L418, Phys SHRA.IMkl;i. and BEN,Ei.~~X S. , Singularities ir, non?ocal in?erface dynamics. fiev A, 3t, 8, 264S-28aE-- 1 1964. Phys. Rev. A, 33, 2, 1302-1308, BENEIM;:: L. , Irsta!zi:ity cf viscxs fingering. 1586. LAVEEST'YE', K.k. ar,dSBXU'T E.V., Methods of the Theory of Functions of a Complex Variable, Nauka, Moscow, 1976. cf similar regions. Uspe6r.iRat. Nauk, 11, SIRYK G.V., Cr.2cr,form21ma;rir.;; 5(71), 57-65, 1956. MUSKHELISHVILI N, ! Sing;;:ar Integral Equations, Nauka, Mcscow, 1968. BEiCTSEW.?‘TSEi 1 S M[ and L: Fkhit'i I.K., Numerica: Methods in Singuiar Integral Equations, Nauka, Mcsccu, iSEE BSRK'E G.2. and H:i;DMAFCu ODE sc:vcrs: a rev;er cf z'drrent and coring .._../. C. , Sriff attractlsns.
J. Cor;lr.
Pr.gs ) ?C, 1, :-62,
IBE”. Translated
U.S.S.R.Co’ll,“rct.Mcri.s..?/~:;..F;;~~.,:’c1.2C,!i~.5,;;.6t-B~,198~
Printed
ir. Great
::‘:-===-‘eg____I
c 199;
Britain
by D.L.
~:C.OGco.oo
Fergamcr.
Press
plc
THE SOLUTION OF PROELEVS IN ELASTICITY THEOi?Y BY COMPLETE-SYSTEV METHODS" E.I.
BECFALC\'A
The main pcinrs of complete-system methods are presented as it app:les to sc:ving prOb:emS concerning the static and free vibrations of inhomogeneous anisotroplc bodies in a variationai se?ting. The characteristic feature of these method is the reduc tion of an initially N-dimensional problem to a system of A! interrelated one-dimensionai problems. IJn:lke the common variational approaches, one no longer has any freedom with regard tc the choice of basis functions with respect to some of the independent variables. Some computationa? aspects of the approach are :?lustrated by a specific example. Introduction The most common. mathematical modeis for the mechanics of deformed bodies are dlffercntlal problems in relation to vector-valued functions of several variables, subject to various boundary conditions in Irregularly-shaped regions; the inhomogeneity of the properties of the elastic medium is represented by different variable parameters. Such problems can be handled by various methods, among them such familiar techniques as projection, varlatlona?, finite-difference, finite-element etc. methods; a recent addition to the list consists of complete-system methods (CSMs) (1, 21 which apply the sophisticated tools now available for solving one-dimensional protl ems to prcblems involving several dlmenslons, *Zh.vychisZ.Mot.mat.Fiz.
,29,9,13A6-1353,1989
5.
6. 7.
On the asymptotic forms of the solution with a transition layer of a NESTERCVA V., Dokl. Akad. Nauk SSSR, 305, 6, perturbed hyperbolic system of equations, singularly 1989. 1350-1353, Singularly Pertubcd Equations in Critical Cases, VASIL’YEVA A.B. and BUTUZOVV.F., Izd. Moskovsk. Gos. Univ., Moscow, 1978. The method of the joining of asymptotic expansions IL’IK A.K. and LELIKOVA E.F., in a rectangle, Matem. Sb., 96, 4, 568-583, du-0(Z. v)u,-l(Z, v) for the equation 1975. Translated
U.S.S.8.Comp~t.Maths.Math.Phys.,Vol.29,No.5,pp.50-60,1989 Printed in Great Britain
by
E.L.S.
0041-5553189 SlO.OO+O.OO 01991 Pergamon Press plc
ANALYSIS OF INTERFACE DYNAMICS IN THE QUASISTATIONARY STEFAN PROBLEM* V.V. GAFIICHUK and L.A.
LUBASHEVSKII
The dynacics of dendritic structure formation are examined cf the quasista?ionary Stefan problem for a closed contour plane. A fie?d 0. defined on r and determining its topology introduced.
in a model F in a is
Introduction The non-equilibrium kinetics governing the formation and evolution of interfaces in systems involving a phase transiticn of the first kind led to one of the most striking - the growth of dendritic structures. Such phenomena crop up phenomena of self-organization chemistry and biology, e.g. the formation of dendritic structures in many branches of physics, in the crystallization of supercooled solutions, aggregation in diffusion-controlled reactions, mathematical terms, such effects are described by the classical probIn formai, and so on. of the interface separating the phases. lem - the propagation As an example, let us consider the problem of describing the propagation of the interas formulated in the electrostatic apprcxiior, fror: an electrolyte, face in electrodeposit In the regicr. of the electrolyte 0 (bounded by a contour F) there is a matlon (Fig. 1). q satisfying Laplace’s equation given electrostatic potentia;
with
boundary
60-0
(1)
VqI ,_,-rl&/f,
(2)
~lr=d,kc,.
