# Interface growth in the channel geometry and tripolar Loewner evolutions

Miguel A. Durán<EMAIL_ADDRESS>Giovani L. Vasconcelos
<EMAIL_ADDRESS>Laboratório de Física Teórica e Computacional, Departamento
de Física, Universidade Federal de Pernambuco, 50670-901, Recife, Brazil.

###### Abstract

A class of Laplacian growth models in the channel geometry is studied using
the formalism of _tripolar_ Loewner evolutions, in which three points, namely,
the channel corners and infinity, are kept fixed. Initially, the problem of
fingered growth, where growth takes place only at the tips of slit-like
fingers, is revisited and a class of exact exact solutions of the
corresponding Loewner equation is presented for the case of stationary driving
functions. A model for interface growth is then formulated in terms of a
generalized tripolar Loewner equation and several examples are presented,
including interfaces with multiple tips as well as multiple growing
interfaces. The model exhibits interesting dynamical features, such as tip and
finger competition.

###### pacs:

68.70.+w, 05.65.+b, 61.43.Hv, 47.54.–r

##  I Introduction

The Loewner equation describes a rather general class of growth processes in
two dimensions where a curve starts from a given point on the boundary of a
domain $\mathbb{P}$ in the complex $z$-plane and grows into the interior of
$\mathbb{P}$. More specifically, the Loewner equation loewner  is a first-
order differential equation for the conformal mapping $g_{t}(z)$ from the
‘physical domain,’ consisting of the region $\mathbb{P}$ minus the curve, onto
a ‘mathematical domain’ represented by $\mathbb{P}$ itself. The specific form
of the Loewner equation depends both on the domain $\mathbb{P}$ and on the
point where the trace ends. The most studied cases are the chordal and radial
Loewner equations review3 , where in the former case the trace ends on a given
point on the boundary, while in the latter it ends on a point in the interior.
Other geometries such as the dipolar case BB2005  and the channel geometry
poloneses  have also been considered in the literature. The Loewner equation
depends explicitly on a driving function, here denoted by $a(t)$, which is the
image of the growing tip under the mapping $g_{t}(z)$. An important
development in the field was the discovery by Schramm schramm  that when
$a(t)$ is a Brownian motion the resulting Loewner evolution describes the
scaling limit of certain statistical mechanics models. This result spurred
great interest in the so-called stochastic Loewner equation (SLE) and the
subject has now been widely reviewed, see, e.g. review1 ; review2 ; cardy ; BB
.

Although the SLE has attracted most of the attention lately, the deterministic
version of the Loewner equation remains of considerable interest both in a
purely mathematical context kadanoff_jsp  and in connection with applications
to growth processes selander ; makarov ; poloneses  and integrable systems
theory zabrodin2 . Indeed, the deterministic Loewner equation has been used to
study Laplacian fingered-growth in both the chordal and radial cases selander
; makarov  as well as in the channel geometry poloneses . In this class of
models, growth takes place only at the tips of slit-like fingers and the
driving function $a(t)$ has to follow a specific time evolution in order to
ensure that the tip grows along gradient lines of the corresponding Laplacian
field. Although this thin-finger model poloneses  was able to reproduce some
of the qualitative behavior seen in experiments combustion , treating the
fingers as infinitesimally thin is a rather severe approximation.

More recently, the growth of “extended fingers” us_pre  or “fat slits”
zabrodin1 , meaning a domain encircled by an interface with endpoints on the
real axis, was considered within the formalism of deterministic Loewner
evolutions in the upper half-plane. In particular, the interface growth model
considered in Ref. us_pre  was shown to exhibit interesting dynamical features
akin to the finger competition mechanism observed in actual Laplacian growth
pelce . Many growth processes, however, take place in a bounded domain, e.g.,
within a channel pelce , for which a Loewner-equation approach was still
lacking. To develop a formalism based on Loewner evolutions to describe the
growth of an interface in the channel geometry is thus the main goal of the
present study. Because the Loewner function $g_{t}(z)$ fixes the points $z=\pm
1$ (the channel corners) and $z=i\infty$ (the channel ‘open’ end), we shall
refer to growth processes in the channel geometry as _tripolar_ Loewner
evolutions, to distinguish it from the dipolar case where two points are kept
fixed BB2005 .

We begin our analysis by revisiting in Sec. II the problem of fingered growth
in the channel geometry, where a curve (finger) grows from a point on the real
axis into a semi-infinite channel. We present a brief derivation of the
corresponding tripolar Loewner and report a new class of exact solutions for
the case of multiple fingers with stationary driving functions. In Sec. III we
then discuss a general class of models where a domain, bounded by an
interface, grows into a channel starting from a segment on the real axis. In
our model, the growth rate is specified at a certain number of special points
on the interface, referred to as _tips_ and _troughs_ , which then determine
the growth rate at the other points of the interface according to a specific
growth rule (formulated in terms of a polygonal curve in the mathematical
plane). A generalized tripolar Loewner equation for this problem is derived
and several examples are given in which the interface evolution is obtained by
a direct numerical integration of the Loewner equation. Our main results and
conclusions are summarized in Sec. IV.

##  II Loewner Evolution for Fingers in the Channel

###  II.1 Tripolar Loewner Equation

Figure 1: The physical $z$-plane and the mathematical planes at times $t$ and
$t+\tau$ for a single curve in the channel geometry. The Loewner function
$g_{t}(z)$ maps the physical domain in the $z$-plane onto the channel in the
mathematical $w$-plane, with the tip $\gamma(t)$ of the curve $\Gamma_{t}$
being mapped to the point $w=a(t)$. The portion (dashed line) of the curve
accrued during a subsequent infinitesimal time interval $\tau$ is mapped under
$g_{t}(z)$ to a vertical slit; see text.

