Transport Phenomena.pdf

§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 x
, v y
, and 2P as functions of x and y. We
have already seen in the previous section that the equation of continuity in two-dimensional
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 convenient,
then, to introduce a velocity potential ф(х, у). Instead of working with the velocity
components v x
and v y
, we choose to work with ф(х, у) and ф{х, у). We then have the following
relations:
дф дф
(stream function) v x
= —r- v u
= — (4.3-6,7)
dy
(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
the real and imaginary parts of any analytic function 3 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
respect to x and Eq. 4.3-11 with respect to у and then adding gives V 2 = 0. Differentiating
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 function
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 velocity
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)
J
az '
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 inverse
function z(w) = х{ф, ф) + гу(ф, ф), in which z(w) is any analytic function of w. Between
the functions х{ф, ф) and у{ф, ф) we can eliminate ф and get
J
dX
F(x,y,