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

174

4

Formula (4) specifies pressure as a linear form, i.e. horizontal pressure gradients are set besides the background pressure. It should be remembered that the pressure is here normalized to the density. 3. Exact solution The exact solution of the system of nonlinear partial equations (1) is sought in the form (Aristov et al., 2014; Aristov et al., 2015; Prosviryakov et al., 2020; Prosviryakov et al., 2018; Aristov et al., 2016; Lin, 1958; Sidorov, 1989; Aristov, 1990):

( ) y V V z = ,

1 ( ) x V U z U z y = + , ( )

0 ( ) ( ) ( ) P P z P z x P z y = + + . 1 2

(5)

Note that formulas (5) describe the inhomogeneous large gradient flow of a viscous incompressible fluid (Aristov et al., 2014; Prosviryakov, 2016). Taking into account the form of the exact solution, we compute the partial derivatives involved in the nonlinear equation system (1):

2 2 V V    = = = = = = =        , 2 2 V V 2 2 2 2 0 y y y y x x x V V V x x y x y x y   



1 x V U y = 



V



U y U 

 = +

,

,

1

x

z  

z

z



2

2 2 y V V z z 



2

2 U y U 

x V

2  = +



,

,

2 =  

1

2   z

2

2

z

z



2 P P y =



1 P P x =



,

,





0 1 2 P P P   

P



x = + +

y

.

z  

z

z

z





The substitution of the obtained expressions into system (1) yields the following equations: ( ) 1 1 1 VU P U yU   = − +  + ,

(6)

2 V P   = ,

0 1 2 P P P    = + + =

P



x

y

g

(7)

.

z  

z

z

z





Hereinafter, the prime means differentiation with respect to the transverse (vertical) z coordinate. Equating the functions at identical powers of y in equations (6) and (7), we arrive at the following system for the determination of the unknown functions: