TWO-LAYER STATIONARY FLOW IN A CYLINDRICAL CAPILLARY TAKINGINTO ACCOUNT CHANGES IN THE INTERNAL ENERGY OF THE INTERFACE Текст научной статьи по специальности «Математика»

DOI: 10.17516/1997-1397-2021-14-4-507-518 УДК 532.5.013.4

Two-layer Stationary Flow in a Cylindrical Capillary Taking into Account Changes in the Internal Energy of the Interface

Victor K. Andreev*

Institute of Computational Modelling SB RAS Krasnoyarsk, Russian Federation Siberian Federal University Krasnoyarsk, Russian Federation

Natalya L. Sobachkina^

Siberian Federal University Krasnoyarsk, Russian Federation

Received 10.03.2021, received in revised form 05.04.2021, accepted 20.05.2021

Abstract. The problem of two-dimensional stationary flow of two immiscible incompressible binary mixtures in a cylindrical capillary in the absence of mass forces is investigated. The mixtures are contacted through a common the interface on which the total energy condition is taken into account. The temperature and concentration in the mixtures are distributed according to a quadratic law, which is in good agreement with the velocity field of the type Hiemenz. The resulting conjugate boundary value problem is nonlinear and inverse with respect to the pressure gradients along the axis of the cylindrical capillary. The tau-method (a modification of the Galerkin method) was applied to this problem, which showed the possibility of the existence of two solutions. It is shown that the obtained solutions with a decrease in the Marangoni number converge to the solutions of the problem of the creeping flow of binary mixtures. When solving the model problem for small Marangoni numbers, it is found that the effect of the increments of the internal energy of the interfacial surface significantly affects the dynamics of flows of mixtures in layers.

Keywords: binary mixture, interface, internal energy, inverse problem, pressure gradient, thermal Marangoni number.

Citation: V.K. Andreev, N.L. Sobachkina, Two-layer Stationary Flow in a Cylindrical Capillary Taking into Account Changes in the Internal Energy of the Interface, J. Sib. Fed. Univ. Math. Phys., 2021, 14(4), 507-518. DOI: 10.17516/1997-1397-2021-14-4-507-518.

Introduction

The specifics of the phenomena occurring at the interface of liquids are related to the existence of the energy and entropy of the surface phase, which are excessive in relation to the bulk phases in the transition layer [1]. However, the energy exchange between the bulk and surface phases has not been sufficiently studied. For ordinary liquids at room temperature, the effect of changes in the internal energy of the interfacial surface on the formation of heat fluxes, temperature fields, and velocities in its vicinity is insignificant in relation to viscous friction and heat transfer . However, at sufficiently high temperatures, when the viscosity and thermal conductivity of ordinary liquids are significantly reduced, as well as for liquids with reduced viscosity (for example, for some cryogenic liquids), the effect of the internal energy increments of the interfacial surface is significant [3].

*andr@icm.krasn.ru

tsobachkinanat@mail.ru https://orcid.org/0000-0002-2025-1785 © Siberian Federal University. All rights reserved

In this paper, we consider a mathematical model describing the two-dimensional stationary thermodiffusion motion of two immiscible incompressible binary mixtures in a cylindrical capillary in the absence of mass forces. The mixtures are contacted through a common interface on which the total energy condition is taken into account. In this geometry, the mechanism of influence of changes in the surface internal energy on the dynamics of binary mixtures is investigated. Without taking into account the effects of thermal diffusion, such a model was studied in the works [4,5].

1. Statement of the problem

We consider a two-dimensional stationary axisymmetric flow of two immiscible incompressible binary mixtures in a cylindrical tube of radius R2, the temperature of which is maintained constant. Binary mixture occupy the field: Qi = {0 ^ r ^ Ri, \z\ < œ} and Q2 = {Ri ^ r ^ R2, \z\ < œ}, where r, z are the radial and axial cylindrical coordinates. Here r = R1 = const is the total interface of binary mixtures, r = R2 = const is the solid wall. The values related to the regions and Q2 are denoted by indexes 1 and 2, respectively. The area of is called the core, and the area is an interlayer or film. It is assumed that its characteristic transverse size is small by compared to the radius of the core, R2 — R1 C R1. Such a geometry corresponds, for example, to the case of displacement of the liquid that originally filled the capillary by another liquid.

