A priori estimates of the conjugate problem describing an axisymmetric Thermocapillary motion for small Marangoni number Текст научной статьи по специальности «Математика»

УДК 532.5.013.4

A Priori Estimates of the Conjugate Problem Describing an Axisymmetric Thermocapillary Motion for Small Marangoni Number

Victor K. Andreev* Evgeniy P. Magdenko^

Institute of Computational Modelling SB RAS Akademgorodok, 50/44, Krasnoyarsk, 660036 Institute of Mathematics and Fundamental Informatics

Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041

Russia

Received 28.02.2019, received in revised form 10.03.2019, accepted 12.06.2019 This paper is devoted to the study of equations solution describing the axisymmetric motion of a viscous heat-conducting liquid. The motion is interpreted as a two-layer flow of viscous heat-conducting liquids in a cylinder with a solid wall and a common movable non-deformable interface. From a mathematical point of view, the arising initial-boundary value problem is nonlinear and inverse. Under certain assumptions concerning to apply the problem is replaced by a linear one. As a result, the unimprovable uniform priori estimates for solutions of the problems posed are obtained.

Keywords: a priori estimates, the conjugate inverse problem, interface, Marangoni number. DOI: 10.17516/1997-1397-2019-12-4-483-495.

Introduction

It is well known that a motion can arise in a non-uniformly heated liquid. In some applications of liquid flows, a joint motion of two or more fluids with surfaces takes place. If the liquids are not soluble in each other, they form a more or less visual interfaces. The petroleum-water system is a typical example of this situation. At the present time modelling of multiphase flows taking into account different physical and chemical factors is needed for designing of cooling systems and power plants, in biomedicine, for studying the growth of crystals and films, in aerospace industry [1-4].

The stationary solution of the Navier-Stokes equations describing the 2D motion of a pure viscid incompressible heat-conducting fluids in the absence of mass forces was found for the first time by [5]. It describes the liquid impingement from infinity on the plane under the no slip condition on it. In the paper [6], this solution for the flow between two plates or for the flow in a cylindrical tube was applied.

The monograph [7] presents the results of specific non-stationary motions studies of a binary mixture with allowance for the effect of thermal diffusion arising in sufficiently long plane and

* andr@icm.krasn.ru

1 magdenko_evgeniy@icm.krasn.ru © Siberian Federal University. All rights reserved

cylindrical layers. The properties of invariant solutions of thermal diffusion equations are considered, when the surface tension on the interface of two mixtures depends linearly on temperature and concentration. A generalization of the Ostroumov-Birich solutions to the motion of mixtures in a cylindrical tube is given.

1. The problem statement

We will consider at the following linear conjugate inverse initial-boundary problem

Vlt = V^V!rr + 1 + fl(t), 0 <r<Ri, (1.1)

V2t = v2[v2rr + 1 V2r^j + f2(t), Rl < r < R2, (1.2)

f>R 1 pR2

vi(Ri,t) = v2 (Ri,t), rvi(r,t)dr +/ rv2(r,t)dr = 0, (1.3)

Jo Jr1

livir(Ri, t) — ii2v2r(Ri,t) = —2eeai(Ri, t), (1.4)

\vi(0,t)\ < to, v2(R2,t)=0, (1.5)

v1 (r, 0)=0, v2(r, 0)=0, (1.6)

f (t) = ft) — (1.7)

Rl

and the closed conjugate problem for functions aj(r,t) is described the following equations:

ajt = Xj ^ajrr + 1 ajr^ , (1.8)

aj(r, 0) = a0(r), \ai(0,t)\ < to, (1.9)

a2(R2,t) = a(t), (1.10)

ai (Ri,t) = a2(Ri,t), kiair (Ri ,t) = k2a2r (R2,t). (1.11)

Here ij is dynamic viscosity coefficient (index j = 1,2 is number of the liquid), Pj is the density, « = —da/dO = const, a(9) is the surface tension coefficient, Xj H kj are the coefficients of thermal diffusivity and conductivity, respectively. The functions vi(r,t), ai(r,t) are limited at r = 0 for all values of time. The system (1.1)-(1.11) describes the two-layer motion of viscous heat-conducting fluids in a cylinder with a solid side surface r = R2 = const and the common interface r = h(t), 0 < h(t) < R2.

The problem (1.1)-(1.11) is obtained from general equations describing the axisymmetric motion of a viscous heat-conducting fluid in the absence of mass forces, if the Marangoni number

M = '<soi^/iixi ^ 0. Here ai = max |a(t)|. The function a(t) is bounded in physical sense

te[o,T ]

for all t from the interval [0, T], where T is constant. The constant aiR2 is the characteristic temperature along a solid wall. In this case, the velocity, pressure, and temperature fields are described by the formulas

Uj = Uj (r,t), Wj = vj (r,t)z, pj = pj (r,z,t), Oj = Oj (r,z,t), (1.12)

where Uj (r, z, t) and Wj (r, z, t) are the projection of the velocity vector on the axis r and z of the cylindrical coordinate system, pj (r,z,t) is the pressure that satisfies the relation

— j pj = dj (r,t) — j ) z"2, djr = Vj ^Ujrr +— Ujr--— Ujt — UjUjr. (1.13)

The temperature field is sought in the form

dj (r,z,t) = aj (r,t)z2 + bj (r,t). (1.14)

The temperature has extreme at the point z = 0. It has a maximum at a(r, t) < 0 and minimum

at a(r,t) > 0.

We note that the problem posed is inverse, since together with Vj (r,t), aj (r,t), bj (r,t), h(t) the functions fj (t) should be found. The functions uj (r,t) are determined by the equalities

1 fr 1 rR2

ui(r,t) =-- rv\(r,t)dr, U2 (r,t) = — I rv2(r,t)dr. (1.15)

r Jo r Jr

With known Uj (r,t), aj (r,t), the problem for functions bj (r,t) is separated. The functions dj (r,t) are reconstructed by quadratures from (1.13). The interface is described by the formula

1 f*

h(t) = fli[1 + M hl(t)\, hi(t) = - — rv1(R1,t)dt. (1.16)

R1 Jo

Remark 1. The second equation in (1.3) and the final formula (1.7) allow us to find the pressure gradients along the axis z, i. e., the functions fj (t).

2. Estimates of the function aj (r,t)

Since the function a1(R1,t) is included in the statement of the problem for Vj(r,t), it is necessary to begin with an estimate aj (r,t) satisfying the initial boundary-value problem (1.8)—(1.11). For smooth solutions, the following matching conditions should be satisfied:

a0,(R2) = a(0), a01(R1) = a0,(R1), (2.1)

kia0ir(R1) = k2a°2r(R1), |a0(0)| < to. (2.2)

Perform a replacement

a(t)(r — R1)2

a2(r,t) = a2(r,t)+ R - ri)2 . (2.3)

Then the boundary condition (1.10) for the function a2(r,t) becomes homogeneous. It satisfies the inhomogeneous equation

( 1 ) 2x2a(t) ( R1) a'(t)(r - R1)2 , , , ,

a2t - *2{ a2rr + ^2r) (2 - r) - R - ri;2 - ^ (2.4)

where the prime denotes differentiation with respect to t. Conditions (1.11) for the functions a2 (r,t), a1(r,t) remain unchanged.

We multiply equation (1.8) (j = 1) by p1cpira1(r,t) and multiply equation (2.4) by p2cP2ra2(r,t), where cpj are the specific heat of liquids at constant pressure. Then, integration the equation obtained over the intervals of definition and adding up, leads to the integral equality (kj = pjcp.Xj)

d Ri R2 R2

— A + k1 ra1r dr + k2 r(i2ir dr = ra2g2(r,t) dr (2.5)

dt o Ri Ri

with function

pRi p R2

A(t) = P^ËL ra\(r, t) dr + P^ a(r,t) dr. (2.6)

2 J0 2 JRi

The solutions of equations (1.8) are sought in the intervals: 0 < r < Ri at j = 1 and Ri <r <R2 at j = 2 at M — 0. We have the inequality [8]

r Ri r R2 / r Ri r R2 \

/ ra1 dr + / ra" dr ^ Mo I ki ra\r dr + k2 ra\r dr I (2.7)

0 Ri 0 Ri

with positive constant M0.

We obtain that using inequality (2.7), the left-hand side of equation (2.5) is greater than or equal to

d 1 ( fRi fR2 \

-A+Mod ra2dr+JRi ra2dr). (2.8)

For the right-hand side we have estimate from above

cR2 ( cR2 \ i/2 / rR2 \ i/2

Therefore, from (2.6) we have

L2r

0 Ri

(2.9)

(2.10)

R2 R2 2 i/2 R2 2

ra2 g2 dr ^ / rg2 dr / rat,2 dr

(2 \ i/2 ( iR2 \ i/2

^ max I - ) I rg2 dr VA = G(t)VA.

j \Pj cpJ \Jri J

Further we obtain the inequality from inequalities (2.8) and (2.9)

dA + 2nA < G(t)VA, n = -tV min(—^ ) = -V mini . dt 1 ^ w ' ' Mo j \pj cpJ Mo j \kj)

Then from (2.10) it follows that

A(t) < (yA + 2 jl G(t)enTdr^j e-2nt, (2.11)

here Ao is value of function A(t) at t = 0:

f R1

A(0) = PiCpi I r(ai)2(r) dr I r(a2)2(r) dr, (2.12)

ao(r — Ri) (R2 — Ri)2

Ri r R2

P^ r(al)2(r) dr + ^ I r(a0)2( 2 0 2 Ri

a°(r) = a"(r) - a°(r - Ri) , ao = a(0). (2.13)

R1 2 R2 2

ra2(r) dr < -A(t), ra2(r) dr < -A(t), (2.14)

Jo Picpi Jri P2cP2

where A(t) satisfies the estimate (2.11). Let us prove the limitations of the integrals

r Ri r R2

ra"r dr, ra"r dr. (2.15)

To do this, we raise equations (1.8) (j = 1), (2.4) to the second degree then multiply them by PicPir, p2cP2r respectively, integrate over their domains r and t, we sum the results. We obtain

another integral identity

t /'Ri

PlCPi / r

>0 J0

a\t + XÏ[ alrr + r air

t f R2

+ ki

-ï 1 ï ( - 1 1 -

aït + Xï\ airr +— aïr 10 JRi V r

f'Ri fR2 f'Ri

-ï 1 i„- / „^l J--i„ / „0 \Ï

+ PlCP2 ra2r dr + ki

dr dt +

dr dt +

R2

0

ra^r dr = kW r(a°r)2 dr + k2 I r(aÏr)2 dr + Ri 0 Ri

t c R2 c t c R2

t R2 t R2

+ Pi^ / rgi(r,t) dr dt + picp2 / rgi(r,t) dr dt = Ai(t). (2.16)

0 Ri 0 Ri

From (2.16) the limitation of the integrals (2.15) follows € [0, T]. Further, taking into account (2.14), (2.16), we have the inequality

\(r,t) =

pR2 pR2 1

(a'i)r dr ^ 2 — \/r \ai\y/r \air\ dr ^ Jr JRi r

<

2

R1 \ JR Hence we obtain the estimate

/ f R2 \ ill / f R2 \l/l 4 /

Ui raÏ dV Uri raÏr dV ^

1

Ri \kïpicp

-A(t)Ai(t)

i/i

(2.17)

\ai(r,t)\ < 2I-A(t)Ai(t))

\RikïPïCP2 J

1/4

(2.18)

uniformly for all r G [Ri,RÏ], t G [0,T].

From the replacement (2.3) we find the estimate

\ai(r,t)\ < \a(t)\ + 2Î-A(t)Ai(t))

\RÎkÏ Pi CP2 J

1/4

(2.19)

Remark 2. If we differentiate with respect to time equations (1.8) (j = 1), (2.4) and the boundary conditions, then in the same way we obtain the estimate

\ait(r,t)\ < \a'(t)\ R

1

Rfkipic

-Ai (t)A3(t)

1/4

(2.20)

P2

where the function A2(t) differs from A(t) in that the function a\(r,t), a2(r,t) should be replaced by ait(r,t), a2t(r,t) in formula (2.6). Similar the function A3(t) differences from A\(t). Also,

\aitt(r, t) \ < \a''(t)\ + 2 (-A4(t)A5(t))

\RtkïPïCP2 J

/4

(2.21)

Here the function A4(t) differs from A(t) in that, in (2.6) their second derivatives with respect to time from functions a 1 (r,t), a2(r,t) are instead of this functions. The function A5(t) is quite similar to A1 (t), however, g2(r,t) from (2.4) contains second derivative a''(t) instead of function a(t), and third derivative a'''(t) instead of function a' (t). The estimates for (2.20) and (2.21) are used later in step 4, where also the expressions for Ak (t), k = 2, 3, 4, 5 are given. Here we note that these functions are continuous for t € [0, T], if the function a(n) (t), n = 0,1, 2, 3 are also continuous in the same interval. Moreover, A2(t), A4(t) satisfy inequalities like (2.11) with their functions G(t).

