On the equations of steady-state convection

ON THE Er)UATIONSOF STEADY-STATE CONVECTION (OS URAVNENIAKA

STATSIONARNOI

PMM Vo1.27, M.R.

No.2,

1. It

December

is known that a heated field

only if

pp.

1963,

UKHOVSKII and V.I. (Rostov-on-Don)

(Received

tational

KONVEKTSII)

fluid

295-300

IUDOVICH

18,

1962)

can be in equilibrium

the temperature

distribution

T(x)

in a gravi-

in it has the

form

T = To (5) = cx, +-co where C and Co denote constants, downwards. However, for is

example,

a stable

steady

the case which we shall

the type just scribed

the axis

the preceding

mentioned,

flow of the fluid

called

notation

vertically

can turn out to be unstable: can come into

examine in the present steady-state

paper.

being.

The flows

natural

convection,

xAT’

= vlvTf,

div v’ =

v’(x)

is

‘lhis of

are de-

[l]

VAV’ = (v’ *V) v’ + Op ’ -+- &T’, where the following

x3 being directed

solution

by the system of equations

(1.1)

is used:

the velocity

of

0

(1.2)

the fluid;

space; T’(x) is the n= (+ x2, x,) is a point in the three-dimensional temperature; p’(x) the pressure; v, x, p, respectively, are the viscosity, thermal conductivity and thermal expansion; g(0, 0, g) is the acceleration due to gravity; the density of the fluid has been set equal to unity. We shall

seek a solution

(v’,

bounded domain Q which satisfies

vo =

0,

7”)

at its

of the system (1.2)

in a

boundary S the set of conditions

T’ Is =cx, + co

V’IS = 0, Tt is evident

p’,

(1.3)

that

T,, = Cs, + Co,

p.

=

-

432

pg ($

Cz,g +

Co&

+

corn%

(1.4)

On the

solves

the problem

v’ = in (1.2) ~Av

(1.2),

(v .V)

of

(4.5)

T’ = T,, f T

we have

v +

with

433

the change of variables

PI = po -I- pg

VP

+

xAT

NT,

?J1s = 0, Together

convection

steady-state

(1.3):Performing

$- v,

VJ

and (1.3),

=

equations

the preceding

= vVT + Cvl, TIs=O

problail,

we shall

div v = 0 (1.6) (1.7)

consider

the linearized

problem vAv

(1.8)

Vp + fIgT,

=

We shall trivial

ask for

solutions

we shall

xAT

div v = 0;

= CvI,

those values

of the problem

reduce the problem

of the parameter

C which lead

(1.6)

In order

(1.6)

in some Hilbert solutions

space H,c The operator to apply

Theorem

operator,

the following

1.1.

theory (1.9).

is

and this

gives

of the bifurcation The system (1.8)

of reduces

acts

The operator

and a corresponding 1.2.

(1)

(1.10)

CBv

the Frechet

differential

of operator

K.

can be established.

R is

selfadjoint,

continuous. It follows that there exists istic values of equation (1.10) (problem

Theorem

equation

(1.9)

I( is continuous,

the general

v = Furthermore,

to the operator

equation

a linear

B,

to non-

to do this

C)

[21 to the study of equation

to the operator where

and (1.7).

and (1.7)

v = K (v us the opportunity

T 1s = 0

v Is =O,

positive

a denumerable (1.8))

complete

system of eigenvectors

Equation

(1.6)

(problem

(1.9),

and completely set of dharacter-

vl,

vpr

(1.7))

...

with C < C,

possesses only a trivial solution. (2) ‘lhe points of bifurcation can to which there only be C,, C,, . . . . (3) Every number C,(k = 1, 2, . ..) corresponds a non-denumerable set of characteristic vectors of equation (1.10)

constitutes

Section ‘Theorems

4 of this

a genuine point paper

contains

of bifurcation

of equation

an example of the application

(1.10). of

1.1 and 1.2.

Sorokin

[3,4]

devotes

to

the

study

of

convection

in

a viscous

fluid.

Reference [31 establishes a variational principle which is satisfied by the eigenvectors and eigannumbers of problem (1.18,. and a Ritz method for their approximate calculation is proposed. However, no proof of the

434

M.R.

Ukhouskii

and

V.I.

Iudovich

existence of the eigenvalues has been given. In the process of proving Theorem 1.1 we derive an alternative variational principle (and an entirely different variant of Ritz’s method). The justification of the variational principle and of the Ritz method given in [31 can be carried out by a method which is closely related to that given here. A proof (1) of Theorem 1.2 is given in 141. In addition, it is proved that uniqueness occurs also for C = Cl, but the proof contains an error*. In spite of this, [41 establishes formal expansions of the solutions to the problem of convection into series of a special kind. 2. In order to derive the operator equations, we shall adopt a method similar to that developed in [51. We define a Hilbert space H, as a closure of a set of smooth solenoidal vectors in space Q which vanish near the boundary S of a norm given by the scalar

s

v dx

rot usrot

(ii, +I, =

(2.1)

n

Let H, be the closure of a set of functions smooth in the domain Q which are equal to zero on the boundary, of a norm given by the scalar product

(f, gh = 1 of. VJg dx cl We shall call the pair integral identities

Y

(v,

(v, T), v EH~,

T E

s

(v. V) vQ, dx -

cD)s, = -

a generalized solution and are arbitrary.

-

1

n

e7Wx--Cl

of problem

H,, which satisfies

the

PiTg@dx

(2.3)

n

n x (T,cp)a,=

(2.2)

(J.6),

(1.7).

0

v,qdx Here OEH,,

(2.4)

q EH2,

From the results in [61 it is easy to prove that the above generalized solution is infinitely differentiable everywhere inside Q and if the boundary of S is sufficiently satisfies equations (1.6) to (1.7);

l

In (2.10) and (1.5) (see [4, p. ent quantities. For this reason is necessary to put aq and the validity. The result itself is,

1991) the symbol P refers to differon the right-hand side of (2.11) it subsequent statements lose their evidently, correct.

On the equations

of steady-state

smooth, then derivatives of an arbitrary, tinuous* in the closed domain Q. Lenuna 2.1.

It

follows

from (2.4)

convection

prescribed

435

will

order

be con-

that

T = CAv

(2.5)

where A denotes an operator operating from H, into Hz which is uniformly continuous and independent of C. For an arbitrary function f EL,,,(Q) we define an operator Lf by the integral identity

(2.6) for all 9 EH2. App 1ying Sobolev’ s canposition theorem [81, we find that the right-hand side of (2.6) represents a linear functional with regard to 9 in H,. It follows from Riesz’ theorem on the general form of a linear functional in a Hilbert space that the operator L is determined; L denotes a linear operator fran Lp(Q) (p >6/5) in Hz which is mmfrom (2.4) that pletely continuous for p > 6/S. It follows

T = LIT = L,T -

&T

CLv8,

= -

L (vVT)

(2.7)

For a fixed v EH1 the operator Lv is completely continuous in H2. We shall prove that for C = 0 (2.7) implies T = 0. Indeed, multi lying (2.7) scalarly by T and applying (2.6) we find at once that 11T i H, =O. According to Fredholm’s theorem it follows further the operator E - Lv exists, and (2.5) is satisfied

that the inverse for

of

Av = - (E - L,)-1Lv3 Let now vn - v in H, for n - 0~. We assume T, = CAV, and prove that T, 3 T=CAv in H,. We write (2.4) for T,,,, vm and T,, vn. Evaluating these relations one from the other and substituting cp = T, - T, we obtain

xllTm--T,II& =\

(vn--Y,)VT,n(Gv-T&ix

-

n -

l

Applying

general solved

C

s n

(v-

-

v,J (T,

-

Z’,,) dz

the methods of E5.61, it is possible to prove problem involving two nonhomogeneous equations “in the whole” (see also [T]).

(2.8)

that a more (1.6) can be

436

M.A.

Ukhovskii

Estimating the +&t-hand equality and the composition x IITm -

T, 11~~ <

and V.I.

Iudovich

side of (2.8) with the aid of ft3der theorem, we obtain

M (11 vm -

vfi IIL,1 II Avm

I/IS, + C IIvm -

vn

in-

It,)

(2.9)

where jw is a constant” which depends only on domain 8. Now, front (2.4) with g, = T, we can easily find that (2.10)

IIAv1l.S G CM1 IIV llr, Making use of the property of complete of ii, into 4 (p < 61, we find that itTN-

T7&3*-““+~

continuity

for m,n.+

of the composition

(2.11)

co

It follows that the continuity lemma is thereby proved.

of

the operator

A is uniform.

We now define an operator KF (p > 6/S) through the identity

for

an arbitrary

vector

v (X, F U+I, =

s

F(x) ELp(ftf

(0 E HS

FEZ,dx

Ihe

(2.12)

n In the same way as was done earlier possible operator.

in relation to operator L it that K, is a linear and completely continuous

to establish

Lemma 2.2.

Every generalized solutian (v, T) of problem (1,6), (2.3), (2.4) satisfies the operator equations

is

(1.7)

in the sense of

v = K (v, C), Kav =

-

(2.13)

K (v* C) = KaY _t C&v

x1 (v, v) v,

Kav =

where I$ and K, are completely continuous tion (v, 7’) of the system (2.f:~) to (2.14) sofution.

-

XI, (BgAv)

(2.14)

in II,. Conversely, every soluconstitutes a generalized

llna justification for Lemma 2,2 follows iaxnediately from Lenma 2.1 and the property of complete continuity of operator I(,.

1

In what follows. pend exclusively

1, hli denote consecutive on tbe domain R.

constants

whose values

de-

equations

On the

Let us first problem fies

(1.8)

determine

of

the generalized

in a form of the pair

the integral

(v,

of the linearized

v EH1,

T E H, which satis-

‘Ihe quest

n (cpE Ha)

for

(2.15)

(2.15)

X(T,cp)a,=-C\v,cpd~

”

PEHl)

Lemma 2.3.

solution 77,

431

identities

v(vA%f,=-8\TB@dr,

in the sense of

convection

steady-state

the generalized

solutions

of problem

to the solution

is equivalent

(1.8)

of the operator

equation (2.16) where the operator constitutes

on the right-hand

a Fr6chet

v = 0. ‘Ihe proof

differential

of this

lenma follows

tors K,,

L and some simple estimates

omit for

the sake of brevity.

3. In order

to prove ‘lheorem 1.1,

2.2 and 2.3,

it

self-adjoint

and positive

‘Ihe fact

side

is sufficient

that R is

and of the following v, w E H,

= 5

\

is completely

of operator

continuous

from the determination in an obvious

we observe

to prove

and

Cl at the point

K(v,

of opera-

way which we shall

that

in view of Lemmas

that operator

Kl@&v,)

hr =

is

in HI.

selfadjoint

is a consequence of

chain of equations

Lv,w, dx = ‘9

(2.6)

which are satisfied

and (2.12) for

arbitrary

(L us),Lw~)H, = (v, Bw)~,

(34

a It

follows

from (3.1)

with the substitution

(Bv, V)H, = ‘Ihis means that the operator Let C, denote the smallest and let

(vr,

T,,

p,)

be its

!+I[&//&>0

is positive. eigenvalue by vl,

(3.2) Theorem 1.1 has been proved.

of the linear

corresponding

plying the first equation (1.8) T, and integrating, we obtain

w = v that

characteristic

problem (1.8), solution.

and the second equation

(1.8)

h%ltiby

(3.3)

4%

WA.

Ukhovskii

and

V.I.

Iudovich

On the other hand, as shown in [41, equations the Kuler equations

of the variational

-

will

constitute

(3.4)

6 = v,T dx

(1.8)

problem

divv

1,

= 0

(3.5)

This means that if there exist values v = v+, T = T+ which render the functional (3.4) a minimum subject to the conditions (3.51, then in the solution of probconjunction with some P = I’+ they will constitute lem (1.8)

for some C = C+. It

is clear

that

vIlv+II~*+XII~+li~,~~in

vllvll&+wu&

” + ‘+= - f0 v*+T+dx Taking into account (3.6) leads to

(3.3)

-%

and the fact

u~T dz!

that C, is a minis,

(3.6)

equation

(3.7)

first (v, p, 7’) be an arbitrary, problem (1.6) which corresponds

Let

linear

the same way as in connection

c=

non-trivial solution to sane C. Proceeding

with the derivation

vU4&+xlI~

II&

v8T dx

Finally,

frcm (3.7)

and (3,8),

we obtain

-

of

(3.31,

I%

of the nonexactly in we obtain

(3.8)

that C, \< C.

of Sections 2 and 3 follows from the results obtained [21 on the basis of Lemmas 2.2 and 2.3. The theorem

The validity by Krasnoselskii

has been proved completely. 4.

