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3 3 3 1

+ (2w1 + w2 + w1 + ?w1 ) cos z2 + s2 sin2 z2 = 0,
3 2 1

w1 + w1 + w2 = 0.
2 3


Hereafter numeration of the reduced systems corresponds to that of the ansatzes in
Subsection 3.1. Subscripts 1 and 2 denote differentiation with respect to the variables
z1 and z2 , accordingly.
?1 ?2
2–3. w1 w1 + w3 w2 ? z1 w2 w2 ? w11 + (1 + ? 2 z1 )w22 ?
1 1 1 1

?2 2
? 2?z1 w2 + s1 = 0,
?1 ?2
w1 w1 + w3 w2 + z1 w1 w2 ? w11 + (1 + ? 2 z1 )w22 +
2 2 2 2

?2 1 ?2 ?1 ?1
+ 2?z1 w2 + 2z1 w2 ? ?z1 s2 + ?z1 = 0, (3.11)
?2 ?2
w1 w1 + w3 w2 ? 2?z1 w1 w2 ? w11 + (1 + ? 2 z1 )w22 +
3 3 3 3

?2 ?3 1 ?2 ?2
+ 2?(z1 w2 )1 ? 2? 2 z1 w2 + (1 + ? 2 z1 )s2 ? ??z1 = 0,
?1
w1 + w2 + z1 w1 + ? = 0,
1 3


where ? = ±3/2 for ansatz (3.2) and ? = 0 for ansatz (3.3). Here and below the
upper and lower sign in the symbols “±” and “?” are associated with t > 0 and t < 0,
respectively.
4–7. For ansatzes (3.4)–(3.7) the reduced equations can be written in the form
wi wi ? wii + s1 + ?2 w2 = 0,
1 1

wi wi ? wii + s2 ? ?2 w1 + ?1 w3 = 0,
2 2
(3.12)
wi wi ? wii + ?4 w3 + ?5 = 0,
3 3

i
wi = ?3

where the constants ?n (n = 1, 5), take on the values
= ±2??, = ?2?µ, = ?(? + 3/2), = ±?,
4. ?1 ?2 ?3 ?4 ?5 = ?.
= ?(? + 3/2), = ±?,
5. ?1 = 0, ?2 = 0, ?3 ?4 ?5 = ?.
= ?2µ, = ??,
6. ?1 = 2?, ?2 ?3 ?4 = ?, ?5 = ?.
= ??,
7. ?1 = 0, ?2 = 0, ?3 ?4 = ?, ?5 = ?.
?4
w1 + (w1 )2 ? z2 (w2 ? ?)2 + z2 w1 w2 ? w22 ?
1 1 1
8.
(3.13)
?1
? 3z2 w2 + z2 s2 = 0,
1


?1 2
w1 + z2 w1 w2 ? w22 + z2 w2 = 0,
2 2 2
(3.14)
?1 3 ?2
w1 + z2 w1 w2 ? w22 ? z2 w2 + z2 (w2 ? ?) = 0,
3 3 3
(3.15)
192 W.I. Fushchych, R.O. Popovych

2w1 + z2 w2 + ?1 /? = 0.
1
(3.16)

w1 ? ?w22 + s2 k + ??1 (ni · w)mi + z2 e = 0, (3.17)
9. t


k · w2 = 0, (3.18)

where y1 = t and

e = e(t) = 2??2 (m1 · m2 ? m1 · m2 )kt ? k + ??2 (2kt · kt ? ktt · k).
t t

Let us study symmetry properties of reduced systems (3.10) and (3.11).

Theorem 3.1 The MIA of (3.10) is given by the algebra ?1 .

Theorem 3.2 The MIA of (3.11) is given by the following algebras:
?2 , ?s , D1 = zi ?i ? wa ?wa ? 2s?s ? = ? = ? = 0;
2
if
a)
?2 , ?s if (?, ?, ?) = (0, 0, 0).
b)

All the Lie symmetry operators of systems (3.10) and (3.11) are induced by
elements of A(N S). So, for system (3.10) the operator ?1 is induced by J12 . For
system (3.11), when ? = 0 (? = ±3/2), the operators D1 , ?2 , and ?s (?2 and ?s )
2

are induced by D, R(0, 0, 1), and Z(1) (R(0, 0, |t|?1/2 ) and Z(|t|?1 )), accordingly.
Therefore, the Lie reductions of systems (3.10) and (3.11) give only solutions that
can be obtained by reducing the NSEs with three-dimensional subalgebras of A(N S)
immediately to ODEs.
Investigation of reduced systems (3.13)–(3.16), (3.17)–(3.18), and (3.12) is given
in Sections 5 and 6.


4 Reduction of the Navier–Stokes equations
to ordinary differential equations
4.1 Ansatzes of codimension three
By means of subalgebraic analysis of A(N S) (see Subsection A.3) and the method
described in Section B one can obtain the following ansatzes that reduce the NSEs to
ODEs:
u1 = x1 R?2 ?1 ? x2 (Rr)?1 ?2 + x1 x3 r?1 R?2 ?3 ,
1.
u2 = x2 R?2 ?1 + x1 (Rr)?1 ?2 + x2 x3 r?1 R?2 ?3 ,
(4.1)
u3 = x3 R?2 ?1 ? rR?2 ?3 ,
p = R?2 h,

where ? = arctan r/x3 . Here and below ?a = ?a (?), h = h(?), r = (x2 + x2 )1/2 ,
1 2
R = (x2 + x2 + x2 )1/2 .
1 2 3

u1 = r?2 (x1 ?1 ? x2 ?2 ), u2 = r?2 (x2 ?1 + x1 ?2 ),
2.
(4.2)
u3 = r?1 ?3 , p = r?2 h,
Symmetry reduction and exact solutions of the Navier–Stokes equations 193

where ? = arctan x2 /x1 ? ? ln r, ? ? 0.

