Diffusion Reaction Interaction for a Pair of Spheres - ETD ...

where and Λ are given by equation (3.16).
Λ10 20
The first step of the derivation to be used here is similar to the development of
equation (2.55) in Section 2.4.1. It suffices to point out that the first equation for n = 0,
from the linear set of equations for for ≥ 0,
can be solved for v in terms of
coefficients for > 0,
and used to eliminate v from the rest of the subset for n ≥ 1
v1n to generate a revised set of linear equations for the coefficients for n ≥ 1.
Subsequently, the first equation
v1n n 10
11
n 10
v1n n = 1 from the first revised set of linear equations
provides an expression for v in terms of v1n
for > 1 that can be used to eliminate v
from the revised equations for n ≥ 2.
This second revision gives a set of linear
v1n equations for the coefficients for n ≥ 2.
As in Section 2.4.1, if this procedure is
v1n repeated one can show by induction that
where the nested, sequential expressions for
v1n n 11
() ()
∑ ∞
i
i
v1 n = M n + Lnmv1m
, i
m=
i
i ≥ 0
( ) 1 −
1−
L
( i+
1)
() i () i () i () i
n ≥ , (3.25)
L = L + L L , (3.26)
nm
nm
( 0)
im
L = L
nm
58
ni
nm
ii
, (3.27)