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T? + (2k + 2)(?(? ) ? 2k ? 1)T k+1 = 0, k = 0, N ? 1,
k N
(5.31)
T? = 0.
Equation (5.31) is easily integrated for arbitrary N ? N. For example if N = 2, it
follows that

f = C3 z 4 ? 4z 2 (?(? ) ? 3)d? + 8 (?(? ) ? 1) (?(? ) ? 3)d? d? +

+ C2 z 2 ? 2 (?(? ) ? 1)d? + C1 .

An explicit form for solution (5.30) with N = 1 is given by (5.29).
Generalizing the solution
f = C0 exp ?z 2 (4? + 2C)?1 + (?(? ) ? 1)(2? + C)?1 d? (5.32)
obtained by means of reduction of (5.22) by the operator G, we can construct solutions
of the general form
N
2k
S k (? ) z(2? + C)?1 ?
f=
(5.33)
k=0

? exp ?z 2 (4? + 2C)?1 + (?(? ) ? 1)(2? + C)?1 d? ,

where the coefficients S k = S k (? ) (k = 0, N ) satisfy the system of ODEs:
S? + (2k + 2)(?(? ) ? 2k ? 1)(2? + C)?2 S k+1 = 0,
k
(5.34)
k = 0, N ? 1, N
S? = 0.
For example if N = 1, then

f = C1 z 2 (2? + C)?2 ? 2 (?(? ) ? 1)(2? + C)?2 d? + C0 ?

? exp ?z 2 (4? + 2C)?1 + (?(? ) ? 1)(2? + C)?1 d? .

Here we do not present results for arbitrary N as they are very cumbersome.
Symmetry reduction and exact solutions of the Navier–Stokes equations 209

Putting g 2 = g 3 = 0 in system (5.24), we obtain one equation in the function g 1 :

g? ? ?z ?1 gz + ?z ?2 g 1 ? gzz + 2gz g 1 ? ?? z ?1 = 0.
1 1 1 1


It follows that g 1 = ?gz /g + (? ? 1)/z, where g = g(?, z) is a solution of the equation

g? + (? ? 2)z ?1 gz ? gzz = 0. (5.35)

Q-conditional symmetry of (5.22) under the operator

Q = ?? + ?gz /g + (? ? 1)/z ?z (5.36)

gives rise to the following

Theorem 5.3 If g is a solution of equation (5.35) and
z
f (?, z) = z g(?, z )dz +
z0
(5.37)
?
z0 gz (? , z0 ) ? (?(? ) ? 1)g(? , z0 ) d? ,
+ ?0

where (?0 , z0 ) is a fixed point, then f is a solution of equation (5.22).

Proof. Equation (5.35) implies

(zg)? = (zgz ? (? ? 1)g)z

Therefore, fz = zg, f? = zgz ? (? ? 1)g and

f? + ?z ?1 fz ? fzz = zgz ? (? ? 1)g + ?g ? (zg)z = 0. QED.

The converse of Theorem 5.3 is the following obvious

Theorem 5.4 If f is a solution of (5.22), the function

g = z ?1 fz (5.38)

satisfies (5.35).

Theorems 5.3 and 5.4 imply that, when ? = 2n (n ? Z), solutions of (5.22) can be
constructed from known solutions of the heat equation by means of applying either
formula (5.37) (for n > 0) or formula (5.38) (for n < 0) |n| times.
Let us investigate symmetry properties and construct some exact solutions of
system (5.19)–(5.20) for ? = 1, i.e., the system

w? ? wzz + ? (? )z ?1 wz = 0,
1 1 1
(5.39)
?

w? ? wzz + (?(? ) ? 2)z ?1 wz + (w1 ? ?(? ))z ?2 = 0.
2 2 2
(5.40)
? ?

If (w1 , w2 ) is a solution of system (5.39)–(5.40), then (w1 , w2 + g) (where g =
g(?, z)) is also a solution of (5.39)–(5.40) if and only if the function g satisfies the
following equation

g? ? gzz + (?(? ) ? 2)z ?1 gz = 0 (5.41)
?
210 W.I. Fushchych, R.O. Popovych

System (5.39)–(5.40), for some ? = ?(? ), has particular solutions of the form
??
N ?1
N
1 k 2k 2
S k (? )z 2k ,
w= T (? )z , w=
k=0 k=0

where T 0 (? ) = ?(? ). For example, if ?(? ) = ?2C1 (?(? ) ? 1)d? + C2 and N = 1,
? ? ?
then

w1 = C1 z 2 ? 2 (?(? ) ? 1)d? + C2 , w2 = ?C1 ?.
?

Let ?(? ) = 0.
?

Theorem 5.5 The MIA of system (5.39)–(5.40) with ?(? ) = 0 is given by the
?
following algebras

wi ?wi , wi (?, z)?wi if ? (? ) = const;
a) ? ?
2? ?? + z?z , ?? , w ?wi , wi (?, z)?wi if ? (? ) = const, ? = 0;
i
b) ? ? ?
2? ?? + z?z , ?? , w1 z ?1 ?w2 , wi ?wi , wi (?, z)?wi if ? ? 0.
c) ? ?

Here (w1 , w2 ) is an arbitrary solution of (5.39)–(5.40) with ?(? ) = 0.
?? ?

For the case ?(? ) = 0 and ? (? ) = const system (5.39)–(5.40) can be reduced
? ?
by inequivalent one-dimensional subalgebras of its MIA. We obtain the following
solutions:
For the subalgebra ?? it follows that

w1 = C1 ln z + C2 ,
w2 = 1 C1 (ln2 z ? ln z) + 1 C2 ln z + C3 z ?2 + C4
4 2

if ? = ?1;
?

w1 = C1 z 2 + C2 ,
w2 = 1 C1 z 2 + 1 C2 ln2 z + C3 ln z + C4
4 2

if ? = 1;
?

w1 = C1 z ?+1 + C2 ,
?

w2 = 1 C1 (? + 1)?1 z ?+1 + C2 (? ? 1)?1 ln z + C3 z ??1 + C4
? ?
? ?
2

if ? ? {?1; 1}.
?
For the subalgebra ?? ? wi ?wi it follows that
1 1
w1 = e?? z 2 (?+1) ? 1 (z), w2 = e?? z 2 (??1) ? 2 (z),
? ?


where the functions ? 1 and ? 2 satisfy the system

z 2 ?zz + z?z + z 2 ? 1 (? + 1)2 ? 1 = 0,
1 1
(5.42)
4?

z 2 ?zz + z?z + z 2 ? 1 (? ? 1)2 ? 2 = z? 1 .
2 2
(5.43)
4?
Symmetry reduction and exact solutions of the Navier–Stokes equations 211

The general solution of system (5.42)–(5.43) can be expressed by quadratures in terms
of the Bessel functions of a real variable J? (z) and Y? (z):
? 1 = C1 J?+1 (z) + C2 Y?+1 (z),
? 2 = C3 J? (z) + C4 Y? (z) + ? Y? (z) J? (z)? 1 (z)dz ? ? J? (z) Y? (z)? 1 (z)dz
2 2

with ? = 1 (? ? 1);
2?
For the subalgebra ?? + wi ?wi it follows that
1 1
w2 = e? z 2 (??1) ? 2 (z),
w1 = e? z 2 (?+1) ? 1 (z),
? ?


where the functions ? 1 and ? 2 satisfy the system
z 2 ?zz + z?z ? z 2 + 1 (? + 1)2 ? 1 = 0,
1 1
(5.44)
4?

z 2 ?zz + z?z ? z 2 + 1 (? ? 1)2 ? 2 = z? 1 .
2 2
(5.45)
4?

The general solution of system (5.44)–(5.45) can be expressed by quadratures in terms
of the Bessel functions of an imaginary variable I? (z) and K? (z):
? 1 = C1 I?+1 (z) + C2 K?+1 (z),
I? (z)? 1 (z)dz ? I? (z)
? 2 = C3 I? (z) + C4 K? (z) + K? (z) K? (z)? 1 (z)dz
with ? = 1 (? ? 1).
2?
For the subalgebra 2? ?? + z?z + awi ?wi it follows that
1 1 1 1
w1 = |? |a e? 2 ? |?| 4 (??1) ? 1 (?), w2 = |? |a e? 2 ? |?| 4 (??3) ? 2 (?)
? ?


with ? = 1 z 2 ? ?1 , where the functions ? 1 and ? 2 satisfy the system
4

4? 2 ??? = ? 2 + a ? 1 (? ? 1) ? + 1 (? + 1)2 ? 1 ? 1 ,
1
(5.46)
4? 4?


4? 2 ??? = ? 2 + a ? 1 (? ? 3) ? + 1 (? ? 1)2 ? 1 ? 2 + 2|?|1/2 ? 1 .
2
(5.47)
4? 4?

The general solution of system (5.46)–(5.47) can be expressed by quadratures in terms
of the Whittaker functions.


6 Symmetry properties and exact solutions
of system (3.12)
As was mentioned in Section 3, ansatzes (3.4)–(3.7) reduce the NSEs (1.1) to the
systems of PDEs of a similar structure that have the general form (see (3.12)):
wi wi ? wii + s1 + ?2 w2 = 0,
1 1

wi wi ? wii + s2 ? ?2 w1 + ?1 w3 = 0,
2 2
(6.1)
wi wi ? wii + ?4 w3 + ?5 = 0,
3 3

i
wi = ?3 ,
where ?n (n = 1, 5) are real parameters.
212 W.I. Fushchych, R.O. Popovych

Setting ?k = 0 (k = 2, 5) in (6.1), we obtain equations describing a plane convecti-
ve flow that is brought about by nonhomogeneous heating of boudaries [25]. In this
case wi are the coordinates of the flow velocity vector, w3 is the flow temperature, s
is the pressure, the Grasshoff number ? is equal to ??1 , and the Prandtl number ? is

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