PSI - Issue 20

Tatiana Fesenko et al. / Procedia Structural Integrity 20 (2019) 284–293 Tatiana Fesenko et al. / Structural Integrity Procedia 00 (2019) 000–000

288

5

Flow potential, which can be determined from (18) and (19), satisfies the equation (2) and the boundary impermeability conditions on all circuits N up to small magnitude order ε . For higher-order approximations determination, we use the following recurrence relations:

k i

k

k



_

_ 1 _

    j i     j i k i ~

,   k

1,2,3...,

 

(20)

j

i

~ ~ , 1  k k

  k 

1,2,3...,

j 

(21)

i

k

k i

_  can be found from boundary problems:

~  and

where

i

_   k i 

0,

(22)

k



k

_ 1

_







i 

  j i

j

 

(23)

,

r r a

r r a

 

 

i

i

i

i

_   k i 

0

,   i r

when

(24)

~   k i 

0,

(25)

k j 



k i 

~ 1

~





  j i

 

(26)

,

r r a

r r a

 

 

i

i

i

i

~   k i 

.  i r

0

when

(27)

Defined this way potential will satisfy conditions of tightness with accuracy to quantities of smallness order 2 1  k  . It is convenient to search flow potential in the form of a decomposition of cylindrical harmonics, which are Laplace equation solutions:

) sin( ) cos( n n и i 

n

sin( cos(

)

i 

  

  

  

  

n

r

,

1,2,3...

.

, 1



i

n

n

r

)

i 

i

i

Then, we will use the following formulas to convert the potential from one polar coordinate system to another:                    0 , , ) ( cos !( 1)! 1)! ( ( 1) cos m ij i ij i m k ij m i k k j j r R i j m m k m k R k m r r k      , (28)     . , , ) ( sin !( 1)! 1)! ( ( 1) sin 0                m ij i ij i m k ij m i k k j j r R i j m m k m k R k m r r k      (29)