PSI - Issue 65

E.Yu. Prosviryakov et al. / Procedia Structural Integrity 65 (2024) 185–190 E.Yu. Prosviryakov, O.A. Ledyankina, L.S. Goruleva / Structural Integrity Procedia 00 (2024) 000–000

187

3

2 y PPPxPyP PxyP P P P P            2 3 2 x y xy 2 3 x y x

0

1

2

3

4

5

6

7

8

9

2

2 3!

2

2

3!

xy

y

x

3 x y

x y

xy

y

2

3

4

2 2

3

4

P P P P   

P

P

P









,

8

9

10

11

12

13

14

2!

3!

4!

3!

4

3!

4!

2 y CC CxCyC CxyC C C C C            2 3 2 x y 2 3 x y x xy

0

1

2

3

4

5

6

7

8

9

2

2

3!

2

2

3!

xy

y

x

3 x y

x y

xy

y

2

3

4

2 2

3

4

C C C C C     

C C 



.

(4)

8

9

10

11

12

13

14

2!

3!

4!

3!

4

3!

4!

i U , i V (

0;2 j  ), k P , k C (

The coefficients

0;5 i  ),

j W (

0;14 k  ) included in equations (4) depend on the

vertical (transverse) coordinate z and time t . The components of mass forces x F , y F , and z F are determined in the following form:

2 y F X XxXyX XxyX       , 2 x

x

0

1

2

3

4

5

2

2

2 y F Y YxYyY YxyY       , 2 x

y

0 1

2

3

4

5

2

2

2 x y F Z ZxZyZ ZxyZ Z Z          2 3 2 x y x

z

0

1

2

3

4

5

6

7

2

2

3!

2!

xy

y

x

3 x y

x y

xy

y

2

3

4

2 2

3

4

Z

Z Z

Z

Z

Z

Z



 









(5)

8

9

10

11

12

13

14

2!

3!

4!

3!

4

3!

4!

We substitute the type of solutions for velocities, pressure, and concentration (4) into the system of Oberbeck– Boussinesq equations (1)–(3). We write down the resulting system taking into account the accepted structure of the source and sink function Q (6) and the vector of mass forces F (5). We write the system of partial differential equations for determining the unknown functions in the form of two subsystems:

  

  

2

U

U



P U U     

X



0

0

,

1

3

5

0

2

t

z





2     1 U P

U



X

1

,

3

1

2

t

z





2     2 U P

U



X

2

,

4

2

2

t

z



