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W.I. Fushchych, Scientific Works 2003, Vol. 5, 240–246.

?
Ansatze of codimension one
for the Navier–Stokes field and reduction
of the Navier–Stokes equation
W.I. FUSHCHYCH, R.O. POPOVYCH, G.V. POPOVYCH
i i (ii) ii-
i i ’–, i i i
i ii ’–. -
i i ’– i.
i ii i .

Finding exact solutions of the Navier–Stokes equations (NSEs) for an incompres-
sible viscous fluid is an actual problem of mathematical physics and hydrodynamics.
There are some ways to solve this problem. One of them is a usage of symmetry
analysis [1–8]. In this article we construct a complete set of inequivalent ansatze of
codimension one for the Navier–Stokes field. Using them, we reduce the NSEs to
systems of partial differential equations in three independent variables and study their
symmetry properties.
It is known that the NSEs
ut + (u · ?)u ? ?u + ?p = 0, (1)
div u = 0
are invariant under the infinite dimensional algebra A(NS) with basic elements
?t = ?/?t, D = 2t?t + xa ?a ? ua ?ua ? 2p?p ,
Jab = xa ?b ? xb ?a + ua ?ub ? ub ?ua , a = b, (2)
R(m(t)) = ma (t)?a + ma (t)?ua ? ma (t)xa ?p , Z(?(t)) = ?(t)?p .
t tt

Here and from now on u = u(t, x) = {ua } is the velocity field of a fluid, p = p(t, x) is
the pressure, x = {xa }, ?t = ?/?t , ?a = ?/?xa , ? = {?a }, ? = ? · ?, ma = ma (t),
? = ?(t) are arbitrary smooth functions of t (for example, from C ? ((t0 , t1 ), R)),
a, b = 1, 3, i, j = 1, 2, repetition of an index signifies a sum.
The set of operators (2) determines the maximal, in the sense of Lie, invariance
algebra of the NSEs [9–11].
Theorem 1. A complete set of A(NS)-inequivalent one-dimensional subalgebras of
A(NS) is exhausted by such algebras:
A1 (?) = D + 2?J12 , ? ? 0;
1) 1
A1 (?) = ?t + ?J12 , ? ? {0; 1};
2) 2
A1 (?, ?) = J12 + R(0, 0, ?(t)) + Z(?(t)) ,
3) 3

where algebras A1 (?, ?) and A1 (?, ?) are equivalent if ? ?, ? ? R, ? ? ? C ? ((t0 , t1 ), R):
3? ?
3

(?, ?)(t) = (e?? ?, e2? (? + ?? ? ? ?))(te2? + ?);
? (3)
?? ?
ii , 1994, 4, P. 37–44.
Ans?tze of codimension one for the Navier–Stokes field
a 241

A1 (m, ?) = R(m) + Z(?) ,
4) (m, ?) = (0, 0),
4


where algebras A1 (m, ?) and A1 (m, ?) are equivalent if ? ?, ? ? R, ? c = 0, ? B ?
4??
4
O(3), ? l ? C ? ((t0 , t1 ), R3 ):

(m, ?)(t) = (ce?? B m, ce2? (? + ? · m ? m · l))(te2? + ?). (4)
? l ?

Theorem 1 is proved by the method described in [12, 13].
With the algebras A1 –A1 from theorem 1 and with the algebra A1 (if some addi-
1 3 4
tional demands are satisfied) one can construct such a set of inequivalent ans?tze of
a
codimension one for the Navier–Stokes field:
u1 = |t|?1/2 (v 1 cos ? ? v 2 sin ? ) + 1 x1 t?1 ? ?x2 t?1 ,
1. 2
u2 = |t|?1/2 (v 1 sin ? + v 2 cos ? ) + 1 x2 t?1 + ?x1 t?1 ,
2
(5)
u3 = |t|?1/2 v 3 + 1 x3 t?1 ,
2
p = t q + 8 t xa xa + 1 ? 2 t?2 r2 ,
?1 1 ?2
2

where
y1 = |t|?1/2 (x1 cos ? + x2 sin ? ),
? = ? ln |t|, r = (x2 + x2 )1/2 ,
1 2
y2 = |t|?1/2 (?x1 sin ? + x2 cos ? ), y3 = |t|?1/2 x3 ;

here and from now on v a = v a (y1 , y2 , y3 ), q = q(y1 , y2 , y3 ), numeration of ans?tze
a
corresponds to that of algebras in theorem 1.

u1 = v 1 cos ?t ? v 2 sin ?t ? ?x2 ,
2.
u2 = v 1 sin ?t + v 2 cos ?t + ?x1 , (6)
u3 = v 3 , p = q + 1 ? 2 r 2 ,
2

where y1 = x1 cos ?t + x2 sin ?t, y2 = ?x1 sin ?t + x2 cos ?t, y3 = x3 .

u1 = x1 r?1 v 1 ? x2 r?1 v 2 + x1 r?2 ,
3.
u2 = x2 r?1 v 1 + x1 r?1 v 2 + x2 r?2 ,
(7)
u3 = v 3 + ?(t)r?1 v 2 + ?(t) arctg x2 /x1 ,
?
p = q ? 1 ? (t)(?(t))?1 x2 ? 1 r?2 + ?(t) arctg x2 /x1 ,
2? 3 2

where y1 = t, y2 = r, y3 = x3 ? ?(t) arctg x2 /x1 .
Remark 1. The expression for the pressure p from the ansatz (7) is indeterminate in
points t ? {t0 , t1 }, where ?(t) = 0. If there are such points t, we will consider the
ansatz (7) in intervals (tn , tn ) that are contained by the interval (t0 , t1 ) and for which
01
one from the conditions
? t ? (tn , tn ) : ?(t) = 0;
a) 01
?(t) ? 0 in (tn , tn )
b) 01

is satisfied. In the last case we consider that ? /? := 0.
?
1
4. With the algebra A4 (m, ?), an ansatz can be constructed only for such a t
wherefor m(t) = 0. If this condition is satisfied, it follows from (2) that the algebra
242 W.I. Fushchych, R.O. Popovych, G.V. Popovych

A1 (m, ?) is equivalent to the algebra A1 (m, 0). An ansatz constructed with the algebra
4 4
1
A4 (m, 0) is
i
u = v i ni + (m · m)?1 v 3 m + (m · x)(m · m)?1 m ? yi n ,
? ?
p = q ? 3 (m · m)?1 ((m · ni )yi )2 ? (m · m)?1 (m · x)(m · x) + (8)
? ?
2
+ (m · m)(m · m)?2 (m · x)2 ,
?

where yi = ni · x, y3 = t,

ni · m = n1 · n2 = 0, |ni | = 1, n 1 · n2 = 0.
ni = ni (t), (9)
?

Remark 2. Vector-functions ni satisfying conditions (9) exist. They can be construc-
ted in such a way: let us fix vector-functions k i = k i (t) for which k i · m = k 1 · k 2 =
0, |k i | = 1 and set

n1 = k 1 cos ?(t) ? k 2 sin ?(t), n2 = k 1 sin ?(t) + k 2 cos ?(t). (10)

? ? ?
Then n 1 · n2 = k 1 · k 2 ? ? = 0 if (k 1 · k 2 )dt.
?
Substituting the ans?tze (5), (6) to the NSEs (1), we obtain reduced systems of
a
PDEs that have the same general form
v a va ? vaa + q1 + ?1 v 2 = 0,
1 1

v a va ? vaa + q2 ? ?1 v 1 = 0,
2 2
(11)
v a va ? vaa + q3 = 0,
3 3

a
va = ?2 ,

where the constant ?i , takes the values
?1 = ?2?, ?2 = ? 2 ,
3
?2 = 3 ,
if if t < 0.
1. t > 0, ?1 = 2?, 2
?1 = ?2?,
2. ?2 = 0.

For the ans?tze (7), (8) reduced equations have the form
a
?1 ?2 1 ?2 2
v1 + v 1 v2 + v 3 v3 ? y2 v 2 v 2 ? [v22 + (1 + ? 2 y2 )v33 + 2?y2 v3 ] + q2 = 0,
1 1 1 1
3.
?1 ?2 2 ?2 1
v1 + v 1 v2 + v 3 v3 + y2 v 1 v 2 ? [v22 + (1 + ? 2 y2 )v33 ? 2?y2 v3 ] +
2 2 2 2

?2 ?1 ?1
+ 2y2 v 2 ? ?y2 q3 + ?y2 = 0,
(12)
?2 3 ?3 1
? ?1
v1 + v 1 v2 + v 2 v3 ? [v22 + (1 + ? 2 y2 )v33 ] ? 2? 2 y2 v3 + 2?y2 v 2 +
3 3 3 3

?1 ?1 ?2 ?2
+ 2?y2 (y2 v 2 )2 + (1 + ? 2 y2 )q3 ? ? ? ?1 y3 ? ??y2 = 0,
?
?1
y2 v 1 + v2 + v3 = 0.
1 3


v3 + v i vj ? vjj + qi + ?i (y3 )v 3 = 0,
i i i
4.
v3 + v j vj ? vjj = 0,
3 3 3
(13)
vj + ?3 (y3 ) = 0,
i


where

?i = ?i (y3 ) = 2(m · m)?1 (m · ni ), ?3 = ?3 (y3 ) = (m · m)?1 (m · m). (14)
? ? ?
Ans?tze of codimension one for the Navier–Stokes field
a 243

Let us study symmetry properties of the systems (11)–(13). All following results
are obtained with the standard Lie algorithm [11, 12]. At first consider the sustem (11).
Theorem 2. The maximal, in the sense of Lie, invariance algebra of (11) is the
algebra
1
?a , ?q , J12 if ?1 = 0;
a)
1
?a , ?q , Jab if ?1 = 0, ?2 = 0;
b)
?a , ?q , Jab , D1 if ?1 = ?2 = 0.
1
c)

Here

Jab = ya ?b ? yb ?a + v a ?vb ? v b ?va , D1 = ya ?a ? v a ?va ? 2q?q .
1

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. 60
( 122 .)



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