PSI - Issue 40

Larisa S. Goruleva et al. / Procedia Structural Integrity 40 (2022) 171–179 Larisa S. Goruleva, Evgeniy Yu. Prosviryakov / Structural Integrity Procedia 00 (2019) 000 – 000

178

8

12 BCh A A  − 

 

2

0

.



 



Thus, with hydrostatic pressure distribution and isobaric fluid flow, there are counterflows in the fluid. It can easily be demonstrated that the counterflows are accompanied by the nonmonotonic velocity profile. Similar studies with the motion of the upper boundary and specification of tangential stresses on it were reported in (Aristov et al., 2014; Aristov et al., 2015; Prosviryakov, 2017; Prosviryakov et al., 2018). With the presence of the pressure gradient 1 S and zero 2 S , the velocity profile is described by the expression

2 4 CB S +       

2

2

2

  

  

BCh

BCh

2 h

(

)

4

3

2 S CB Z A +

U Z =

Z

h Z A +

−

+

− +

1 2

1

12

3







Since the inertia forces are taken into account in the problem statement, the effect of the gradient 1 S causes no additional counterflow zones. The restrictions imposed on the parameters whose combination causes the polynomial to have one root, are as follows:

2  − −        2 1 S h 12 2 BCh A A 

0

The second root of polynomial (17) can appear at a nonzero value of the horizontal gradient 2 S . This means that the flow of a viscous incompressible fluid is stratified into three zones. Consequently, the boundary value problem parameters can always be selected for the adjustment of the thicknesses of the counterflow zones necessary for engineering devices. Note that the specific kinetic energy: ( ) ( ) 2 2 2 2 2 2 2 2 1 1 1 2 2 x y E V V U U y V U V UU y U y  = + = + + = + + + has a complex profile at a fixed value of the y -coordinate, which is nonmonotonic and can have up to two zero values. 6. Conclusion A new exact solution to the Navier – Stokes equations for the steady-state flow of a viscous incompressible fluid has been presented and studied. This solution describes a fluid flow in a thin layer, which can be treated as a large flow of a vertical vortex fluid without prerotation. The structure of the obtained solution has been studied. It has been shown that, for isobaric flows and hydrostatic distribution of pressure, the velocity field is stratified into two zones (one stagnation point accompanied by a counterflow). In the presence of pressure gradients, the velocity field can be stratified into three zones, and this means that the specific kinetic energy has two zeros. References Ershkov, S.V., Prosviryakov, E.Yu., Burmasheva, N.V., Christianto, V., 2021. Towards understanding the algorithms for solving the Navier– Stokes equations. Fluid Dynamics Research 53, 044501. Aristov, S.N., Knyazev, D.V., Polyanin, A.D., 2009. Exact solutions of the Navier–Stokes equations with the linear dependence of velocity components on two space variables. Theoretical Foundations of Chemical Engineering 43, 642–662. Drazin, P.G., Riley, N., 2006. The Navier–Stokes Equations: A classification of flows and exact solutions. Cambridge Univ. Press, Cambridge, pp. 196. Pukhnachev, V.V., 2006. Symmetries in Navier–Stokes equations. Usp. Mekh 4, 6–76. Broman, G.I., Rudenko, O.V., 2010. Submerged Landau jet: exact solutions, their meaning and application. Physics - Uspekhi 53, 91–98.