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?? ? ??? = 0.
2 1


?1 ?1 ? ? ?4 ?2 ?2 + ??1 ?1 ? ?1 ? 3? ?1 ?1 + ? ?1 h? = 0,
3–4. ? ?? ?
?1 2
?? ?? ? ??? + ? ?? + ?1 = 0,
12 2
(4.11)
??1 ?3 + ?1 ?3 + ?? ?2 ?2 ? ?3 ? ? ?1 ?3 + ?2 = 0,
? ?? ?
1 1
2? + ??? + ?2 = 0,
where
?2 = (? + 3 )
3. if t > 0,
?1 = ?, 2
?1 = ??, ?2 = ?(? + 3 ) if t < 0.
2
4. ?1 = ?2 = ?.

?3 ?1 ? ?1 ? µ1i ?i + c11 + c21 ? = 0,
5–6. ? ??
? ?? ? ?2 ? µ2i ?i + c12 + c22 ? + ?2 a2 ?3 = 0,
32
??
(4.12)
? ?? ? ?3 + ?1 a2 ?2 + h? = 0,
33
??
3
?? = ?,
Symmetry reduction and exact solutions of the Navier–Stokes equations 195

where µ11 = ?B1 , µ12 = ?B2 ??1 a1 , µ21 = ?b21 +?1 a1 , µ22 = ?b22 , ? = ?1 ?B1 ?b22 .
?1 + ?1 ?1 + ?2 ?2 ? (? 3 )?1 ?3 ?2 + (? 3 )?2 ?1 = 0,
7. ?
?2 ? ?2 ?1 + ?1 ?2 + (? 3 )?1 ?3 ?1 + (? 3 )?2 ?2 = 0,
?
(4.13)
?3 + ?t (? 3 )?1 ?3 = 0,
3
?
2?1 + ?t (? 3 )?1 = 0.
3


?? + ??1 (nb · ?)mb = 0,
8. t
(4.14)
n · mt = 0.
a a



4.3 Exact solutions of the reduced systems
1. Ansatz (4.1) and system (4.9) determine the class of solutions of the NSEs (1.1)
that are called the steady axially symmetric conically similar flows of a viscous fluid
in hydrodynamics. This class of solutions was studied in a number of works (for
example, see references in [16]). For ?2 = 0 it was shown, by N.A. Slezkin [34], that
system (4.9) is reduced to a Riccati equation. The general solution of this equation
was expressed in terms of hypergeometric functions. Later similar calculations were
made by V.I. Yatseev [38] and H.B. Squire [35]. The particular case in the class of
solutions with ?2 = 0 is formed by the Landau jets [24]. For swirling flows, where
?2 = 0, the order of system (4.9) can be reduced too. For example [33], an arbitrary
solution of (4.9) satisfies the equation

?2 ?2 sin2 ? ? sin ?(?? sin?1 ?)? + 2?? cot ? + 2? = const,

where ? = (?3 ? 1 ?3 ?3 ) sin2 ? ? ?3 cos ? sin ?, and the Yatseev results [38] are
? 2
completely extended to the case ?2 sin ? = const.
2. System (4.10) implies that

?2 = ??1 + C1 ,
h = ?(1 + ? 2 )?1 + (2? 2 + 2 ? ?C1 )?1 + C2 ,
?
(1 + ? )??? + (4? ? C1 )?1 + ?1 ?1 + 4?1 +
2 1
(4.15)
?
2 ?1
+ (1 + ? ) (C1 + 2C2 ) = 0,
2

(1 + ? 2 )?3 ? (C1 ? 2?)?3 + (1 + ?1 )?3 = 0.
?? ?

If ?3 = 0, the solution determined by ansatz (4.10) and formulas (4.15) coincides with
the Hamel solution [18, 23]. In Section 6 we consider system (6.14) which is more
general than system (4.10).
3–4. Let us integrate the last equation of system (4.11), i.e.,

?1 = C1 ? ?2 ? 1 ?2 . (4.16)
2

Taking into account the integration result, the other equations of system (4.11) can
be written in the form

h? = ? ?3 ?2 ?2 + C1 ? ?3 ? 1 ?2 ?,
2 2
4

?2 ? ((C1 + 1)? ?1 ? 1 ?2 ?)?2 = ?1 ,
?? ?
2
196 W.I. Fushchych, R.O. Popovych

?3 ? ((C1 ? 1)? ?1 ? 1 ?2 ?)?3 ? ?1 ?3 = ?? ?2 ?2 + ?2 . (4.17)
?? ?
2

Therefore,
? ?3 ?2 ?2 d? ? 1 C1 ? ?2 ? 1 ?2 ? 2 ,
2 2
(4.18)
h= 2 8
2
1
|?|C1 +1 e? 4 ?2 ? d? +
?2 = C2 + C3
(4.19)
2 2
1 1
|?|C1 +1 e? 4 ?2 ? |?|?C1 ?1 e 4 ?2 ? d? d?.
+ ?1

If ?1 = 0, it follows that
2
1
|?|C1 ?1 e? 4 ?2 ? d? +
?3 = C4 + C5
(4.20)
2 2
1 1
|?|C1 ?1 e? 4 ?2 ? |?|?C1 +1 e 4 ?2 ? (?2 + ?? ?2 ?2 )d? d?.
+