A/

Fig. 1. The scheme of the flow region

Binary mixture is characterized by constant thermal conductivities kj, specific heat capacities cpj, dynamic viscosities Uj, densities pj; let Xj = kj/pj cpj is the thermal conductivity, Vj = Uj/pj is the kinematic viscosity (here and further, j = 1,2). The influence of gravity is not taken into account, which may be justified, for example, if the tube it is quite narrow to the capillaries.

The system of equations of motion, continuity, internal energy balance and concentration transfer has the following form [6]:

UjUjr + WjUjz + — Pjr = Vj (^Auj - ,

Uj Wjr + Wj Wjz +--Pjz = Vj Awj,

Uj pj n (1)

Ujr + r + Wjz =0, v '

Uj @jr + Wj 0jz = Xj ASj, Ujcjr + Wj cjz = dj Acj + aj dj A0j,

where Uj, Wj are projections of the velocity vector on the r, z axis of the cylindrical coordinate system; pj is the pressure in the layers; 6j, j are deviations of temperature and concentra-

tion from their equilibrium values; dj, aj are the diffusion and thermal diffusion coefficients, respectively; A = d2/dr2 + r-1 d/dr + d2/dz2 is the Laplace operator.

The linear dependence of the interfacial tension coefficient on temperature and concentration is assumed:

a(d,c) = ao - iei(0 - do) - ^(c - c0). (2)

Here > 0 is the temperature coefficient, r<e2 is the concentration coefficient of the surface tension (normally '<s2 < 0, because the surface tension increases with increasing concentration); 90, c0 are the temperature and concentration on the interfacial surface in the as balance. The solution to the problem is sought in a special form:

Uj = Uj (r), Wj = ZVj (r), Pj = Pj (r, z),

(3)

dj = aj(r)z2 + bj(r), cj = hj(r)z2 + gj(r).

A solution of the form (3) is called a solution of the type Hiemenz [7], in which the velocity field is linear with respect to the transverse coordinate. Thus, the temperature dj takes an extreme value at the point z = 0: the maximum at aj (r) < 0 and the minimum at aj (r) > 0. We get a similar interpretation for the concentration cj, only instead of aj (r) the function hj (r) is considered.

After substituting the special form (3) into the equations of motion (1) we will have the following system with unknown functions Uj (r), Vj (r), pj (r), aj (r), bj (r), hj (r), gj (r):

1 = ( 1 Uj \

UjUjr + ~Pjr = VjyUjrr + r Ujr - ), (4)

z(UjVjr + V2) + — Pjz = Vjz(vjrr + 1 Vjr^j , (5)

Ujr +—- + Vj =0, (6)

1

a—rr \ a— r

1

u- a-r + 2v-a- = X^a-rr +— a-^j, (7)

u- b-r = bjrr + i b-r + 2a-j, (8)

Uj— -r \ 2v- - — d*- ^h-rr \ -r^ \ ^^-d*- ^Qi-rr \ a1-r^, (9)

u-9-r = d^g-rr \— g-r \ \ a- d- ^b-rr \— b-r \ . (10) From the equations (4), (5), we express the pressure gradients (p-r ,p-z):

( 1 U- (-,-,)

P-r — P- V- ^U-rr H- U-r 2 J P- U- U-r, (11)

P-Z = z P- V-iv-rr \ 1 V-r) - P- (U- V-r \ V-) , (12)

Conditions for the compatibility of the equations (11), (12) are satisfied identically: P-rz = P-zr = 0. It follows that the pressure in the layers will be restored by the formula:

z2

P- = -P- f- y\ s- (r), (13)

where the derivative of the variable r from the functions s- (r) is exactly the right-hand side of the equation (11). Integrating this equation, we obtain for the functions s- (r) the following view:

S- (r) = P-v-^U-r \ r U^ - 1 P- u2 \ s-0, s-0 = const. (14)

In turn, the functions Vj (r) are defined from the equation:

1

UjVjr + v2 = Vj{ Vjrr + - Vjr) + Jj

)+ fj, (15)

where fj = const. The flow in the layers is induced by the longitudinal pressure gradients fj. These are unknown constants that are subject to by definition. Therefore, the problem is reversed.

