Page 86 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Dynamics of Inviscid Fluids 71

3) is a direct consequence of the equality lim & = 1 which is always
|Z|00
valid for a canonical conformal mapping.
But we have already established a function F(Z) answering these
questions; hence, the function f(z) that we seek is given by f(z) =
F(h(z)) where, of course,

Ree oa
F(Z) = Vo(Ze** + ) — 57 log Z.
Z

It is shown that the thus determined function f(z) is, up to an ad-
ditive constant without importance, the unique’ function satisfying the
conditions 1), 2) and 3). The fundamental problem of the theory of
profiles is thus reduced to the problem of determination of the canonical
conformal mapping of the domain (d) — the exterior of the profile —

onto the outside of the circular disk.
If the fluid flow past a circular disk has some singularities (sources,
point vortices, doublets, etc.) an important result which allows the de-
termination of the corresponding complex potential is the “circle (Milne—
Thompson) theorem” which states the following:
The function f(z) which is analytic in D — the exterior of the cir-
cumference |z| = R — except at finite number of singular points E c D,

whose principal parts with respect to these singularities is fo(z) and
which is continuous on D\E, will satisfy the requirement Im f (z)lj=r =
2
0 only if F(z) = fol2) +7 (4 ) +a, a being a real constant.
Zz
Some remarkable extensions of the circle (Milne~Thompson) theorem
are given by Caius Iacob [69].
The Blasius formulae [52] allow us to evaluate directly the global
efforts exerted on the profile by the fluid flow. We will limit ourselves to
the determination of the general resultant of these efforts, which comes

to the “complexforce” F given by the formula (Blasius—Chaplygin) [52]
. 2
F= $f (£) dz , (c) being considered in a direct sense.
(c)
To calculate this integral we remark that it is possible to continuously
deform the integration contour (c) into a circular circumference of an
arbitrarily large radius, centered at the origin, gf. being analytic and
uniform in the whole outside of (c), that means in (d); on the other

°This result is a consequence of the uniqueness of the solution of the external Dirichlet
problem for a disk with supplementary condition of a given non-zero circulation. See, for
instance, Paul Germain, “Mécanique des millieux continus”, pag. 325, Ed. Masson, 1962
[52].