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

177

7

3 40 S B CB S A h h z A h  +    − − + +           . 2 1 2 2 4

In the latter expressions the background velocity U is described by a quintic polynomial, and the spatial velocity gradient 1 U is described by the Couette profile (Ershkov et al., 2021; Aristov et al., 2009; Drazin e al., 2006; Pukhnachev et al., 2006). 5. Velocity field analysis To study the obtained gradient flow velocities, we rewrite the expression for velocity (14) in a form more convenient for studying,

 

  

 

2

2

2

2 S h

2 S h

2 S h

2

C  

(

)

2

1 Z Z 

.

(16)

V

Z

С − + 

Z C

=

+ =

− −

 

2

2

2

2

2 S h







A dimensionless transverse coordinate Z z h = is introduced, which will be used in what follows. This factorization of the velocity V is possible only for gradient fluid flows. In this case there are counterflows in the fluid. The boundary of the reverse flows is defined by the equation of plane as:

2 2 2 Z C S h  =

2

2 C h S

,

.

0

 



2

Curiously enough, fluid counterflows will be observed when the specified velocity C on the boundary and the pressure gradient 2 S are of the same sign. In this case the solution represented by expression (16) describes the homogeneous Couette flow (Ershkov et al., 2021; Aristov et al., 2009; Drazin e al., 2006; Pukhnachev et al., 2006), ( ) 1 V C Z = − .

Let us now study the characteristic properties of the velocity

1 ( ) x V U z U z y = + . The study of the properties of ( )

1 U is trivial; therefore, we focus our attention on studying the spectral properties of the

the linear function

polynomial U , which is rewritten for convenience as

  

  

  

  

4

4

4

2

2

2

2 S Bh

2 S Bh

2 S Bh

BCh

BCh

2 h

(

)

5

4

3

2

U

Z

Z

Z

S CB Z +

= −

+

+

−

+

+

−

1

2

2

2

12

3

40

12

12













4 40 S Bh CB S A h Z A    +  − − + +           . 2 2 1 2 2 4

(17)

0 S S = = , then the background fluid velocity is described by a quadric polynomial as:

Assume that 1

2

 

  

2

2

2

2

BCh

BCh

B Сh

BСh

4

3

2 Z A 

U Z =

Z

Z A +

.

−

+

− +

12

3

2

4









By virtue of the boundary conditions, this polynomial has the root the interval ( ) 0;1 the function becomes zero if the following inequality is fulfilled:

1 Z = . The study of root localization shows that on