58
3. Analytic Solutions of Hydrodynamic Dispersion Equations
Taking the Laplace transform of both sides of Eq. (3.2.1), we obtain
02C*
oC*
DL ox2 - V ox - (A. - p)C* = o.
(3.2.9)
This equation may be solved as a second-order ordinary differential equation
and its solution is expressed as
(
Vx
x
+ B(p)exp 2D + In
L
yDL
(3.2.10)
After imposing the boundary condition (3.2.4), we must have C*( 00, p) = 0
when x = 00, C = O. Consequently, B(p) = 0, and in terms of the boundary
condition (3.2.3), when x = 0, C = Co, we have
C*(O,p) =
Coe-ptdt =~.
J
oo
C
o
P
Substituting this equation into Eq. (3.2.10) yields A(p) = Co/po Thus, Eq.
(3.2.10) becomes
* Co ( Vx
x J( V 2 ) )
C = pex p 2D L -.,JD;. 4D L + A. + P .
(3.2.11)
After C* is obtained, the original solution, C, can also be obtained by using
the inverse of Laplace transformation. Letting L -1 indicate the inverse transformation, we then have
where
X
x 2
a = - -
b 2 = -
+ A..
.,JD;.'
4DL
(3.2.13)
From the Table of Laplace Transforms, we know that
and