(3)
conditions
8. is the characteristic Here 00 is a constant proportionai to the external potential, dr is the capillary length and k is the curvar is the radius-vector, surface potential, ture of I. The propagation of r is described by the normal translation velocity: v.=8(~$.n)lr. where
(4)
g is a constant of proportionality and n is the unit vector normalto I’. Formulae (l)-(O) in closed form determine the dynamics of motion of the interface. The distinctive feature of this motion is that a spontaneous local increase in the curvature k of I Induces a corresponding increase in the field gradient and (Vv,n) Ir thereby a local increase in the velocity of the Interface. It is the latter that determines the further evolution of such small perturbations. Thus, even in the context of this simple model the regular motion of the interface is unstable to a broad spectrum of perturbations, resulting in the formation of a complicated interface structure of the dendrltic type. In fact, even in this case, despite the apparent simplicity of the problem and its long research history 11, 21, we have practically no satisfactory non-linear theory of the formstion of even a single non-selfintersecting surface. Moreover, even the direct approach Of numerical simulation of problem (l)-(4) has not proved to be very efficient. *Zh.vychisZ.Nat.~t.Fiz.
,29,9,1331-1345,1989
51
FIG.
1
At present there are basically two approaches to the solution Of this mathematical problem. The first involves numerical simulation of the processes in a two-dimensional region, either by numerical solution of the partial differential equation itself [31, or by a Monte-Carlo technique describing the microscopic features of, say, aggregation in diffusioncontrolled reactions (see, e.g. [PI). The second approach considers the quasistationary Stefan problem (l)-14), using conformal mapping methods to analyze lnterfsce dynamics in a plane region (5, 61. In this paper, using methods of complex function theory, we define a certain effective field 1) on the interface. This field completely defines both the geometry and the velocity un of the interface. We shall thus be able to construct the effective surface dynamics of the interface and thus reduce the number of dimensions involved.
FIG. 2 1. Equations of motion To obtain the equations cf motion of the interface, we consider a closed COEtC'Jr r in the plane of z = z + iy, corresponding to the boundary of the regicn Q containing t.?e point at infinity of the plane (Fig. 1). The motion of this contour in the 2 plane is spccifled by the velocity u (s, t) along the normal to r (S is 2 natural parameter). Let f(o) be a conformal mappln”f of the exterior of a circle Q* (lol>R) in the plane of 09u+iv onto Q (see Fig. 1). witn a pcle of order 1 at the point 0-m such that dJ;dlLl.,.=l. Then the function In(df!do)-lnIdfidoI+irt(o). value of its real part on the boundary (1.1). satisfies the relation
Define
a field
n(6)
in the
interval
which r’= (0-R
(1.1) is regular exp(r8))
in Q:?, uniquely and, in addition,
defines the by condition
(0,2n]: q(e)-((djidolr.)-’
(1.2!
52
r under represents the dilatation of the boundary rl(0) lo identical with and the imaginary part Imh(&/do)(,-cp@) the tangent vectors to r and to r’ at corresponding points (see f0: ows immediately from the Schwarz lntcgral for the exterior of see, e.g. [71j that
The magnitude mapping /-‘. angle between addition, it (lo!>&
cf
the conformal the directed Fig. 1). In the disk
&=f(w,) and Thus the field uniquely defines the curve r given any initial point q(e) the equation of motion of the conformal mapping in fact reduces to the equation of motion of the fleid ?j(e, i) together with some distlngulshed point O-(2,(0, n(0). rl(R 1) we consider two positions To construct the equation of motion of the field where c is an infinitesimal of the contour I’ at close instants of time t and t+c, time increment (Fig. 2). and r, along the normal The distance between the contours r,+, n=n Therefore, if the conformal mapping + in3 to r, at a point 5 Is ev.(s, f). CC-&, 1s extended by continuity to a small neighbourhood of rr, then f,-’ carries into the curve r-(6-r(B)exp(ie)), r, into a circle (o'lR(t), and the contour r,+. In addition, the motion of the where, to within c, we have @)=R(t)+cV.(~(@), t)q(& 2). fixed point 0 is naturally determined by the correspondence of points Ot and O,,, along the normal to r,; hence under the conformal mapping 1-l their coordinates are &(I)-R(f) and e,(t)=e,u+t): o’=df(w) Of the bounded
by
Schwarz within
integral E
To construct a conformal mapping r,(t+r)-Ro(t)+ C&(8‘. t)q(e,, 1). onto the region of the o’ plane exterlor of the disk lUl>R(fte) and satisfying a condition analogous to (1.11, we can use the Repeating the arguments of [Bl almost word for word, we see that, to
r,,,
and
Here i.e.
and the
El=R(f-e)exp(l@J
angular
<, > denote
brackets
averaging
ever
the
ar.g:e
8.