In general

eigenvalues operator B, a given

it

to establish the multiplicity of the in view of the self-adjointness of of problem (1.81, i.e. the number of characteristic functions which corresponds to

eigenvalue

is

impossible

cannot be determined.

We now adduce an example for which this the preceding

considerations

of bifurcation We shall

lead

can be done, and for which

to the demonstration

of the existence

points.

seek solutions

v,

I’, p of system (1.6)

which are periodic

On the

equations

of

steady-state

439

convection

in x1, x2 and .x3 with periods 2a, 26 and By which render UJ denumerable with respect to x2, x3 and non-denumerable with respect to nJ; v2 is denumerable with respect to x2, x3 and non-denumerable with respect to x3; U3’ T are denumerable with respect to x1, n2 and non-denumerable with respect to x3, F is denumerable with respect to xJ, x2, n3. It is not difficult to show that ‘lheorans 1.1 and 1.2 are satisfied also in this case because the requirement of periodicity completely replaces the preceding boundary conditions. We shall lem reduces

now remark further that the corresponding linearized itself to the determination of 7’ from the equation AsT = ‘g

if v and ]? are excluded, icity and denumerability. 5 kmn

conditions of periodproblem are

k$

ps -(~+~+~,‘(~+~)-1

(k*+m*#O;

to which there

(4.1)

+ T-)

subject to the preceding ‘lhe eigenvalues of this vXn4

=

(T,,

prob-

(4.2)

k,m=O,l,

correspond

2,.

2, . ..)

..; n=l,

the characteristic

functions

T k,,,,, k cos

(4.3)

We remark that A,_,, >/v *x4&g = A,, the preceding solution of problem (1.6) of a, 6 and y. If to the eigenvalue characteristic function,

and this means that for C < A,, does not exist for any values

hknn there were to correspond more than one then AA,,,,, = Ak R n for some other choice of

a and 6. 7hen y Aa: &e uniquely expressed in that the multiciplim, R, k,, ml, ral, a and 6. It is clear city of such gammas as correspond to all possible choices of k, m, n, kl, ml and nJ cannot be higher than denumerable. llence, for arbitrary values y, except perhaps for some denumerable set, all eigenvalues will be simple and each of them will constitute a point of bifurcation, k 18 ml,

nl. terms of k,

Let us fix

BIBLIOGRAPHY 1.

Landau, L.D. and Lifshits, E.M., hlekhanika of Continua). GITTL, 1954.

2.

Krasnoselskii, integrai’nykh linear

Integral

M.A.,

Topoligicheskie

uravnenii Equations).

(Topological

netody

sploshnykh

v teorii

Methods

GITTJI, 1956.

sred

in

(Mechanics

nelineinykh the

Theory

of

Non-

M.R.

440

Ukhovskii

and

V.I.

Iudovich

3.

Sorokin. V. S., Variatsionnyi metod v teorii konvektsii method in convection). PMMVol. 17, No. 1, 1953.

4.

Sorokin.

0 statsionarnykh

V. S.,

dvizheniiakh

vaemoi snizu (On steady-state motions low). PMUVol. 18, No. 2, 1954. 5.

(Variational

v zhidkosti,

of a fluid

Vorovich, I. I. and Iudovich, V.I., Statsionarnoe zhidkosti (Steady motion of a viscous fluid). S!%SR Vol. 124, No. 2. 1959.

podogre-

heated

techenie Do&l.

from be-

viazkoi

A&ad.

Nauk

8.

Vorovich, I. I. and Iudovich, V.I., Statsionarnoe techenie viazkoi neszhimaemoi zhidkosti (Steady motion of a viscous incompressible fluid). Matem. sb. Vol. 53, No. 4, 1961.

7.

Ladyzhenskaia, maemoi

0. A.,

zhidkosti

Incompressible

8.

Sobolev, natenat in

Nekotorye

icheskoi icaZ

primeneniia

jizike Phys

(Some its).

voprosy Problens

Fizmatgiz,

Flows),

S.L.,

Mathemat

Matematichcskic (Mathematical

the

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neszhi-

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Analysis

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Translated

by

J.K.