On one two-dimensional stationary flow of a binary mixture and viscous fluid in a plane layer Текст научной статьи по специальности «Физика»

УДК 532.5

On One Two-dimensional Stationary Flow of a Binary Mixture and Viscous Fluid in a Plane Layer

Marina V. Efimova*

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

Russia

Received 12.10.2015, received in revised form 10.11.2015, accepted 20.12.2015 Nonlinear model of convection in Oberbeck-Boussinesq approximation describing the flat joint motion of a binary mixture and viscous fluid with a common interface is investigated. It is important that the longitudinal temperature gradient and the concentration is quadratic dependence on the coordinate x. Stationary solution of the system is built.

Keywords: Oberbeck-Boussinesq equations, convective motion, binary mixture, steady flow. DOI: 10.17516/1997-1397-2016-9-1-30-36.

Convection is one of the most common of hydrodynamic phenomena in nature. Study of convection is an important part of the theoretical fluid mechanics. Natural convective motions occur in an inhomogeneous field of mass forces caused by the ununiform heating of the liquid. The area of practical applications of this phenomenon is very wide. Convective processes influence the thermal conditions in the oil storage tanks, chemical process technology and others. Theoretical study of natural convection usually deals with equations of motion in the Oberbeck-Boussinesq approximation. Problems for thermal convection are very complex because of the diversity of cavities and thermal boundary conditions for the nonlinear system of partial differential equations. Solutions of the Oberbeck-Boussinesq equations with a linear dependence of temperature on one of the space coordinates firstly were studied by G. A. Ostroumov [1]. The exact solution described plane stationary flow in a strip under action of longitudinal temperature gradient and transversal gravity field, was obtained by R. V. Birikh [2]. Some generalizations of this solution taking into account concentration of liquid mixture are described in [3]. The existence of solutions with nonlinear dependence of density on temperature and concentration is proved in [4] where two boundary value problems with exponential temperature distribution on the walls are solved. In [5-7] the exact solutions of the three-dimensional convection problem for two immiscible viscous, incompressible fluids in a channel with a rectangular cross section in the presence of the interface and under the influence of a longitudinal temperature gradient are studied. In this paper we are built the exact solution of the two-layer convection system with longitudinal temperature gradient at the solid walls and shear force of gravity.

1. Problem formulation

Let us consider the joint motion of a binary mixture and viscous fluid with a general interface. Suppose that Qi = { |x| < to, 0 < y < l1} is the region occupied by a binary mixture and

* [email protected] © Siberian Federal University. All rights reserved

Q2 = {|x| < ro, 11 < y < 12} is the region with a viscous fluid. The system is bounded by solid walls y = 0 and y = 12 with given temperature distribution on them.

For description of the motion of both region Qj (j = 1, 2) we use the Boussinesq approximation. We assume that the temperature and the concentration differ only slightly from constant mean values therefore the Oberbeck-Boussinesq approximation is valid

Pj = poj(1 - je - j),

where p0j is the characteristic medium density corresponding to the mean values of the temperature and concentration in the layer j, e and c are the deviations from their mean values (c corresponds to the light component), yj and yj are the temperature and concentration expansion coefficient; yc = 0. Then the equations of binary mixture convection can be written in the form

Ujt + (Uj • V)uj = -—Vpj + VjAuj - (0j - Ooj) + jcj - c0j)), p0j

6jt + Uj • Vdj = Xj AOj, cit + ui • Vci = diAci + adiAdi, div u,- = 0.

(1)

Here uj is the velocity field, pj is the pressure measured from the hydrostatic pressure corresponding to p0j, p0j is the density, Vj is kinematic viscosity, Xj is the temperature conductivity, d1 is the diffusivity, ad1 is the thermal diffusion coefficient. Normal thermal diffusion corresponds to the value of a < 0, and for the anomalous a > 0.

We introduce the coordinate system with the x axis aligned with the lower boundary of the layer 1 and the y axis directed vertically upward (Fig. 1).

////////////////z y /////////////////

l2

h

\\\\\\\\\\\\\\\\\ ' o

Fig. 1. The scheme of two-layer flow between the rigid walls with interface y = 1i

Let us define the boundary conditions. On solid walls are put no-slip conditions and the temperature distribution and the absence of mass flux through the walls are written as

y = 0: ui = 0, 0 = 0io(x), ciy + aOiy = 0; y = ¿2 : u2 =0, 0 = 02o(x).