I.r A(ejd6
Then the so that
reqcired
zonfcrmai
mapping
f;,‘,
is defined
by the
composition
I;!, (:)=b,f-‘-I,-‘(z).
(1.4)
and
0, f+e) to cvalLa:e the field corresponding to the contour r,+,,
Therefore,
to within
c
the derivatives in (1.4) are evaluated at the point namely at the point zr,,,+en(s. L)u.(s, 1). %,I+.:-
we have
I-iii-dig-’ I
11.5)
x E10rl(e’,
thtg y,eq.
0
where
8 and
Hence,
by (1.31,
the arguments to within e:
of complex
0 are
l
(J-o’
_*_c[u.(e)~(e)-(u.(e)?(e))l
dbf-'
Ido’ I
numbers
-‘;
0
R(r)
and
o such
that
@-6j(o),
(1.6)
53
Using C,
we
expressions find the
(?.3), required
the (1.5) and ‘1.6) and expanding equation oi mction for The field
ae
81 -rl(e,t)
i
i
PI-
[ q(e,t)Lkce,t)-G
q(B(B, e))
of
in powers
I”
aq(e.t) _-f R(f) 1
atl(e.0
function 9 (e. 0 :
2n D
c.(e’,f)~(e*,
!.lj2des-
tbtg
(1.7)
’ 3nre,,~~“.(e,,~,,e,]+ j!
+q(e,r~~..ce,f)$&
h r1(e’,thg2
e’-e
de,_
. 0 -q(e.
and the
equal:=:
In addition, if along the circle
for
the
the field r, is,
0 $k_
7 L‘. (8’. 0 q(e’. tktg A!+} . ,
radius:
see. t) by (?.3‘,
is knowr., the
equation
of mot;or? of the
fixed
point
8~
(1.9)
and SC, in the
phys;cal
z plane, d:,;d:=c.fe,.
The set of equations (1.7)-C1.101, of motion of the conformal mapFlng
2.
f)exp(ily(&)+6,)1
combined f, given
with (1.2!, the velocity
I.
il.iOi
constitutes h(e,
the
required
equations
1).
One-sided model
We first consider equations (1.7), (l.B), of and t: ri.8
a version of the Stefa:. prcblem C:)-(P), we have tc know the normal velocity of the th-ue.
Tc close contour
the system of r, as a function.
q. 2).
It should substantial localized
be noted that the formulation of the problem represented by (l)-(4) involves a simplification, in that vn is determined exclusively by the potential v of 6 to one side of F. (That is why we have called this the “one-sided model.“) must be harmonic, Since the function we can introduce a complex potential cp(r* V) and express condition (4) as Y 0)-Q@, u)+t@(+. I/) (2.1)
n-esp[i(e+$)] where is the complex representation of the normal vector the interior of Q. By the definition (1.23 of q (8. I) and the properties mapping f, Eq. (2.1) is equivalent tc
~~(8.t. q)-q(e.t)--since
p dY R(t) 38
r,
’
n pointing
of the
into conformal
54
0 are used to determine the curvawhen the boundary conditions (2), (3) for the potential k-(d/&s) (8+~+-x/2)-~(l-cd~/~8)/R ture of the ccntour the Schwerz integral for the exterior of the disk enables us to express the imaginary part of the complex potential ImY in expiicit fcrrr, as a functional cf expresde, t) and Q, and thus to reduce the required sion for the normal velocity ‘Jn to the form
c.(e,~,f)-p-
q
(8, t) R(t)
i
I”
a i t Tic;, f fl w, tktg dngoR(l)z
1
$2
de’-
(2.2)
solves the prctle? formuiated in this part of the paper, closing the equations (1.8’ cf fne field s(@. I). We *;11 analyse these eqJ8tlOns. In themselves, Eqs. (1.7) and !1.8! are not directly relareC tS tne piys::al mechanism determining the motion of the interface in real spa:e ; rat.ner , they deterr:ne the variation o f the law governing the mapping of a circle inrc t.ne contour r, given the normal veiocity V n of the contour. Thus the equation has a purely geometrical meaning, whereas the equation for v reflects the physical mechanism of
This forzla >f m-*. -r v.,_vl,
I.?),
the mo?ion cf r,. As a result, if the physical nature ofnthe system is changed, there is no need to change Eqs. (1.7) and 11.8), which retain their fcrm; what must change is expression (2.21 for vn. Nevertheiess, nctwithstanding its generality, this approach offers several substantial advantages in anaiysing interface dynamics in the two-dimensional Stefan problem. First, the governing equaticn (1.71, (1.8) and (2.2) of the field q(e. t) are one-dimensional, i.e. they concerr, surfaces, ur.?ikt the usual formulation of S?efan-type problems. Second, irrespective cf the spatial propagation and geometrical deformation of r,. the segment E)E[@. 