UNSTEADY FLOW OF TWO BINARY MIXTURES IN A CYLINDRICAL CAPILLARY WITH CHANGES IN THE INTERNAL ENERGY OF THE INTERFACE Текст научной статьи по специальности «Физика»

DOI: 10.17516/1997-1397-2022-15-5-623-634 УДК 517.9

Unsteady Flow of two Binary Mixtures in a Cylindrical Capillary with Changes in the Internal Energy of the Interface

Natalya L. Sobachkina*

Siberian Federal University Krasnoyarsk, Russian Federation

Received 10.09.2022, received in revised form 10.11.2022, accepted 20.12.2022 Abstract. The problem of two-dimensional unsteady flow of two immiscible incompressible binary mixtures in a cylindrical capillary in the absence of mass forces is studied. The mixtures are contacted through a common interface on which the energy condition is taken into account. The temperature and concentration of mixtures are distributed according to the quadratic law. It is in good agreement with the velocity field of the Hiemenz type. The resulting conjugate boundary value problem is a nonlinear problem. It is also an inverse problem with respect to the pressure gradient along the axis of the cylindrical tube. To solve the problem the tau-method is used. It was shown that with increasing time the solution of the non-stationary problem tends to a steady state. It was established that the effect of increments of the internal energy of the inter-facial surface significantly affects the dynamics of the flow of mixtures in the layers.

Keywords: non-stationary solution, binary mixture, interface, energy condition, internal energy, inverse problem, pressure gradient, tau-method, thermal diffusion.

Citation: N.L. Sobachkina, Unsteady Flow of two Binary Mixtures in a Cylindrical Capillary with Changes in the Internal Energy of the Interface, J. Sib. Fed. Univ. Math. Phys., 2022, 15(5), 623-634. DOI: 10.17516/1997-1397-2022-15-5-623-634.

Introduction

The energy exchange between volume phases and a transition layer between them can lead to the inhomogeneous temperature distribution along the inter-facial surface. The mechanism of formation of the Marangoni stresses has been known for quite a long time. It implies that temperature gradient along the interface can arise and be maintained due to local increments of the internal energy of the inter-facial surface [1]. The temperature gradient, in turn, leads to the concentration gradient in the liquid mixture (these are the effects of thermodiffusion) [2].

For many liquids the surface tension is well approximated by a linear function. In this case energy condition is simplified to the following form [3,4]:

k2 m - ki m• u (i)

where x = — Oaf 89, a = a(9,c) is the surface tension coefficient that depends on temperature and concentration; k1, k2 are the coefficients of thermal conductivity; n is a unit vector of the normal to the interface of liquids r directed into the second liquid; Vr = V — (n • V)n is a surface gradient; 9 = 91 = 92, u = u1 = u2 are pairwise coincident values of temperatures and velocity vector of both liquids on r; Vr • u = divr u is a surface divergence of the velocity vector

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

u. According to the condition (1), changes in the surface internal energy induce corresponding changes in heat flows through the interface r.

For ordinary liquids at room temperature the effect of changes in the internal energy of the inter-facial surface on the formation of heat fluxes, temperature fields and velocities in its vicinity is insignificant in relation to the viscous friction and heat transfer [1]. However, at sufficiently high temperatures when the viscosity and thermal conductivity of ordinary liquids decrease significantly and for liquids with low viscosity increments of the internal energy of the inter-facial surface can have a significant impact on the dynamics of the flow [5]. The influence of changes in the internal energy of the inter-facial surface on the movement of liquids was studied [6-9].

A mathematical model that describes the two-layer unsteady thermodiffusion motion of binary mixtures in a cylindrical capillary in the absence of mass forces is considered in this paper. The mixtures are in contact through a common interface on which the energy condition is taken into account. The mechanism of influence of changes in the surface internal energy on the dynamics of the flow of binary mixtures in layers is studied. A similar geometry in the case of steady motion of mixtures was studied [10]. The non-linear stationary problem was reduced to a system of non-linear algebraic equations which was solved by the Newton method.

