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

79

5

  1

  2

  1

  2

  1

  2

z h dU dU dz dz  

du

du

dV

dV

,

,

.

(12)





dz

dz

dz

dz

z h 

z h 

z h 

z h 

z h 

1

1

1

1

1

1

To understand whether it is adequate to use the smoothness condition for solutions on the common boundary, consider the viscous stress tensor. According to Newton's law (Truesdell, 1977), the stress tensor in the fluid is defined as follows:

           

           

  i

   

 

  i

  i

  i

  i

     

V



V

V

V V 







y

2

p  

i 

i 

i 

 

z  

x

x

x

      

 

x

y

x

z

x











  i

  i

  i

      

    

  i

  i

V

V

V







V

V





  i

y

y

y

2

p    



i   

i 

i 

z  

.

(13)

x

y

x

y

z

y











  i

 

  

  i

  i

  i

  i

V



V V 

V

V







y

2

p    

i 



i 



x

z

z

z

i    

z

y

y

z

z











Here, p is the value of hydrostatic pressure in the fluid and  i is the dynamic viscosity coefficient for the i -th layer. We calculate the values of the components of this tensor (13) for the selected class of exact solutions (5) as

          

          

  i

  i

   

   

dU du

  i

p

u  

y





i

i

dz

dz

  i

dV

  i

  i

.

u

p







i 

i 

dz

  i

  i

  i

   

   

dU du y 

dV

p



i 

i 

dz

dz

dz

Thus we see that the off-diagonal components of the viscous stress tensor (shear stresses) prove to be related to the corresponding derivatives of the velocity field components,

  i

  i

  i

xy yx i u      ,

  i

  i

  

   

dU du y dz dz 

  i

  i

,

xz zx       i

  i

dV

  i

  i

.

(14)

    

yz

zy

i

dz

Based on expressions (14), we can conclude that, if we take the smoothness condition (11) (or its version (12)) as the remaining condition for determining the integration constants c i in the obtained solution (8), the shear stress  xz ( i ) ,  xy ( i ) ,  yz ( i ) becomes discontinuous on the common boundary z = h 1 :

  10

  1

  1

  2

  2

  2

,

u

u

u





1 



1 



2 





xy

xy

z h 

z h 

z h 

z h 

z h 

1

1

1

1

1