2.11 remains fixed, SC that one can expect further development of numerical and analytic methods for sclving prctlens cf this type. Third, and finally, the method has a very attractive feature: twc ;cin:s that are close together cr. the active side cf an evolving front rl, but net necessarily close together if the distance is measured along the contcur itself, correspond to nearby points on the 6 axis. This point is of paramount importance, as will become evident later when we determine inhomogeneous distributions of interfaces in the form of dendritic structures. To investigate the properties of the solutions of Eqs. (1.7) and (1.81, we consider the linear stage of instability of a cylindrical interface. Consider a perturbation of the q==l+-by esp(ine), where n = +(1,2,3,...). Linearizing Eqs. (1.7) and field 0. 1) w we obtain the following dispersion equation: (1.8) and formula (2.2),
COB y--gy [ n-2 where
-2
(n’-l)n].
(2.3)
+q,‘bq.
As is clear from Eq. (2.3:, if d,-U the main instability is due to shortwave corresponding to the prevalent physical notion of interface instability in perturbations, Stefan-type problems 12, 31. For small values of d,+O the system is unstable to perturbations over a broad spectrum of wavevectors, which explains the complex fractal structure of dendrltic formations. The surface ‘energy (capillary length) imposes an upper bound on the instability region of the wavevectors and dictates the minimum radius of curvature r of a dendrite. This damping effect of the surface energy on interface evolu1 tlon has two Important consequences: first, it determines the relatively regular structure of the Interface on scales of the order of rz, and, second, it implies that to a first approximation one can legitimately ignore the surface energy for the spatial structure of a dendrite on scales significantly exceeding r2. As will be shown below, the first o f these two features will enable us to establish a fairly simple relationship between the one-sided and two-sided Stefan problems. The second will give us a way of allowing for the effect of surface energy on dendritic growth in the quasilocal approximation.
55
Two-sided model
3.
In many physical problems, such as the propagation of the crystallization front in a supercooled liquid, the dynamics of the phase interface r
depend
regions
on the
on both
distribution
sides
of the field
of r.
In such
9
in
situations,
in the quasistationary approximation of the Stefan problem, the field q satisfies Laplace’s equaon r and at tion (1) at all points of space; the FIG.
point
at
infinity
the
described by (2) and city is defined by
3
distribution
(3),
but
of
now the
p is
normal
velo-
(3.1)
~.-B[(~O’~n)-(VO-.“)lIr,, where
q-
9’
Q-1.
(or
to
assumed keep
(or
CJ-) is
the
value
In addition,
the
be significantly
within
of the potential in the L, of the contour,
exterior
length
greater
than
rl,
which,
even
(resp.,
at the
properly
interior)
initial
speaking,
time
makes
it
of
r:
t = 0,
is
possible
to
the
bounds of the quasistationary Stefan problem. As already observed, the surface energy of the interface determines the growth of dendritic structures mainly on a characteristic minimal scale r It is thanks to this feature that we shall be able, still 1’ using the quasistationary approximation, to establish a simple relationship between the one-sided and twz-sided Stefan probiens. When the one-sided selfintersecting
we have
d,-0.
Stefan protlerr. cor~toilr
and problem
a--o (l)-,A;.
Since
a small
parameter,
Vn as a sum: for
the
one-sided
speciai
aspects
with (3),
Stefan
BP=0
in the first
enables
problen
with
(3.1)
is
exactly
mean curvat’ure
the
of
equivalent
a closed
to
non-
k(s’)d.s’=Zn.‘L:
we can
rewrite
terr
1.X n
is
the
seccnd,
d&J.
problem,
boundary
formula
corresponds
(3.1)
s:m;ly
to
the
which the
for
the
previous
essentially
set
normal
velocity
formtiia
of equations
(2.2)
re;resents
the
fli-‘31,
!3.1)
equations
Cl)-
condition.
consider us to
it
The first
of the two-sided
We will (3.1)
charge vectors of r,,
and neglecting
10IA “,==C. $.