In order to set the stage for the remainder of the paper and to establish the
relevant notation, we begin our discussion by considering the simplest Loewner
evolution in the channel geometry, namely, that in which a single curve grows
from a point on the real axis into a semi-infinite channel whose side walls
are placed at $x=\pm 1$ and the bottom wall is at $y=0$; see Fig. 1. Here the
relevant domain, $\mathbb{P}$, for the Loewner evolution is

$\mathbb{P}=\left\\{{z=x+\mbox{i}y\in\mathbb{C}:y>0,x\in]-1,1[}\right\\}.$

The curve at time $t$ is denoted by $\Gamma_{t}$ and its growing tip is
labeled by $\gamma(t)$. Now let $w=g_{t}(z)$ be the conformal mapping that
maps the ‘physical domain,’ corresponding to the upper half-channel
$\mathbb{P}$ in the $z$-plane minus the curve $\Gamma_{t}$, onto the upper
half-channel $\mathbb{P}$ in an auxiliary complex $w$-plane, referred to as
the ‘mathematical plane,’ i.e., we have

$g_{t}:\mathbb{P}\backslash\Gamma_{t}\rightarrow\mathbb{P},$

with the curve tip $\gamma(t)$ being mapped to a point $w=a(t)$ on the real
axis in the $w$-plane; see Fig. 1.

The mapping function $g_{t}(z)$ is required to satisfy the so-called
‘hydrodynamic condition’ at infinity, namely,

$g_{t}(z)=-iC(t)+z+O(\frac{1}{|z|}),\qquad{{\rm Im}(z)\rightarrow\infty},$ (1)

where $C(t)$ is a real-valued, monotonously increasing function of time
satisfying the condition $C(0)=0$. Condition (1), which implies
$g_{t}^{\prime}(i\infty)=1$, together with the requirement $g_{t}(\pm 1)=\pm
1$, uniquely determines (up to a reparametrization of the time $t$) the
mapping $g_{t}(z)$. The time parametrization is specified by fixing the
function $C(t)$, which in turn is related to the growing rate of the curve;
see below. The Loewner function $g_{t}(z)$ also satisfies the initial
condition

$g_{0}(z)=z,$ (2)

since we start with an empty channel.

We consider the growth process to be such that growth occurs only the tip of
the curve and in such a way that the accrued portion from time $t$ to time
$t+\tau$, where $\tau$ is an infinitesimal time interval, is mapped under
$g_{t}(z)$ to a vertical slit of (infinitesimal) height $h$ in the
mathematical $w$-plane; see Fig. 1. For convenience of notation, we shall
represent the mathematical plane at time $t+\tau$ as the complex
$\zeta$-plane, so that the mapping $\zeta=g_{t+\tau}(z)$ maps the physical
domain at time $t+\tau$ onto the upper half-channel $\mathbb{P}$ in the
$\zeta$-plane, with the new tip $\gamma(t+\tau)$ being mapped to the point
$\zeta=a(t+\tau)$.

In order to derive the Loewner equation for the problem formulated above it is
necessary to write $g_{t}$ in terms of $g_{t+\tau}$, i.e.,
$g_{t}=F(g_{t+\tau})$, where $w=F(\zeta)$ is the conformal mapping between the
mathematical $\zeta$\- and $w$\- planes, and then consider the limit $\tau\to
0$. Alternatively, one can transform the growth problem defined in the channel
geometry into a corresponding Loewner evolution in the upper half-plane. To do
that, we apply the following transformations:

$\tilde{z}=\phi(z),\qquad\tilde{w}=\phi(w),\qquad\tilde{\zeta}=\phi(\zeta),$
(3)

where

$\phi(z)=\sin\left(\frac{\pi}{2}z\right),$ (4)

so that the channels in the $z$-, $w$-, and $\zeta$\- planes are mapped to the
upper half-plane in the auxiliary complex $\tilde{z}$-, $\tilde{w}$-, and
$\tilde{\zeta}$\- planes, respectively. Let us also introduce introduce the
following notation:

$\tilde{g}_{t}=\phi\circ g_{t}\circ\phi^{-1}.$ (5)

Next, consider the mapping $\tilde{w}=\tilde{F}(\tilde{\zeta})$ from the upper
half-$\tilde{\zeta}$-plane onto the upper half-$\tilde{w}$-plane with a
vertical slit, so that one can write

$\tilde{g}_{t}(\tilde{z})=\tilde{F}(\tilde{g}_{t+\tau}(\tilde{z})).$ (6)

The slit mapping $\tilde{F}(\tilde{\zeta})$ can be easily computed (see Ref.
poloneses  for details), and after taking the limit $\tau\to 0$, one obtains
the following Loewner equation

$\dot{\tilde{g}}_{t}(\tilde{z})=\frac{\pi^{2}}{4}d(t)\frac{1-\tilde{g}_{t}(\tilde{z})^{2}}{\tilde{g}_{t}(\tilde{z})-\tilde{a}},$
(7)

where

$\tilde{a}(t)=\sin\left[\frac{\pi}{2}a(t)\right].$ (8)

and the growth factor $d(t)$ is defined by

$d(t)=\lim_{\tau\to 0}\frac{h^{2}}{2\tau}.$ (9)

By taking the limit $\tilde{g}_{t}(z)\to\tilde{a}(t)$ in Eq. (7) in an
appropriate sense poloneses , one finds that the driving function $a(t)$ is
determined by

$\dot{\tilde{a}}=-\frac{\pi^{2}}{8}d(t)\tilde{a}.$ (10)

