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2
Journal of Siberian Federal University. Mathematics & Physics 2023, 16(1), 26—34
EDN: GZTDQG УДК 532.6
Construction of an Exact Solution of Special Type for the 3D Problem of Thermosolutal Convection in Two Layered System
Marina V. Efimova*
Institute of Computational Modelling SB RAS Krasnoyarsk, Russian Federation
Received 10.07.2022, received in revised form 20.08.2022, accepted 31.10.2022 Abstract. A three-dimensional joint flow of a liquid and a binary mixture with common interface is considered. It is assumed that the temperature field in the layers has a quadratic distribution. An exact solution of certian model problem is constructed, explicit expression for all the required function are obtained using a specific closing relation.
Keywords: Oberbeck-Boussinesq approximation, surface energy, binary mixture.
Citation: M.V. Efimova, Construction of an Exact Solution of Special Type for the 3D Problem of Thermosolutal Convection in Two Layered System, J. Sib. Fed. Univ. Math. Phys., 2023, 16(1), 26-34. EDN: GZTDQG.
The theory of the motion of liquid media with interfaces or with a free boundary attracts a lot of attention, due to the numerous technological applications. Thermocapillary flows induced by the surface tension forces arising at the interface can influence the movement of the liquid in the volume. To take into account various factors affecting the fluid dynamics, it is necessary to use new mathematical models and to formulate initial-boundary value problems. Therefore, there is a need to construct nontrivial exact solutions, to study stability issues, and to develop efficient numerical algorithms for such models.
Work [1] presents various formulations of problems on the motion of two immiscible liquids with a common interface. Possible generalizations and consequences of the formulations of the arising initial-boundary value problems are discussed. Exact solutions obtained in the frame of the different statements of the problem are the useful tool to study features of convection in fluidic systems. The works [2,3] describe the construction of exact solutions of the classical convection equations which describe flows with evaporation, in the two-dimensional and three-dimensional cases. Basic characteristics obtained with the help of the exact solutions allowed one to analyze the impact of different factors affecting the convective regime structure. In [4], a three-dimensional flow of a viscous incompressible fluid in a single-layer system with a non-uniform temperature distribution at free boundaries was studied.
In this paper, we construct an exact solution to the problem describing a three-dimensional flow in a liquid-binary mixture system with a common interface. The Navier-Stokes equations in the Oberbeck-Boussinesq approximation are used as a mathematical model. Thermal and diffusion processes are described by heat and mass transfer equations.
* efmavi@mail.ru https://orcid.org/0000-0002-5023-2470 © Siberian Federal University. All rights reserved
1. Problem statement and form of exact solution
We consider flow of two viscous incompressible media (liquid and binary mixture) filling the plane channel and having the common interface r. The domain occupied by the liquid is denoted by Q1 = {(x,y,z) : \x\ < œ, \y\ < œ, -l1 <z< 0} and Q2 = {(x,y,z) : \x\ < œ, \y \ < œ, 0 <z<l2} is domain filled by the binary mixture.
For description of the motion in regions Qj (j = 1, 2) we use the Boussinesq approximation. Indexes j = 1 and j = 2 refer to the lower liquid and upper binary mixture, respectively. We assume that the temperature and the concentration slightly differ from constant mean values therefore the Oberbeck-Boussinesq approximation is valid. The state equation is taken in the following form
Pj = poj(1 - je - jc),
where p0j is the characteristic density of j the medium corresponding to the mean values of the temperature and concentration in the layer, e and c are the functions giving deviations of the temperature and concentration, respectively, from their mean values (c corresponds to the concentration of light component in the binary mixture), (j- and (c are the temperature and concentration expansion coefficients; (1 = 0. Then, the equations describing the convective flows in the two-layer system incompressible media can be written in the form
Ujt + (Uj • V)uj = --LVpj + VjAuj - g((j(ej - eoj) + (cc - c-)), p0j
°jt + Uj • Wj = Xj , ct + u2 •Vc = DAc + aDA92,
(1)
div Uj = 0.
where Uj = (uj,vj,wj) is the velocity vector, pj is the pressure deviation from hydrostatic pressure, g = (0,0, -g) is vector of the gravity acceleration, Vj = pj /pj is the kinematic viscosity, Xj is the thermal diffusivity, D is the coefficient diffusion and a is the thermal diffusion parameter. All thermophysical parameters are assumed to be constant and correspond to the average values temperature and concentration.