On a solid wall r = R2, the boundary conditions are satisfied:

U2(R2)=0, V2 (R2) = 0, 02(^2) = a20, b2(R2 ) = &20,

(16)

h2r (R2) + a2a2r (R2) = 0, g2r (R2) + a2&2r (R2) = 0,

with the given constants a20, b20. Note that when a20 < 0 the wall temperature has a maximum value at the point 2 = 0, and for a20 > 0 — minimal.

On the interface r = Ri, given the dependence (2), we will have the following conditions:

cui(Ri)= U2(Ri), vi(Ri)= V2 (Ri), (17)

ai(Ri) = a2(Ri), bi(Ri ) = b2(Ri), (ig)

hi(Ri) = h2(Ri), gi(Ri) = g2(Ri), ( )

H2V2r(Ri) - MiVir(Ri) = -2«iai(Ri) - 2«2^i(Ri), (19)

di[hir (Ri) + aiair (Ri)] = d2[h2r (Ri) + a2a2r (Ri)], (20)

k2a2r(Ri) - kiair(Ri) = «iai(Ri)Vi(Ri), (21)

k2b2r(Ri) - kibir(Ri) = ®ibi(Ri)Vi(Ri).

The relation (21) is called the energy condition on the interface of two binary mixtures [8-10]. It means that the jump in the heat flow in the direction of the normal to the surface section r = Ri is compensated by a change in the internal energy of this surface. In turn, this change is associated with both a change in temperature (and with it the specific internal energy) and a change in the area of the interface.

For a complete statement of the problem to the relations (17)-(21), it is necessary to add the boundedness of the functions on the axis of the cylindrical capillary at r = 0:

|ui(0)| < to, |vi(0)| < to, |si(0)| < to, |ai(0)| < to,

|bi(0)| < to, |hi(0)| < to, |gi(0)| < to. ( )

2. Transformation to a problem in dimensionless variables

For what follows, it is essential that the equations (6), (7), (9), (15) are independent of the others and form a closed subsystem for defining the functions Vj (r), aj (r), hj (r) and the constants fj (j = 1,2). After solving it, the functions bj(r), gj(r) are found from the equations (8), (10), and sj(r) is uniquely restored by the formula (14). If we integrate the continuity equation (6) and exclude functions Uj(r) in the equations (7), (9), (15) with given the conditions of boundedness (22) and sticking on a solid wall (16), the problem is reduced to the conjugate boundary value problem of finding only the functions Vj (r), aj (r), hj (r) and the constants fj. We introduce dimensionless variables and functions by equalities:

r R2 R2i Vj

^ = Ri , R = Ri > 1, Vj = Mavi ,

i i 4 i (23)

Aj = j , Hj = hhj , Fj = ^

j a20 ' j c0 ' j Ma vf '

where a20, c0 are the characteristic temperature and concentration.

As the defining parameters of the problem under consideration, we choose the following:

«ia2oRf A/r «2C0 R Ma = -, Mc = -

V, Pr, = — , Xj Se, = Vl dj ' Srj = a, a,2o co

ki k2 d = di d2 M= Mc Ma = œ2Co œia2o

M2 Vi V2Vi A.J -j -u (24)

Vl V = Vl

M2 ' V2

Here Ma is the thermal Marangoni number, Mc is the concentration Marangoni number, Pr^ are the Prandtl numbers, Sc^ are the Schmidt numbers, Sr^ are the Soret numbers.