The Loewner evolution in the original channel geometry can then be obtained
from Eqs. (5) and (7), yielding

$\dot{g}_{t}(z)=\frac{\pi}{2}d(t)\frac{\cos\left[\frac{\pi}{2}g_{t}(z)\right]}{\sin\left[\frac{\pi}{2}g_{t}(z)\right]-\sin\left[\frac{\pi}{2}a(t)\right]},$
(11)

with

$\dot{a}=-\frac{\pi}{4}d(t)\tan\left(\frac{\pi}{2}a\right).$ (12)

From the boundary condition (1) it follows that
$C(t)=(\pi/2)\int_{0}^{t}d(t^{\prime})dt^{\prime}$. Without loss of generality
we shall take $C(t)=\pi t/2$ in this section, implying that $d(t)=1$. Note
that the Loewner equation (11) indeed fixes the points ${z}=\pm 1$ as well as
the point at infinity, in the sense of Eq. (1). Hence we shall refer to Eq.
(11) as the tripolar Loewner equation, in analogy with the dipolar case BB2005
where only the points $x_{\pm}=\pm 1$ are kept fixed. The tripolar Loewner
equation in the upper half-plane given in Eq. (7) also fixes the points
$\tilde{z}=\pm 1$ and the point at infinity. More specifically, the boundary
condition at infinity reads:
$\tilde{g}_{t}(\tilde{z})\approx\exp[-\pi^{2}t/4]\tilde{z}$, for
$|\tilde{z}|\to\infty$.

The Loewner equation given in Eq. (7) can be readily extended to the case of
multiple curves growing simultaneously in the channel, yielding

$\dot{\tilde{g}}_{t}=\frac{\pi^{2}}{4}(1-\tilde{g}_{t}^{2})\sum_{i=1}^{n}\frac{d_{i}(t)}{\tilde{g}_{t}-\tilde{a}_{i}(t)},$
(13)

where

$\dot{\tilde{a}}_{i}=-\frac{\pi^{2}}{8}d_{i}(t)\tilde{a}_{i}+\frac{\pi^{2}}{4}(1-\tilde{a}_{i}^{2})\sum_{\stackrel{{\scriptstyle
j=1}}{{j\neq i}}}^{n}\frac{d_{i}(t)}{\tilde{a}_{i}-\tilde{a}_{j}}.$ (14)

We shall assume for simplicity that the growth factors are all constant, i.e.,
$d_{i}(t)=d_{i}$, subjected to the condition $\sum_{i=1}^{n}d_{i}=1$, as
follows from our choice above for $C(t)$. Other growth models have been
considered poloneses  where the normal velocity is of the form
$v_{n}\sim|\vec{\nabla}\phi|^{\eta}$, with the growth factors being given by
$d_{i}(t)=|f_{t}^{\prime\prime}(a_{i}(t))|^{-\eta/2-1}$, where $f_{t}(w)$ is
the inverse of $g_{t}(z)$. In this case, the analysis of the Loewner equation
is much more difficult because of the implicit dependence of the growth
factors $d_{i}(t)$ on the Loewner function $g_{t}(z)$. By considering the
simpler (but still very interesting) case where the growth factors are
constant in time, we are able both to find exact solutions for multifingers
and to easily integrate the Loewner equation (13) on the computer.

###  II.2 Exact solutions

An exact solution for the Loewner function $g_{t}(z)$ for a single curve in
the channel geometry can be found explicitly for a stationary driving
function, i.e., $a(t)=0$, corresponding to a finger growing vertically along
the channel centerline poloneses . Here we extend such stationary solutions to
the case of multifingers. For simplicity we consider only the case of two
fingers, but the procedure outlined below applies, in principle, to any number
of fingers (although the calculations become increasingly more difficult).

For two fingers, Eq. (14) becomes

$\dot{\tilde{a}}_{1}=-\frac{\pi^{2}}{8}\left[d_{1}\tilde{a}_{1}-2d_{2}\frac{1-\tilde{a}_{1}^{2}}{\tilde{a}_{1}-\tilde{a}_{2}}\right],$
(15)
$\dot{\tilde{a}}_{2}=-\frac{\pi^{2}}{8}\left[d_{2}\tilde{a}_{2}-2d_{1}\frac{1-\tilde{a}_{2}^{2}}{\tilde{a}_{2}-\tilde{a}_{1}}\right],$
(16)

whose stationary solution (fixed point), $\tilde{a}_{1}^{0}$ and
$\tilde{a}_{2}^{0}$, are given by

$\tilde{a}_{1}^{0}=-\left[\frac{2d_{2}^{2}(1+d_{1})}{1+d_{2}}\right]^{1/2},\qquad\tilde{a}_{2}^{0}=\left[\frac{2d_{1}^{2}(1+d_{2})}{1+d_{1}}\right]^{1/2}.$
(17)

Note that if $d_{1}=d_{2}$, one has $\tilde{a}_{1}^{0}=-1/2$ and
$\tilde{a}_{2}^{0}=1/2$, and the solution corresponds to two identical fingers
growing vertically. This symmetrical two-finger solution is, of course,
related to the single-finger solution described in Ref. poloneses  by a simple
transformation, since in this case we can view each finger as growing along
the centerline of a channel with half the width of the original channel. Thus,
we are mainly interested here in asymmetrical solutions where $d_{1}\neq
d_{2}$. In the remainder of this section we shall omit, for convenience of
notation, the upperscripts from $\tilde{a}_{1}^{0}$ and $\tilde{a}_{2}^{0}$.