1. Statement of the problem

A two-dimensional unsteady axisymmetric flow of two immiscible incompressible binary mixtures in a cylindrical tube of radius R2 is considered. The temperature of the system is kept constant. Binary mixtures occupy domains Qi = {0 < r < Ri, \z\ < to} and Q2 = {R1 < r < R2, \z\ < to}, where r, z are radial and axial cylindrical coordinates. Here, the common interface of binary mixtures is at r = R1 = const, and the solid wall is at r = R2 = const. Values related to the regions and are denoted by the indices 1 and 2, respectively. It is assumed that the characteristic transverse size of domain is small compared to the radius of domain Q1 so R2 - R1 C R1 (Fig. 1).

Fig. 1. Schematic of the flow domain

Binary mixture is characterized by constant conductivity kj, specific heat capacity cpj, dynamic viscosity Hj, density pj, coefficient of thermal conductivity Xj = kj/pj cpj, kinematic viscosity Vj = Hj/pj (hereinafter, j = 1,2). The influence of gravity is not taken into account. It can be justified for a narrow capillary.

Taking into account the effect of thermal diffusion, the defining equations of motion, heat and mass transfer have the form

Ujt + UjUjr + WjUjz +--pjr = Vj ^Auj — -jj

1

Wjt + Uj Wjr + Wj Wjz +--Pjz = Vj AWj

Uj

j Pj (2) Ujr + r + Wjz = 0,

djt + Uj djr + Wj Ojz = Xj AOj,

cjt + Uj Cjr + Wj Cjz = dj Acj + aj dj AOj,

where Uj, Wj are projections of the velocity vector on r, z axis of the cylindrical coordinate system, respectively; pj is the pressure in the layers; 6j, Cj are deviations of temperature and concentration from the average values e0,c0; A = d2/dr2 + r-1 d/dr + d2/dz2 is the Laplace operator; dj, aj are diffusion and thermal diffusion coefficients, respectively. Generally speaking, these coefficients depend on temperature and concentration. However, using assumptions given above, one can consider that they have constant values corresponding to the average values of temperature and concentration. Let us note that the diffusion coefficient is always positive. The thermal diffusion coefficient can be either positive or negative. It depends on the type of a mixture and on the average temperature and average concentration of the selected component [11]. It is assumed that c in system (2) is the concentration of a light component.

It is assumed that inter-facial tension coefficient depends linearly on temperature and concentration:

a(e,c) = oq — &i(6 — e0) — &2(c — co). (3)

Here «i > 0 is the temperature coefficient, r<e2 is the concentration coefficient of the surface tension (usually «2 < 0, since the surface tension increases with increasing concentration). The solution of the problem is taken in the special form:

Uj = Uj (r,t), Wj = zvj (r,t), Pj = Pj(r,z,t),

(4)

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

In this representation, the velocity field corresponds to a solution of the Hiemenz type [12]. In this case, temperature ej takes an extreme value at the point z = 0: the maximum value when aj(r,t) < 0 and the minimum value when aj(r,t) > 0. A similar interpretation was obtained for the concentration of cj. One should only consider function hj (r,t) instead of aj (r,t).

Substituting (4) into equations of motion (2) and separating the variable z, one can obtain the following system for unknown functions Uj (r,t), Vj (r,t), pj (r,t), aj (r,t), bj (r,t), hj (r,t),

gj(r,t) ^

Ujt + Uj Ujr + Pjr Vj (Ujrr + Ujr j), (5)

Pj r r2

z(Vjt + Uj Vjr + V2) + — Pjz = zVj (Vjrr + - Vjr ), (6)

Pj r

U

Ujr + — + Vj =0, (7)

r

ajt + Uj ajr + 2Vj aj = Xj (ajrr +— ajr), (8)

bjt + Ujbjr = Xj(bjrr + - bjr + 2aj), (9)

hjt + Uj hjr + 2Vj hj = dj (hjrr +— hjr) + aj dj (a,jrr +— ajr), (10)

gjt + Ujgjr = dj (gjrr + - gjr + 2hj) + aj dj (bjrr + - bjr + 2aj). (11)

