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

179

3

2

  

2

2

2

   

( ) k  

 

  

 

2       

 

( ) k V V

( ) k

( ) k

( ) k

( ) k

( ) k

V

V

V









c 



3 3  1 1 j i  

j

( ) k

Q

2

2











i

3

1

2

  

( ) k

( ) k

( ) k

( ) k

( ) k

( ) k

( ) k

x

x

c

x

x

x

2











 

 

p

j

i

p

1

2

3

2

2

2

2

2

2

  

  

  

  

  

      

  

  

  

  

  

( ) k

( ) k

( ) k

( ) k

( ) k

( ) k

V

V

V

V

V

V

























1

2

1

3

2

3

( ) k

( ) k

( ) k

( ) k

( ) k

( ) k

x

x

x

x

x

x













2

1

3

1

3

2

  

  

  

  

  

  

( ) k V V  

  

  

  

  

  

( ) k

( ) k

( ) k

( ) k

( ) k

V

V

V

V











2

2

2







3

3

1

2

1

2

 

( ) k

( ) k

( ) k

( ) k

( ) k

( ) k

x

x

x

x

x

x













2

1

3

1

3

2

.

(2)

Here, in (2) the physical coefficient ( ) k

p c is the specific heat capacity of the liquid at constant pressure [Ershkov et

al. (2023)]. System (1), taking into account the Rayleigh dissipative function (2), consisting of 5 n equations, is written in coordinate form as

        

  

( ) k

( ) k    P

2 ( ) k V V V   2 ( ) k

2 ( ) k

V





( ) k







1

1 k

1 k

1 k

( ) k

( )2

( )2

( )2

t

x

x

x

x











1

1

2

3

,

 

( ) k

( ) k    P

2 ( ) k V V V   2 ( ) k

2 ( ) k

V





( ) k







2

2

2

2

( ) k

( )2 k

( )2 k

( )2 k

t

x

x

x

x











 ,

2

1

2

3

  

( ) k

2 ( ) k V V V   2 ( ) k

2 ( ) k

( ) k    P

V





( ) k







3

3 k

3 k

3 k

( ) k

( )2

( )2

( )2

t

x

x

x

x











3

1

2

3

,

  

  

( ) k

2 ( ) k

2 ( ) k

2 ( ) k

T

T

T

T









( ) k









( )2 k

( )2 k

( )2 k

t

x

x

x









1

2

3

( ) k    

2

2

2

 

  

 

2       

 

( ) k

( ) k

( ) k

V

V

V







2

2





3

1

2

( ) k

( ) k

( ) k

( ) k

c

x

x

x







  

 

p

1

2

3

2

2

2

2

2

2

  

  

  

  

  

      

  

  

  

  

  

( ) k

( ) k

( ) k

( ) k

( ) k

( ) k

V

V

V

V

V

V

























1

2

1

3

2

3

( ) k

( ) k

( ) k

( ) k

( ) k

( ) k

x

x

x

x

x

x













2

1

3

1

3

2

  

  

  

  

  

  

( ) k V V  

  

  

  

  

  

( ) k

( ) k

( ) k

( ) k

( ) k

V

V

V

V











2

2

2







3

3

1

2

1

2

 

( ) k

( ) k

( ) k

( ) k

( ) k

( ) k

x

x

x

x

x

x













2

1

3

1

3

2

,