
1006 HIGHER-ORDER PARTIAL DIFFERENTIAL EQUATIONS
Example 11.1. Consider the problem of a slow (Stokes) inflow of a viscous fluid into a ha lf-
plane thro ugh a slit of width 2a at a constant velocity U that ma kes an angle β with the normal to
the boundary (the angle is measured off from the nor mal direction counterclockwise).
With the stream function w introduced by the relations v
x
= −
∂w
∂y
and v
y
=
∂w
∂x
(v
x
and v
y
are
the fluid velocity c omponents), the problem is reduced to the specia l case of the previous problem
with
f(x) =
(
U cos β fo r |x| < a,
0 for |x| > a,
g(x) =
(
U sin β for |x| < a,
0 for |x| > a.
Dean’s solution:
w(x, y) =
U
π
(x − a) cos β + y sin