PSI - Issue 40

Natalya V. Burmasheva et al. / Procedia Structural Integrity 40 (2022) 75–81 Natalya V. Burmasheva at al. / Structural Integrity Procedia 00 (2022) 000 – 000

77 3

is the fluid velocity vector field in the i -th layer ( i =1,2);  i ,  i are the density and dynamic viscosity of the i -th layer, respectively ( i =1,2);           , i i i i x y z V V V x y z                 V

is a convective derivative;

x y z       i j

κ

   

is the Hamilton operator; i , j , k are the unit vectors of the axes Ox , Oy , Oz ;

2 2 2 x y z       2 2 2

   

is the Laplace operator. In coordinate notation, the system (1), (2) takes the form

  i

  i

  i

  i

  i

  i

  i

           

i          

   

2

2 V V V   2

V

V

V

V











  i

  i

  i

V

V

V

,

i 











x

x

x

x

x

x

x

x

y

z

2

2

2

t

x

y

z

x

y

z















  i

  i

  i

  i

  i

  i

  i

i           i          

   

2

2 V V V   2

V

V

V

V











  i

  i

  i

y

y

y

y

y

y

y

,

V

V

V











i 

x

y

z

2

2

2

t

x

y

z

x

y

z















  i

  i

  i

  i

  i

  i

  i

   

2

2 V V V   2

V

V

V

V











  i

  i

  i

,

(3)

V

V

V

i 











z

z

z

z

z

z

z

x

y

z

2

2

2

t

x

y

z

x

y

z















  i

  i

  i

y x z V V V x y z        

0 = .

(4)

We will seek a solution to system (3), (4) in the class of exact solutions linear along some of the coordinates (Burmasheva et al., 2020; Burmasheva et al., 2020; Burmasheva et al., 2021; Burmasheva et al., 2021):

  i z V = . 0

  i     i i x V U z u z y  = ,    

  i     i y V V z = ,

(5)

Class (5) describes vertical vortex shear flows with an arbitrary dependence on the vertical coordinate z . After the substitution of expression (5) into system (3), (4) and simple transformations, we obtain the following system of equations:

  2 i

  i 2 U u 

  i

   

   

V z

2





0

,

.

(6)

i 



Vu   

y



i

i

2

2

2

z

z







Taking into account the principle of indefinite coefficients, system (6) can be represented as