After de-dimensionalization, we obtain a nonlinear inverse boundary value problem in the domain with respect to the spatial variable £, which, for j = 1 varies between 0 and 1, and when j = 2 — in the range from 1 to R. For 0 < £ < 1 we will have:

1 Ma f^

K1(V1, F) = Vi,, + -Vi, + — Vi, / xVi(x) dx - MaV2 + F = 0 , (25)

s ç J0

Si(Vi, Ai) = Pi (Ai,, + -A^ + M Ax 0 xVi(x) dx - 2MaAV =0; (26)

TiW, Ai, Hi) , sCi (Hi,, + -Hi,) + Sci + -^ +

sCi ( Hi,,+- h*)+SCi K+-

Ma f, + — H^ xVi(x) dx - 2MaHiVi = 0.

s Jo

For 1 < Ç < R, we have:

,-R

(28)

(30)

K2V2F2) = 1 (v2,, + -V2^ - M V2, j, xV2(x) dx - MaV22 + F =0, S2V2A2) = p-v (a2& + - A2^ - M A2, J* xV2 (x) dx - 2MaA2V2 = 0; (29)

T2 (V2, A2,H2) ^ sC^ (H2,, + -H2^ + SCV (A2,, + ^ ~

Ma fR --H2, xV2(x) dx - 2MaH2V2 = 0.

ç J,

Then, on a solid wall Ç = R, the conditions are met:

Vî(R) = 0, A2(R) = - H2, (R) + Sr2 A2, (R) = 0. (31)

On the interface Ç = -

V^) = V^), J xVi(x) dx = 0, J xV2(x) dx = 0, (32)

Ai(-) = A2 (-), Hi(-)= H2 (-), (33)

V2,W - |Vi,(-) = -2Ai(-) - 2MHi(-), (34)

d(Hi, (-) + Sri Ai, (-)) = H2, W + Sr2A2, (-), (35)

(1) - kA^(1) = EAi(1)Vi(1), (36)

where E = x^a20Rl/^2k2 is a parameter that determines the effect of the internal energy of the interface on the dynamics of the movement of liquids inside the layers. On the axis of symmetry, the conditions of boundedness are set:

|Vi(0)| < |Ai(0)| < to, \Hi(0)\ < to. (37)

Remark. The integral redefinition conditions in (32), meaning the flow closure conditions, are necessary to find the unknown longitudinal pressure gradients Fj in the layers of binary mixtures, j = 1, 2.

3. Solving of the conjugate problem for small Marangoni numbers

We will assume that the thermal Marangoni number Ma C 1 (a creeping motion), and Ma ~ Mc, that is, the thermal and concentration effects on the interface £ = 1 of the same order. Formally decomposing the functions Vj, Aj, Hj in a series of Ma, we obtain for the first approximation the problem (25)-(27), (28)-(30) with Ma = 0. In the equations of momentum, energy, and concentration transport, the convective terms are discarded. As for the nonlinear boundary condition (36), it is remains unchanged. To do this, we must assume that E = O(1).

Then the conjugate inverse boundary value problem for small Marangoni numbers becomes linear:

Vi55 + 1 Vi5 = -Fi, (38)

Ai^ \ 1A^ = 0, (39)

s

#1ÇÇ \ 1 Hiç = 0, 0 < s < 1; (40)

V2K \ 1 V2ç = -F2V, (41)

A2ÇÇ \ 1A2Ç = 0, (42)

H2ÇÇ \ 1H2Ç =0, 1 <£<R; (43)

with the boundary conditions (31)-(37).

