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

78

4

  2 i V z  2

  2 i u z 2

  2 2 i U Vu z     i







0

0

,

,

.

(7)







i

The general solution to system (6) can easily be written as

  i i i u c z c   ,   i    

 

 

i i V c z c   ,

1

2

3

4





   

   

    i i c c c c  1 4 2 3

    i i

    i i

    i i

c c

c c

i   i

  i

  i

  i

4

3

2 z c z c  

U

z

z

.

(8)







1 3

2 4

5

6

12

6

2

The resulting exact solution (8) satisfies the original system of equations (1), (2) (this fact is verified by direct substitution); however, at the same time, it describes a lot of similar flows. In order to use solution (8) to determine the flow characteristics, it is necessary to determine the specific type of parameters c j ( i ) ( i =1,2; j =1,…,6). To do this, it is necessary to set the appropriate boundary conditions. Note that the condition for velocities is set on the lower boundary z =0 of the two-layer fluid. Taking into account the structure of solution (5), it is equivalent to three relations with respect to the components U (1) , u (1) , V (1) of the velocity vector V (1) . Both the adhesion condition (in the case of a hydrophobic surface bounding the layer from below) and the slip condition (in the case of a hydrophilic surface) can act as the discussed condition. Similarly, at the upper boundary z = h 1 + h 2 of the layer, three more ratios are set for the components U (2) , u (2) , V (2) of the vector V (2) . In this case, twelve integration constants are involved in the exact solution (8). Thus it is necessary to set six additional conditions on the boundary z = h 1 common for the two layers. 3. Discussion of the conditions at the common boundary in a two-layer fluid A natural and fairly obvious option would be to use the condition of the equality of the velocities at the common boundary z = h 1 of two layers during the transition from one layer to the other: Substituting the selected structure of solution (5) into condition (9), we obtain three more relations for the velocity vector components:         1 2 1 1 U h U h  ,         1 2 1 1 u h u h  ,         1 2 1 1 V h V h  . (10) In fact, condition (9) is a condition for the continuity of the solution for the velocity field. Paired with the continuity condition, the smoothness condition is traditionally considered. Note that solutions (8) constructed for the each layer are smooth. Therefore, the smoothness on the common boundary z = h 1 means the coincidence of the values of these derivatives,   1   2 1 1 x x z h  z h  V V  ,   1   2 1 1 y y z h  z h  V V  . (9)

  1

  2

  1

  2

V

V





V

V





y

y

,

,

(11)





x

x

z

z

z

z









z h 

z h 

z h 

z h 

1

1

1

1

or in the component form,