(2)

At the interface y = li the conditions of equality of the velocities, kinematic and dynamic conditions are written as

ui = U2, vi = V2 = 0,

P2V2U2y - PiViUiy = -œiOix - œ2cix.

X

The condition of temperature continuity and the equality of heat fluxes are as follows

Oi = O2, k1d1y = k262y. (4)

In addition, the condition of absence of mass flux through the interface is

ciy + aOiy = 0. (5)

Here kj = XjPjcPj are the thermal conductivities, a = a(O, c) is the coefficient of surface tension. For many mixtures, the linear law provides a good approximation of this dependence

a(O, c) = a0 - œi(O - O0) - œ2(ci - c0),

where œi > 0 is the temperature coefficient and œ2 is the concentration coefficient (usually œ2 < 0 since the surface tension increases with concentration). Constants O0, c0 are the temperature and concentration values of arbitrary point on interface.

We should to add the initial conditions: uj = 0, Oj = Ojj(x,y), ci = c0(x,y). All the physical characteristics of the system are assumed to be constant and correspond to the mean temperature and concentration.

2. Exact solution of the two-dimensional problem

We find the form of the solution describing the convective flow in the system of liquids with the interface in the form

Uj = Uj (y,t)x + Wj (y,t), Vj = Vj (y,t); Oj = Aj (y,t)x2 + Bj (y,t), ci = Hi (y,t)x2 + Ei(y,t), (6)

Pj = P (x,y,t).

The substitution of solution (6) into equations of motion (1) gives the relations

Vj Ujyy - Ujt - Uj2 - Vj Ujy = 2g I' (j Aj + j Hj )dy + sj (t),

JQj

1 j x2 —Pj = (vjUjyy - Ujt - U2 - Vj Ujy^ + hj (7)

j \ jjyy Jt j j jy/ 9 pj 2

j Vjy + g(j bj 1 yj EJ

hjy = Vj Vjyy - Vjt - Vj Vjy + g(j Bj + jEj ),

Wj =0, Vjy = -Uj.

The equations for determining of the temperature and the concentration field take the form

Ajt + 2Uj Aj + Vj Ajy = Xj Ajyy ,

Bjt + VjBjy = Xj (2Aj + Bjyy) ,

(8)

Hit + 2UiHi + ViHiy = di (Hiyy + aiAiyy),

Eit + ViEiy = di (2Hi + Eiyy + ai(2Ai + Biyy)). The following boundary condition at solid walls are held

Ui(0,t)=0, U2(l2,t)=0, Ai(0,t)= Aw(t), A2(l2,t) = A20(t);

Bi(0,t) = Bi0(t), B2(l2,t)= B20; (9)

Hiy (0,t)+ aiAiy (0,t) =0; Eiy (0,t) + aiBiy (0,t)=0.

The boundary conditions at the interface y = li are:

U = U2, P2V2^2y - piviUiy = -2«iAi - 2^Hi, Ai = A2, kiAiy = k2^2y; Bi = B2, kiBiy = k2B2y; Hiy + aiAiy =0; Eiy + aiBiy = 0; rh fl'2

/ Ui(y,t)dy = 0, / U2(y,t)dy = 0. (11)

JO ./¡1

The first from conditions (11) is a consequence of the kinematic conditions, when the interface is stationary, and the second one is the slip condition for velocity components V2(y,t) on the wall of y = /2.

The initial data are written in the form

Uj(y, 0) = 0, Vj(y, 0) = 0, Aj(y, 0) = a°(y), Bj(y, 0) = b0(y), Hi(y, 0) = H0(y), Ei(y, 0) = E0(y).

Note that this problem is nonlinear and inverse. Because of the function sj(t) remains unknown as well as functions Uj, Vj, Aj, Bj, Hi, Ei.