Common solutions of systems (38)-(43) are easily found (the boundedness conditions (37) are taken into account):

Vi(S) = Ci - F-s2, Ai(S)= C2, Hi(S)= C3; (44)

V2(S) = C4 \ C5 ln s - Ffe , A2(S) = Ce \ C7 ln s, H2(S) = Cs \ C9 ln s, (45)

with the constants Ci,... ,C9, which are determined from the boundary conditions (31)-(36). Exactly,

Fi 8

fi i fi fi fi fi

Ci = Y, C2 = Ce = 8-ËF\ïnR, C3 = Cs, (

C = 2F2V - Fi C = 2F2V(R2 - 1)\ Fi C = EFi C = C {)

C4 = 8 , C = 8inR , C7 = EFmR-8, C9 = -SrC

As for the constant C3, from the boundary condition (34) it is defined as follows:

= F2V - FiV - 4C2 - 2C5 (47)

3 4M ' V ;

But such a representation for C3 makes it difficult to further search for the pressure gradients Fi, F2 along the layers when solving the inverse boundary value problem. On the other hand,

this constant can be found if you set the average concentration over the cross section z = 0, so

i

J £Hi(£) d£ = 0. From where we get that C3 = 0 and, therefore, C8 = 0.

0

The pressure gradients Fi, F2 are related by the relation F2 = FiN(R), where the function N(R) is defined by the formula:

N (R)= R2 - 2 in R -1 (48)

In addition, the functions Uj (£)are recovered from the continuity equation (6):

Ui(£) = §£(£ - 1)(£ +1), U2(£) = Fk\(R2 - £2)(8C4 - 4C5 - F2V(R2 + £2))+ 8C5(R2 lnR - £2 ln£)\.

(49)

If the expression for the constant C3 from (47) vanishes, then after some calculations a quadratic equation arises with respect to the unknown pressure gradient Fi:

EL(R)ln RF2 - 8L(R)Fi - 128 ln R = 0, (50)

where L(R) is defined by the formula:

L(R) = 4vlnR(p - N(R))+2vN(R)(R2 - 1) + 1. (51)

Of interest are the cases related to the number of solutions of the equation (50).

1. If E = 0, we get the equation: -8L(R)Fi - 128 ln R = 0, which has a unique solution Fi = -16lnR/L(R). The pressure gradient F2 is easily determined from the ratio (48).

2. If R ^ 1, then we have the equation: -8L(R)Fi = 0, which has the only solution Fi = 0. Here it is taken into account that the function L(R) takes positive values on the interval (1, +to). Then it follows from (48) that F2 = 0. The equality of the pressure gradients to zero means that there is no source of motion of the mixtures in both layers. Thus the mixtures are at rest.

Next, we find the discriminant of the quadratic equation (50):

D = 64L(R)(L(R) + 8Eln2 R), (52)

depending on the sign of which the equation has a different number of roots.

3. If D > 0, we get: E > -L(R)/8ln2 R. In this case, the square equation has two roots:

Fi'2 ^ '-. (53)

4L(R) ± 4^JT2{K)+8EL{Kyin2R ~EURj\nR

4. The discriminant vanishes at E = -L(R)/8ln2 R, (L(R) = 0). Then the equation will have a unique solution: Fi = -32 ln R/L(R). Note, what is the expression L(R)/8ln2 R >0 when R e (1, +to). Therefore, the parameter E takes negative values. This is possible with a20 < 0, since E depends on this parameter.

5. The negative sign of the discriminant corresponds to the condition: E < -L(R)/8ln2 R, which is equivalent to the absence of real roots of the square equation.

Thus, the number of solutions to the equation (50) depends more on the parameter E. In other words, the energy of interfacial heat transfer has a significant effect on the processes occurring in the contacting liquids.

4. Model problem

We present the quantitative results of solving the problem for the model system formic acid (mixture 1) — transformer oil (mixture 2). According to the tabular data, the physical constants are as follows:

3 kg 4 kg 6 m2 6 m2

u1 =1.78 • 10-3^- , u2 =198.1 • 10-4—2- , v1 =1.46 • 10-6— , v2 =22.5 • 10-6— , m • s m • s s s

m2 m2 Wt Wt

X1 = 1.057 • 10-7 — , x2 = 7.55 • 10-8 — , k1 = 0.267-, k2 =0.1106-,

s s m • K m • K

cto = 37.58 • 10-3 N , = 1.2826 • 10-4 .

m m • K

The following parameter values were also used: R = 1.5, R1 = 10-9 m, E = 0.7 (a20 > 0). As a result of the calculations, two solutions were obtained for the longitudinal pressure gradients in the layers: F/ = -1.78305, F1 = -71.22054 and F? = 29.96938, F22 = 1197.06399. It can be seen that for the second solution, the gradient values in both mixtures are too high, which is unphysical.

