PSI - Issue 40

Natalya V. Burmasheva et al. / Procedia Structural Integrity 40 (2022) 82–89 Natalya V. Burmasheva, Evgeniy Yu. Prosviryakov / Structural Integrity Procedia 00 (2019) 000 – 000

85

4

Here, the prime denotes the derivative with respect to the z coordinate. Note that the nonlinear equation of system (5) can be easily integrated in view of the fact that the first two equations of this system are linear and isolated,

2 1 u c z c = + ,

4 3 V c z c = + ,

 

  

4

3

2

1

z

z

z

(

)

.

(6)

2 4 U c c = 

2 3 1 4 c c c c

c c

c z c

+ +

+

+ +

1 3

5

6

12

6

2



The general solution to system (6) is polynomial. The constants c i ( i =1,…,6) resulting from the integration of this system are determined from the boundary conditions. Let us further formulate the corresponding number of boundary conditions. Suppose that the slip condition (the Navier condition (Navier, 1823)) is satisfied on the lower solid impenetrable boundary z =0 of the layer and that the upper solid impermeable hydrophilic boundary z = h moves relative to the lower one at a known velocity,

0 z =   V V n  =

( ) 0

, ( )

h h = V V ,

(7)

Here, α is the slip length ( α >0), n is the outer (with respect to the solid surface, bounding the layer under consideration from below) normal, which coincides with the positive direction of the vertical axis Oz . Note that the second condition in system (7) can be interpreted as the velocity of the upper hydrophilic surface relative to the lower hydrophobic one. Moreover, if we put α =0 in the first condition of system (7), we obtain the classical no-slip condition and hence the case of a fluid flow between two movable hydrophilic surfaces. If we put α→∞ in conditions (7), the corresponding condition will describe the perfect slip of the fluid along the lower hydrophobic surface. Taking into account the structure (4) of the exact solution, conditions (7) can be rewritten in the following form (equivalent within this class): ( ) ( ) ' 0 0 U U =  , ( ) ( ) ' 0 0 u u =  , ( ) ( ) ' 0 0 V V =  , ( ) h U h U = , ( ) h u h u = , ( ) h V h V = . (8)

Note that the resulting number of boundary conditions (8) coincides with the number of integration constants in the general solution (5). Thus it is easy to find a solution to the posed boundary value problem:

(

)

(

)

u u h + =

z

V V h + =

z

+



+



h

h

,

,





) ( 3

)

2 2 h U U h z h hz =

2 2 h z + + + + + +      2 3 2

(

h



+

h h u V

) ( 3

4 h z hz h h z − + − − 4 4 3 4

3 4 + −      3 2 4 4 hz z h +

+

−

(

12

h

 

+

)

2 2 2 2 + + − +      . 3 2 2 3 2 3 6 4 6 6 h z hz z h z 6 −

(9)