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

180

4

( ) k     ( ) k 3 2 V 

( ) k V V x x  1 ( ) k



0

( ) k

( ) k

x





.

(3)

1

2

3

For the system of equations (3), we will then find an exact solution; we will construct an exact solution to the nonlinear system of equations (3) in partial derivatives (3), using the Lin-Sidorov-Aristov class of exact solutions as a basis [Prosviryakov et al. (2024), Goruleva et al. (2022), Burmasheva et al. (2022), Burmasheva et al. (2021)]. Let us represent the velocity field in linear forms relative to two horizontal (longitudinal) coordinates ( ) 1 k x and ( ) 2 k x with functional arbitrariness:       ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 3 1 3 1 2 3 2 , , , k k k k k k k k k V U xtU xtx U xtx    ,       ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 3 1 3 1 2 3 2 , , , k k k k k k k k V WxtW xtx W xtx    ,   ( ) ( ) ( ) 3 3 , k k k V w x t  . (4)

( ) 1 k U ,

( ) 2 k U ,

( ) 1 k W ,

( ) 2 k W , and

( ) k U ,

( ) k W ,

( ) k w which depend on the

To determine the unknown functions vertical (transverse) coordinate ( ) 3

k x and t time and form the structure of hydrodynamic fields (4), it is necessary to supplement them with expressions for the pressure field and the temperature field. They are written as follows:

  ( ) 2 1 k

( ) k x P PxtPxtx Pxtx Pxt      ( ) k ( ) k ( ) k ( ) k ( ) k ( ) k ( ) k ( ) k ( ) k ( ) k , , , ,

















0

3

1

3

1

2

3

2

11

3

2

  ( ) 2 2 k x









( ) k P x t x x P x t   ( ) k ( ) ( ) k k ( ) k ( ) k , ,

12

3

1 2

22

3

2

,

  ( ) 2 1 k

( ) k x T TxtTxtx Txtx Txt      ( ) k ( ) k ( ) k ( ) k ( ) k ( ) k ( ) k ( ) k ( ) k ( ) k , , , ,

















0

3

1

3

1

2

3

2

11

3

2

  ( ) 2 2 k x









( ) k T x t x x T x t   ( ) k ( ) ( ) k k ( ) k ( ) k , ,

12

3

1 2

22

3

2

.

(5)

To determine the type of functions that define hydrodynamic fields (4) and (5), in what follows we will perform a standard procedure characteristic of polynomial classes of exact solutions with functional arbitrariness for the equations of hydrodynamics of incompressible media.

3. The class of exact solutions

Substituting expressions (4) and (5) into the system of equations (3), we obtain the following defining relations: