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

176

6

Having integrated the latter second-order ordinary differential equation with polynomial inhomogeneity, we obtain the following exact solution:

S c

c c

c c

S c

c c

2 4 S c c   2 2

  

4       z

3  

  

.

(14)

5

2 z c z c + +

2 3 U z =

z

1 3

2 3

+

+

1 4 + +

1 + +

2 4

 

 

5

6

2

2

12

6 6

40

24



 





The symbols 1 c , 2 c , 3 c , 4 c , 5 c , and 6 c in formulas (12) – (14) denote the integration constants, which will be then computed with the use of the boundary conditions (11). Substituting the boundary conditions, we arrive at the following system to determine 1 c and 2 c .

S

( )

0

V h

2 2 h c h c = + + =  1 2 2

.

( ) 2 0 V c C = = .

The integration constants 1 c and 2 c are found as follows:

2 2 S h С 

c

, 2 c C = .

h = − −

1

The exact solution of the boundary value problem composed of formulas (8) and (11) for the velocity y V V = is obtained as:

S

S h С

  

 

(15)

2 2 V V z

z C +

= = − +

2

y

2

2

h



 

Formula (15) describes the Poiseuille exact solution. Let us now find the integration constants 3 c , 4 c , 5 c , and 6 c for the velocity x V . By repeating the computations for the determination of the free parameters 1 c , and 2 c (solution of the linear equations), we obtain the following solution to the boundary value problem (9), (11) for the functions (13) and (14):

B h = − , 4 c B = ,

3 c

3 60 S B CB S c A h h h  −  = − + −       2 1 5 2 2 4

, 6 c A = .

1 U and U have, respectively, the forms:

Thus the expressions for

1   =  −    z h

,

U B

1

2 S B

2 S B

2 S Bh

1

BC

BC

  

4       z −

 

(

)

5

3

2

U

z

z

S CB z +

= −

+

+

+

+

−

1

2

2 12 12 

2

2

3

2

40

12

h

h

h

 







