Page 144 - Bird R.B. Transport phenomena

128 Chapter 4 Velocity Distributions with More Than One Independent Variable

Similar elimination of ф gives
G(x, у,ф) = (4.3-14)
Setting ф = a constant in Eq. 4.3-13 gives the equations for the equipotential lines for
some flow problem, and setting ф = constant in Eq. 4.3-14 gives equations for the stream-
lines. The velocity components can be obtained from

_dz_ v x + (4.3-15)
dw
•3
Thus from any analytic function w{z), or its inverse z{w), we can construct a flow net
with streamlines ф = constant and equipotential lines ф = constant. The task of finding
w(z) or z(w) to satisfy a given flow problem is, however, considerably more difficult.
Some special methods are available 45 but it is frequently more expedient to consult a
table of conformal mappings. 6
In the next two illustrative examples we show how to use the complex potential w{z)
to describe the potential flow around a cylinder, and the inverse function z(w) to solve
the problem of the potential flow into a channel. In the third example we solve the flow
in the neighborhood of a corner, which is treated further in §4.4 by the boundary-layer
method. A few general comments should be kept in mind:
(a) The streamlines are everywhere perpendicular to the equipotential lines. This
property, evident from Eqs. 4.3-10,11, is useful for the approximate construction
of flow nets.
(b) Streamlines and equipotential lines can be interchanged to get the solution of
another flow problem. This follows from (a) and the fact that both ф and ф are
solutions to the two-dimensional Laplace equation.
(c) Any streamline may be replaced by a solid surface. This follows from the
boundary condition that the normal component of the velocity of the fluid is
zero at a solid surface. The tangential component is not restricted, since in po-
tential flow the fluid is presumed to be able to slide freely along the surface (the
complete-slip assumption).

EXAMPLE 4.3-1 (a) Show that the complex potential
Potential Flow around (4.3-16)
a Cylinder
describes the potential flow around a circular cylinder of radius R, when the approach veloc-
ity is v x in the positive x direction.
(b) Find the components of the velocity vector.
(c) Find the pressure distribution on the cylinder surface, when the modified pressure far
from the cylinder is 0^.

SOLUTION (a) To find the stream function and velocity potential, we write the complex potential in the
form w(z) = ф(х, у) + гф(х, у):

R 2 R 2
w(z) = —v x (4.3-17)
x 2
x +
J. Fuka, Chapter 21 in K. Rektorys, Survey of Applicable Mathematics, MIT Press, Cambridge, Mass.
5
(1969).
6
H. Kober, Dictionary of Conformal Representations, Dover, New York, 2nd edition (1957).