The Loewner equation (13) for two fingers reads

$\dot{\tilde{g}}_{t}=\frac{\pi^{2}}{4}\frac{(1-\tilde{g}_{t}^{2})[\tilde{g}_{t}-(d_{1}\tilde{a}_{2}+d_{2}\tilde{a}_{1})]}{(\tilde{g}_{t}-\tilde{a}_{1})(\tilde{g}_{t}-\tilde{a}_{2})},$
(18)

where we have used the fact that $d_{1}+d_{2}=1$. This equation can be easily
integrated, yielding an implicit solution for the mapping $\tilde{g}_{t}$ in
the form

$\left[\frac{\tilde{g}_{t}+1}{\tilde{z}+1}\right]^{\alpha_{1}}\left[\frac{\tilde{g}_{t}-1}{\tilde{z}-1}\right]^{\alpha_{2}}\left[\frac{\tilde{g}_{t}-(d_{1}\tilde{a}_{2}+d_{2}\tilde{a}_{1})}{\tilde{z}-(d_{1}\tilde{a}_{2}+d_{2}\tilde{a}_{1})}\right]^{\alpha_{3}}=e^{-\frac{\pi^{2}}{2}t},$
(19)

where

$\alpha_{1}=\frac{(1+\tilde{a}_{1})(1+\tilde{a}_{2})}{1+(d_{2}\tilde{a}_{1}+d_{1}\tilde{a}_{2})},\qquad\alpha_{2}=\frac{(1-\tilde{a}_{1})(1-\tilde{a}_{2})}{1-(d_{2}\tilde{a}_{1}+d_{1}\tilde{a}_{2})},$
(20)

and $\alpha_{3}=2-\alpha_{1}-\alpha_{2}$.

The tip trajectories $\gamma_{i}(t)$, for $i=1,2$, can now be obtained by
setting $\tilde{z}=\tilde{\gamma}_{i}$ and
$\tilde{g}(\tilde{\gamma}_{i})=\tilde{a}_{i}$ in Eq. (19), where
$\tilde{\gamma}_{i}=\sin(\pi\gamma_{i}/2)$. One then obtains

$\left[\frac{\tilde{\gamma_{1}}+1}{\tilde{a}_{1}+1}\right]^{\alpha_{1}}\left[\frac{\tilde{\gamma_{1}}-1}{\tilde{a}_{1}-1}\right]^{\alpha_{2}}\left[\frac{\tilde{\gamma_{1}}-(d_{1}\tilde{a}_{2}+d_{2}\tilde{a}_{1})}{d_{1}(\tilde{a}_{1}-\tilde{a}_{2})}\right]^{\alpha_{3}}=e^{\frac{\pi^{2}}{2}t},$
(21)
$\left[\frac{\tilde{\gamma_{2}}+1}{\tilde{a}_{2}+1}\right]^{\alpha_{1}}\left[\frac{\tilde{\gamma_{2}}-1}{\tilde{a}_{2}-1}\right]^{\alpha_{2}}\left[\frac{\tilde{\gamma_{2}}-(d_{1}\tilde{a}_{2}+d_{2}\tilde{a}_{1})}{d_{2}(\tilde{a}_{2}-\tilde{a}_{1})}\right]^{\alpha_{3}}=e^{\frac{\pi^{2}}{2}t},$
(22)

where the roots for $\tilde{\gamma}_{1}$ and $\tilde{\gamma}_{2}$ must be
chosen in the upper half-plane. The asymptotic trajectories in the limit
$t\to\infty$ can be found by writing $\gamma_{i}=x_{i}+iy_{i}$ and considering
the limit $y_{i}\to\infty$. After a straightforward calculation one obtains

$\gamma_{1}(t)\approx\frac{\alpha_{1}-\alpha_{2}-\alpha_{3}}{2}+i\frac{\pi}{2}t,\qquad\gamma_{2}(t)\approx\frac{\alpha_{1}-\alpha_{2}+\alpha_{3}}{2}+i\frac{\pi}{2}t,\qquad
t\to\infty.$ (23)

We thus see that for large times the two fingers approach the vertical axes

$x_{1,2}=(\alpha_{1}-\alpha_{2}\mp\alpha_{3})/2,$ (24)

respectively, and both of them move asymptotically with the constant speed
$v=\pi/2$.

An example of such a solution is shown in Fig. 2(a), where we used $d_{1}=2/3$
and $d_{2}=1/3$, which from (17) implies $a_{1}^{0}=-0.3534$ and
$a_{2}^{0}=0.6387$. In this case, the fingers should approach the vertical
axes $x_{1}=-0.3712$ and $x_{2}=0.6099$, as predicted by (23), and this is
indeed observed in the figure. Note that this asymptotic behavior holds for
any initial conditions $a_{1}(0)$ and $a_{2}(0)$, on account of the fact that
the fixed point given in (17) is stable. This is illustrated in Fig. 2(b)
where we show a solution with initial conditions $a_{1}(0)=-0.6$ and
$a_{2}(0)=0.4$. To generate the curves shown in Fig. 2 we integrated the
Loewner equation given in Eq. (13) using the numerical scheme described in
us_pre .

(a) (b)

Figure 2: Evolution of two fingers with $d_{1}=2/3$ and $d_{2}=1/3$. The
initial conditions are: $a_{1}(0)=a_{1}^{0}=-0.3534$ and
$a_{2}(0)=a_{2}^{0}=0.6387$ in (a), and $a_{1}(0)=-0.6$ and $a_{2}(0)=0.6$ in
(b). In both cases the trajectories approach the axes $x_{1}=-0.3712$ and
$x_{2}=0.6099$.