ï

ï

However, such arguments are not suitable for estimate |ai(r,t)|. From inequality (2.19) and first equality (1.11) we have the estimate

lai(Ri,t)| = la"(Ri,t)l < |a(t)| + 2\

1

Rk P2Cp2

-A(t)Ai(t)

i/4

For ai(r,t) we obtain the problem at 0 < r < Ri

ait = Xi (airr + - ai^ , ai(Ri,t) = a2(Ri,t), |ai(0,t)| < œ, ai(r, 0) = ai(r).

(2.22)

(2.23)

(2.24)

(2.25)

The initial boundary value problem for equation (2.23)-(2.25) at given ai(Ri,t) with esti-mate(2.22) has the solution [9]

2xi ft œ

ai(r,t) = -^r ai(Ri,T)E:

Ri Jo n=i

2 r Ri œ

+ W Za0(Z

Ri Jo n=i

£nJo(£nr/Ri)

Ji(n

■ exp

Xi£(t - T)

Jo (£n r/Ri)Jo(tnC/Ri)

J2(n

exp -

Ri

Xiti t

dT+

(2.26)

where are the roots of equation J0(£) = 0. From estimate (2.22) and from formula (2.26) there is the limitation \ai(r, t)\ for all r G [0, Ri] and t G [0, T]. Indeed, the first term in (2.26) is less than or equal to

2

1

max |a(t)| + -—k

te[o,T] 1 (R2kP2cp2)i/4 te[o,T]

max (A(t)Ai(t))i/4

Under derivation of expression (2.27) we use inequality (2.22) and the relation [10]

Zjg^- 0 <'<*■

Second summand in (2.26) does not exceed

max |ao(r) I.

re[o,Ri] i

(2.27)

(2.28)

(2.29)

We have proved

Lemma 2.1. The solution of the initial-boundary value problem (1.8)—(1.11) is limited for all r G [0, Ri] (j = 1) and r G [Ri, R2] (j = 2), and t G [0, T]. The estimates are the following

a^t) < 2

max ^(t) + 77^—1-rU4

te[o,T]< y (R2k2P2Cp2)i/4 te[o,T]

^t) < at) +2 (R

1

R2lk2P2C

A(t)A1(t)

max (A(t)A1(t))1/4 e[o,'T11

1/4

+ max ao(r) , re[o,Rir

(2.30)

P2

with function A(t), Ai (t) from (2.6) and (2.16) respectively.

3. Estimates of the function Vj (r,t)

We turn to obtaining a priori estimates of the functions vj (r,t) that satisfy the equations (1.1), (1.2), boundary conditions (1.3)-(1.5) and initial data (1.6). In order to make boundary condition (1.4) homogeneous, we make the change of the function v2(r,t)

< +\ - < +\ 2ma 1 (R1,t) r, ^ (31)

V2(r,t) = V2(r, t)--P4 (r). (3.1)

M2

The polynomial of the fourth order P4(r) satisfies the following conditions: 1) P4(R1) = 0,

R 2

P4(R2) = 0; 2) dP4/dr =1 at r = R1; 3) J rP4(r)dr = 0. We take

Ri

P4(r) = D2rj? 1 DJr2 - (Ri + R2)r + RiR2)(r2 + Cir + C2) (3.2)

R1 (Ri - R2)

with constants

C (Ri + R2)(2Ri + 2R2 + R1R2) C RC

Ci =--(R2 - Ri)(3R2 +2Ri) , C2 = -RiCi (3.3)

We explain the construction of the polynomial P4(r) briefly. The polynomial P4 = g(Ri, R2)x x (r-a)(r-b)(r2 +C1r+C2) follows from the condition 1). Selecting g(R1,R2) = [R\(R1 -R2)]-1 from condition 2) we obtain the equality R1C1 + C2 = 0. The another equation for C1 and C2 follows from equality 3):

(3R2 + 3R2 + 4RiR2)Ci + 5(Ri + R2C = -(Ri + R2)(2Rj + 2Rj + R2R1),

whence the formulas (3.3) can be obtained.

