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

80

6

  1

  1

  2

  2

  2

  2

   

   

   

   

   

   

  12

dU du

dU du

dU du

  1

  2

y

y

y



1 

1 

2 

















,

xz

xz

dz

dz

dz

dz

dz

dz

z h 

z h 

1

1

z h 

z h 

z h 

1

1

1

  1

  1

  2

  2

  2

  2

   

   

   

   

   

   

  12

dU du

dU du

dU du

  1

  2

y

y

y



1 

1 

2 

















.

(15)

xz

xz

dz

dz

dz

dz

dz

dz

z h 

z h 

1

1

z h 

z h 

z h 

1

1

1

Taking into account the fact that stresses characterize the distribution of internal forces in a continuous medium, we find that the use of the smoothness condition (11) (or its version (12)) for velocities contradicts the idea of a continuous medium. In terms of the physics of the process, the condition of the equality of stresses on the common boundary is more suitable for determining the integration constants in the exact solution (8),

  1

  2

,   1 xz 

  2

,   1 yz 

  2

.

(16)















xy

xy

xz

yz

z h 

z h 

z h 

z h 

z h 

z h 

1

1

1

1

1

1

Due to the structure of solution (5), expressions (16) take the following form:

  1

  2

dU

dU

  1

  2

u

u

,

.

1 



2 

1 



2 

dz

dz

z h 

z h 

1

1

z h 

z h 

1

1

  1

  2

  1

  2

du

du

dV

dV

,

.

(17)

1 



2 

1 



2 

dz

dz

dz

dz

z h 

z h 

z h 

z h 

1

1

1

1

Note that the condition of the smoothness of the velocities (12) proves to be equivalent to the condition of the continuity of stresses (17) at the common boundary of two layers in the case of a two-layer fluid, when the dynamic viscosities of both layers coincide,

1 2    .

(18)

Note also that the fulfillment of condition (18) does not mean that the fluid in the upper layer and the fluid in the lower layer coincide. The fact is that the layers (this was discussed above) can also differ in density. Examples of such pairs of fluids (with the same dynamic viscosities but different densities) are water and ethyl alcohol (96%) at the zero temperature, as well as methyl alcohol (methanol) and chloroform. Taking into account the boundary conditions (10), (17) and the corresponding conditions on the upper and lower boundaries of the two-layer fluid, from the general solution (8), we obtain an exact solution to system (1), (2), which corresponds to a specific boundary value problem. Problems of this kind were considered, for example, in (Burmasheva et al., 2020; Burmasheva et al., 2021). 4. Conclusion The paper has discussed the selection of boundary conditions for describing the flows of stratified viscous fluids using the example of a two-layer fluid model with a piecewise constant density function and a piecewise constant dynamic viscosity function. It has been shown that the boundary conditions should most naturally reflect the features of the considered fluid medium, first of all, its continuity. The paper also provides an example of the exact solution describing the shear isothermal flow of a viscous incompressible two-layer vertical vortex fluid.