###  II.3 Tripolar Loewner Chains

In the spirit of chordal Loewner chains BB , it is clear that the tripolar
Loewner equation (13) can be extended to describe the growth of more general
domains in the channel geometry, not restricted to a curve or a set of curves.
The corresponding tripolar Loewner chain thus reads

$\dot{g}_{t}(z)=\left(1-g_{t}(z)^{2}\right)\int_{-1}^{1}\frac{\rho_{t}(x)dx}{g_{t}(z)-x}\,,$
(25)

where the density of singularities $\rho_{t}(x)$ can be viewed as a measure of
the growth rate at a point $z$ at the boundary of the growing set that is the
preimage of $x$ under $g_{t}(z)$. Since the shape of the growing domain is
fully encoded in the map $g_{t}(z)$, equation (25) specifies the growth model
once the density $\rho_{t}(x)$ is known. As discussed elsewhere us_pre , the
formalism of Loewner chains is of limited practical use, except when the
density is a sum of Dirac $\delta$-functions, in which case we are back to
multiple growing curves. In this context, it is important to consider more
explicit models to describe the growth of a domain bounded by an interface.
One such model was first introduced in Ref. us_pre  for the upper half-plane,
and in the next section it is extended to the channel geometry.

##  III Loewner Evolutions for Interfaces in the Channel

Figure 3: The physical and mathematical planes for a growing interface in the
channel geometry. The mapping $g_{t}(z)$ maps the interface $\Gamma_{t}$ to a
segment on the real axis, while the new interface $\Gamma_{t+\tau}$ is mapped
to a tent-like shape.

###  III.1 Tripolar Loewner Equation for Interfaces

Here we consider the problem of an interface starting from a segment
$[z_{1},z_{2}]\subset(-1,1)$ on the real axis in the $z$-plane and growing
into the upper half-channel, as indicated in Fig. 3. In our growth model we
assume, for simplicity, that the growing interface has certain special points,
referred to as _tips_ and _troughs_ , where the growth rate is a local maximum
and a local minimum, respectively, while the interface endpoints $z_{1}$ and
$z_{2}$ remain fixed. Let us denote by $\Gamma_{t}$ the interface at time $t$
and by $K_{t}$ the growing region delimited by $\Gamma_{t}$ and the real axis.
It is assumed that the curve $\Gamma_{t}$ is simple so that the physical
domain $\mathbb{P}\backslash K_{t}$ is simply connected, hence $K_{t}$ is a
hull. As before, the Loewner function $g_{t}(z)$ maps the physical domain in
the $z$-plane onto the upper half-channel $\mathbb{P}$ in the mathematical
plane $w$-plane, that is,

$g_{t}:\mathbb{P}\backslash K_{t}\rightarrow\mathbb{P},$

Under the action of $g_{t}(z)$, the interface $\Gamma_{t}$ is mapped to an
interval on the real axis in the $w$-plane, with the images of the tips,
troughs and end points being denoted by $a_{i}(t)$, $i=1,...,N$, where $N$ is
the total number of such special points; see Fig. 3. The mapping function
$g_{t}(z)$ is again required to satisfy satisfy the hydrodynamic condition (1)
and the initial condition (2). Here, however, we shall not fix the function
$C(t)$ a priori.

Figure 4: The transformed physical and the mathematical planes for a growing
interface. The ‘tilded’ planes are obtained from the original domains shown in
Fig. 3 by applying the transformation
$\phi(z)=\sin\left(\frac{\pi}{2}z\right)$; see text.

The growth dynamics in our model is specified by requiring that tips and
troughs grow along gradient lines in such a way that the interface
$\Gamma_{t+\tau}$ at time $t+\tau$, for infinitesimal $\tau$, is mapped under
$g_{t}(z)$ to a polygonal curve, as shown in Fig. 3. The exterior angle at the
$i$-th vertex of our polygonal curve is denoted by $\pi(1-\alpha_{i})$, with
the convention that if the angle is greater than $\pi$ the corresponding
parameter $\alpha_{i}$ is negative. In other words, the parameter
$\alpha_{i}$’s are negative for tips and positive for troughs and end points.
From a trivial trigonometric relation it follows that

$\sum_{i=1}^{N}\alpha_{i}=0.$ (26)

If we denote by ${h}_{i}$ the heigth of the $i$th vertex of the polygonal
curve in the $w$-plane, then as $\tau\to 0$ the parameters $h_{i}$ and
$\alpha_{i}$ all go to zero, and in this limit one finds the following
relation:

$\sum_{i=1}^{N}a_{i}\alpha_{i}=0.$ (27)

As before, the mathematical plane at time $t+\tau$ is represented by the
complex $\zeta$-plane, so that under the mapping $\zeta=g_{t+\tau}(z)$, the
interface tips, troughs, and end points at time $t+\tau$ are mapped to the
points $\zeta=a_{i}(t+\tau)$, as indicated in Fig. 3. To derive the tripolar
Loewner equation for the growing interface we first need to obtain the mapping
$w=F(\zeta)$, so that we can write $g_{t}(z)=F(g_{t+\tau}(z))$, and then take
the limit $\tau\to 0$. As discussed in Sec. II, it is more convenient however
to work with the tripolar Loewner equation in the upper half-plane by applying
the transformations given in Eq. (3). The corresponding transformed domains
are shown in Fig. 4. In the $\tilde{w}$-plane, we obtain (in the limit of
$\tau\to 0$) a polygonal curve whose vertices are located at the points
$\tilde{a}_{i}+i\tilde{h}_{i}$, where $\tilde{a}_{i}=\sin(\pi a_{i}/2)$ and
the heigth $\tilde{h}_{i}$ in the $\tilde{w}$-plane is given by

$\tilde{h}_{i}=\frac{\pi}{2}\cos\big{(}\frac{\pi}{2}a_{i}\big{)}h_{i},$ (28)

and the exterior angle at the $i$th vertex is denoted by
$\pi(1-\tilde{\alpha}_{i})$. Relations analogous to (26) and (27) are
obviously valid for $\tilde{\alpha}_{i}$ and $\tilde{a}_{i}$.