The pressure gradients (pjr,pjz) can be expressed from equations (5), (6):

Pjr = Pj Vj (Ujrr + 1 Ujr - uj ) - Pj (Ujt + Uj Ujr ), (12)

Pjz = z[pj Vj (vjrr + 1 Vjr ) - Pj (vjt + Uj Vjr + vj)]. (13)

Conditions for the compatibility of equations (12), (13) are satisfied identically: pjrz = pjzr =0. Hence it follows that pressure in the layers can be restored:

z2

pj = -Pj fj (i)y + Sj (r,t), (14)

where the derivative of function Sj (r, t) with respect to r is exactly the right hand side of equation (12). Integrating this equation, we obtain the following representation of functions Sj(r,t)

1 id fr 1 \

Sj (r, t) = Pj Vj (Ujr + r,Uj) - Pj ( dt J Uj dr +2 u2J + Sj (t). (15)

Functions Vj(r,t) are determined from the equation:

Vjt + Uj Vjr + vj = Vj (Vjrr + 1 Vjr ) + fj (t). (16)

It follows that flow in the layers is induced by longitudinal pressure gradients fj(t), j = 1,2. These are unknown functions to be defined along with functions Vj, aj, bj, hj, gj. Therefore, we have an inverse problem.

Conditions on the solid wall (r = R2) are

U2(R2,t) = 0, Vj(Rj,t) = 0, aj(Rj ,t)= aj(t), bj(Rj,t) = bj(t); (17)

hjr (Rj ,t) + «2a2r (Rj,t) = 0, gjr (Rj,t)+ ajbjr (Rj,t) = 0, (18)

with the specified functions a2(t), b2(t).

The inter-facial surface is assumed to be cylindrical and non-deformable. To do this, it is enough to require that Weber number We ^ to. Then taking into account this requirement and relation (3), we have the following boundary conditions at r = R1:

U1R1, t) = Uj (R1,t), V1R1, t) = Vj(R1,t); (19)

a1(R1,t)= aj(R1,t), b1(R1 ,t) = bj(R1,t); (20)

h1(R1 ,t) = hj(R1,t), g1(R1,t)= gj(R1,t), (21)

M2V2r(R1,t) - M1 V1r(R1,t) = -2[«1a1(R1,t) + ®2h1(R1,t)]; (22)

d1[h1r (R1,t) + a1a1r (R1,t)] = dj[hjr (R1,t) + ajajr (R1,t)]; (23)

d1[g1r (R1,t) + a.1b1r (R1,t)] = dj[g2r (R1,t) + ajbjr (R1,t)]; (24)

k2a2r(R1,t) - k1a1r(R1,t) = «1a1(R1,t)v1(R1,t); (25)

k2b2r(R1,t) - k1b1r(R1,t) = «1b1(R1,t)v1(R1,t). (26)

Relations (25), (26) are energy condition on the interface of two binary mixtures. It can be interpreted as follows: a jump in the heat flow in the direction of the normal to the interface (at r = R1) is compensated by the change in the internal energy of this surface. In turn, this change

is associated with both the change in temperature (specific internal energy) and the change in the area of the interface [13].

In addition, it is necessary to require the boundedness of functions on the axis of the cylindrical capillary (r = 0):

|wi(0,t)| < œ, bi(0,t)| < œ, |si(0,t)| < œ,

(27)

K(0,t)| < œ, |61(0,t)| < œ, h(0,t)| < œ, |#1(0,t)| < œ. Initial conditions at t = 0 are

uj(r, 0) = uj0(r), vj(r, 0) = Vj0(r), aj(r, 0) = ajo(r), bj(r, 0) = bj0(r);

(28)

hj(r, 0) = hjo(r), gj(r, 0) = gjo(r), Sj(r, 0) = Sjo(r), fj(0) = j = const.

Let us note that functions uj0 and vj0 should be constrained according to continuity equation (7); functions hj0, aj0 should be constrained according to conditions (18), (23); functions gj0, bj0 should be constrained according to conditions (18), (24); functions vj0, a10, h10 should be constrained according to condition (22); functions v10, aj0 should be constrained according to condition (25), and functions v10, bj0 should be constrained according to condition (26). Thus, the compatibility conditions are fulfilled.

2. Formulation of the problem in dimensionless variables

It should be noted that equations (7), (8), (10), (16) are independent of the others and they form a closed subsystem for defining functions Vj (r,t), aj (r,t), hj (r,t) and fj (t) (j = 1, 2). After solving it, functions bj(r,t), gj(r,t) can be determined from equations (9), (11), and functions sj(r,t) can be uniquely determined from relation (15). Taking into account conditions (27) and adhesion on the solid wall (17), we integrate continuity equation (7) and exclude uj(r,t) in equations (8), (10), (16). Then one needs to find only functions Vj(r,t), aj(r,t), hj(r,t) and fj (t). We introduce dimensionless variables and functions

e = -

e R1

R = R >1 ■

Rr »s

R-jvj

Ma v1 ,

A- = 01

Hj =

C0

Fj

Rifj Ma

(29)

where 00, c0 are the characteristic temperature and concentration. The main similarity criteria in the problem under consideration are

Ma =

M

M2 V1 M2

Mc = "2C°R? M2V1 , Pr, = j , Xj Scj = —j dj ' Srj = aj O0 dj C0

v = —1, k = V2 k1 d = d1 d2 M Mc = œ2C0

k2 = Ma

(30)

Here Ma is the thermal Marangoni number, Mc is the Marangoni concentration number, Prj is the Prandtl number, Scj is Schmidt number, Srj is the Soret number.

We obtain a nonlinear inverse initial-boundary value problem in the domain of the spatial variable £. When j = 1 it varies from 0 to 1, and when j = 2 it varies from 1 to R. For 0 < £ < 1 we have

1 Ma f^

K1(V1, F1 ) = V1K + - V1Ç - V1T + — V1Ç (e, T) de - Ma»!2 + F1 (t)

e e J0

0,

(31)

Si(Vi,A1) - J- (A^ + 1 Ai^ - AlT + M A1( 0 çvi(ç,t ) dÇ - 2MaA1V1 = 0; (32) T1(V1, Ai, Hi) - çL (^Hitt + 1 ffi^ + SCI (Ai« + 1 Ai^ - Hit+

(36)

1 ( a 1 * \ * Ma rt

PÏT [Aitt + iA()- AiT +T

sCi (hitt + 1 Hit) + £ (Ai{{ + Ç ^

Ma it V '

+~T Hia VÇ t) dÇ - 2MaHiVi = 0.

Ç Jo

For 1 < Ç < R we have

K2(V2, F2) - V (v2tt + 1V2^ - V2t - M V2t J ÇV2(Ç, t) dÇ - MaV22 + F2(t) = 0, (34) S2(V2,A2) - J- (A2tt + 1A2^ - A2t - M A2tj ÇV2(Ç, t) dÇ - 2MaA2V2 = 0; (35)

^ ^ H2) - Sc2V (^t + ^H*) + S|2V ^2tt + ^A*)

Ma /'R -— H2t J ÇV2 (Ç, t) dÇ - 2MaH2 V2 = 0.

Then the following conditions are satisfied on the solid wall (Ç = R)

V2(R,T )=0, A2(R,T ) = , H2t (R,T ) + Sr2A2t (R,T )=0. (37) On the interface (Ç = 1) we have

VI(1,t) = VI(1,t), 0 ÇVI(Ç,t) dÇ = 0, j* ÇV2(Ç,t) dÇ = 0, (38)

Ai (1,t ) = a2(1,t ), hI(1,t ) = h2(1,t), (39)

V2t(1,t) - tiVit(1,t) = -2[aI(1,t) + MhI(1,t)], (40)

d[Hit (1, T ) + Sri Ait (1, t )] = H2t (1, T ) + Sr2A2t (1, t ), (41)

