J. theor. Biol. (1982) 95,409-411
LETTERS TO THE EDITOR
Solutions for Water Transport
through
Plant Tissue
In an article in this journal, Molz (1976) models water transport through plant tissues assuming two pathways, namely the apoplasm and the symplasm pathways. By a careful analysis Molz (1976) obtains the following system of partial differential equations,
where T(X, t) and +(x, t) are the water potentials in the apoplasm and symplasm pathways respectively. The constants a, b, D1 and D2 are given by 1 1 D&z, (2) a==, b=RC,’ 2
2
where Ax is the length, width and height of each cell through which the water flows, RI is the water flow resistance per cell of apoplasm pathway, R2 is the water flow resistance per cell of symplasm pathway, R is the water flow resistance between pathways, Ci is the water capacity per cell of apoplasm pathway and C2 is the water capacity per cell of symplasm pathway. We wish to point out that a similar concept of two pathways has been developed by Aifantis (1979a, b) in the context of diffusion of ions in metals in the presence of high diffusivity paths. Further we note that the associated mathematical results are relevant to the coupled system (1) (see for example Aifantis & Hill, 1980; Hill & Aifantis, 1980; and Hill, 1979, 1980, 1981). In particular it can be shown that the general solutions of (1) for the initial conditions
4x9 0) = f(x),
44x70) = g(x),
(3)
409
0022-5193/82/060409+03
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@ 1982 Academic
Press Inc. (London)
Ltd.
410
J.
M.
HILL
AND
A.
I.
LEE
are given by 7(x, t) = eparhl(x, Dlt)
+ a 1’2~ohh(x, 81 d& $(x, t) = epbfh2(x, l&t)
where hI(x, t) and h2(x, t) are solutions of the classical diffusion equation with unit diffusivity, namely ah -=-
a2h
(5)
at ax2’
such that hl(x, t) and h2(x, t) satisfy the same initial conditions as r(x, t) and rL(x, t) respectively. Furthermore y, A and 7 are defined by
(6)
7)= (;““‘;‘) [(Dlt-.g(r-D*t)]1’2, 1 2
and I0 and II denote the usual modified Bessel functions. Thus, for example, the tissue immersion problem discussed by Molz (1976), namely 7(x, 0) = 6,
w,
7(0, t) = 0, E (L, t) = 0,
OCx
0) = 8,
t>o,
$(O, t) = 0, z (L, t) = 0,
(7)
t>o,
reduces to the problem of solving the classical diffusion equation (5) subject to the conditions h(x,0)=6,
h(0, t)=O,
g (L, t) = 0,
(8)
which has the solution (see Carslaw & Jaeger, 1959, p. 101) 12n+l)h2r/4Lz
h(x,t)=;
:
n=l
e
(2n + 1)
sin
(2n +1)%X 2L
1
(9)
LETTERS
TO
THE
411
EDITOR
Finally we remark that it can be shown from (1) that both r and I++(and therefore the total water potential 4 = T + 4) satisfy the fourth order equation g+(a+b)$=(aD;+bD,)$+(Dl+D*)-$&DlD2g$.
(10)
Further, for large times (equilibrium) this equation is dominated classical diffusion equation (see Aifantis & Hill, 1980),
and we may identify
an equilibrium
or large time diffusivity
D given by (12)
D=
On simplification
by the
from (2) we obtain D
= hW2(K RIR2(Cl+
+Rd
(13)
C2)’
which is in complete agreement with the result given in Molz (1976) which he obtained by elementary reasoning. Thus our approach gives an independent derivation of (13) and an alternative insight into the concept of local equilibrium. We have attempted here to give the main mathematical results applicable to the model defined by (1). The reader is referred to the references for further details. Department of Mathematics, The University,of Wollongong, Wollongong, N.S. W., Australia (Received
30 June
JAMES ALEXANDER
M. HILL I. LEE
1981) REFERENCES
AIFANTIS, E. C. (1979a). J. appl. Phys. 50, 1334. AIFANTIS, E. C. (19796). Acta Metallurgica 27, 683. AIFANTIS, E. C. & HILL, J. M. (1980). Q. J. Mech. Appl. Math. 33,l. CARSLAW, H. S. &JAEGER, J. C. (1959). Conduction of Heat in Solids. 2nd edition. Oxford University Press. HILL, J. M. & AIF~NTIS, E. C. (1980). Q. J. Mech. Appl. Math. 33, 23. HILL, J. M. (1979). Scripfa Metallurgica 13, 1027. HILL, J. M. (1980). J. Austral. Math. Sot. (Series B) 22,58. HILL, J. M. (1981). J. Inst. Math. Applies, 27, 177. MOLZ, F. J. (1976). J. theor. Biol. 59, 277.
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