Axisymmetric thermocapillary motion in a cylinder at small Marangoni number Текст научной статьи по специальности «Математика»

УДК 517.532

Axisymmetric Thermocapillary Motion in a Cylinder at Small Marangoni Number

Evgeniy P. Magdenko*

Institute of Computational Modeling SB RAS Akademgorodok, 50/44, Krasnoyarsk, 660036

Russia

Received 21.04.2015, received in revised form 05.05.2015, accepted 20.06.2015 The solution to the linear problem of axisymmetric thermocapillary motion of two non-miscible viscous fluids in a cylindrical tube is presented. Their common interface is fixed and undeformable. This problem is an inverse problem because pressure gradients are unknown functions. The solution of the non-stationary problem is presented in the form of analytical expressions. They are obtained with the use of the method of Laplace transformation. If the wall temperature is stabilized then the general solution tends to the stationary solution as time increases. Numerical calculations confirm the theoretical results.

Keywords: thermocapillary motion, inverse problem, Laplace transform, fluid-fluid interface. DOI: 10.17516/1997-1397-2015-8-3-303-311

1. Problem statement

The axisymmetric motion of viscous thermally conducting fluid in a cylindrical system of coordinates is described by the Navier-Stocks equations

uit + uiuir + vi пи + 1 pr = v (Ami - Uf) , (1-1)

V11 + uivir + viviz + 1 pz = v Avi, (1.2)

P

uir + 1 Ui + viz = 0, (1.3)

r

9t + uiOr + vi0z = xA9, (1.4)

where ui(r, z,t), vi(r, z,t) are the projections the velocity vector on the axes r, z; p(r,z,t) is the pressure; 9(r,z,t) is the deviation of the temperature from the equilibrium value; A = d2/dr2 + r-id/dr + d2/dz2 is the Laplace operator, p, v, x are density, kinematic viscosity and thermal diffusivity, respectively.

System of equtions (1.1)-(1.4) admits of subgroup of four-dimensional continuous transformations [1]. They are generated by the operators (dz, tdz + dvi, dp, dg}. Their invariants are t, r, u. Therefore, partially invariant solutions of rank 2 and 3 should be sought in the form [2]

ui = ui(r,t), vi = vi(r,z,t), p = p(r,z,t), 9 = 9(r,z,t). (1.5)

* magdenko_ evgeniy@ icm.krasn. ru © Siberian Federal University. All rights reserved

In this case, it follows from the equation of conservation of mass (1.3) that vi is a linear function of z:

Vi = w(r,t)z + Wi(r,t). (1.6)

Moreover, we have

rw + (uir)r = 0. (1.7)

The momentum equations (1.1), (1.2) give us the following relations

wt + uiwr + w2 = v ^wrr +— w^j + h(t), (1.8)

1 p = l(t) - vwr - U- - °/Ul(r,t) dr - ^ z2 (1.9)

with arbitrary functions h(t) h l(t).

We use solutions (1.5)—(1.9) to describe the motion in a cylindrical tube of radius b with the fluid-fluid interface at a < b. We assume for simplicity that w1 = 0 in (1.6). If to write down the problem in dimensionless form, then the nonlinear term will stand Marangoni number M = ie9a2/pv\. It is assumed that M ^ 1, that is last performed in thin layers or a very high viscosities. As a result, we obtain the following problem

wit = vi ^wirr +— wir^ + hi(t), 0 < r < a, (1.10)

w2t = V2 (w2rr + 1 w2r^ + h2(t), a <r <b, (1.11)

wi(r, 0) = wio(r), w2(r, 0) = w2o(r), (1.12)

wi(a,t) = w2(a,t), (^w2r - Miwir)z = , (1.13)

dz

d9\

0i(a, z,t) = 02 (a,z,t), [ki-ß1 - ^d2

(1.14)

Assume that the surface tension linearly depends on temperature:

a = a0 - sr(0 - d0), & = const > 0. (1.15) It is obvious that for the temperature we have following representations

0j (r,z,t) = aj(r,t)z2 + bj (r,t), j = 1, 2. (1.16) Taking into account (1.15), the second boundary condition (1.13) can be rewritten as

