CC BY
6
УДК 532.51
Influence of the Interfacial Internal Energy on the Thermocapillary Steady Flow
Victor K. Andreev*
Institute of Computational Modelling Siberian Branch of the Russian Academy of Sciences Akademgorodok, 50/44, Krasnoyarsk
Institute of Mathematics and Fundamental Informatics
Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041
Russia
Received 10.05.2017, received in revised form 10.06.2017, accepted 20.07.2017 Two-dimensional creeping motion of a two immiscible viscous heat-conducting fluids on the interface for which the surface tension depends linearly on the temperature is investigated. On solid walls the temperature has extreme values and this degrees well with the velocity field of the Hiemenz’s type. At small Marangoni numbers an exact solution of arising inverse boundary value problem is found. The estimation of degree of influence of the interfacial internal energy on the stationary flow is given.
Keywords: interface, thermocapillary, interfacial internal energy, inverse problem.
DOI: 10.17516/1997-1397-2017-10-4-537-547.
1. Statement and transformation of the problem
The system of the two-dimensional stationary motions of viscous heat-conducting fluids in the absence of mass forces has the form
u1u1x + u2u1y + p px v (u1xx + u1yy ); (1.1)
u1u2x + u2u2y + ppy v(u2xx + u2yy); (1.2)
u1x + u2y 0; (1.3)
u1ex + u2ey x(exx + eyy ^ (1.4)
where u1(x, y), u2(x, y) are the components of the velocity vector, p(x, y) is the pressure, d(x, y) is the temperature, p > 0, v > 0, x > 0 are the density, the kinematic viscosity, the thermal diffusivity, respectively. The values of p,v,x are represented by constants.
Suppose, that ui = u1(x,y), u2 = v(y), p = p(x,y), в = 9(x,y) is solution of the system (1.1)-(1.4). Substitution of this solution in equations (1.1)-(1.3) leads to relations
ui = w(y)x + g(y), w + vy =0,
wt + vwy + w2 = f + vwy
1 d( ) fx2
-pp = d(y),
dy = vvyy - vvy, vgy + wg = 0
(1.5)
with an arbitrary constant f.
© Siberian Federal University. All rights reserved
Equation (1.4) for the temperature is rewritten as
(wx + g)0x + v0y = x(Qxx + 0yy )■
Among its solutions there are quadratic one relatively of the variable x
0 = a(y)x2 + m(y)x + b(y). (1.6)
Below, for simplicity, we assume that g(y) = 0 and m(y) = 0. The latter means that the temperature field has an extremum at the point x = 0, more precisely, at a(y,t) < 0 it has a maximum and at a(y,t) > 0 it has a minimum. Let us apply the solution (1.5), (1.6) to describe the two-layer motion of the viscous heat-conducting fluids in the flat layer with solid walls y = 0, y = h and common interface Г y = l(x), see Fig. 1.
/ / / / / / / 1 У v\\\\\\\\
liquid 2
у =i(x) Y c= cr1]-Ee(01-0o)
^ h
liquid 1 X
////// / / // // // /
Fig. 1. Schematic diagram of liquid flows in a horizontal layer with interface
Let us introduce index j = 1, 2, fixing the fluid. Then in the domain 0 < y < l(x) the functions wi(y), vi(y) satisfy the equations
Upon that
VlWiy + w1 = VlWlyy + fi, wi + Viy =0.
1 d ( ) fix2 d
— Pi = di(y)----— , diy = viviyy - viviy.
Pi 2
Similarly, in the domain l(x) < y < h yields
v2w2y + w2 = V2w2yy + f2, w2 + v2y = 0;
1 d ( ) f2x2 d
— P2 = d2(y)------v— , d2y = V2v2yy - v2v2y.
P2 2
(1.7)
(1.8)
(1.9)
(1.10)
Besides, in the same domains of definition (j = 1,2) the unknowns aj, bj satisfy the equations
On the interface y
2wj aj + vj ajy = Xj ajyy; (1.11)
vj bjy = Xj bjyy + 2xj aj. l(x) the following conditions are imposed [1]: (1.12)
wi(l(x)) = w2(l(x)), vi(l(x)) = v2(l(x)); (1.13)
xwi (l(x))lx = vi(l(x)); (1.14)
ai(l(x)) = a2(l(x)), k2 - ki= xaiwi; on on (1.15)
- 538 -
bi(l(x))
b2(l(x)),
db2 dbi
---M——
dn dn
&biwi.
(1.16)
Here k1, k2 are the constant coefficients of heat conductivity and normal to curve y = l(x) is the vector n = (1 + lX)-i/2(—lx, 1).