When change (3.1) is used, boundary conditions (1.3), (1.5) remain the same (of course, we should use the function V2 instead of function v2). The equation for the V2(r, t) is inhomogeneous and has the form

( 1 \ 2v2 » ( 1 \ 2»

V2t = V2\ V2rrr + - V2rr--ai(Ri,t) P4rr + ~ P4r +--aU(Ri,t)P4 (r) + f2(t) =

V r J P2 V r J P2

= V2 (v2rrr + ^V2rr^ + f2 (t) + Q2(r,t).

(3.4)

Taking into account the second condition in (1.6) we find the initial data for the function V2:

2»

V2(r, 0) = — a0i(Ri)P4(r) = V%(r). (3.5)

P2

We multiply equation (1.1) by rp1v1, equation (3.4) by rp2v2, integrate them through domain and sum the results. We obtain

dE rRl 2 J rR2 2 J rR2 ^ J 2mai(Ri,t) fRl , —- + Hi l rvirdr + H2 rv<2rdr = P2 j rv2Q2dr---- rvidr,

dt J0 JRi JRi Ri Jo

P fRi p R2

E(t) = ~i rv^dr rv\ dr. (3.6)

2 o 2 Ri

The first condition in (1.3)-(1.5), and equality (1.7) are used to derive identity (3.6).

The left-hand side of (3.6) is greater than or equal to

dE 1 dt + M

( i R1 2 f R2 2 \ / rv^dr + rv<2r dr

(3.7)

with the constant M = R2(iixX0)-i, where x0 is the smallest positive root of the described above transcendental equation (2.5) with y2 = \Jli/l2 = /|. The right-hand side of (3.6) does not exceed

i/2

iR2 \ 2 — y/2P2\ R rQ2dr\ + — \ai(Ri,t)\

From (3.7) and (3.8) we obtain the inequality

y/Et).

(3.8)

dE + 2SE < 2/E dt

(RrQldr]

1/2

+

m

yfpi

\ai(Ri,t)\

= 2\fEHi (t) (3.9)

with the constant 6 = M i min(P- , p- ). From (3.9) we find the estimate of function E(t):

E(t) <

+ f Hi (r)eSrdr

o

„-2St

(3.10)

where, according to the first equality in (1.6) and (3.5),

R2 2

P fR2 2—2 P R

E(0) = rv20(r)dr =-P2 (a0(Ri ))2 rP^(r)dr. (3.11)

2 JR1 P-2 J R1

Thus, the estimates of the quantities vi and v2 in L2-norms Vt G [0,T] follow from (3.10) where a(t) and a (t) are given.

To evaluate the derivatives of vir, v2r in L2-norm we multiply (1.1) by rPivit and equation (3.4) by rP2v2t, integrate by their domains and sum up the results. We obtain identity

PU vltdr + P2 vlt

o R1

d + — d

v2tdr + 2 dt

R1 R2 li rv2rdr + 12 i rv\rdr

o R1

i'r2 2m fR1

P2 rv2tQ2dr — — ai(Ri,t) rvitdr,

R1 Ri o

where Q2(r,t) is defined in (3.4). We estimate the right-hand side of (3.12)

P2 i -2

2

rv,

P2 i2

m

2

2td

dr + TT rQ2dr + -— ax(Ri,t) + Pi rvudr.