^2w2r (a,t) — ^i wir (a,t) = 2&ai(a,t). (1.17) Functions aj (r,t), bj (r,t) satisfy the following equations

ait = Xi ^ai„. + , 0 < r < a, (1.18)

a2t = X2 (a2rr + 1 a2rj , a<r<b, (1.19)

0

r=a

(1.20)

(1.21) (1.22)

It is necessary to add conditions of the boundedness of w1(r,t), a1(r,t), b1(r,t) at r = 0 and no-slip conditions at r = b:

r b

(r,t) dr = 0. (1.23)

bit = X1 [birr + r bi rj + 2xiai, 0 <r < a, b2t = Xi ^b2rr + 1 b2^ + 2x2a2, a <r < b,

aj(r, °) = aj0(r), bj(r °) = bj0(r).

pa i'b

W2(b,t)=0, / rwi(r,t) dr = 0, / rw2(r,t) dr = 0. j 0 ja

The last two relations follow from equation (1.7). They allow us define the functions hi(t) and h2(t) if function a1(r,t) is known. So at first we find functions aj(r,t). Taking into account (1.16) and boundary conditions (1.14), we write

dai(a,t) da2 (a,t) ai(a,t) = a2(a,t), ki--- = K2~

(1.24)

dr dr

Functions bi(r,t) and b2(r,t) admit similar conditions. In addition, on the solid wall r = b the temperature is given: 62(b,z,t) = a(t)z2 + ß(t), where functions a(t) and ß(t) are known. This means that

a2(b,t) = a(t), b2(b,t) = ß(t). (1.25)

2. The stationary solution

Let us assume that all functions do not depend on time. Then from (1.18)-(1.20), (1.24), (1.25) we find that a\ = a2 = const = as and

where

wS (r)

2

aœas (1 r2\ a

fi(S){-2 - 02), 3 = i2 < 1

wS(r) =

fi(3) = fs(3) =

h{ =

2aœas

1

M2 f3(3)

(1 - 3)2 ^ ,r\ _ r2

1 - 3 + 3In3 nvb) + - b2

23f2(3)

fs(3) '

(1 - 3)2

1 - 3 + 3 In 3

4œv3fi(3)as

f2(3)=

31

2 +

(1 - 3) In 3 T-T+sinS

- 23 + 2p3f2(3), M = —

Pi M2 ' 8œ3as

ap2f3(3) ' " V27 ' 2 The dependence of velocity components on r is given by formulas

vi ,s _

v = — , h2 =-—— .

V2 ap2j3(o)

a2œas 4M2

1--2

a2

(2.1) (2.2)

(2.3)

(2.4)

(2.5)

(2.6)

a2 œas

(1 - 3)2

M2fs(3)\r) [2(1 - 3 + 3In3)

r2 1

1n

a2

3r2

- 1n 3 +1 - ^

r2 3 r4 3

+------1+—

+ a2 2 a4 1+ 2

(2.7)

us =

1

a

us =

2

2

a

2

a

Function bs(r) are

b1 (r)

bs2 (r)

where k = ki/ks.

sbs

Srs

1 + (1 - k)ölnS--

sbs

Srs Srs

1 - sr- ^ - k)S lni S—

ss

+ ßs, 0 < r < a,

+ ßs, 0 < a < b,

(2.8) (2.9)

3. Solution of the problem by the method of the Laplace transform

To solve linear adjoint problems one can use the Laplace transform [3]. It is defined as follows

aj (r,s) = aj (r,t) j 0

aj (r,t)e-st dt, j = 1, 2.

(3.1)

Then the problem is reduced to a boundary value problem for ordinary differential equation

(3.2)

(3.3)

1 ~ s

chirr +— ~alr--ai = —aio(r), 0 < r < a,

r Xi

1 ~ s ~ ! \ 1

asrr + - asr--as = -a;o(r), a < r < b,

r Xs

(3.4)

dai(a,s) 3a2(a,s)

ai(a,s) = a,2(a, s), ki--- = k2-~-,

or or

a,2(b, s) = a(s).