We introduce the characteristic length scales, time functions Uj, Vj ,Pj, Aj, Bj, Hi, Ei, hj, Sj respectively

x = lie, t = -t, U,. = u; , v,. = v.* , p = ffiiAA/iP;,

Vi PiVi j PiVi j j

ft9 A A ft9 A A/2

Aj = AAA*, Bj = AA/2B*, Hi = ^ H, Ei = E*, (12)

Pi Pi

= aiAA/ i = ffiiAA *

h j — h j, s, — s j,

j pi j j pi/i j

where AA = max |A20(t) - A i0(t)| > 0. If A20(t) = A i0(t) than AA = maxmax |Aj0(y)| > 0.

j y

We have the multiplier

m=aiAAA/i (13)

P iv2

called the Marangoni number at the nonlinear summands in equations (7)-(8) written with dimensionless variables. Let us mention here also Prandtl number Prj, Schmidt number Scj, parameter Gj = Grj/M (where Grj is Grashof number), parameter w and split ratio

Prj = j, Scj = Vj, Gj = g^, w = ^i, * =

3. Stationary solution

In the present section we describe stationary solution of (7)-(8). Assume that the motion is creeping in one layers so M C 1 and parameter Gj = Grj/M = O(1). These conditions can be valid in either thin layer or very viscous fluid according to the formula (13). In such case the steady state problem (7)-(8) has a special form

jj = 2Gj/ (Aj + §Hj)dn + sj vi ./n, Pi

A _ 0 R __(14)

Ajnn 0 Rjnn 2Aj ,

Hi nn = 0, Ei nn = —2Hi ;

j = -Uj, P = X2 + h-

Pj 2v1 Ujwx + hj , (15)

Vj Pj

hjn = ~ Vjnn + Gj Bj + GiEj. vi pi

Integrating (14) provides a solution to the problem as the

Ai = min + , A2 = m^n + m4,

mi 3 2 m3 3 2

Bi = -—n - m2n + m5n + me, B2 = - — n - m^n + m7n + m8,

TT n m9 3 2

Hi = m9n + mio, Ei = —— n - mwn + mnn + mn, (ig)

„ n fmi+mg 4 m2 + mio 3\ Si 2 .

Ui = Gi {-12-n +--3-n ) +2n + mi3n + mu,

„ (m3 4 m4 3\ VS2 2

U2 = vG2y—n + -3-n ) + n + mi5n + mi6.

Constants mi, i = 1,16, s^ s2 in (16) determined from the boundary conditions (9)-(11) and have the form

l(A20 - Aio) A20 l - kmi mi = ~l-kl + k , m2 = Aio, m3 = kmi, m4 =-j-, m6 = Bio,

m5 =

3l2(2kl-2k - l)Ai0-3l (l-1)2A20 + 3 l3(Bio -B20)+ mi (l (kl2 + 3kl- l2 - 6 k) +2k)

mr =

3l2 (kl - k - l)

3Ai0 kl3 + 31 (kl2 - 212 - k)A20 + 3 kl3 (Bi0 - B20) + k (kl3 - 3 kl2 - l3 + 612 +2 k) mi

3l2 (kl - k - l) :

3Ai0kl2 + 3l (kl - k - 2l + 1) A20 + 3l2 (B20l(1 - k) - Biok)

m8 =

3l2 (kl - k - l) k (kl2 - 3 kl - l2 +2 k + 61 - 2) mi

3l2 (kl - k - l)

mg = 'mi, mio = 'Aio, mii = 'm5, mi4 = 0,

(' + 1)((7pv (l - 1) - 41) mi + 5 Aio (3pv (l - 1) - 21)) Gi

mi3 =-7x7-;-n--+

60(pv l - pv - l)

+ V (l - 1)3 (3 k (l - 1) mi +5 A2o l) G2 vp ('u + 1) (l - 1) (Aio + mi)

mi5

60l3 (pvl - pv - l) pvl - pv - l '

lp (' + 1)(l + 2) (5 Aio + 3 mi) vGi lp (' u + 1) (l + 2) (Aio + mi) v

60(lvp - pv - l)(l - 1) (lvp - pv - l)(l - 1)

5 (2 v (l - 1) (l2 +41 + 1) p - 312 (l + 3)) A2ovG2 _ 60l2 (lvp - pv - l)

kmi (2 p (l - 1) (213 + 312 - 121 - 3) v - l2 (712 +61 - 33)) v G2

60l3 (lvp - pv - l)

_ vp + 1)(21 + 1)(5 A10 + 3 mi) Gi - 60vp (^w + 1) (21 + 1) (Ai0 + mi) mi6 = 120 (1vp - pv - 1) (1 - 1) +

vG2A20 (1 - 1) (201 (12 - 1) pv - 51 (612 +31 - 1)) + 12013 (1vp - pv - 1) (1 - 1) +

vG2 (1 - 1) k (2 (1 - 1) (412 - 31 - 6) p v - 1413 + 1512 + 121 - 3) m1 + 12013 (1vp - pv - 1)(1 - 1) '

(^ + 1) ((1 - 1) (25 Aio + 9 mi) p v - 21 (10 A10 + 3 mi)) Gi

si = —

20 (pvl - pv - 1)

v (1 - 1)3 (3 k (1 - 1) mi + 5 A20 l) G2 - 6013vp (^w + 1) (1 - 1) (Ai0 + mx)

2013 (pvl - pv - 1) '

12p (^ + 1) (5 Aio + 3mi) Gi - 6012p (^w + 1) (A10 + mi)

S2 =---h

2 20(1 - 1)(1pv - pv - 1)

(1 - 1)2 (3 k (1 - 1) (2 pv (1 - 1) - 31) mi +5 A20 1 (4 pv (1 - 1) - 51)) G2 + 20(1 - 1) (1pv - pv - 1) 12 '

The expression for Vj, Pj can be obtained with help of formulas (15) and have not shown here because cumbersome.

On Fig. 2a we represent the velocity profiles. In the upper layer the vertical velocity component V for parameters Ai0 = 0.1, A20 = -0.3, Bi0 = 25, B20 = 20, Gi = 1.05, G2 = 0.98, M = 0.034 is positive and the horizontal component of the velocity changes sign. The fluid moves vertically when x = 0 along the y axis and vertically downward in the lower layer and symmetrically rotated about the axis y (Fig. 2b).

The work received financial support from RFBR (14-01-00067).

References

[1] G.A.Ostroumov, Free convection under the conditions of the internal problem, Washington, National Advisory Committee for Aeronautics, 1958.

[2] R.V.Birikh, Thermo capillary convection in a horizontal layer of liquid, J. Appl. Mech. Tech. Phys., 5(1966), 69-72.

[3] V.K.Andreev, Birikh solution of convection equations and some their generalizations, Preprint of ICM SB RAS, Krasnoyarsk, 2010 (in Russian).

[4] V.K.Andreev, I.V.Stepanova, Ostroumov-Birikh solution of convection equations with nonlinear buoyancy force, Applied Mathematics and Computation, 228(2014), 59-67.

[5] V.V.Pukhnachov, Group-theoretical nature of the Birikh solution and its generalizations. Symmetries and Differential Equation, Trudy rossiiskoi konferentsii po simmetrii i differ-entsial'nym uravneniyam, Krasnoyarsk, Inst. Comput. Modeling, Sib. Branch Russian Acad. of Sci., 2000, 180-183 (in Russian).

[6] O.N.Goncharova, O.A.Kabov, V.V.Pukhnachov, Solutions of special type describing the three dimensional thermocapillary flows with an interface., Int. J. Heat Mass Transfer, 55(2012), no. 4, 715-725.

[7] V.V.Pukhnachov, Non-stationary Analogues of the Birikh Solution, Nauchnyi zhurnal teo-reticheskih iprikladnyh issledovanii. Novosti Altaiskogo Gos. Universiteta, 69(2011), no. 1-2, 62-69 (in Russian).

Двумерное стационарное течение бинарной смеси и вязкой жидкости в плоском слое

Марина В. Ефимова

В настоящей статье рассматривается нелинейная модель конвекции в приближении Обербека-Буссинеска, описывающая плоское совместное движение бинарной смеси и вязкой теплопроводной жидкости с общей поверхностью 'раздела. Важно, что продольный градиент температуры и концентрации имеет квадратичную зависимость от координат х. Построено стационарное решение системы.

Ключевые слова: уравнения Обербека-Буссинеска, конвективное движение, бинарная смесь, установившееся течение.