PSI - Issue 65

E. Yu. Prosviryakov et al. / Procedia Structural Integrity 65 (2024) 177–184 E. Yu. Prosviryakov, O. A. Ledyankina, L. S. Goruleva / Structural Integrity Procedia 00 (2024) 000–000

182

6

2

( ) k  

  

( ) k W W W x     ( ) k ( ) k 1

( ) k



( ) k

x



2

 

1

2

( ) k

( ) k

( ) k

( ) k

c

x

x

x







p

3

3

3

,

( ) k      ( ) k 2 ( ) 3 k w x

( ) k U W

0

1

.

(6)

In the system of equations (6), we open the brackets and group the expressions at the same degrees of the horizontal coordinates   1 k x and   2 k x . Next, we apply the method of undetermined coefficients and obtain a system of equations for determining the unknown functions for hydrodynamic fields (4) and (5):

( ) k

2 ( ) k U 1

U





( ) k   P 11

( ) k



1 t

2

x





,

3

( ) k

2 ( ) k U 2

U





( ) k   P 12



2

( )2 3 k

t

x





,

( ) k

2 ( ) k U

U





( ) k   P

( ) k



1

( )2 3 k

t

x





,

( ) k

2 ( ) k W

W





( ) k   P 12

( ) k



1

1

( )2 3 k

t

x





,

( ) k

2 ( ) k W

W





( ) k   P 22

( ) k



2

2

( )2 3 k

t

x





,

( ) k

2 ( ) k W

W





( ) k   P

( ) k



2

( )2 3 k

t

x





,

( ) k

( ) k

( ) k

( ) k

( ) k

P x

P x

P x

P x

P x











0

0

0

0

0











1

11

2

22

12

( ) 3 k

( ) 3 k

( ) 3 k

( ) 3 k

( ) 3 k











,

,

,

,

,

( ) k

( ) k

2 ( ) k w

P

w







0  



( ) k

( )2 k

t

x

x







,

3

3

( ) k

w







( ) k U W

( ) k

 

1

2

x



,

3