Next consider the mapping, $\tilde{w}=\tilde{F}(\tilde{\zeta})$, from the
upper half-$\tilde{\zeta}$-plane onto the upper half-$\tilde{w}$-plane with a
polygonal cutout region; see Fig. 4. Since the domain in the $\tilde{w}$-plane
can be viewed as a degenerate polygon, the mapping function $\tilde{F}$ can be
easily obtained from the Schwarz-Christoffel transformation CKP , which yields

$\tilde{g}_{t}=\tilde{F}(\tilde{g}_{t+\tau})=K\varphi(\tilde{g}_{t+\tau})+C,$
(29)

where

$\varphi(\tilde{\zeta})=\int^{\tilde{\zeta}}\prod_{i=1}^{N}{[z-\tilde{a}_{i}(t+\tau)]^{-\tilde{\alpha}_{i}}}\,dz.$
(30)

The constants $K$ and $C$ are determined so as to guarantee that
$\tilde{F}(\pm 1)=\pm 1$, and hence we can rewrite Eq. (29) as

$\tilde{g}_{t}=\frac{2\varphi(\tilde{g}_{t+\tau})-[\varphi(1)+\varphi(-1)]}{\varphi(1)-\varphi(-1)}.$
(31)

The integral in Eq. (30) cannot, in general, be performed explicitly. It is
thus convenient to expand the integrand in powers of the infinitesimal
quantities $\tilde{\alpha}_{i}$ and then proceed with the integration term by
term. Doing this up to the first order in the $\tilde{\alpha}_{i}$’s, one
finds

$\varphi(\zeta)=\zeta-\sum_{i=1}^{N}\tilde{\alpha}_{i}(t)[\zeta-\tilde{a}_{i}(t+\tau)]\ln[\zeta-\tilde{a}_{i}(t+\tau)].$
(32)

If one now inserts Eq. (32) into Eq. (31), expand the resulting expression up
to first order in $\tau$, and then take $\tau\to 0$, one obtains the following
Loewner equation

$\dot{\tilde{g}}_{t}(z)=\sum_{i=1}^{N}\tilde{d}_{i}(t)\left\\{[\tilde{g}_{t}-\tilde{a}_{i}(t)]\ln[\tilde{g}_{t}-\tilde{a}_{i}(t)]-A_{i}(t)\tilde{g}_{t}+B_{i}(t)\right\\},$
(33)

with

$A_{i}(t)=\frac{1}{2}\left\\{[1+\tilde{a}_{i}(t)]\ln[1+\tilde{a}_{i}(t)]+[1-\tilde{a}_{i}(t)]\ln[1-\tilde{a}_{i}(t)]\right\\},$
(34)
$B_{i}(t)=\frac{1}{2}\left\\{[1+\tilde{a}_{i}(t)]\ln[1+\tilde{a}_{i}(t)]-[1-\tilde{a}_{i}(t)]\ln[1-\tilde{a}_{i}(t)]\right\\},$
(35)

where the ‘tilded’ growth factors $\tilde{d}_{i}(t)$ are defined by

$\tilde{d}_{i}(t)=\lim_{\tau\to 0}\frac{\tilde{\alpha}_{i}}{\tau}.$ (36)

The time evolution of the driving functions $\tilde{a}_{i}$ is determined by
requiring that the points $\tilde{a}_{i}(t+\tau)$ be taken under the mapping
$F$ to the points $\tilde{a}_{i}(t)$. In other words, we impose that

$\tilde{a}_{i}(t)=F(\tilde{a}_{i}(t+\tau)).$ (37)

Applying the procedure used above to derive the Loewner equation Eq. (33), it
immediately follows that the differential equations governing the dynamics of
the driving functions $\tilde{a}_{i}$ can be obtained by simply setting
$\tilde{g}_{t}=\tilde{a}_{i}$ in Eq. (33), which yields

$\dot{\tilde{a}}_{i}=\sum^{N}_{j=1}\tilde{d}_{j}(t)\left\\{[\tilde{a}_{i}-\tilde{a}_{j}(t)]\ln\left|\tilde{a}_{i}-\tilde{a}_{j}(t)\right|-A_{j}(t)\tilde{a}_{i}+B_{j}(t)\right\\}.$
(38)

Figure 5: Loewner evolution for a symmetric interface growing in the channel
geometry. The solid curves show the interface at various times $t$, starting
from $t=0.5$ up to $t=2.0$ with a time separation of $\Delta t=0.3$ between
successive curves.

The tilded growth factors $\tilde{d}_{i}$ defined in Eq. (36) can be related
to the growth factors, ${d}_{i}(t)=\lim_{\tau\to
0}\left({{\alpha}_{i}}/{\tau}\right)$, defined for the original channel
geometry by virtue of trivial trigonometric relations. After some calculation,
one finds

$\tilde{d}_{i}=\frac{{C}_{i+1}-{C}_{i}}{\tilde{a}_{i+1}-\tilde{a}_{i}}+\frac{{C}_{i-1}-{C}_{i}}{\tilde{a}_{i}-\tilde{a}_{i-1}},\qquad
2\leq i\leq N-1,$ (39)

where

$C_{i}=\frac{\pi}{2}\cos\left(\frac{\pi}{2}a_{i}\right)\sum_{j=1}^{i-1}d_{i-j}(a_{i}-a_{i-j}).$
(40)