u1 = x1 |t|?1 ?1 ? x2 r?2 ?2 + 1 x1 t?1 ,
3. 2
u = x2 |t| ? + x1 r ? + 1 x2 t?1 ,
?1 1 ?2 2
2
2
u3 = |t|?1/2 ?3 + (? + 1 )x3 t?1 + ?|t|1/2 t?1 arctan x2 /x1 , (4.3)
2
p = |t| h + 8 t R ? 1 ? 2 x2 t?2 +
?1 1 ?2 2
3
2
+ ?1 |t|?1 arctan x2 /x1 + ?2 x3 |t|?3/2 ,

where ? = |t|?1/2 r, ?? = 0, ?2 ? = 0, ?1 ? 0, ? ? 0.

u1 = x1 ?1 ? x2 r?2 ?2 ,
4.
u2 = x2 ?1 + x1 r?2 ?2 ,
(4.4)
u3 = ?3 + ?x3 + ? arctan x2 /x1 ,
p = h ? 1 ? 2 x2 + ?1 arctan x2 /x1 + ?2 x3 ,
3
2

where ? = r, ?? = 0, ?2 ? = 0, and for ? = 0 one of the conditions

? = 1, ?1 ? 0; ? = 0, ?1 = 1, ?2 ? 0; ? = ?1 = 0, ?2 ? {0; 1}

is satisfied.
Two ansatzes are described better in the following way:
5. The expressions for ua and p are determined by (2.1), where

v 1 = a1 ?1 + a2 ?3 + b1i ?i ,
v 2 = ?2 + b2i ?i ,
(4.5)
v 3 = a2 ?1 ? a1 ?3 + b3i ?i ,
p = h + c1i ?i + c2i ??i + 1 dij ?i ?j .
2

In formulas (4.5) we use the following definitions:

?1 = a1 y1 + a2 y3 , ?2 = y2 , ? = ?3 = a2 y1 ? a1 y3 ;
ai = const, a2 + a2 = 1; a2 = 0 if ?1 = 0;
1 2
?1 = ?2?, ?2 = ? 2 if t > 0 and ?1 = 2?, ?2 =
3 3
if t < 0.
2

bai , Bi , cij , and dij are real constants that satisfy the equations

b1i = a1 Bi , b3i = a2 Bi , c2i + a2 ?1 b2i = 0,
b21 Bi + b22 b2i ? ?1 a1 Bi + d2i = 0,
(4.6)
B1 Bi + B2 b2i + ?1 a1 Bi + d1i = 0,
(B1 + b22 )(B2 + a1 ?1 ? b21 ) = 0.

6. The expressions for ua and p have form (2.2), where v a and q are determined
by (4.5), (4.6), and ?1 = ?2?, ?2 = 0.

Note 4.1 Formulas (4.5) and (4.6) determine an ansatz for system (2.7), where
equations (4.6) are the necessary and sufficient condition to reduce system (2.7)
by means of an ansatz of form (4.5).
194 W.I. Fushchych, R.O. Popovych

u1 = ?1 cos x3 /? 3 ? ?2 sin x3 /? 3 + x1 ?1 (t) + x2 ?2 (t),
7.
u2 = ?1 sin x3 /? 3 + ?2 cos x3 /? 3 ? x1 ?2 (t) + x2 ?1 (t),
(4.7)
u3 = ?3 + ?t (? 3 )?1 x3 ,
3

p = h ? 1 ?tt (? 3 )?1 x2 ? 1 ?tt ? j (? i ? i )?1 r2 ,
j
3
3
2 2

where ? = t,
? a ? C ? ((t0 , t1 ), R), ? 3 = 0, ? i ? i = 0, ?t ? 2 ? ? 1 ?t ? {0; 1 },
1 2
2
i i j j ?1 j j ?1
? = ?t ? (? ? ) , ? = (?t ? ? ? ?t )(? ? ) .
1 2 12 12


u = ? + ??1 (na · x)ma ,
8. t
(4.8)
p = h ? ? (mtt · x)(na · x) + 1 ??2 (mb · ma )(na · x)(nb · x),
?1 a
tt
2

where ? = t, ma ? C ? ((t0 , t1 ), R), ma · mb ? ma · mb = 0,
tt tt

? = ?(t) = (m1 ? m2 ) · m3 = 0 ?t ? (t0 , t1 ),
n1 = m2 ? m3 , n2 = m3 ? m1 , n3 = m1 ? m2 .

4.2 Reduced systems
Substituting the ansatzes 1–8 into the NSEs (1.1), we obtain the following systems of
ODE in the functions ?a and h:
?3 ?1 ? ?a ?a ? ?1 ? ?1 cot ? ? 2h = 0,
1. ? ?? ?
?3 ?2 + ?2 ?3 cot ? ? ?2 ? ?2 cot ? + ?2 sin?2 ? = 0,
? ?? ?
(4.9)
?3 ?3 ? ?2 ?2 cot ? ? ?3 ? ?3 cot ? + ?3 sin?2 ? ? 2?1 + h? = 0,
? ?? ? ?
1 3 3
? + ?? + ? cot ? = 0.

(?2 ? ??1 )?1 ? (1 + ? 2 )?1 ? ?1 ?1 ? ?2 ?2 ? ?h? ? 2h = 0,
2. ? ??
(? ? ?? )?? ? (1 + ? )?2 ? 2(??2 + ?1 ) + h? = 0,
2 1 2 2
?? ? ?
(4.10)
(? ? ?? )?? ? (1 + ? )??? ? ? ? ? ? ? 2??3 = 0,
2 1 3 2 3 13 3
?

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