General solution of equations (3.2), (3.3) can be represented in the form (the condition of boundedness of a1 at r = 0 is taken into account)

ai(r, s) = Ciio

— r) +

Xi

Xi-MJXIr

- I^I— r)Ko

aio(r) dr, (3.5)

as(r, s) = CsIo

— r ) + C3Ko

— r) +

Xs

io\j—akjj —r\~

-^ ^ 'HJ X;r

aso(r) dr, (3.6)

where I0(x), K0(x) are modified Bessel functions of the 1st and 2nd kind. The quantities CÎ, C2 and C3 are determined from boundary conditions (3.4)

ci = Â

fi(s) Io(z) Ko(z)

fs(s) -Io(y) -Ko (y)

fs(s) -VXii(y) VXKi(y)

a

2

a

2

r

o

o

r

C = A

= -T

3 A

0 fi(s) Ko(z) Io(x) f2(s) -Ko(y) kIi(x) h(s) JXKi(y)

(3.7)

0

Io(x)

Io(z) -Io(y)

fi(s)

f2(s)

kIi (x) -VXIi(y) fs(s)

A

0

Io (x)

Io(z) -Io(y)

Ko(z) -Ko (y)

kIi(x) -VXIi(y) VXKi(y)

where x = ay/s/xi, y = a^Jsjx2, z = by/s/x2, X = Xi/X2, k = ki/k2,

fi(s)= i t

a

f2(s) = - f\

Io(z)Ko(, 1x2 t) - IoU ^ t)Ko (z)

a2o(T) dT + a(s),

IoU— T)Ko(x) - Io(x)KoU— t

fs(s) = k ^ t Io^Jxi t)ki(x)+ Ii(x)Ko(yJxi t)

aio (t) dT,

aio(T) dT.

(3.8)

When t ^ 0 we have Io(t) - 1 +12/4, Ko(t)--1n(t/2), Ii(t) - t/2+ t3/16, Ki(t)

1/t +11n(t/2)/2 and

A(s) - 1 { f 1nn (a) -vx

Therefore from (3.6)-(3.8) we obtain

1+H a)+^

1im sà,j(s) = 1im sa(s) = as = const.

(3.9)

(3.10)

This means that function aj (r,t) tends to constant value as time increases [3]. Let us turn to the definition of the functions Wj (r, t). The motion arises only under the action of thermocapillary forces, that is, initial conditions (1.12) are zero: Wjo(r) =0, j = 1,2. Then for the image Wj (r, s) we have the following boundary value problem

1 ~ s _ hi(s)

wirr +— wir--wi =--, 0 < r < a,

r vi vi

, 1 ~ s ~ h2(s)

W2rr + - W2r--W2 =--

r v2 v2

Wi (a, s) = W2(a, s),

a < r <b,

(3.11)

(3.12)

(3.13)

M2W2r(a, s) - MiWir(a, s) = 2ψi(a, s),

(3.14)

o

/• a /• b

W2(b,s)=0, / rwi(r,s) dr = 0, / rw2(r,s) dr = 0.

JO J a

Here, the function a1 (a, s) is already known from equation (3.5) and 1^(0, s)| < to. The solution of equations (3.11), (3.12) can be represented as

(3.15)

Wi — DiI0[ J — r +

hi (s)

W2 — d2iO[j —r + D3K0ij —r +

h2(s)

(3.16)

The boundary conditions (3.13), (3.14) and the first condition (3.15) allow us to find the values of D1, D2 and D3:

Di —

sA,

-h2 h-2 - hi

Io(zi) Ko(zi)

-Io(yi) -Ko(yi)

№2

ai(a,s) -Ii(yi) Ki(yi)

D2

sA-i

0

Io(xi)

-h2 h2 - hi

Ko(zi ) -Ko(yi)

D3

sAi

-^Ii(xi) -a-(a,s) Ki(yi)

Vv №2

0 Io(zi) -h2

Io(xi) -Io(yi) h2 - h-

№ t t \ T < \ - t Ii(xi) -Ii(yi)--a-(a,s)

Vv №2

A

1 —

0

Io(zi) Ko(zi)

Io(xi) -Io(yi) -Ko(yi)

-^Ii(xi) -Ii(yi) Ki(yi) v

where x- — a^Js/vy- — a^Js/v2; z- — by/s/i2; № — №1/^2; v — 11/12. Since [4]

rIo\\ — r ) dr — J — aIi[* — a

I rIo (r) dr — v [ziIi(zi) - yiIi(yi)],

J rKo ^y!!^ dr — -2 [yiKi(yi) - z-Ki(zi)].