Fig. 2-4 demonstrates the function Vj (£) and the velocity profile Uj (£) depending on the various defining parameters of the model.

b)

Fig. 2. The behavior of the function Vj (£) and the velocity profile Uj (£): a) for the first solution, b) for the second solution

Fig. 2 shows the functions Vj(£) , Uj(£), corresponding to the two solutions {F/,F2] and {F?, F22}.

Fig. 3 shows that as the parameter E increases, the values of the functions Vj(£), Uj(£) in absolute value decreases significantly. You can choose such values of E, at which the model problem will have a single solution. So, for E = 0 (a20 = 0) we get: F1 = -1.89641, F2 = -76.27046. By E « -2.6 (a20 = -3.46 • 1023) we have: F1 = -3.79282, F2 = -152.54093.

The increase of the parameter R is strongly influenced by the velocity profile Uj (£) and the function Vj (£). Fig. 4 shows that the absolute values of the functions increase. This is due to the fact that for a fixed R1 , the radius of the outer cylinder increases, since R — R2/R1. It is also important to trace how the change in the radius of the inner cylinder R1 affects the flow pattern in the layers. It turned out that with the growth of R1, the values of the functions Vj (£), Uj(£) in absolute value decreases. This is due to the fact that with an increase in the radius of the inner cylinder at fixed R and E, the influence of a constant temperature set on the surface of the outer cylindrical tube weakens.

Fig. 3. The dependence of the functions Vj(£), Uj(£) on the parameter E: 1 — E = 0.05, 2 — E = 0.2, 3 — E = 0.7

Fig. 4. The dependence of the functions Vj(£), Uj(£) on the parameter R: 1 — R = 1.5, 2 — R = 1.7, 3 — R =2.0

Fig. 5 shows the "temperature" and "concentration" functions Aj (£), Hj (£), corresponding to the first solution {F^F^}. In the first layer, these functions are constant. In the second layer Aj (£) increases and Hj (£) decreases, which corresponds to the phenomenon of abnormal thermal diffusion.

Fig. 5. The behavior of functions Aj (£), Hj (£) in the case of the first solution

Thus, the effect of changes in the internal energy of the interfacial surface on the two-layer flow of two immiscible binary mixtures in a cylindrical capillary is studied. It is found that with an increase in the parameter E, which is responsible for the influence of changes in the surface internal energy on the dynamics of liquids in layers, the absolute values of the functions Vj (£), Uj (s) decreases.

5. Derivation of a finite-dimensional system of nonlinear algebraic equations

To solve the nonlinear problem (25)-(37), the tau method is used, which is a modification of the Galerkin method [11]. For the future, it is essential to replace the variables: = £ with j = 1 and = (£ — R)/(1 — R) when j = 2 and re-assign ^ An approximate solution is sought in the form of sums:

Vn(£) = £ vR'^iO, An(£) = £ AjRf,i](£), Hn(£) = £ HjRf'i)(£), (54) 1=0 1=0 1=0

где R^^iO are the shifted Jacobi polynomials. In general, they are defined in terms of the Jacobi polynomials Pjа'в)(у) as follows (a > — 1, в > —1) [12]:

R{k'p)(y) = P(a'e)(2y — 1), y e [0,1]. (55)

Coefficients Vj, Aj, Hj and constants Fj are found from the Galerkin approximation system, namely:

f Kj(VJ1, FjR^^) ÇdÇ = 0, (56)