(1:1-i3i,
Lt”rZ,
the
restate
latter the
is distributed on a contour illustrating the identity measured, e.g. from a point
situation.
two-sided
In this
problerr
at
the
case
se?
of
follows: a certain I‘, (the circie in Fig. 3 shows the distribution of (3.5)) with surface density q(s\ !S is a natural parameter 4) satisfying the integral eqaatior.
q-0
as
(3.2;
and,
since
r2/Lt
is small.
the
totai
charge
is,
to
a first
approximation,
4 I 0 The normal
velocity
Ye introduce
u (”
of
r,
in this
q ($1ds=O.
case
by
expression
(3.3)
is
for
the
defined
by
u,=4x;q(s).
annew variable j q(s’)ds’.
p(s)=
which,
is
(3.3)
a well-defined potentials
function q-* cc-
q(r) --
0
of s
2
on
1‘,.
be written
can
In terms as
of
this
new variable
jdsl-dlz.-zl 127-11
the
(3.4)
r,
follows
For further manipulations of the integral (3.4), directly from the geometry of the problem (see
we need Fig. 3):
a system
of
identities
which
56
d(z,-zJ
1
-li-
(s,-r)ds
IL--iI’
Il.-2)
(3.5a)
(z,-z)ds-Rel(z,-2)~~-i(2.-z)Xdsl-Re[(z~)ds],
where zs-z is the radius-vector corresponding complex number, along
ar,d
r,
the
between
ccrresponding
complex
functions
, we can associate’ Y-: Re V--othe in O-; thanks tc ‘t.41, (2.5’ and
the
ds and
and with
the
the
potentials
Z;-Z is the conjugate of the an infinitesimal displacement Since 0’ and v- are harmonic
Yy-: Re Y*-q+
in
to
q(S)
potentials
of
Identity
of
vector
respectively.
complex
relaticnship the
are
number,
them
S and zs,
points
ds
(3.5b)
the
by a Cauchy
Is,-tl’-(L.-Z)(~).
Y(z)--
2
variable
Q* and is
defined,
integral
jP(S)A, .
(3.6)
r, in
the
sense
va:ue
The except of
that,
cf the
pci.nts
the
Y-(z)
function
rcle
f ST the
of
pfs)
contour
2
the point Y-(:) or
potential
hclocorphic
tf,ar
as
according
complex
wii: (see
then
lies
in
in Q-
can
be played
2 c in
Q-.
is a real ccnstant. In order to est;mate 6s
reiaticnsh:; the
will
be
between
tangent
the 7
vectcr
d:,=dsexp(lQ).
:ke
evident
real frc~
Se(s)
to
the cne
constant the
c.ur;‘atore
On the
also a
by
the
be
integral
(3.6)
gives
by a Cauchy
represented
Certain
d:,+e 2.-z
Tt c
Q-,
the
purely
imaginary
integral,
function
g(S)
IS;): V’_(z)=2i
where
Q’ or
Y-(z).
of
kis!
contour hand,
c,
sequel, and
the
we consider
a charazteristlc whence
g(s)-d,k(s)q,. ccntcur ar,d
the
tke
(3.7)
’
axis,
03
integrai
tne
that
(3.7)
mean it
angle
point
fellows,
using
8-e-Ij-X/2 ds.
k(s) s-d!!
the
betweer.
and
mcreover
yields
Re Y-(z.)-d,q,!R:-c. vhere
Rc is
potential, +he c
Thus
a ckaracteristic of
real
the
same
constant
linear crder
cf
unit
a small
c is
assumed
here we car; put C = G. Comparing the integrals
points
of C-:
on
magnitude
parameter
(3.6)
On the
r:.
as
and
is
of type
P /L,
(3.“;,
that
the
enables
us
an explicit sion
exFresrion in
relation
explicitly
to
expression
immediately By (3.6!,
yields the
is
p(s)-q(r) to
cur
determine
for
g as
tcundary
lim:t
the
to of
expression
condition
of
ar. identity
the
value
of
valid
at
all
in
c-+
interior
for
p(s)
geometrical the
the
and
of
r,
g!s);
velocity
function
of u
r,,
of
Rr Y’--qn
(see if
this
where
we can
(3.5t!
and
the
is the angle between express ,3.6‘ as
aI
geome trical
?he
situation
radius-vector
#(s’)dazI
-5
r,
l.~-.,Wr,-
S
- s and
4q.k
(~1.
expres-
can
be
written
(3.6)
illustrated
2
we have
r,.
