# F(4")G(4)d= 0
ABCDA (4-25)
As both F,(4) and G(4) have a period of 2;r, the integral along BC balances the integral along
DA. Hence the sum of integrals along path AB and CD is zero,
I exp(2np(x))G (x)(p'(x)dx + J exp(2nf (x))G' (x)/'(x)dx = 0
S-(4-26)
Equations 4-22, 4-23 and 4-26 form a complete set of equations to solve G((x) and
GP(x). The detailed solving procedure is detailed in Appendix C. G(4) is subsequently
obtained from Equation 4-20, and O(x,y) is determined from Equation 4-19.
Since the electric field is defined asE = -VV, the tangential electric field strength along
walls, which is used in solving stream function, is obtained from
d 1
E(x) = 2 3[G (x)] -
dx 2r (4-27)
and
d 1
2 3[G (x)]-
E (x)= dx 2;
Fl+a sin (x) (4-28)
4.3.2 Stream Function
The same approach is also used to obtain the solution of the biharmonic equation
(Equation 4-12). Using the periodic boundary condition (Equation 4-13), a general solution to
the biharmonic equation is written as,
V(x, y)= By2 +Cy + (R(S)+R())+y (()+ Q()) (4-29)
whereR(") and Q(;) are arbitrary periodic analytical functions satisfying the periodic
boundary conditions in the x-direction R(y/2 +(x- )i/2)= R(y/2 +(x + )i/2) and
Q(y/2+(x- r)i/2)= Q(y/2 +(x+)i/2);B and C are real-valued constants.96'102
Using Equations 4-14, 4-21 and 4-29, the pressure can be expressed as