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

175

5

2 V P   = ,

(8)

0 U  = ,

1 1 U VU P   = + ,

(9)

1

0 P g z =

1 P P z z



   

0

.

(10)

,

2 = =



4. Boundary value problem solution To solve the boundary value problem, we write the boundary conditions (2) – (4) in the form consistent with the structure of the exact solution (5),

( ) 0 U A = , ( ) 1 0 U B = , ( ) 0 V C = , ( ) 0 U h = , ( ) 1 0 U h = , ( ) 0 V h = , ( ) 0 P h S = , ( ) 1 1 P h S = ( ) 2 2 P h S =

(11)

We then integrate the system of ordinary differential equations (8) – (10). This system is divided into two isolated subsystems. Equations (10) can be easily integrated, and by virtue of the boundary conditions, the solutions are written as follows: ( ) 0 P g z h S = − + ,

1 1 P S = , 2

2 P S = .

0 P is distributed according to the

Thus it follows from the obtained solution that the background pressure

hydrostatic law and that the horizontal pressure gradients 1 P and 2 P are constant across the thickness. We substitute the solutions for the longitudinal pressure gradients 1 P and 2 P into equations (8) and (9) and integrate the isolated second-order ordinary differential equation (8) to find the velocity V . Double integration gives the following Poiseuille exact solution (Ershkov et al., 2021; Aristov et al., 2009; Drazin e al., 2006; Pukhnachev et al., 2006):

2 2 2 V S z c z c = + +  , 1 2

(12)

The further integration of equation (9) allows us to obtain an expression for spatial acceleration (horizontal velocity gradient) 1 U , which is determined by the Couette-type exact solution (Ershkov et al., 2021; Aristov et al., 2009; Drazin e al., 2006; Pukhnachev et al., 2006):

(13)

1 3 4 U c z c = +

To find the background velocity, we substitute expressions (12) and (13) into the differential equation (9),

( 2 U S z c z c c z c S     = + + + +      . ) 2 2 1 2 3 4 1