A2t(1,t) - kAit(1,t) = EAi(1, t)Vi(1,t), (42)

where parameter E = 'œ2Q0R2/t2k2 characterizes the significance of the process of release or absorption of heat at local increments of the area of the inter-facial surface for the development of convective motion near this surface. The mechanism of local change in the internal energy of the interface should be taken into account for most conventional liquids at elevated temperatures or for liquids with reduced viscosity, for example, for some cryogenic liquids. Calculations carried out for physical parameters of various liquids and phase interfaces showed that E = O(1) is quite realistic [5].

The conditions of boundedness of functions are set on the axis of symmetry:

\Vi(0,t)| < œ, Ai(0,t)| < œ, \Hi(0,t)| < œ. (43)

The initial conditions at t = 0 are

Vj(Ç, 0) = Vjo(Ç), Aj(Ç, 0) = Ajo(Ç),

Hj(Ç, 0) = Hjo(Ç), Fj(0) = Fjo - const.

Note that the integral redefinition conditions in (38) are used as additional ones when solving the inverse problem and they are nothing more than a closed flow condition. They play an important role in finding unknown longitudinal pressure gradients Fj (t) in layers of binary mixtures.

Let us consider the creeping unsteady flow of binary mixtures (this is a linear problem).

Let us assume that the thermal Marangoni number Ma C 1 (a creeping motion) and Ma ~ Mc, that is, thermal and concentration effects on the interface £ =1 are of the same order. Then the equations of momentum, energy and concentration transfer are simplified by discarding convective acceleration. As a result, the conjugate inverse initial-boundary value problem becomes linear. However, such problem cannot be solved consistently because of the non-linearity of energy condition (42).

3. Derivation of a finite-dimensional system of first-order ordinary differential equations

To solve non-linear problem (31)-(44) the tau-method is used. It is a modification of the Galerkin method [14]. For further consideration, it is essential to replace the variables: £' = £ when j = 1 and £' = (£ — R)/(1 — R) when j = 2 and re-assign £' ^ £. Taking into account (43), an approximate solution is taken in the form

n

vn(£,T ) = E vk (t )Rfm

k=0 n

An(£,T ) = e Ak (t )R{k'1)(£), (45)

k=0

Hjn(£,T ) = e H (t )R(k'1)(£),

k=0

where Rk°'1)(£) are shifted Jacobi polynomials. In general, they are defined in terms of Jacobi polynomials F'ka'3)(y) as follows (a > — 1,3 > —1) [15]

Ra'%)= P(a'P)(2y — 1), y G [0,1]. (46)

It is known that shifted Jacobi polynomials R^'^y are orthogonal on the segment [0,1] with the weight (1 — y)ay3. Then

r 1

/ (1 — y)VR(a'%)Rla'«(y) dy _ Skmhm, (47)

Jo

h _ r(a + m + 1)r(3 + m + 1) _ _( 1, k _ m,

hm — 7} j 7) j ^ j TTiT/ i n i i 7T , * km

{

m!(a + / + 2m + 1)r(a + / + m + 1) ' \0, k _ m,

where r(x) is the Euler gamma function.

In addition, polynomials R^^y) form a basis in L2(0,1) with the weight (1 — y)ay3 and they satisfy the following properties [16]

R(a,P) (0) = ( —1)k (p + k)! R(a,p)(1) = (a + k)! . (48)

R (0) = -M- ' Rk (1) = ^!kT' (48)

dm R(*,P)(y) = r(k + m + a + p + 1) R(a+m,p+m)(y) ( )

dym R (y) = r(k + a + p +1) Rk-m (y). ( )

Functions Vj (r), Aj(r), Hj (r), Fj (r) are determined from the system of Galerkin approximations

r-1

f Kj(VJ1, FjRm^iO ÇdÇ =0, (50)