-doq.k(s).
identity
i21).
result
hence,
parameters
normal
this
r-c-4-r,
Using
the
accuracy
the
hclomcr~h~r
geometry
between
the for
(3)
mear.
tc w:ttir
a fnnrticn
describe
relationship
a function
des ired
the
the
namely
goal,
SC
2
the
Rt ~((;,)-dc~,(k)-do~,!L,.
d---IL-o, :,--:
I Ip(s)-igb)]
so
hand,
q(&)-
we ottain
T,
Considered
other
T‘(” ),
:r.
the
Fig.
0~ axls.
3,
we have
Using
this
formuia
(3.9)
as
57
a+ n
b
a FIG. Isolating
le,-q* of
on the
term
the
integrating over a part the equivalent equation
of
(3.5~
due to
contained
r,
4
in
the
an
singularity
in the
2ng(s)-2! f(s’)da, r,
behaviour
nelghbourhood
infinitesimal
of
of s,
Qz and
we obtain
(3.10)
-d,q,k(s),
It should be noted that in this which the integral tern EC longer has a singularity. Eq. (3.10) is considerably simpler thar. the equivalent equation (3.2), since it is essentialiy a Fredholm integral equaticr cf the second kind, whereas (3.2) is a Fredholm eqilaticr. cf the first kind. If the interface is ir fact a circle, i.e. the characteristic curvature of the contour kar-‘, then, since the cara-eter r .:5 :s small, the first term in (3.10) is iS SUCh that 1 g(s)=(2n)-‘d&(s). dominant and in
case
New ccnsider
the
quar.ri:v
A,-[;rk(s)l-‘Sk(s’)da.
(3.11)
r,
for a developed $eomeTric strdct’we r2 linked by straight lines cf lengtt the
integral
For
“flatter”
(3.11)
character;st::
interfaces,
developed
strongly the
interface
ing
larger
is
for
the
denditic
geometry of
values
In view a solution
FCirtS
absclute
illustrated A
valce
structure
(see
Fig.
aforesaid, of the cquatlor.
of
IF:&. La‘, :, w5ere
the
e.g. a ccl:ectior. cf se?lcir::es Cirert by ass;mFtion 1 >> r 1’ .L=O.i6 8_ and F shows that
of radius evaiuat:or.
Than
fcr
cI
one has in Fig
4c),
but
.i
is
/.i_iai,
zess
;,
and
moreover,
as may be verified
4b.
Naturally,
s;ich
formations
a first apprcximation in the form (3.10):
of
there are the
exist rare
in
solution
a
by analys:ng.\, topologies denditic under
for
cfr,,yieldgrowth.
consideration
g(si=~(2rr)-‘d,~,k(s), where
the
parameter
form-factor value
A,
constant of F-((l-.1.)-‘).
F la certain by the
formula
the
of
A,=-1.
and
(3.12) order of unity) We also note
is related to the that the correction
mean terms
to
formula (3.12) for the solution of Eq. (3.10), due to its non-local nature, are oscillato the value of the function ptr, need not have a cardi; hence their contribution nal effect on the expression obtained using (3.12). Summing up, we see that (3.12) provides an approximation for the normal velocity of p(s)+i2b‘gr*(s) where the contour function y(d) (t)+(dlds)P(s). is the limit the contour
tory
of a certain cimplex function holomorphic analogous to the equivalent assertion for
length
region
Q*.
one-sided
Stefan
This
assertion
problem
with
is
completely
capiliary
2Fd,. In other
problems
in the the
are
words,
equivalent,
in the
approximation
subject
to
of
a suitable
(3.12)
the
one-sided
renormalization
and two-sided
of the
capillary
Stefan length:
A--2Fk.
the
Quasilocal approximation 4. the We now formulate an approximation that establishes normal velocity un and the field n for the quasistationary
local
relationship
Stefan
problem.
between This
approxi-
simulation of dendritic structures, considerably enhances the prospe,. -7s for numerical since the computation of the double integrals in (2.2) is a lengthy affair. In investigating the effect of surface energy on dendritic growth dynamics, we limit ourselves as before to studying the least characteristic scale r2 on which the interface we consider the one-sided geometry is relatively regular geometry. To simplify matters, that, as shown in the previous section, the results can Stefan problem, noting, however,
mation
USSR
29-5-E
58
be immediately generalized cut Ijefore serting the physical background.
to the two-sided problem. the mathematical theory of quasilocal Consider a relatively regular interface
where
integer
Ri
-2’r
say that
P
of
the
with
1’ is
interface
radius
fi.
