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We can then, after dividing by t , write the mass conservation equation in the
form
%.q.a; t /
q.b; t//
C
Z
b
a
%f
dx
D
Z
b
a
%
@c
@t
dx
:
(7.14)
Assuming that % is constant, the terms on the left-hand side can be transformed to
an integral over ˝ by using integration by parts. We have
Z
b
@x
dx
D
Z
b
%
@q
q
@%
@x
dx
C
%Œq
a
;
a
a
that is,
%.q.a; t /
q.b; t//
D
Z
b
a
%
@q
@x
dx
:
Collecting the two integrals, we can write the mass conservation principle in the
form
@x
f
dx
D
0:
Since this equation must be valid for any arbitrarily chosen interval Œa; b,the
integrand must be zero. The rigorous proof of this assertion is mathematically com-
plicated, but a rough justification goes as follows. Think of the integrand being
non-zero in a small interval and then choose Œa; b to be this interval to obtain a
non-zero (i.e., incorrect) value of the integral; the integrand must vanish.
Using the fact that the integrand must vanish gives us the PDE for mass conser-
vation:
Z
b
%
@c
@t
C
@q
a
@t
C
@q
@c
@x
f
D
0:
(7.15)
There are two unknowns here, c and q, and one equation. Thus, we need an
additional equation relating the concentration c to the velocity q. This equation is
called Fick's law:
q
D
k
@c
@x
:
(7.16)
Fick's law states that the velocity of the substance depends on variations in the con-
centration, and that the flow is directed where c decreases, i.e., the substance flows
from regions of high concentration to regions of low concentration. This sounds rea-
sonable. The parameter k depends on the substance and the medium in which the
substance diffuses and must be measured in physical experiments. In most appli-
cations of the diffusive transport of a substance in a fluid, k can be regarded as a
constant. Thinking of ink in water or sugar in coffee, it seems reasonable to assume
that the diffusive transport ability is the same throughout the entire medium (water
or coffee), i.e., k
D
const.
By inserting Fick's law (7.16) in the mass conservation equation (7.15), we can
eliminate q and get a PDE with only one unknown function, c:
