Information Technology Reference
In-Depth Information
@c
@t
Crq
f
D
0:
(7.18)
We have here assumed that % is constant.
Fick's law reads, in the three-dimensional case,
q D
k
r
c;
(7.19)
The physical principle is that the substance moves from regions of high concentra-
tion to regions of low concentration, the direction being determined by the greatest
decrease in c, which is in the direction of the negative gradient (
r
c). Inserting
(7.19)in(7.18) eliminates
q
and gives the three-dimensional diffusion equation
for c:
@c
@t
D
k
r
2
c
C
f.x;y;
z
;t/:
(7.20)
Here we have used
rr
c
Dr
2
c:
@
@x
;
@c
D
@
2
c
@x
2
C
@
2
c
@y
2
C
@
2
c
@
z
2
@
@y
;
@
@
z
@x
;
@c
@y
;
@c
r
2
c:
@
z
Boundary Conditions
The common boundary conditions are typically
-
Prescribed concentration at parts of the boundary,
-
Impermeable boundaries, i.e., no normal flow,
@c
@x
D
0;
q
D
0
or by Fick's law
in one-dimensional problems and
@n
D
0.
@c
@c
q n D
0
or by Fick's law
@n
r
c
n
/
in three-dimensional problems, or
-
Prescribed inflow Q at a boundary,
or, by Fick's law,
k
@c
q n D
Q
@n
D
Q:
More information about such boundary conditions is given at the end of Sect.
7.3.2
.
The initial condition for diffusive transport is simply a specification of the concen-
tration at initial time (t
D
0): c.x; 0/
D
I.x/,whereI.x/ is a known function.
