Page 76 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Dynamics of Inviscid Fluids 61

® = constant and the streamlines yw = constant form, at any point of
the fluid flow, an orthogonal network, the inner product grady- grad ®

being zero. At the same time we also have

dz Oz Ox Oy Oy

The function a = u — iv is also an analytic function of z, called
the complex velocity of the flow and which will be denoted by ¢ ; the
modulus and the argument of ¢ define, respectively, the magnitude gq of

the velocity and the angle w, with changed sign, made by the velocity
vector with the axis Oz, as

_F
=u-iv=qe ™.
¢= dz
We conclude that the kinematic description, the whole pattern of the
considered flow, could be entirely determined by knowing only the ana-
lytic function f(z;t),the complex potential of this flow at the considered

moment 7.
In the previous considerations we have seen that, to any incompress-
ible potential plane fluid flow it is possible to associate a complex poten-
tial. It is important to find out if, conversely, any analytic function of
z can be seen as a complex potential, i.e., it determines an incompress-

ible irrotational plane flow of an inviscid (ideal) fluid. To answer this
question we recall that, from the physical point of view, it is necessary
to choose the function f such that its derivative, the complex velocity,
is not only an analytic function but also a uniform function in the con-
sidered domain (DP), so that, at any point of (D), ¢ = gf takes only one
value.
Once accomplished this requirement, due to the analyticity of the
function at any point of (DP), the conjugate harmonic functions u(z, y)

and —v(z,y) (the real and the imaginary part of ¢) satisfy the Cauchy—
Riemann system, that is ie = — 3 , oe = oe ; but such a fluid flow
should be an incompressible irrotational plane flow of an inviscid fluid.
On the other hand, if the domain (DP) is simply connected, we will also
deduce that f(z) is analytic and uniform too, which means a holomor-
phic function in (D). Really, zo being the affix of point of (D), we have
a
z&
f(z) = f(zo) + f ¢dz, the integral being taken along an arbitrary arc
20
connecting the points Mp and M (or zp and z). The Cauchy—Goursat
theorem proves, ¢ being uniform and (DP) simply connected, that the
above expression for f(z) does not depend on the chosen are and con-
sequently f(z) is uniform. It will not be the same if the domain (PD) is