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

183

7

  

  

( ) k

2 ( ) k

( ) k

( ) k

( ) k

( ) k

( ) k

T

T

U

U

W

W

2















( ) k









1

1 k

1

1

( ) k

( ) k x x

( ) k

( ) k x x

( ) k

( )2

t

x

c





 

 

p

3

3

3

3

3

,

   

  

2

2

  

      

  

( ) k

2 ( ) k

( ) k

( ) k

( ) k

T

T

U

W











( ) k







11

11

1

1

( )2 k

( ) k

( ) k

( ) k

t

x

c

x

x









 ,

p

3

3

3

     

     

( ) k

2 ( ) k

( ) k

( ) k

( ) k

( ) k

( ) k

T

T

U

U

W

W

2















( ) k









2

2 k

2

2

( ) k x x

( ) k

( ) k x x

( ) k

( )2

t

x

c





 

 

p

3

3

3

3

3

,

( ) k

2 ( ) k

( ) k

( ) k U U W W     ( ) k ( ) k

( ) k

T

T

2







( ) k









12

12

2

1

2

1

( )2 k

( ) k

( ) k x x

( ) k

( ) k x x

( ) k

t

x

c





 

 

p

3

3

3

3

3

,

   

  

2

2

  

      

  

( ) k

2 ( ) k

( ) k

( ) k

( ) k

T

T

U

W











( ) k







22

22

2

2

( ) k

( ) k

( ) k

2 x c

t

x

x









 ,

p

3

3

3

( ) k

2 ( ) k

T

T









( ) k

( ) k T T 11

( ) k  

( ) k









0

0 k

22

( )2 3

t

x





 ( ) ( ) 2 k k

 





   2 ( ) 1 k U W U W     2 ( ) k ( ) k 2 2



2

( ) k





1

c

p

 

2

2

2

  

      

      



( ) k

( ) k

( ) k

w

U

W







2



   

( ) k

( ) k

( ) k

x

x

x







3

3

3

.

System (7) consists of inhomogeneous partial differential equations of parabolic type with a convective term and gradient type equations to satisfy the condition of fluid incompressibility, to determine horizontal pressure gradients ( ) 1 k P , ( ) 2 k P , and functions ( ) 2 k P , ( ) 12 k P , ( ) 22 k P that determine the curvature of the hydrodynamic field. 4. Conclusion An exact solution to the Oberbeck–Boussinesq equations for describing slow flows of multilayer fluids with the Rayleigh function in the heat conduction equation has been constructed. The construction of the exact solution is based on the Lin-Sidorov-Aristov velocity field. In the announced exact solution the velocity field depends on two spatial coordinates (horizontal or longitudinal). The form factors are functions of the third coordinate (vertical or transverse) and time. The pressure field and the temperature field are described by quadratic forms with a similar functional dependence. To determine the unknown coefficients, a system of linear partial differential equations with gradient equations has been obtained.