Page 104 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Dynamics of Inviscid Fluids 89

the relation 7

Bi =O, + 3

from where
sin 2; = cos @;, cos B; = — sin 8; .

After derivation with respect to the normal we get

ly = [ (2; = 23) 008 Bi + (yi ~ vs) Sin Bi 5
ayo j
0 (x; — 2;)” + (yi — yy)?

where
Ly = Xj + 8;0080;,y; = Y; + 5; sin8; .

By replacement, the integral becomes

[2 Cs; +D
lj =
0 85 249As;+B

where
A=-—(%- Xj) cos 85 — Y;) sin 6; ,
B= (x; — X;)° gers
C= sin(6; — 6;),
D = — (4; — Xj) sin 0; + (yi — Y;) cos 6; .

But the denominator of the integrand is of the form

(sj + A)? + B- A® = (sj; + A)? +E? >0

where
E = (x; — Xj) sin 8; — (yi; — Y;) cos 6;

thus, consequently,

2
ie
Ii = 4 sin (6; — 8;) In f 4 teas
(2.11)

— cos (6; — 8;) Jarctg (5334) —arctg (4)|

By using the system (2.10), with the introduction of the dimensionless
. . , d;
(undimensional) variables rj = 577, We get

m
) Ti5X5 = sin 6;,2 = 1,....m
j=l

where J;; are given by (2.11), excepting J;; = a for every 7.