Page 143 - Bird R.B. Transport phenomena

§4.3 Flow of Inviscid Fluids by Use of the Velocity Potential 127

We want to solve Eqs. 4.3-3 to 5 to obtain v , v , and 2P as functions of x and y. We
y
x
have already seen in the previous section that the equation of continuity in two-dimen-
sional flows can be satisfied by writing the components of the velocity in terms of a
stream function ф{х, у). However, any vector that has a zero curl can also be written as the
gradient of a scalar function (that is, [V X v] = 0 implies that v = —Уф). It is very conve-
nient, then, to introduce a velocity potential ф(х, у). Instead of working with the velocity
components v x and v , we choose to work with ф(х, у) and ф{х, у). We then have the fol-
y
lowing relations:
дф дф
(stream function) v x = —r- v u = — (4.3-6,7)
dy J dX
(velocity potential) v x = ~ v y = ~ (4.3-8,9)

Now Eqs. 4.3-3 and 4.3-4 will automatically be satisfied. By equating the expressions for
the velocity components we get
дф дф дф дф
-г- = -т- and — = - — (4.3-10,11)
дХ ду ду дХ
These are the Cauchy-Riemann equations, which are the relations that must be satisfied by
3
the real and imaginary parts of any analytic function w{z) = ф(х, у) + iiftix, у), where z =
x + iy. The quantity w(z) is called the complex potential. Differentiation of Eq. 4.3-10 with
2
respect to x and Eq. 4.3-11 with respect to у and then adding gives V</> = 0. Differentiat-
ing with respect to the variables in reverse order and then substracting gives Vfy = 0.
That is, both ф(х, у) and ф{х, у) satisfy the two-dimensional Laplace equation. 4
As a consequence of the preceding development, it appears that any analytic func-
tion zv(z) yields a pair of functions ф{х, у) and ф{х, у) that are the velocity potential and
stream function for some flow problem. Furthermore, the curves ф(х, у) = constant and
ф{х, у) = constant are then the equipotential lines and streamlines for the problem. The ve-
locity components are then obtained from Eqs. 4.3-6 and 7 or Eqs. 4.3-8 and 9 or from
^T=-v x + iv 4 (4.3-12)
az ' J
in which dw/dz is called the complex velocity. Once the velocity components are known,
the modified pressure can then be found from Eq. 4.3-5.
Alternatively, the equipotential lines and streamlines can be obtained from the in-
verse function z(w) = х{ф, ф) + гу(ф, ф), in which z(w) is any analytic function of w. Be-
tween the functions х{ф, ф) and у{ф, ф) we can eliminate ф and get
F(x,y,<W = 0 (4.3-13)

3
Some knowledge of the analytic functions of a complex variable is assumed here. Helpful
introductions to the subject can be found in V. L. Streeter, E. B. Wylie, and K. W. Bedford, Fluid
Mechanics, McGraw-Hill, New York, 9th ed. (1998), Chapter 8, and in M. D. Greenberg, Foundations of
Applied Mathematics, Prentice-Hall, Englewood Cliffs, N.J. (1978), Chapters 11 and 12.
4
Even for three-dimensional flows the assumption of irrotational flow still permits the definition of
a velocity potential. When v = -Уф is substituted into (V • v) = 0, we get the three-dimensional Laplace
2
equation У ф = 0. The solution of this equation is the subject of "potential theory/' for which there is an
enormous literature. See, for example, P. M. Morse and H. Feshbach, Methods of Theoretical Physics,
McGraw-Hill, New York (1953), Chapter 11; and J. M. Robertson, Hydrodynamics in Theory and
Application, Prentice-Hall, Englewood Cliffs, N.J. (1965), which emphasizes the engineering applications.
There are many problems in flow through porous media, heat conduction, diffusion, and electrical
conduction that are described by Laplace's equation.