2

R1 2 R1 2P1

Therefore, from (3.12) we obtain the required inequality

R2

R1 R2

■•2r dr + 12I rv\r

o R1

l i rv lr dr + 12

dr < 12 I r(v°r) dr+ R1 t (■ R2

2

P t R2 m2 t + — rQ2 (r,t)drdt +— a2(Ri,t)dt = H2(t), (3.13)

2 J0 JR1 Pi JO

(3.12)

2

R

R

2

R

o

whence the limitation of the derivatives v1r, v2r follows in L2-norms Vt £ [0,T]. Also, as in 2., we have

2 ( f R2 \1/V , R2 2 V/2

/ rv^dr / rv^r dr

\Jr( 2 ) \JR, )

(T f (^->y

<

" Ri

2

1 \ Jr1 J \ JR

2 ( H2(t> \1/2 ( 2

v2 (r,t> = —2 I V2v2rdr ^ r- ^ J rv2,dr j ^ j rv^rdr ) ^

1/2 / 0 \ 1/2

Thus, Vt £ [0,T], r £ [R1, R2] the estimate is valid

rr ( 2 \1/ 4

\V2(r,t>\ W — -H2(t>E(t>) , (3.14)

V R1 \P2P2 J

where H2(t> is the right-hand side of inequality (3.13) and the function E(t> is estimated by expression (3.10). Taking into account the replacement of (3.1), we obtain the estimate

py~ ( 2 \1/4

\v2(r,t>\ < — K—1 ,t>\ max \P4(r>\ W— -H2(t>E(t>) . (3.15)

P2 r€[Ri,R2] V —1 \P2P2 J

Similarly, differentiating problem for v1(r,t>, v2(r,t> by t, we get a priori estimate of form (3.15):

( 2 \1/4

\v2t(r,t>\ < — \a1t(—1,t>\ max \P4(r>\ W— -H3(t>E1(t> , (3.16)

P2 re[RuR2] V —1 \P2P2 J

where E1(t> is different from E(t> that v1, v2 should be replaced by v1t, v2t; H3(t> is different from H2(t> that a1 (R1,t>, a1t(R1,t> should be replaced by a1t(R1,t>, a1tt(R1,t>, respectively. These estimates, in view of the first equality in (1.6) and the inequalities (2.20), (2.21), are already known.

To estimate \v1(r,t>\, we proceed as follows. We consider the problem

v1t = V^[v1rr + + f1(t>, 0 <r<—1, (3.17)

v1(R1,t>= v2(R1 ,t>, \v1(0, t)\ < to, v1(r, 0>=0, (3.18)

assuming v2(R1,t> to be known, satisfying estimate (3.15). The solution of this problem is given by formula [11]

, 2v1 ^ J(U/—1> r m , ( »¿I(t — t> N 1(r,t> = —1 ^ J1(tn> Jo v2(—1,T>eXH--—1 )

v1(r,t> = ^^™ T::\ I v2—1,T>exp( — ^^-^ )dT+

n=1

2/

2 ^ Jo(Znr/R1> f V1&(t — t >\j („10)

+ Jo f1(T >eXP{--— JdT, (3.10)

£n are the roots of the Bessel function J0(£n>=0.

Note that f1(t> is unknown; we find its relationship with v1. Multiplying the equation (3.16) by r and integrating from 0 to R1, we find

2 d fRl 2 fr2

f1(t> = —2v1v1r (R1,t> + rv1dr = —2v1v1r (R1,t> — rv2tdr, (3.20)

R1 dt J 0 R1 J R1

where the second equality (1.3) is used. In equality (3.20) we do not have an estimate for the first summand, the second can be estimated using inequality (3.16). We differentiate equation (3.17) with respect to r and introduce a new function V(r,t) = v1r (r,t). We obtain the equation for it following:

1 1

Vt = vW Vrr + -rVr - ^ V

(3.21)

where \V(0,t)| < ro. The second boundary condition for V is found from the consideration of the integral

cRi fRi fRi

r2Vdr = / r2vir dr = R2vi(Ri,t) — 2 rv\dr = ) J 0 J 0

/■ R2

= R2v2(Ri,t) + 2 rv2dr = g(t) J Ri

(3.22)

with known a priori estimate function g(t), g(0) = 0 (we use the second equation applied (1.3). We change the function V:

V(r, t) = V(r, t) + (V — 6 Rir3^ g(t).