(3.17)

(3.18)

s

s

1

1

1

a

o

Then taking into account (3.16), (3.18) and the second and third relations (3.15), we find

D = _ ahi(s) 1 Ii(XiV h 2 2 (3.19)

D2 [zili (zi) - yili (yi)] + D3 [viKi (yi) - ziKi (zi)] = - h2(s)f - ^) .

2v2

Substitution of Di, D2, D3 from (3.17) into (3.19) allows us to define hi(s) and h2(s):

• h2M=Af-AI • (3*20)

where

Ai = ^Trh + Io(zi)Ki(yi) + Ko(zi)Ii(yi),

2I1(X1)

A2 = - - Io(zi)Ki(yi) - Ii(yi)Ko(zi), y1

Fi = 2^ai(a,s)yi [Io(yi)Ko(zi) - Io(zi)Ko(yi)], aP2

ii = -^Ii(xi) [1 - yiIi(yi)Ko(zi) - yiIo(zi)Ki(yi)}. v

i2 = [ziIi(zi) - yiIi(yi)]

Ii(xi)Ko(yi) + Io(xi)Ki(yi)-

v

- -^Ii(xi)Ko(zi) v

+ [yiKi(yi) - ziKi(zi)]

Ii(xi)Io(zi)+

v

+ Io(xi)Ii(yi) - Ii(xi)Io(yi) v

(1 - s) 2a

+ ^ y2A -

F2 = i(a, s) yiIo(xi) [1 - y^o(zi)Ii(yi) - yiK i(yi)Io(zi)] . aP2

After some complicated mathematical treatment the limiting equalities

lim swj (r, s) = wS(r), j = 1, 2,

are proved. This means that the solution tends to the stationary solution as time increases.

Figs. 1, 2 show the dimensionless function Wj = a2wj/vi for silicon-water system at temperature of 20oC. Fig. 1 presents the case when a(r) = sin(10-2T), where t = a?t/vi - is the dimensionless time that is a(T) does not have limit by t ^ <x>. Thus, the solution with time growth does not converge to stationary. Fig. 2 shows the case when a(T) = 1 + e-T sin (t).

Remark 1. The problem of determination of the image aj(r, s) is similar to problem (3.2)-(3.4). One need to replace -ajo(r) with -bjo(r) - 2aj(r, s) and a(s) with /a(s). Thus, these functions can be found with the use of (3.5)-(3.8).

The author is grateful his scientific advisor, doctor in physical and mathematical sciences V.K.Andreev for formulating the problem and for valuable advice. The work was supported by the Russian Foundation for Basic Research, grant 14-01-00067.

Fig. 1. Dimensionless function profiles Wj at а(т) = sin(10-2T); curve 1: т = 200; curve 2: т = 400; curve 3: т = 700; curve 4: stationary solution. Curves 1-3 with time growth does not converge to curve 4

Fig. 2. Dimensionless function profiles Wj at a(r) = 1 + e T sin (t); curve 1: t = 20; curve 2: t = 50; curve 3: t = 100; curve 4: stationary solution

References

[1] V.K.Andreev, Mathematical modeling of convective flows, study guide, Krasn. Gos. Univ., Krasnoyarsk, 2006 (in Russian).

[2] L.V.Ovsyannikov, Group analysis of differential equations, Academic Press, New York, 1982.

[3] M.A.Lavrentiev, B.V.Shabat, Methods of the theory of functions of a complex variable,

Nauka, Moscow, 1973 (in Russian).

[4] G.Bateman, A.Erdelyi, Higher transcendental functions. 2. Bessel functions, the parabolic cylinder function, orthogonal polynomials, McGraw-Hill, 1953.

Осесимметрическое термокапиллярное движение в цилиндре при малых числах Марангони

Евгений П. Магденко

В статье решена линейная задача об осесимметрическом термокапиллярном движении двух несмешивающихся вязких теплопроводных жидкостей в цилиндрической трубе. Их общая поверхность раздела фиксируема и недеформируема. Задача является обратной, так как градиенты давлений есть искомые функции. В изображениях по Лапласу решения находятся в виде квадратур. Доказано, что если температура на стенке трубы стабилизируется со временем, то решение также с ростом времени стремится к стационарному режиму. Проведённые численные расчёты хорошо соотносятся с теоретическими результатами.

Ключевые слова: термокапиллярность, обратная задача, преобразование Лапласа, поверхность 'раздела.