Jo

f Sj (jA)!®^) ÇdÇ =0, (51)

o

f1 Tj (Vn,Anj,Hn)R{m,1)(Ç) ÇdÇ = 0, m = 0,... ,n - 3, j = 1, 2. (52)

o

Taking into account conditions (38) and property (47), we obtain that V°(t) = V§(t) = 0. Taking into account properties (48) and (49), boundary conditions take the form

n n / \

Y^(-1)k (k + 1)V£ (t ) = 0, ]T (-1)j (k + 1)Aj (t ) = , (53)

k=o k=o o

YJ(-1)k-1h(h + 1)(k + 2)[Hk (t ) + Sr2A2fc(r )] = 0. (54)

k=i

£ Vk(t) = £ V2(T), £ A2(T) =Z A2(T),

2=0 2=0 2=0 2=0

£ H(t)=Z h2(t),

2=0 2=0

(55)

nn

]T k(k + 2)(V22 (t ) - V (t )) = -2y,(A2 (t ) + MH2(t )). (56)

2=1 2=0

dYk(k + 2)[H2(t) + SriA2(t)] = Vk(k + 2)[H2(t) + Sr2A2(t)], (57)

2=1 2=1

Y1^j2k(k + 2)(Ak (t ) - kAk(T)) = Ej^Ak (t )£ vj (t ). (58)

1 - R k=1 k=o k=o

The finite-dimensional system of Galerkin approximations for functions Vj (t), Aj (t), Hj(T) (k = 0,... ,n, j = 1,2) and the calculation of the resulting definite integrals over various products of shifted Jacobi polynomials are described in detail in [17].

The system of integro-differential equations can be transformed to a closed system of firstorder ordinary differential equations with respect to unknown functions Vj (t), Aj (t), H j (t), F j(T) (k = 0,... ,n - 3, j = 1,2). It involves rather cumbersome treatment and it is not given here. The initial conditions follows from (44) and (45):

Vjk(0) = Vjo, Aj (0) = Ajo, Hkj(0) = H jo, Fj (0) = F jo - const, (59)

where constants Vjo, Ajo, Hjo are the coefficients of the expansions of functions Vjo(Ç),Ajo(Ç),Hjo(Ç) in terms of the shifted Jacobi polynomials Rj°,Î^(Ç).

4. Numerical solution of the non-linear problem

The system of ordinary differential equations of the first order was integrated numerically using the Runge-Kutta method of the fourth order of accuracy. Note that when using the tau method in order to ensure the exact fulfilment of the boundary conditions it is necessary to use a sufficient number of coefficients in the trial solution. In this case calculations were performed for n = 10, 11, 12 in Galerkin approximations. At the same time, with an increase in n a rapid increase in the accuracy of the solution is observed.

Some results of numerical solution are presented for the model system that consists of an aqueous solution of formic acid (mixture 1) - transformer oil (mixture 2). According to tabular data , the dimensionless parameters of the specified system are as follows

¡ = 0.0898, v = 0.0649, x = 1-4, k = 2.41, d = 0.0152, Pri = 13.8, Pr2 = 298, Sci = 963, Sc2 = 225, Sri = 6, Sr2 = 7, Ma = 20, Mc = 15.

The following parameter values were also used: R =1.5, E = 0.7.

Fig. 2 shows the results of calculations of the velocity field. Function Vj (£,r) and the radial velocity profile Uj (£,r) are shown at various points in time. Analysing this result, we came to the conclusion that solution of a non-stationary problem with increasing time tends to the stationary mode obtained by solving the non-linear problem by the Newton method [10]. In turn, the pressure gradients Fj (t) in the layers are stabilized with time and they converge to the values F1 = —1.78305, F2 = —71.22054. Calculations show that the pressure gradient in the second layer significantly exceeds the pressure gradient in the first layer in absolute value. This is because transformer oil is more viscous compared to the aqueous solution of formic acid. The greater is viscosity of the liquid the smaller is its mobility. Consequently, much greater pressure forces are required to cause movement in the second layer.