(3.23)

The problem for V(r, t) takes the form (it is nonclassical)

Vt = vi (Vrr + 1 Vr — v) + v^ 15r2 — yRi^j g(t) + Rir3 — r4) gt(t), (3.24) _ r Ri _ _

V(r, 0)=0, r2Vdr = 0, |V(0,t)\ < to. (3.25)

Lemma 3.1. Solution (3.24), (3.25) has form

v (r,t) = £ V n(t)jJ R-Y

n=1 ^ 1'

where Zn are the positive roots of equation J2(Z) = 0 and we obtain

V n(t)= g(t)h2n + (vihl — h

R2

0t

g(r)exp

f (t — T )

dr.

where hn and hn are defined by formulas

RiJ'l(Zn)

/'(15'-3—? Ri ) R)d

C (6 Rr—'l) Ji ( Ir)

Proof. Since the formula is valid [9]

f rkJk-i(r)dr = zkJk(z), ■J 0

(3.26)

(3.27)

(3.28)

here k > 1 is integer. Then the solution of problem (3.24), (3.25) should be searched for as a Fourier series (we have k = 2) (3.26), since [12] ZJ1(Z) — J1(Z) = -ZJ2(Z), the roots of the

0

i

h

n

2

h

n

equation J2(Z> = 0 are the roots of function (J1((> — J1((>. The presentation of (3.26) takes place [12].

Substitution of (3.26) in (3.24), (3.25) leads to the Cauchy problem _ v z 2___

Vnt + V n = V1g(t>hn + gt(t>h2n, Vn(0> = 0, (3.20)

—1

where h^, hhn are coefficients of Fourier series of functions 15r2 — 48R1r/7, 6R1 r3/7 — r4 respectively, defined by formulas (3.28). From (3.20) we find that the solution of problem (3.24), (3.25) has form (3.27). □

Taking into account replacement (3.23), we obtain an expression for the function v1r(R1,t):

1 4 1 4 2

vlr(Ri,t) = V(Ri,t) + 7 R\g(t) = ^ Ri + £hlMCn) j g(t)+

+ it^vihi - h^ Ji(Zn) 0 g(T)exp

iRt (t - t )

dT. (3.30)

The series in (3.30) and series (3.26) converge uniformly. Really we have

= JiT (15-- 7 *) ¿H R r) )*

rjÛT Mk')dr' (3-31)

hn=Jrd* ' ("4 - 7RJ2-{ k r)dr (332)

Since the equality Jk-1(z) + Jk+1 (z) = 2kz-1Jk(z) is valid [9], so we obtain J3((n) = — J1(Zn) (we recall, that J2((n) = 0) and so the expression for the function hn and h2n have following the form

h1 = h2 = hn = » , hn = » , Zn Zn

where ^, ^ are coefficients of Fourier series of functions — 15R1r and 3R1(r3 — 4R1r2/7) when

œ

they are decomposed by function J2(R'-1Çnr). The series J2 (j)2 converge, and then, by virtue

n=1

œ œ

of inequality \hJn\ < 2-1 [(j)2 + 1/ZV\, the series \hJn\ also converge. The series Z-2

n=1 n=1

converge, since (n ~ nn at n ^ 1. Moreover we obtain [11] J2 1/n2n2 = 1/12.

n=1

As for the second term in (3.30) its is less or equal then to

R21( f + f)m

n=1

with obviously convergent series.

Remark 3. In the preceding arguments we used the well-known inequality \ Jk (z) \ ^ 1, where the constant k > 0 is integer [9].

Remark 4. Following the monographs [10,12], we can show that the function V(r,t) is the sum of series (3.26) and it has derivatives of all orders on r and t at t ^ e > 0. In particular, the solution of problem (3.24), (3.25) is classical.

From (3.16), (3.20) and (3.30) we find estimate fi(t) at t e [0,T]:

'K\ , \h2n\

\fi(t)\ < 2vi

~R\ +

OO \ CO /

IXl +2r2Y,( + R2

n=i ) n=i \

z

R2

max I g (t ) I

te[0,T] 1

+

22 R2 — Ri

Ri

2œ

— max \ait(Ri,t)\ max \P4(r)\ +

m te[0,T] m u re[Ri,R2i

2 f 2 \V4 + \l— max [ -H3(t)Ei(t)\

Ri te[0,T]\p2m2

The estimate f2(t) follows from (1.7)

lf2(t)\ < p\fi(t)\ +

2œ

P2Ri te[0,T]

max \ai(p2Ri,t)j,

(3.33)

(3.34)

where p = p1/p2 and f1 (t) satisfies inequality (3.33).

