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U θ

x

Figure 10.3.1 A uniform flow, U, impinges on a cylinder of radius R. The polar

coordinate frame is shown.

The problem posed by the flow configuration in Figure 10.3.1, assuming that the motion

is incompressible and irrotational, is to find a solution of Laplace’s equation that has zero
radial flow on the surface of the cylinder and approaches the uniform, oncoming flow as

r goes to infinity. The uniform flow has a velocity potential, ! = Ux so that, in polar

coordinates, the problem is,

1 ! # !" & + 1 ! "
2
%
(
r !r $ r !r ' r !) 2 = 0,
2
!"
= 0, r = R, (10.3.3 a, b, c)
!r

" * Ur cos), r * +

It is a simple matter to check that a the solution satisfying the equation and all the
boundary conditions is,

# R &
2
! = U cos" r + r ' ( , r ) R (10.3.4)
%
$
so that the velocity components are,

!" % R (
2
u = = U cos# 1$
(r) ' 2 *
!r & r )
% R ( (10.3. 5 a, b)
2
u = 1 !" = $U sin# 1+
2 *
&
(# ) r !# ' r )

Chapter 10 14