The boundary conditions on solid walls are
z = -li : ui = 0, 61 = 6io(x, y, t);
z = I2 : U2 =0, 62n = 0, cn = 0. On the interface surface z = 0 the following conditions are set:
ui = U2, u • n = Vin, (P2 - Pi)n = 2aHin + ▽na;
61 = 62, k262n - Min = ®i6 Vii U2, Cn + a62n = 0,
where kj is coefficient of thermal conductivity, n is unit vector normal to the interface r and it directed into the domain to il2, Vin is velocity of motion of the surface r by n, Pj = -pj E + 2pj Vj Dj is the stress tensor, E is unit tensor, Hi is the mean curvature of the surface r, Vn = V - (n • V)n is the operator of surface gradient, a = a(6, c) is the coefficient of surface tension at the interface. For most mixtures, the linear law provides a good approximation
of this dependence a(ei, c) = a0 — «i(e — e0) — «2(c — c0) with the constants a0 > 0, > 0, «2.
Let us assume that solution of systems (1) has the form [4]
Uj = (fj (z,t) + hj (z,t))x, Vj = (fj (z,t) — hj (z,t))y, Wj = —2 fj (z1,t)dz1;
J zo
ej = aj (z,t)x2 + bj (z,t)y2 + ej (z,t), (2)
c = M(z, t)x2 + N(z, t)y2 + C(z, t).
Substitution of solution (2) into system of equations (1) leads to the following system of equations describing heat transfer in layers::
(4)
ajt + 2a j (fj + hj) — 2ajz fj (zi,t)dzi = Xj ajzz,
J 0
bjt + 2b j (fj — hj) — 2b jz fj (zi ,t)dzi = Xj bjzz, (3)
J z 0
j — 2ejz fj (zi ,t)dzi = Xj ejzz +2xj (aj + bj).
0
From the equation of momentum and continuity, we obtain
fjt + j + hj — 2fjzi fj (zi, t)dzi + sji(t) = z / ) = Vj fjzz — g J (j (aj (zi,t) + bj (zi,t)) + Sj pc(M (zi,t)+ N (zi,t))J dzu
hjt + 2fj hj — 2hjz fj (zi,t)dzi + Sj2(t) = ■Jo
= Vjhjzz — g J^ (pej (aj (zi,t) — bj (zi,t))+ Sj¡3c(M(zi,t) — N(zut))^ dzu
where Sji(t), Sj2(t) are arbitrary functions. Physically, they represent additional pressure gradients. Here and below, we assume that Si = 0.
Equations for determining the functions describing the distribution of concentration in a layer with a binary mixture have the following form:
Mt + 2M(f2 + h2) — 2Mz i f2(zi,t)dzi = DMzz + aDa2zz,
J0
Nt + 2N (f2 — h2) — 2Nz j f2(zi,t)dzi = DNzz + aDb2zz, (5)
■Jo
Ct — 2Cz J f2(zi,t)dzi = DCzz + 2D(M + N) + aD^zz + 2(a2 + h^j .
0
The pressure functions in the layers are determined by the formulas: pj Pj = x2[g (j aj (zi,t)+ Sj (3cM (zi,t))dzi + Uj^j +
z
2 e c
(g J ß bj (zi,t) + SjßcN(zi,t))dzi + Uj^j -- 2vj fj - gz + g J ßS(zi,t) + SjßcC(zl,t))dzl - 2 ^ J fj (zut)dz^ + j
qjo are arbitrary constants.
Substituting solution (2) into the boundary conditions, we obtain the following relations being the result of the no-slip condition for velocities
rl 2
/iHi) = hiHi)=0, f2(l2) = h2(l2 ) = / f2(z1,t)dz1 =0. (6)
■J 0
We assume that the upper wall is thermally insulated and impenetrable, and at the lower boundary the temperature distribution has the form ai = ai0x2 + bi0y2 + Ti0. Then,
z = -li : ai = ai0, bi = b№ di = Tw, z = I2 : a2Z = b2Z = h* = 0, Mz = Nz = Cz = 0. Boundary conditions at the interface z = 0 are written as:
fi = /2, hi = h2, ai = a2, bi = b2, Si = S2,
(7)
P2V2f2z - Pivifiz = + b2j - œ^M + N
p2v2h2z - pivihiz = -œi ( a2 - b^ - œ^ M - N
(8)
(9)
k2a2z - kiaiz = 2œiaifi, k2b2z - kibiz = 2œibifi,
k2S2z - kiëiz = 2œiëifi,
Mz + aa2z =0, Nz + ab2z =0, Cz + aS2z = 0.
The kinematic condition on the immovable and non-deformable interface is equivalent to the integral equality
f fi (zi,t)dzi =0. J-i 1
-h
For the complete definiteness of the problem posed, it is necessary to set additional integral conditions
/0 rl2
hi(zi,t)dzi =0, / h2(zi,t)dzi = 0. (10)
-h J0
Boundary conditions (6)—(10) correspond to the initial conditions at t = 0
Uj (x, 0) = u0j (x), 0j (x, 0) = Û0j (x), c(x, 0) = c0(x)
with the given u0j (x), 00j (x), c0(x) on their domains. For a smooth solution, these functions must satisfy the matching conditions.
2. Solution of a stationary problem
The stationary case of problem (3)-(10) is considered. We introduce the nondimensional variables x = Çlj, y = nlj, z = Çlj,
fj = Fj, hj = lf Hj, aj = a*Aj, bj = a* Bj, = Tj,
-, = f Sij, M = ßO- Kl, N = ßßf- K2, C = ß2^K3, Pj = gßj a'l* pp.
Here, a* = max{|ai( —1)|, |&i( —1)|} > 0, 9* is the characteristic temperature at the point x = 0, y = 0, z = —1.
Then, system of equations (3)-(5) in dimensionless variables takes the following form:
2Aj(Fj + H3) — 2AKfC Fj(Ci)d(i = f j, Jo Xi
2Bj(Fj — Hj) — 2BjJ Fj(Zi)dZi = X1 BjK, (11)
Jo Xi
—2TKf Fj(Zi)dZi = jTjzz + 2XXdi(Aj + Bj). Jo Xij Xi
Ff + H, - 2Fjz Jo F,(Ci)dCi + Sji = jjFjzz - G, ^ (ä, + Bj + S,(K + K2)^jdÇu
0 2 °rZf \ (12)
2Fj Hj - 2HjZ IF (C1)d(1 + Sjf == j HjCC - Gj ^ [A, - Bj + S, (K - K2)j dÇu
2K\(F<2 + H,) - 2KiC J Ff(Zi)dZi = Lel2 (kkc + ^A2CC 2Kf(Ff - H,) - 2Kfz 0 Ff((i)d(i = Lel2 (K2CC + №2«^), -2K3Z 0 F,(Zi)dZi = Le(l2K3<< + 2(Ki + K) + é Tfçç + 2(A2 + B2^ .
(13)
/0 0
i Fi(Ci)dCi = 0, J i Hi(Ci)d(i = 0; (14)
Ai = ai, Bi = a.2, Ti = a3; (15)
Z = 1: F2 =0, H2 =0, ( F2(Ci)dCi =0, ( H2(Ci)dCi =0; (16)
00
A2Z =0, B2Z =0, T2Z =0, Kiz =0, K2Z =0, K3C = 0; (17)
Z = 0: Fi = F2, Hi = H2, Ai = A2, Bi = B2, Ti = T2, (18)
é
Kiz + éA2C = 0, K2Z + éB2Z = 0, K3Ç + éT2C = 0, (19)
di
IA2Z - kAiz = 2AiFi, IB2Z - kBiz = 2œx Bi Fi, (20)
IT2Z - kTiz = 2œ^TiFi, (21)
lF2Z - pvFiZ = -Mpv(A2 + B2) - Mpvuj(Ki + K2), (22)
lH2Z - pvHiZ = -Mpv(A2 - B2) - Mpvuj(Ki - K2). (23)
Here, the following dimensionless complexes are introduced in the problem: the Prandtl number Pr., the Marangoni number M, the Grashof number G., the Lewis number Le and other parameters and relationships determined by the formulas
vL -a* li g(j liL Pi D
PrL = vL, m = —-1, GL = MPriLj, Ll = ——, Le = —,
Xj PiviXi -i Xi
= -2^2 d= a*H ^ = ad i = i± = Vi = Pi k = —
" = -i(c, 1 = 9* , P = (I , = l2 , V = V2 , P = P2 , = k2 ' To reveal the characteristic features of the thermocapillary flow, we consider an approximate analytical solution in each of the layers. To do this, we construct an asymptotic solution of the problem in the form of a power series in the parameter M << 1:
Fl = MF0 + M2 Fi, HL = MH0 + M2 Hl, A. = A0 + MA], Bj = B0 + MBi, Tj = T0 + MTl, (24)
K l = K0 + MK ], K2 = K0 + MK]; K3 = K0 + MK3i, Sij = mS0l + M2 Sj.
Substituting (24) into (11)-(13) and neglecting the terms with the parameter M, we obtain a linear system of equations. The desired functions that determine the fields of velocities, temperatures, and concentrations are found by simple integration (here and below, we omit the upper index of 0 denoting the first term of the expansion). As a result, we have
A i = CiZ + C2, B i = C3C + CA, Ti = -di(Ci + C3)c3 - di(C2 + C4)c2 + C5c + C6, Fi = L4(Ci + C3)Z4 + lT(C2 + C4)Z3 + Sri z2 + CrC + C8, Hi = (Ci - Ci)Z4 + L-(C2 - C4)Z3 + 2PriZ2 + CgZ + Ci0; A2 = CiiZ + Ci2, B2 = Ci3Z + Ci4,
Ti = -(Cii + Ci3)Z3 - ^(Ci2 + Ci4)Z2 + Ci5Z + Ci6, Ki = Ci7z + Ci8, K2 = Ci9Z + C20,
K3 = - 312 (Ci 7 + C i 9)Z3 - 12 (Ci 8 + C2o)Z2 + C2 i Z + C22,
F2 = VL2 (Ci i+ Ci 3 + Ci 7 + Ci 9)Z 4 + VLL2 (Ci 2 + Ci4 + Ci 8 + C20)z3 + 2S\l ^ + C23Z + C24,> H2 = vL2 (Ci i - Ci 3 + C i 7 - Ci 9)Z 4 + VL (C i 2 - C i 4 + Ci 8 - C20)Z 3 + ^p^^ Z 2 + C25Z + C26.
The unknown constants Cn, as well as the functions S. are found from the boundary conditions. Note that conditions (20), (21) taking into account the effect of energy on the interfaces under the above assumption will be rewritten in the following form:
lA2Z - kAiz = 2Edi AF, lB2Z - kBiz = 2EdXBF, (25)
IT2Z - kTK = 2Ed1T1F1, (26)
œ2d*
where E = —. Conditions (22), (23) take the form: p1v1k2
lF2C - pvFK = -pv(Â2 + B2 - + K2^, (27)
lHK - pvHiz = -pv(Â2 - B2 - u(Ki - K2)). (28)
3. Algorithm for computing the integration constants
To determine the integration constants, we use boundary conditions (14)-(19), given conditions (25)-(28).
From (15), (17), (18) we obtain that Cn = C13 = C\7 = Cw = 0, C24 = C8, C26 = C10, C12 = C2, C14 = C4, Ci6 = CQ, Ci = C2 - ai, C3 = C4 - a2. The third condition in (15) determines the connection
C5 = CQ--^(C2 + C4) - -31 (ai + a2) - a3.
Taking T2z =0 on the wall Z = 1, we have
C15 = —-T (C2 + C4).
Further, we consider conditions (19). Two of them are fulfilled identically. The third condition —^
gives relation C21 = —^ (C2 + C4).
l2
From the condition K3q(1) =0 we define C18 = -^(C2 + C4) - C20.
