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

84

3

x y z       i j

κ

 = + +

is the Hamilton operator; i , j , and k are the unit vectors of the Ox , Oy , and Oz axes;

2           2 2 2 2 2 x y z 

 = + + 

is the Laplace operator. Equation (1) (the Navier-Stokes equation) is the law of momentum conservation, and equation (2) (the incompressibility equation) is the law of mass conservation for an incompressible fluid. Let us further consider the class of flows for which the vertical component of the velocity vector is identically zero, i.e.

0 z V = .

That is, throughout what follows, we will consider only shear flows. In technical devices flows of this kind arise when a fluid moves through a slit with a slowly varying gap thickness. Within the framework of this assumption, equations (1), (2) take the form

2 2 V V V + +       , 2 2 2 2 x x x x y z        2 V V V + +       , 2 2 2 2 y y y x y z          2

x y V V  +



x V V x 

=

x

y



V

V





y

y

x V V x + 

=

y

y



y x V V x y    

0

+ =

.

(3)

The resulting system (3) consists of three nonlinear partial differential equations, from which two nonzero components of the velocity vector should be found. In other words, the system is overdetermined, i.e. it is necessary to obtain the consistency condition for solutions to equations (3) and make sure that it is satisfied. Consistency conditions for systems of the form (3) were obtained, for example, in (Baranovskii et al., 2021; Burmasheva et al., 2020; Burmasheva et al., 2020; Burmasheva et al., 2020; Burmasheva et al., 2021; Burmasheva et al., 2021). We will further seek a solution in the following linear (in horizontal coordinates) forms: ( ) ( ) x V U z u z y = + , ( ) y V V z = . (4) The solution of the form (4) removes the problem of the overdetermination of the original system of equations (3) since, for expressions (4), incompressibility is equalized automatically. Substituting dependences (4) into the first two equations of system (3), taking into account the independence of the longitudinal and vertical coordinates, we obtain the following system of ordinary differential equations:

0 u  = , '' 0 V = , U Vu  =  .

(5)