V4 = 0 (4-12)
For the given solution domain as shown in Figure 4-2a, the resulting non-dimensionalized
boundary conditions include the following equations,
v~(x r, y) = y(x + y), (4-13)
Po (L/2jl)2
p(x- ,y)= p(x + ,y)+
pv o (4-14)
t- V ,D walls = 0 (4-15)
,.V//js (4-15)
V wallss = V'walls, (4-16)
(x r, y)= (x + r, y)- 1, (4-17)
and
,Vol, =0 (4-18)
where (i, h) are the unit vectors in the tangential and normal directions with the respect to the
walls.
4.3 Formulation of Solutions
Since q(x, y) is uncoupled from yf(x, y), it can be solved separately from Equation 4-9
with boundary conditions given by Equations 4-17 and 4-18. Subsequently, Vy(x, y) can be
obtained by solving Equation 4-12 with the boundary conditions (Equations 4-13 to 4-16).
4.3.1 Electric Potential
Using complex functions,102 a general solution to the Laplace equation (Equation 4-9) is
written as,
(x, y) = Ax + 23[G((r)] (4-19)
where 4 = y/2 + ix/2 is a complex variable; 9t and 3 denote the real and imaginary parts of a
complex value; A is a real constant; G(,;) is an arbitrary analytical periodic complex function
satisfying the periodic boundary conditions in the x-direction G(y/2 +(x- 'r)i/2)