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
86
5
Note also that the velocity V y = V retains its sign; therefore, the rotation of the fluid during its flow is possible only along the Ox axis. Let us analyze the velocity V x = U+uy further in detail. 3. Analysis of the exact solution For the convenience of the further analysis, we introduce two substitutions:
Z z h h = . In this case, the exact solution (19) for the velocity components U , u , and V takes the form = , a
h u Z a
(
)
(
)
u
V V h + =
z
=
+
+
h
,
,
1
a
+
6
8
h U h
h h u V h
) (
) (
) (
)
(
) ( Z a a
)
(
2
2 1 4 6 1 a a + − − − + + Z a
2 2 a Z Z Za a
2 4 6 + + =
1 + + +
U
=
( 12 1
)
(
)
3
3
1
a
a
+
+
(
)
8
2
1 4 6 + +
h h u V h
a a
6
8
h U h
h h u V h
(
)
(
)
2 2 Z Z Za a + + 4 6
2
Z a
=
−
+ +
.
(10)
(
)
( 12 1
)
( 12 1
)
3
2
1
a
+
a
a
+
+
It can be seen from expression (10) that the velocity U profile is determined by the competitive interaction of two flows, one of which has a linear profile and the other is nonlinear. Moreover, if the coefficient in front of the nonlinear term in (10) is zero, the velocity
( 1 h U h U Z a a = + + ) ( 6
)
takes on values of only one sign. Let us see under what conditions the velocity can take on zero values. To do this, we select the nonlinear term:
(
)
8
2
1 4 6 + +
h h u V h
a a
8
) ( ) 1 h U h a + 6
8
h h u V h
h h u V h
(
(
)
2 2 Z Z Za a
2 4 6 + + +
U
Z a
=
−
+ =
( 12 1
)
( 12 1
)
( 12 1
)
2
3
2
a
a
a
+
+
+
8
( 12 1 h h u V h Z Z Za a k Z a a = + + + + + , ) ( ) 2 2 2 2 4 6
(
)
where
(
)
( 12 1
)
2
2
2
1 4 6 + +
h h h a U u V h −
a a
+
k
=
.
(
) a u
2
1
V h
+
h h
The locus of points whose coordinates satisfy the condition
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