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CANADIAN APPLIED
MATHEMATICS QUARTERLY
Volume 5, Number 1, Winter 1997
A CHEMICAL DIFFUSION PROCESS WITH REACTION TAKING PLACE AT FREE BOUNDARY SHIJIN DING, YOUSHAN TAO AND HONG-MING YIN
ABSTRACT. In this paper we derive a mathematical model
which describes a solid-liquid chemical reaction-diffusion process with reaction taking place only at the solid-liquid interface. The force generated by nonequilibrium mass flux across
the interface drives the interface to move. The normal velocity of the moving boundary is assumed to be proportional t o
the rate of the reaction. We show that this free boundary
problem has a short time solution in the classical sense.
1. Derivation of the mathematical model. For a conventional
chemical reaction-diffusion process, the diffusion and reaction take
place throughout the region. However, many experimental observations
indicate that some chemical reactions may take place only a t some
local sites, due to the effect of certain chemical catalysts (see [17,
51). In this paper we consider a chemical substance which has solid
and liquid states. Moreover, we assume that the reaction takes place
only at the solid and liquid interface. Experiment shows that the
chemical concentration is continuous throughout the region, but the
mass flux has a jump at these local sites due to the localized reaction.
The magnitude of the jump typically depends on the concentration
[I]. Moreover, because of this nonequilibrium force produced by the
reaction, the solid-liquid interface will be moving during the reaction.
How fast the interface moves often depends on how strong the reaction
is.
To describe the above physical model, we let R be a bounded domain
in Rn with boundary S = dR E C2+Qfor some a E ( 0 , l ) . Let robe a
C3+a-hypersurface in R, which divides the domain R into two regions
R1 and Rg with dR = 801, ro= dR2 and BRl 7
r 802 = 4. Consider a
chemical material which has solid and liquid phases at the beginning.
Moreover, the liquid phase occupies the region R1 while the solid phase
~ c c e ~ t e d f publication
or
on June 5, 1997.
Key words and phrases. Diffusion with localized reaction, free boundary problem, solvability.
AMS (MOS) Subject Classifimtion. 35R35,35K57.
Copyright 01997 Rocky Mountain Mathematics
consortium
50
S. DING. Y. TAO AND
H.-M.YIN
occupies the region R2. Let u(x,t) denote the concentration of the
chemical substance in QT = R x (0, T] and rt be the interface at time
t. Let us assume this moving interface between the solid and liquid
phases can be expressed by @(x,t) = 0. Namely,
As formulated in [6], the local reaction on the interface I? can be
expressed by
[vv(z,t)Hsolid]f
(21)1
where u(x, t) is the inward normal direction at (x, t) on I? and
is
the usual characteristic function which equals 1 in the solid phase and
0 in the liquid phase.
Assume that the diffusion coefficients are {Aij(x,t,u)) in the region. Fickian's law and the conservation of mass lead to the following
reaction-diffusion equation
One should understand the above equation only in the weak sense since
Vu(z,t)Hwud is only defined in the weak sense.
To find the conditions on the interface, we assume that the concentration of the chemical substance is continuous across the interface I?.
Moreover, we assume that the normal velocity of the free boundary
(interface) is proportional to the reaction:
where Vv(,,t)represents the normal velocity at (x, t) E
rt.
Let Q$ and Q$ denote the liquid and solid phases, respectively. It is
clear that u(x, t) satisfies
Moreover, on the interface r, since u(x, t) is continuous in QT, we have