Page 77 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

62 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

multiply connected. Let (D), for example, be the domain sketched in

Figure (2.1) where (Z,) and (£2) are two arcs joining Mp and M oriented
z
as it is shown; by calculating the integral f Cdz along Ly and then along
z0
Lz, we will get distinct values whose difference is equal to the integral,
of function ¢, calculated along the closed contour ZL = Ly UL. On the

Figure 2.1. The case of a multiply connected domain

other hand it is known that the difference is equal to m (T+ iD), where
m. is a positive, negative or null integer® while T+ iD is the number

given by

r+iD= f cde = f (ude + vdy +i (udy ~ vde)),

(C) (C)

(C) being a closed contour of (PD), encircling once, in the direct sense,

the domain ( A ) of boundary (C,). We remark that T = f[ v-dr is
(C)
the circulation of the velocity vector when we contour once, in a direct

sense, the curve (C) and D = f v-nds is the flux across (C), as we
(C)
have already made precise. _

But then the function reip log(z—a), where a is the affix of an inside
point A of (A), has exactly the same nonuniformity properties as f(z),
which means, by deplacing along the same (L) the difference between

©The modulus of the integer m is the number which expresses how many times the respective
contour encircles the simply connected domain (A) of boundary (Cj); m is negative if the
contour is encircled, |m| times, in an inverse sense and it is positive if the encircling is in a
direct sense (in the case of Figure 2.1, m = —1).