From the Eqs. (26) and (27) it also follows that the growth factors $d_{i}$
satisfy the following relations

$\sum_{i=1}^{N}d_{i}=0,$ (41) $\sum_{i=1}^{N}a_{i}d_{i}=0.$ (42)

In view of the relations above, we can express the parameters $d_{1}(t)$ and
$d_{N}(t)$ associated with the end points in terms of the other growing
factors $d_{i}(t)$, so that our growth model is completely specified by
prescribing the functions $d_{i}(t)$ at the tips and troughs of the interface.
Thus, once the growth factors $d_{i}$ are given, one can compute the tilded
growth factors $\tilde{d}_{i}$ from Eq. (39), proceed with the integration of
the Loewner equation given in Eq. (33), and then invert the transformation (5)
to obtain the Loewner mapping $g_{t}(z)$. Next we discuss some examples of
interface growth described by the above model.

###  III.2 Examples

####  III.2.1 Single Tip

For an interface with a single tip (i.e., $N=3$), the factors $d_{1}(t)$ and
$d_{3}(t)$ can be written in terms of $d_{2}(t)$, so that we can set
$d_{2}=-1$ without loss of generality, since this amount to a mere rescaling
of the time coordinate. Let us consider first the case in which the interface
is symmetric with respect to the channel centerline: $a_{2}(t)=0$ and
$a_{3}(t)=-a_{1}(t)\equiv a(t)$. This imply, in particular, that
$\tilde{d}_{1}=\tilde{d}_{3}=-\tilde{d}_{2}/2$. Introducing the parameter
$\tilde{d}(t)=|\tilde{d}_{2}(t)|$, the Loewner equation (33) then becomes

$\dot{\tilde{g}}_{t}(z)=\frac{1}{2}\tilde{d}(t)\left\\{[\tilde{g}_{t}+\tilde{a}(t)]\ln[\tilde{g}_{t}+\tilde{a}(t)]+[\tilde{g}_{t}-\tilde{a}(t)]\ln[\tilde{g}_{t}-\tilde{a}(t)]-2\tilde{g}_{t}\ln\tilde{g}_{t}-A_{s}(t)\tilde{g}_{t}\right\\},$
(43)

where

$A_{s}(t)=[1+\tilde{a}(t)]\ln[1+\tilde{a}(t)]+[1-\tilde{a}(t)]\ln[1-\tilde{a}(t)],$
(44)

with the dynamics for $\tilde{a}(t)$ being given by

$\dot{\tilde{a}}(t)=\tilde{d}(t)\left(2\ln 2-A_{s}\right)\tilde{a},$ (45)

In this case, one can show from Eq. (39) that $\tilde{d}(t)$ becomes

$\tilde{d}(t)=\frac{\pi}{2}\tan\left[\frac{\pi}{2}a(t)\right]=\frac{\pi}{2}\frac{\tilde{a}(t)}{\sqrt{1-\tilde{a}^{2}(t)}},$
(46)

where we have used the fact that $|d_{2}|=1$.

Figure 6: Loewner evolution for an asymmetric interface growing in the channel
geometry. The solid curves show the interface at several times from $t=0.5$ up
to $t=1.7$, with $\Delta t=0.3$ between successive curves, while the dashed
curve represents the path traced by the tip.

An example of a symmetric interface in a channel is shown in Fig. 5. In this
figure, the solid curves represent the interface at various times $t$,
starting from $t=0.5$ up to $t=2.0$ with a time separation of $\Delta t=0.3$
between successive curves. To generate the curves shown in Fig. 5 we
integrated the Loewner equation (43) backwards in time from ‘terminal
conditions’ $\tilde{g}_{t}=\tilde{w}$, for
$\tilde{w}\in[-\tilde{a}(t),\tilde{a}(t)]$, to obtain the respective initial
values $\tilde{g}_{0}$ from which we determine the corresponding points $z$ on
the interface: $z=({2}/{\pi})\arcsin\left(\tilde{g}_{0}\right)$. One sees from
the figure that as time proceeds the interface expands and tends to occupy the
entire channel. It is worth mentioning, however, that it is hard to go past
the latest time shown in Fig. 5 because the function $a(t)$ becomes very close
to unity, rendering the numerical integration of the Loewner equation very
difficult after this point.

In the case of an asymmetric interface, we need to integrate the full Loewner
equation given in Eq. (33) with $N=3$. In Fig. 6 we show a solution where the
solid curves represent the interface at various times $t$ (see caption for
details), while the dashed curve indicates the path traced by the tip
$\gamma_{t}=g^{-1}_{t}(a_{2}(t))$. As seen in Fig. 6, the tip is “repelled” by
its image with respect to the closest channel wall, so that an asymptotically
symmetrical shape is expected for sufficiently long time. (It is difficult,
however, to integrate the Loewner equation for larger times for reasons
already mentioned.)

Figure 7: Symmetrical growing interface with two tips in the channel for
$d_{2}=d_{4}=-2$ and $d_{3}=1.5$. The solid curves represent the interface at
times from $t=0.5$ to $t=1.7$, with $\Delta t=0.3$ between successive curves,
whereas the dashed lines indicate the trajectories of the tips and the trough.

####  III.2.2 Multiple Tips

Figure 8: Asymmetrical growing interface with two tips in the channel. The
growth factors are the same as in Fig. 7 but the interface is initially
located off center.

