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

89

8

V V

 

z + =  

y

V  . 

(11)

= =   

z      y 

yz

zy

Taking into account the exact solution (9), the expressions  xy ,  yz take the form

u z

V h +

(

)

,

.

xy   =

+

yz =  



h

h

h

+





Based on these expressions, it can be argued that there are no tangential stresses (i.e. there are only tensile/compression stresses in the fluid) only at V h =u h = 0. In this case, due to the exact solution (9), the stress  xz becomes equal to the following expression:

(

) 2

U

U

) ( 3

)

(

)

.

2 2 2

2 + + =  

2 h h

h h









=

+ +

h

h

(

(

)

xz

3

h

h





+

+

Therefore, this stress takes on the zero value only at U h =0. Thus, for any value of the slip length α , the relative immobility of two solid boundaries determining the geometry of the considered extended layer is a necessary and sufficient condition for the absence of shear stresses in the fluid under study. 4. Conclusion A new exact solution has been obtained that describes the shear isobaric flow of a viscous fluid with slipping. It has been shown that the shape of the profile of the flow velocity vector components and the magnitude of the shear stress field components depend on the value of the slip length. In this case, according to the constructed exact solution of the Navier-Stokes equations, the slip length does not have any effect on the conditions for the existence of only longitudinal stresses in the fluid. References Baranovskii, E.S., Burmasheva, N.V., Prosviryakov, E.Yu., 2021. Exact Solutions to the Navier – Stokes Equations with Couple Stresses. Symmetry 13, 1355. Burmasheva, N.V., Prosviryakov, E.Yu., 2020. Exact Solution of Navier – Stokes Equations Describing Spatially Inhomogeneous Flows of a Rotating Fluid. Trudy Instituta Matematiki i Mekhaniki URO RAN 26, 79 – 87. Burmasheva, N.V., Prosviryakov, E.Yu., 2020. A Class of Exact Solutions for Two – dimensional Equations of Geophysical Hydrodynamics with Two Coriolis Parameters. The Bulletin of Irkutsk State University. Series “Mathematics” 32, 33– 48. Burmasheva, N.V., Prosviryakov, E.Yu., 2020. Isothermal Layered Flows of a Viscous Incompressible Fluid with Spatial Acceleration in the Case of Three Coriolis Parameters. Diagnostics, Resource and Mechanics of Materials and Structures 3, 29 – 46. Burmasheva, N.V., Prosviryakov, E.Yu., 2021. Exact Solutions for Steady Convective Layered Flows with a Spatial Acceleration. Russian Mathematics 65, 8 – 16. Burmasheva, N.V., Prosviryakov, E.Yu., 2021. Exact Solutions to the Oberbeck – Boussinesq Equations for Shear Flows of a Viscous Binary Fluid with Allowance Made for the Soret Effect. The Bulletin of Irkutsk State University. Series “Mathematics” 37, 17– 30. Dytnersky, Yu. I., 1995. Processes and Devices of Chemical Technology. Part 1. Theoretical Foundations of Chemical Technology Processes. Chemistry, Moscow, pp. 400. Gershuni, G.Z., Zhukhovitskii, E.M., 1976. Convective Stability of Incompressible Fluids: Israel Program for Scientific Translations. Keter Publishing House, Jerusalem, pp. 330. Kochin, N.K., Kibel, I.A., Roze, N.V., 1964. Theoretical Hydromechanics. Wiley Interscience, pp. 577. Navier, С.L.M.H. , 1823. Mémoire sur les Lois du Mouvement des Fluides. Mémoires de l’Académie des Sciences 2, 389 – 440.