APPENDIX C
SOLUTION TO ELECTRIC POTENTIAL AND STREAM FUNCTION*
This appendix describes the details in solving the periodic functions G (x), G (x),
R1(x), RP(x), Q1(x), and Q (x) that are used for calculating G(4), R(4) and Q(4). The
symmetry of the channel geometry ensures that for Stokes flow 0(x) = 0(-x) so that
O(x) + 0(- x) O(x) 0(-x)
9[)[(x)] = and 3[0(x)] =
2 2i
C.1 Electric Potential Solution
Equation 4-22 and Equation 4-23 in Chapter 4 can be reduced to
n(g + gm)=0 (C-l)
and
m(g,8 + ~ = a (m, ) (C-2)
4(i
Similarly, Equation 4-26 becomes
exp(nh)gl g E, = 0, (C-3)
-,0- a m(,1 m 1) n 0
where E, n =)2 3 is Kronecker delta, andln(z) is the nth order
In (-na) n 0
1
modified Bessel function of first kind, In (z) f exp(zcosO)cosnOd Equations C-1, C-
2, and C-3 form a complete set for solving coefficients {g}) and {g,~, m eZ \ {0} .
Since the electric field is E= -V0, as a result, the tangential electric field strength along
the walls (Equation 4-27 and Equation 4-28) can be evaluated using:
+co 1
Ef (x)= 21n.g exp(-inx)-- (C-4)
m 2;r
and
1
n mr(g g m)]exp(-imx) -
Ef(x) = ( -2 2 (C-5)
1+a2 sin2(x)
C.2 Stream Function Solution
For the stream function, Equations 4-39 to 4-42 give
n(ri +r e)+ mh(q, + q)= 0, (C-6)
2hm,,5B m (r r )+ (q + q,,, )- h (qlm- ,,)