Jo

f Sj(Vn,Aj)rtyXï) ÇdÇ = 0, (57)

o

f1 Tj(VJ1, Aj, Hji)R(;<0'1\O Çdç = 0, m = 0,... ,n - 2, j = 1, 2. (58)

o

It follows from the integral redefinition conditions of (32) that V0 = V20 = 0. The boundary conditions are transformed as follows:

E(-1)1 V2 =0, E (-1)lAl2 = 1, (59)

l=0 l=0

n

Y^(-1)l-1l(l + 1)(l + 2)[Hl2 + Sr2 A2] = 0. (60)

l=1

n n n n n n

T,vl = £Vl ^Afi = 53A2, Y^Hl = £H2, (61)

l=0 l=0 l=0 l=0 l=0 l=0 nn

Y,l(l + 2)(V2 - ^Vi ) = -2^(А + MH1 ). (62)

l = 1 l=0 nn

d^l(l + 2)[Hl + Sri Ai] = £ l(l + 2)[Hl + Sr2A2], (63)

l=i l=i

n n n

Y^l(l + 2)(Al2 - kAli) = -E^AY; Vl. (64)

l= l=0 l=0

Verbose output finite-dimensional system galerkins approximations for the coefficients Vj, Aj, Hj, l = 0,.. .,n, j = 1, 2, and also the calculation of definite integrals from different product of shifted Jacobi polynomials are present in the work [13].

As a result, the system of integro-differential equations are converted to a closed system of nonlinear algebraic equations unknown coefficients Vj, Aj, Hj and gradients of pressure Fj, where l = 0, ...,n, j = 1,2. Its solution was used Newton's method with a given accuracy e = 10~5. As an initial approximation, the results obtained in solving the model problem were taken.

Applied to a nonlinear inverse boundary value problem (25)-(37) the tau-method showed the possibility of existence of two solutions for the longitudinal pressure gradients and, accordingly, for the rest of the desired functions of the problem. Calculations were performed for n = 10,12 in Galerkin approximations. As the number of n increases, a rapid increase in the accuracy of the solution is detected.

Fig. 6 shows the dependence of the functions Vj (£) , Uj (£) on different values of the thermal Marangoni number, obtained for the first solution: Fi = —1.78355, F^ = —71.73149. We conclude that the solutions found with a decrease in the Marangoni number converge to solutions of the problem of the creeping flow of binary mixtures.

Fig. 6. The dependence of the functions Vj(£), Uj (£) of the thermal Marangoni number: 1 — Ma =15, 2 — Ma = 3, 3 — Ma = 0.5, 4 — Ma = 0.28, 5 — a creeping current

The work was supported by a grant from the Russian Foundation for Basic Research no.

20-01-00234.

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Двухслойное стационарное течение в цилиндрическом капилляре с учетом изменения внутренней энергии поверхности раздела

Виктор К. Андреев

Институт вычислительного моделирования СО РАН Красноярск, Российская Федерация Сибирский федеральный университет Российская Федерация

Наталья Л. Собачкина

Сибирский федеральный университет Российская Федерация

Аннотация. Изучена задача о двумерном стационарном течении двух несмешивающихся несжимаемых бинарных смесей в цилиндрическом капилляре в отсутствие массовых сил. Смеси контактируют через общую поверхность раздела, на которой учитывается полное энергетическое условие. Температура и концентрация в смесях распределены по квадратичному закону, что хорошо согласуется с полем скоростей типа Хименца. Возникающая сопряженная краевая задача является нелинейной и обратной относительно градиентов давлений вдоль оси цилиндрического капилляра. К этой задаче применен тау-метод (модификация метода Галеркина), который показал возможность существования двух решений. Показано, что полученные решения с уменьшением числа Маран-гони сходятся к решениям задачи о ползущем течении бинарных смесей. При решении модельной задачи при малых числах Марангони установлено, что влияние приращений внутренней энергии межфазной поверхности существенно сказывается на динамике течения смесей в слоях.

Ключевые слова: бинарная смесь, поверхность раздела, внутренняя энергия, обратная задача, градиент давления, тепловое число Марангони.