(Fig.
tionai
tt
the
3!,
and,
radius
We no*
whose absolute
regular
containing R.
on these
surface
energy,
to
inhomogenelry
of
I-,
w:tr,;n of
s scaled
the tc
value
scales,
a characteristic second,
that
the
length
(see
Fig.
3).
-/?.
gradient
the
tounds
s-ale
Rir
Ri
is
is
of the order
we mean,
inhomogeneity
s.
es:imatez~hetnorkal
to
at
i an
reiatively
si
of of
‘?*p
of
the
present
approximately
approximation, on several
first,
is
that
part
entire
in a disk
the
of
When we
unity. the
localized
potential
the
of
that
we describe scales Ri,
contour
p on the
model.
The limiting
d&k,.
where ki
part of
is
propor-
interface
potential
is
the
due due
ql
mean curvature
i.e., ,+a,
5 k(s’)ds’
k,-R,-’
Therefore, value
the
of
the
contribution
rorma;
ccrrespcnding
gradient
is
to
each
all
and the
is
(V,q),--d,q,k,/R,
these
contributions:
scale
by summing
cttained
total
(4.1) ,-I,
Since
“rL
by its
15
functlcn
ex~oneztla;
ar.
continuo,us
of
i,
we can replace
the
estimating
expression
(4.1)
limit:
lA.2!
Corpar;ng identical :f
are
which
is
‘4.3; with we identify
leg;timate,
pra:tically
the
since
same for
or
v
of
mation: gral
is
.
the
the
sign,
quantity assigning
analogous
we obtain
contcur
to
the
the
it
undertaking
r
direct?
y tc
derive energy
rl(8.i the
in the
integral
value
Pcincare-Bertrand
required
This
surface
fixed
relation
of
from
‘i.2’,
t.he contc,‘~r
Is’-si:r;,
due tc
5. Simulation Equations (1.71, this
scaies
pclnts
quasilocal approximation. We will now proceed city
;ts rigcrc~s ana;og,;e tne integrals
I-, is
the (Z.2:
that
1s in fart
in far:
required
and its
the
formula
:wt
ds,d6
ass;mpt:on
between
the
expressicns
and
reg;:ar
the
cf
ve:o-
apprcxi-
thrcugh
forwart
fcllcw;;.g
is
the
norma;
quasiloca:
i .7 tr.e
car: be crc:gnt using
the
basic
re;atio-
gecmetr::.
Then,
n(@). comm’utatioa
we see
the
ider.tity,
inte-
wr.:ch
:G:,
(2.2):
dendritic
structures
(1.8) and (4.3) was the approximation
been investigated numericaliy. Underlying integrands f by trigonometric pc?ynomia?s [lo]:
have
of zr-,
1.W;~
!(edsin[n(e-8,)
e-e, (5
]ctg-_,
._I
where 8,=nk n. k-0. 1,. . ..2n-1. Expansions of this type make it possible quadrature formulae to evaluate singular integrals [lo]. As a result, tions (1.7), (1.8) and (4.3) reduces to a system of ordinary non-linear ~,-o,(n~
.Q.._,. R)
with
full
Jacobian;
this
system
was
integrated
to
use
wei;-known
the system equations using
of
a program
equa-
59
FIG.5 do=lO-‘. r+,-~-1. I,-O. fr-750. f,-z 1 lo’. f,-3.4.10’, ~(Io!-1-0.1 COSi38), R(f,)_lO
FIG. 6. Initial perturbations ~(9)~,~~1-10-~~~~(39): &-9.i. opp~.1; lo-O. R(t,)=R,=10. 11-667. RV,)-R,-38, r,=z 10’. R(I*)=&= 69. r,-3 lo’, R(W-&-94; graph c - field distribution at time f0 and 13 from [ill. The required accuracy was achieved by suitable choice of the number N = 2n and the local error e. In the actual numerical experiments we took A’ = 61 and c=IO-“, the required computing time on the EC-1060 computer was generally at most one hour. Knowing the values of and and the appropriate radii R(t), and applying the tl(6, t) ct(6, f), transformations dy--
R(tld6 sinlB+rC(B,t)+n/2], l-Ice*t)
was possible to reproduce geometrically defined interfaces in parametric form. Figure 5 and 6 illustrate the characrerlstic form of non-uniform distributions of the field (graphs a) and the COrrespOnding interfaces (graphs b). As examples, l(e, t) UC determined different snow-flake shapes, depending on the form of the initial perturbatlon. The reader should observe that relatively smooth variations of the interface correspond to quite marked variations in the field s(e, i) and hence also in the angle O(8, t). Thus, relatively small errors in determining do not cause marked changes in the n(g. 1) intsrface geometry. In addition, as is evident from the figures, relatively even long sections of the interface contract when mapped into the interval 10, 2x1, whereas pronounced changes in the contour f. conversely, are etretched out. In this sense such mapit
60
with respect to information about the detailed are “adaptive” pings cf r onto [O. 2x1 peometlc struct’ure of the interface. as these solutions Correspond to existing physical ideas, In conclusicc, we note that, there seem tc be good prospects for further success in the purposeful numerical modelling of when the system involves the effect of uncorrelated sources of noise non-uniform structures, on the phase interface; it should aiso be possible to analyze the behaviour of the solutions This will make it possas a function of the surface energy and anisotropies in the system. to reconstruct the entire picture of self-OrganizatIonal phenoible, in the finai analys:s, mena in the interface. Incidentally, the universal nature of the governing equations as derived in this paper indicates that they should be applicable in many problems of physics, chemistry and biology. the ast5ors wccld ;ike to thank 1.1. Lazurchak and I.V. Tyslyuk for their Finally, and also A.A. Dorodnltsyn for his constant help ln carrying out the numericai experiments, interest. REFERENCES
1.