Thus, the functions fj (t) are bounded and continuous at t e [0, T].

The limitation of the function v1(r,t) at r e [0, R1], t e [0,T] follows from it representation in the form (3.19) (the equality (2.28) is used)

2Ri O 1

\vi(r,t)\ < Ri max \v2(Ri,t)\ +-- max \fi(t)\V

te[0,T]< vi te[0,T] ^ \Ji(&

(3.35)

where the functions modules \v2(Ri,t)\, \fi(t)\ satisfies estimates (3.15) and (3.33). The series in (3.35) converge, since Çn ~ nn, \ Ji(Çn)\ ~ 1/\fn at n ^ 1. Thus, we have

Theorem 3.1. The solutions of the initial-boundary value problem (1.1)—(1.7), (1.15) vj(r,t) and the function fj(t) are limitation for all r G [0, RjJ (j = 1) and r G [Ri, R2] (j = 2), and t G [0,T]. Estimates (3.15), (3.35) are valid for function vj (r,t) and estimates (3.33), (3.34) take place for function fj (t).

Conclusion

One partially invariant solution of the equation describing the axisymmetric motion of a viscous heat-conducting liquid is studied. As a result, the unimprovable uniform priori estimates for the solutions of the conjugate problem posed are obtained for small Marangoni number.

References

[1] V.K.Andreev, V.E.Zahvataev, E.A.Ryabitskii, Thermocapillary Instability, Novosibirsk, Nauka, (2000) (in Russian).

[2] A.Nepomnyashii, I.Simanovskii, J.-C.Legros, Interfacial Convection in Multilayer System, New-York, Springer, 2006.

[3] R.Narayanan, D.Schwabe, Interfacial Fluid Gynamics and Transport Processes, Berlin, Heidelberg, New-York, Springer-Verlag, 2009.

[4] R.Kh.Zeytovnian, Convection in Fluids, Dordrecht, Heidelberg, London, New-York, Springer, 2009.

[5] K.Hiemenz, Die grenzschicht an einem in den gleichformigen Flussigkeitsstrom eingetauchten graden Kreiszylinder, Dinglers Polytech. J., 326(1911), 321-440.

[6] J.F.Brady, A.Acrivos, Steady flow in a channel or tube with an accelerating surface velocity, J. Fluid Mech., 112(1981), 127-150.

[7] V.K.Andreev, N.L.Sobachkina, The motion of a binary mixture in planar and cylindrical regions, Krasnoyarsk, Siberian Federal University, 2012.

[8] V.K.Andreev On the Friedrichs inequality for compound domains, J. Sib. Fed. Univ. Math. and phys., 2(2009), 146-157 (in Russian).

[9] G.Bateman, A.Erdein, Higher transcendental functions. Bessel functions, parabolic cylinder functions, orthogonal polynomials, Moscow, Nauka, 1974 (in Russian).

[10] S.G.Mikhlin, Linear partial differential equations, Moscow, High school, 1977 (in Russian).

[11] A.P.Prudnikov, Y.A.Bychkov, O.I.Marichev, Integrals and series. Special functions, Moscow, Nauka, 1983 (in Russian).

[12] V.S.Vladimirov, Equations of mathematical physics, Moscow, Nauka, 1976 (in Russian).

Априорные оценки сопряжённой задачи, описывающей осесимметричное термокапиллярное движение при малых числах Марангони

Виктор К. Андреев Евгений П. Магденко

Институт вычислительного моделирования СО РАН Академгородок, 50/44, Красноярск, 660036 Институт математики и фундаментальной информатики Сибирский федеральный университет Свободный, 79, Красноярск, 660041

Россия

Данная работа посвящена исследованию решения уравнения, описывающего осесимметричное движение вязкой теплопроводной жидкости. Оно интерпретируется как двухслойное движение вязких теплопроводных жидкостей в цилиндре с твёрдой стенкой и общей подвижной неде-формируемой поверхностью 'раздела. С математической точки зрения, возникающая начально-краевая задача является нелинейной и обратной. При некоторых (часто выполняющихся в практических приложениях) предположениях задача заменяется линейной. Для неё получены априорные оценки.

Ключевые слова: априорные оценки, сопряжённая обратная задача, поверхность раздела, число Марангони.