In the joint solution of equations (14), the following constants are determined:
C7 = 4C8 + Li( + 0i±0
7 1 ( —0 30
Co ( C2 - C4 ai - a2 C10 = —--Li
4 V 80 1—0
5,, = 6C8Pr, ± + 3a±«>Y
S12 = 3CoPri + PriLi( 11(C2 - C4) + 01-02
12 — 9 1 1 1V 40 10
Further, conditions (16) complete the following relation:
C23 = -4C8 - —2(C2 + C4)(^ - 1);
C25 = -4Cio - —2 (—C20 + HC2 + C4) - (C2 - C4)) ;
6l2 1
S21 = C8Pr 1 + — PriL2(^ - 1)(C2 + C4);
S22 = PriL2C20 + 6PV^Ci0 + (^(C2 + C4) - (C2 - C4))
Dynamic condition projection (28) results in relation
C = -^L^24^ C20 + C - CA - + CA]) +
6l(l + pv) l + pv \ )
+ 60ITH (3(C - ^ + 2a - - 12^ + ^ - (C- •
Condition of heat transfer (26) at the interface Z = 0 determines the constant C6
C 2di (3 + kl)(C2 + C4) kdi(ai + a2) ka3
6 = 3l(k + 2EdiC8) + 3(k + 2EdiC8) + k + 2EdiC8 •
Finally, we determine the constants C2,C4,C8,C20,C22 when substituting into boundary conditions (25), (27). These conditions are not enough. Therefore, we introduce the condition that determines the distribution of the average concentration in the layer
i
2
'0
From this, we obtain the following relations
f (Ki£2 + K3)dC — 71 e2 + 73. Jo
C20 — 72, C22 — (71 + 72) — 73, C2 — —1C4, 3l2 a2
C4 — — a2 (71 + 72) c8 — — k f1 + + "2)'
L + ^(ai + «2^
V 7i + 72 )'
^(«1 + a2y 2Edi\ 71 + 72
In addition, the following relationship between the physical parameters of the system results
a1 + a2 =------Y1 + 72---- (120kld(l + pv)+
2 2l^(60k^(l + pv) - pv EL1 d1(7l + l2))\ ' r J'
+5vdiE(^ - 1)(71 + 72) + 3pvld1E(L1 + 20(w^ - 1))(71 + 72)) .
So, all constants have been defined. The functions that determine the field of velocities, temperatures, and concentrations (2) are written out explicitly.
Conclusion
In this paper, an exact solution of three-dimensional Oberbeck-Boussinesque equations is found that describes the stationary flow of a two-layer system of liquid media with a common interface in a channel bounded by solid walls. The construction of exact solutions is of particular value in the study of mathematical models of fluid dynamics in domains with interfaces. The solutions in the close formulas make it possible to determine the role of different mechanisms in the formation of certain types of flows.
The solution obtained can be used in the development of experiments to study joint flows of liquid media in a closed channel. In further work, it is planned to analyze the physical parameters of the system and to select fluids that meet the conditions of the problem, to elucidate the effect of thermocapillary, gravitational and other forces on the nature of the flow, and to consider special cases of heating a solid substrate. In particular, when there is radial heating at a10 = 610 (7) or the case when a10 + 510 = 0. Moreover, the constructed solution is planned to be used as a test at finding of corresponding non-stationary solution during a modeling of evolution process
of heat and mass transfer with parabolic temperature field. Also using the obtained solution as a test one, it is planned to simulate a non-stationary process.
This work received financial support from RFBR (20-01-00234).
The author is grateful to V. K. Andreev for formulation of the problem and useful discussions.
References
[1] V.K.Andreev, Yu.A.Gaponenko,O.N.Goncharova, V.V.Pukhnachov, Mathematical models of convection (de Gruyter studies in mathematical physics), Berlin/Boston: De Gruyter, 2012.
[2] V.B.Bekezhanova, O.N.Goncharova, I.A.Shefer, Analysis of an Exact Solution of Problem of the Evaporative Convection (Review). Part I. Plane Case, J. Sib. Fed. Univ. Math. Phys., 11(2018), no. 2, 178-190. DOI: 10.17516/1997-1397-2018-11-2-178-190
[3] V.B.Bekezhanova, O.N.Goncharova, I.A.Shefer, Analysis of an Exact Solution of Problem of the Evaporative Convection (Review). Part II. Three-dimensional Flows, J. Sib. Fed. Univ. Math. Phys., 11(2018), no. 3, 342-355. DOI: 10.17516/1997-1397-2018-11-3-342-355
[4] V.V.Pukhnachov, Model of a viscous layer deformation by thermocapillary forces, European Journal of Applied Mathematics, 13(2002), 205-224. DOI: 10.1017/S0956792501004776
Построение точного решения специального вида для трехмерной задачи термоконцентрационной конвекции в двухслойной системе
Марина В. Ефимова
Институт вычислительного моделирования СО РАН Красноярск, Российская Федерация
Аннотация. Рассмотрено трехмерное совместное течение жидкости и бинарной смеси с общей границей раздела. Предполагается, что поле температуры в слоях имеет квадратичное распределение. Построено точное решение некоторой модельной задачи. Получены явные выражения для всех искомых функций с помощью определенного замыкающего соотношения.
Ключевые слова: приближение Обербека-Буссинеска, поверхностная энергия, бинарная смесь.