2. 3. A.
6. 7.
8. 9. 1C
li
Fror,t,
Interfaces and Patterns, Fhyslca. D12, NOS. 1-3, 1984. LANGEF J.S., Instabiiitiesand pattern formation in crystal growth. Revs. Mod. Phys., 52, 1, l-26, 198f. lJMAN”=EV A .,,F. VIN3GKAXC V.V. and B%!SW V.T., Simulation of the evolution of .Krisrallographiya, 3i, 5, 1002, 1986. dendritic s*ructure. c Monte-Carlo approach to dendritic growth. J. Phys. SZEP J., CSEFTI 2. and KEIESi J., 1965. A, i8, 8, Liz3-L418, Phys SHRA.IMkl;i. and BEN,Ei.~~X S. , Singularities ir, non?ocal in?erface dynamics. fiev A, 3t, 8, 264S-28aE-- 1 1964. Phys. Rev. A, 33, 2, 1302-1308, BENEIM;:: L. , Irsta!zi:ity cf viscxs fingering. 1586. LAVEEST'YE', K.k. ar,dSBXU'T E.V., Methods of the Theory of Functions of a Complex Variable, Nauka, Moscow, 1976. cf similar regions. Uspe6r.iRat. Nauk, 11, SIRYK G.V., Cr.2cr,form21ma;rir.;; 5(71), 57-65, 1956. MUSKHELISHVILI N, ! Sing;;:ar Integral Equations, Nauka, Mcscow, 1968. BEiCTSEW.?‘TSEi 1 S M[ and L: Fkhit'i I.K., Numerica: Methods in Singuiar Integral Equations, Nauka, Mcsccu, iSEE BSRK'E G.2. and H:i;DMAFCu ODE sc:vcrs: a rev;er cf z'drrent and coring .._../. C. , Sriff attractlsns.
J. Cor;lr.
Pr.gs ) ?C, 1, :-62,
IBE”. Translated
U.S.S.R.Co’ll,“rct.Mcri.s..?/~:;..F;;~~.,:’c1.2C,!i~.5,;;.6t-B~,198~
Printed
ir. Great
::‘:-===-‘eg____I
c 199;
Britain
by D.L.
~:C.OGco.oo
Fergamcr.
Press
plc
THE SOLUTION OF PROELEVS IN ELASTICITY THEOi?Y BY COMPLETE-SYSTEV METHODS" E.I.
BECFALC\'A
The main pcinrs of complete-system methods are presented as it app:les to sc:ving prOb:emS concerning the static and free vibrations of inhomogeneous anisotroplc bodies in a variationai se?ting. The characteristic feature of these method is the reduc tion of an initially N-dimensional problem to a system of A! interrelated one-dimensionai problems. IJn:lke the common variational approaches, one no longer has any freedom with regard tc the choice of basis functions with respect to some of the independent variables. Some computationa? aspects of the approach are :?lustrated by a specific example. Introduction The most common. mathematical modeis for the mechanics of deformed bodies are dlffercntlal problems in relation to vector-valued functions of several variables, subject to various boundary conditions in Irregularly-shaped regions; the inhomogeneity of the properties of the elastic medium is represented by different variable parameters. Such problems can be handled by various methods, among them such familiar techniques as projection, varlatlona?, finite-difference, finite-element etc. methods; a recent addition to the list consists of complete-system methods (CSMs) (1, 21 which apply the sophisticated tools now available for solving one-dimensional protl ems to prcblems involving several dlmenslons, *Zh.vychisZ.Mot.mat.Fiz.
,29,9,13A6-1353,1989