Let us now illustrate some of the patterns that emerge from our growth model
for interfaces with two tips and one trough (i.e., $N=5$), in which case tip
competition can arise. Here we shall consider for simplicity only cases in
which $d_{i}(t)=d_{i}={\rm const.}$ We expect nevertheless that the
qualitative behavior seen in this simpler situation (see below) should also be
valid in more general cases. As a first example, we show in Fig. 7 a
symmetrical growing interface, where the dashed lines represent the
trajectories of the tips and the trough. In this figure, we started with
symmetrical initial conditions, namely, $a_{5}(0)=-a_{1}(0)=0.3$,
$a_{4}(0)=-a_{2}(0)=0.2$, and $a_{3}(0)=0$, and chose the growth factors of
the two tips to be the same, $d_{2}=d_{4}=-1$, so as to preserve the initial
symmetry, with the trough growth factor being $d_{3}(t)=0.6$.

If we break the symmetry, either by choosing an asymmetric initial condition
or using different growth factors for the tips, then inevitably of the tips
will grow faster and “screen” the other tip. For instance, in Fig. 8 we show a
situation where the growth factors are the same as in Fig. 7 but the initial
segment from which the interface grows is off-center, namely, $a_{1}(0)=0.2$,
$a_{2}(0)=0.3$, $a_{3}(0)=0.5$, $a_{4}(0)=0.7$, and $a_{5}(0)=0.8$. In this
case, the tip closest to the right channel wall grows slower and falls behind
the other tip. Here, however, the screening is partial in the sense that the
slower tip continues to grow but with a velocity that is a fraction of that of
the faster tip. (Total screening, whereby the slowest tip eventually stops
growing altogether, can be observed if the faster tip has a sufficiently large
growth factor.)

####  III.2.3 Multiples Interfaces

Figure 9: Loewner evolution for two symmetrical surfaces growing in the the
channel geometry for $d(t)=1$ for each interface. The interfaces start
respectively at $[-3,-1]$ and $[1,3]$. The different curves represent the
interfaces at time intervals of $\Delta t=0.3$.

The generic Loewner equation given in Eq. (33) can also describe the problem
of multiple growing interfaces in the channel geometry. In this case each
interface $\Gamma^{i}_{t}$, for $i=1,...,n$, where $n$ is the number of
distinct interfaces, will be mapped under $g_{t}(z)$ to a corresponding
interval on the real axis in the $w$-plane. Similarly, each advanced interface
$\Gamma^{i}_{t+\tau}$ is mapped by $g_{t}(z)$ to a polygonal curve in the
$w$-plane. One can readily convince oneself that the generic Loewner evolution
defined by (33) and (38) applies to this case as well, where $N$ is now the
total number of vertices corresponding to the sum of the number of vertices
for each interface. Note that for each interface we have conditions analogous
to (41) and (42), but involving only the growth factors $d_{i}$ of the
respective interface.

Figure 10:  Loewner evolution for two asymmetric growing fingers. Here the
growth factors are the same as in Fig. 9 but the initial segments from which
the interface starts to grow were shifted to the right.

In Fig. 9 we show numerical solutions for two growing symmetrical interfaces,
with one tip each, where the interfaces are the images of one another with
respect to the $y$ axis. Here the interfaces start to grow from the intervals
$[-3,-1]$ and $[1,3]$, respectively, with the corresponding tips starting at
symmetrical points $a_{4}(0)=-a_{2}(0)=2$. (The interfaces have the same
growing factors, meaning that $d_{2}=d_{4}=-1$.) Notice that, as time goes by,
the inner sides of the two interfaces move towards one another leaving a
narrow channel between them. For sufficiently large time, the width of such a
channel becomes infinitesimally small so that for all practical purposes the
resulting evolution will look like a single, symmetrical interface.

If the symmetry between the two interfaces is broken, then one of them will
eventually move ahead of the other one. For instance, in Fig. 10 we show the
case in which the growth factors are the same as in Fig. 9 but the initial
positions of the interfaces are no longer symmetric with respect to the
channel centerline. In this case, the interface closer to the centerline is
the “winner,” in the sense that its tips moves ahead of the tip of the second
interface. This competition between the two growing interfaces resembles the
so-called “shadowing effect” in fingering phenomena, whereby the longer
fingers grow faster and hinder the growth of the shorter ones in their
vicinities.

##  IV Conclusions

We have discussed a class of growth models in the channel geometry where the
growth dynamics is described in terms of tripolar Loewner evolutions. In the
tripolar geometry, the Loewner mapping fixes the points $z=\pm 1$ and the
point at infinity. The growing domain starts from a point (in the case of a
curve) or from an interval (in the case of a domain bounded by an interface)
on the real axis, ending at infinity (for infinite time). In particular, a
novel class of exact solutions of the tripolar Loewner equation for multiple
curves was obtained for the case of stationary driving functions. As for the
more difficult problem of interface growth in the channel geometry, we
introduced a generalized tripolar Loewner equation and presented several
numerical solutions, including the case of an interface with multiple tips as
well as the case of multiple interfaces. It was argued that the behavior seen
in the model is reminiscent of the phenomenon of tip/finger competition
observed in fingering experiments.

A natural extension of the work presented here would be to consider the
problem of tripolar stochastic Loewner equation (SLE) and investigate its
connection with statistical mechanics models and conformal field theory.
Although stochastic growth processes were not considered in this paper, it is
nonetheless hoped that a better understanding of deterministic Loewner
evolutions in the channel geometry, which was the main thrust of our study,
may be of some help in tackling the more difficult problem of tripolar SLEs.
Another possible direction for future work would be to consider more general
geometries, such as “quadripolar” Loewner evolutions where four points are
kept fixed. This geometry would be of interest in connection, for instance,
with growth processes in doubly periodic domains.

###### Acknowledgements.

This work was supported in part by the Brazilian agencies FINEP, CNPq, and
FACEPE and by the special programs PRONEX and CT-PETRO.

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