Let ?1 = 0 (and, therefore, ? = 0). Then, if ?2 = 0, the general solution of equation
(4.17) is expressed in terms of Whittaker functions:
2
1 1 ?1
?3 = |?| 2 C1 ?1 e? 8 ?2 ? W (??1 ?2 + 1 C1 ? 1 , 1 C1 , 1 ?2 ? 2 ),
4 24 4

where W (?, µ, ? ) is the general solution of the Whittaker equation
4? 2 W? ? = (? 2 ? 4?? + 4µ2 ? 1)W. (4.21)
If ?2 = 0, the general solution of equation (4.16) is expressed in terms of Bessel
functions:
1
?3 = |?| 2 C1 Z 1 C1 (??1 )1/2 ? ,
2


where Z? (? ) is the general solution of the Bessel equation
? 2 Z? ? + ? Z? + (? 2 ? ? 2 )Z = 0. (4.22)

Note 4.2 If ?2 = 0, all quadratures in formulas (4.18)–(4.20) are easily integrated.
For example,
?
? C2 + C3 ln |?| + 1 ?1 ? 2 if C1 = ?2,
? 4
C2 + C3 1 ? 2 + 1 ?1 ? 2 (ln ? ? 1 )
?2 = if C1 = 0,
? 2 2 2
? ?1
C2 + C3 (C1 + 2)?1 |?|C1 +2 ? 1 ?1 C1 ? 2 if C1 = ?2, 0.
2

5–6. Let ? = 0. Then the last equation of system (4.12) implies that ?3 = C0 =
const. The other equations of system (4.12) can be written in the form
h = ??1 a2 ?2 (?)d?,
(4.23)
?i ? C0 ?i + µij ?j = ?1i + ?2i ?,
?? ?

where ?11 = c11 , ?21 = c21 , ?12 = c12 + ?2 a2 C0 , ?22 = c22 . System (4.23) is a linear
nonhomogeneous system of ODEs with constant coefficients. The form of its general
solution depends on the Jordan form of the matrix M = {µij }. Now let us transform
the dependent variables
?i = eij ? j ,
Symmetry reduction and exact solutions of the Navier–Stokes equations 197

where the constants eij are determined by means of the system of linear algebraic
equations
eij µjk = µij ejk
? (i, j, k = 1, 2)
?
with the condition det{eij } = 0. Here M = {?ij } is the real Jordan form of the matrix
µ
i
M. The new unknown functions ? have to satisfy the following system
??? ? C0 ?? + µij ? j = ?1i + ?2i ?,
i i
(4.24)
? ? ?
?
where ?1i = eij ?1j , ?2i = eij ?2j . Depending on the form of M, we consider the
? ?
following cases:
?
A. det M = 0 (this is equivalent to the condition det M = 0 ).
0?
? , where ? ? {0; 1}. Then
i. M =
00
?1 ?1 ?1
? 2 = C1 + C2 eC0 ? ? 1 ?22 C0 ? 2 ? (?12 ? ?22 C0 )C0 ?,
2? ? ?
?1 ?1 ?1
? 1 = C3 + C4 eC0 ? ? 1 ?21 C0 ? 2 ? (?11 ? ?21 C0 )C0 ? +
2? ? ?
(4.25)
?2 ?1 ?2
+ ? ? 1 ?22 C0 ? 3 ? 1 (?12 ? 2?22 C0 )C0 ? 2 +
6? 2? ?
?1 ?2 ?1 ?1
+ C1 + (?21 ? 2?22 C0 )C0 C0 ? ? C2 C0 ?eC0 ?
? ?

for C0 = 0, and
? 2 = C1 + C2 ? + 1 ?22 ? 3 + 1 ?12 ? 2 ,
6? 2?
(4.26)
? 1 = C3 + C4 ? + 6 (?21 ? C2 )? 3 + 1 (?11 ? C1 )? 2 ? ?
1 1 1
?5 ?4
? 2? 120 ?22 ? 24 ?12 ?

for C0 = 0.
?1 0
? , where ?1 ? R\{0}. Then the form of ? 2 is given either by
ii. M =
00
formula (4.25) for C0 = 0 or by formula (4.26) for C0 = 0. The form of ? 1 is given
by formula (4.28) (see below).
?
B. det M = 0 (this is equivalent to the condition det M = 0).
?1 0
? , where ?i ? R\{0}. Then
i. M =
?2
0
? ?1 ? ?1 ?1
? 2 = ?22 ?2 ? + (?12 ? C0 ?22 ?2 )?2 + C1 ?21 (?) + C2 ?22 (?), (4.27)
?
? ?1 ? ?1 ?1
? 1 = ?21 ?1 ? + (?11 ? C0 ?21 ?1 )?1 + C3 ?11 (?) + C4 ?12 (?), (4.28)
?
where
v v
?
1 1
?i1 (?) = exp ?i2 (?) = exp
Di )? ,
2 (C0 2 (C0 + Di )?
if Di = C0 ? 4?i > 0,
2
v v
1 1
?Di ? , ?i2 (?) = e 2 C0 ? sin ?Di ?
1 1
?i1 (?) = e 2 C0 ? cos 2 2
if Di < 0,
1 1
?i1 (?) = e 2 C0 ? , ?i2 (?) = ?e 2 C0 ?
if Di = 0.
198 W.I. Fushchych, R.O. Popovych

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