Fig. 2. Plots of functions Vj (£,t) and Uj(£,t) at various points in time: 1 t = 0.4, 3 - t = 1.21, 4 - t = 1.42, 5 - t = oo

0.08, 2

It is of interest to consider how the change in the internal energy of the interface affects the flow pattern of binary mixtures. As a result of calculations it was found that with an increase in the energy parameter E at a fixed t the absolute values of function Vj (£, t) decrease (see Fig. 3). One can be concluded that the effects associated with the heat of formation of the inter-facial surface contribute to a decrease in the intensity and laminarization of the flow near this surface.

Note that function Vj (£,t) when passing through zero on the intervals 0 < £ < 1 and 1 < £ < 1.5 changes sign. This means that flows of binary mixtures change the direction of

Fig. 3. The relationship between functions Vj(£,t), Uj(£,t) and parameter E: 1 - E = 0.05, 2 - E = 0.2, 3- E = 0.7

movement. Thus, return flow zones appear in liquid layers near the interface. This happens not only due to the gradient of surface tension but also due to the non-stationary pressure drop in the layers that occurs during heating.

Let us consider the obtained numerical results for other functions. Due to the formation of heat function Aj (£, t) increases in both layers. As for "concentration", function Hj(£, t) decreases (see Fig. 4). One should take into account the Soret number Srj. This dimensionless parameter has a great impact on the concentration distribution in mixtures. Depending on the thermal diffusion coefficient aj, the Soret number can be either positive or negative. If the Soret number for both mixtures is negative then the directions of the temperature gradient and the diffusion mass flow are the same. As a result, light components move to the more heated area, and the heavy ones stay in areas with reduced temperature. This corresponds to the phenomenon of normal thermal diffusion. For positive Soret numbers, the direction of movement of components changes to the opposite. At the same time, the corresponding effect of thermal diffusion is abnormal. The results of numerical calculation allow one to conclude that abnormal thermal diffusion takes place in this model.

Fig. 4. Plots of functions Aj(£,t), Hj(£,t) at fixed t

Conclusion

A study of the unsteady two-layer flow of binary mixtures in a cylindrical capillary was carried out with consideration for changes in internal energy on the inter-facial surface. The resulting conjugate initial-boundary value problem is non-linear and inverse with respect to pressure gradients along the axis of the cylindrical capillary. To solve the problem the tau method was used. Shifted Jacobi polynomials were taken as basis functions. As a result, the system of integro-differential equations was reduced to a closed system of ordinary differential equations of the first order. To solve the system of equations the Runge-Kutta method of the fourth order was used. It was shown that with increasing time the solution of the non-stationary problem tends to the stationary mode. As a result of calculations for the model problem it was found that when energy parameter increases the characteristic convection velocity changes and intensity decreases. The increase of the energy parameter also promotes laminarization of the flow near the inter-facial surface.

The work was supported by the Russian Foundation for Basic Research (grant no. 20-0100234).

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Нестационарное течение двух бинарных смесей в цилиндрическом капилляре с учетом изменений внутренней энергии поверхности

Наталья Л. Собачкина

Сибирский федеральный университет Красноярск, Российская Федерация

Аннотация. Исследуется задача о двумерном нестационарном течении двух несмешивающихся несжимаемых бинарных смесей в цилиндрическом капилляре в отсутствие массовых сил. Смеси контактируют через общую поверхность раздела, на которой учитывается энергетическое условие. Температура и концентрация в них распределены по квадратичному закону, что хорошо согласуется с полем скоростей типа Хименца. Возникающая сопряженная начально-краевая задача является нелинейной и обратной относительно градиентов давлений вдоль оси цилиндрической трубки. Для ее решения применяется тау-метод. Показано, что с ростом времени численное решение нестационарной задачи выходит на стационарный режим. Установлено, что влияние приращений внутренней энергии межфазной поверхности существенно сказывается на динамике течения смесей в слоях.

Ключевые слова: нестационарное решение, бинарная смесь, поверхность раздела, энергетическое условие, внутренняя энергия, обратная задача, градиент давления, тау-метод, термодиффузия.