The order of the relation of the right side of equality (1.15) (or (1.16)) to the first term of its left-hand side is estimated by the parameter E = юг9*/p2k2 (piki should be put for the second case), where 9* is characteristic temperature on the interface. These parameters for ordinary liquids at room temperature are small [2]. So, in the experiments for the air-ethanol system at 9* = 15oC we have E ~ 5 • 10-4. Therefore, the right-hand side in (1.15) and (1.16) is often omitted and it is said about the equalities of the heat flux across the interface. However, for liquids with low viscosity these terms must be taken into account. Calculations [3] carried out for the bubbles motion in various liquids show that values E = 0(1) are reached at sufficiently high temperatures. This means that the viscosity rapidly decreases with increasing temperature. Besides, the same fact occurs for some cryogenic liquids, for example, for liquid C02. The maximum values of E near the critical points are reached. So for the water E ~ 0.02 at 9 = 303.15 K; E ~ 0.6 at 9 = 573.15 K; E ~ 0.7 at 9 = 623.15 K (critical point for water 9кр = 647.30 K). In the present work the influence of the right-hand side of (1.15) on flow dynamics will be taken into account in the framework of the creeping flow model.
Dynamic condition on the interface has the following form [1]
(pi — P2)n +2[p2D(u2) - ^iD(ui)]n = 2aKn + Viia, pj = pjVj, (1.17)
where a(9i) is the surface tension coefficients, K is the average curvature of the interfaces, Vii = V — n(n • V) is the the surface gradient, D(u) is the velocity-strain tensor. Further, we suppose that (see Fig. 1)
a(9i)= a0 — ®(9i — 90), (1.18)
a0 > 0, ж > 0 are the constants, 90 is the temperature in the some point of the interface.
Projecting (1.17) on the tangential directions т = (1 + lX)-i/2(1, lx) and using dependence (1.18) we obtain
[p2D(u2) — piD(ui)]n • т = —*Vii9i • т (1.19)
at y = l(x), uj = (xwj(y),vj(y)). In our case
D(uj)
x
/ Wj — Wjy
j 2 j y
x
\ 2 wjy vjy
(1.20)
Now we rewrite condition (1.19) taking into account representation (1.6) and (1.20) for the temperature at m = 0
lx [^2 (v2y — W2) — pi(viy
wi)] + I (1 — lX)(PXW2y — Piwiy) =
= — *(9ix + lx9iy) = — *[2aix + lxXaiy x2 + biy)]. (1.21)
Projection (1.17) on the normal n with use of formulae for the pressure from (1.5) results in the equality
Pi di — p2d2 +
[P2f2 — Pifi]x2 2
+ 2[p2D(u2) — piD(ui)]n • n =
= [a0 — ^(aix2 + bi)]
(1 + lX)3/2'
(1.22)
xx
Boundary conditions on the solid walls are the following
wi (0) = 0, ui(0)=0, w2(h)=0, V2(h)=0;
(1.23)
ai(0) = aio, a2(h) = a2o; (1-24)
bi (0) = bio, b2(h) = 620 (1-25)
with specified constants aj0, bj0, j = 1, 2.
Conditions (1-24), (1.25) correspond to the temperature on solid walls is given. Another condition can be specified, for example, the top wall is thermally insulated: a2y(h) = 0, b2y(h) = 0.
Note the following features of the problem. It is strongly nonlinear and inverse, since constants fj are unknowns also. It is easy to understand this, if we exclude Vj (y) from the second equations in (1.7), (1.9). Then the problem reduces to the conjugate problem for functions wj(y), aj(y) and l(x). The problem for functions bj(y,t) separates at the known functions Vj(y,t) and aj(y,t). The functions dj(y,t) can be restored by quadratures from (1.8), (1.10) up to time functions. The second boundary condition in (1.13) and the last condition in (1.23) are helpful for determining of the constants fj, j = 1,2.
In real situations, for many liquid media the value a0 is very large. Therefore, the relation (1.21) gives lxx = 0, i. e. l(x) = ax + l and at a0 ^ <x the interface can be straight only. Further, we assume that it is parallel to the solid walls y = 0, y = h, therefore a = 0, l = const. The solution of the problem is found in the following form
ewj1 + e2wf'> + ...,
b1)
aj = eaj1 + e2d'2') + ..., bj = ebj1 + e2bj2') + ..., fj = ej + e2 fj2 + ...
evjji) + e2vf) + ..., (i)
j
w
V
j
j
where e is the formally small parameter. Substituting these expressions into the corresponding equations and boundary conditions and passing to the limit at e ^ 0, we obtain for wj11, vj11,
aj1^, bj1, fj1 the linear problem. The boundary conditions (1.15), (1.16) will be homogeneous for the problem, i.e. the effect of interfacial energy on the motion is absent.
(1.26)
In the first approximation the problem has the form
wjyy = - f (i ) j_2 w( 1) + vj y) = 0,
jyy vj '
a(1) = =0, b(1) = -2a(12
jyy jyy - j
with boundary conditions
w(1)(0) = 0, v(i)(0) = 0, w(2i)(h)=0, v2i)(h)=0,
a(i)(0) = ai0 , a2 1)(h) = a20 , b[i)(0) = bi0, b2i)(h) = b20:
k2a2iy\l) = kia\y (l), k2b2y (l) = kib\y (l), ai> (l) = a%> (l),
0,
b(i)(l)
,(i)f
-(1)i
b2\l), v[1\l) = v2r)(l)
№w2i(l) - Miwiy(l) = -2&aii;(l),
,,(i)
wi1)(l) =
(1h
(1.27)
pl pn
/ w^y dy = 0, / w21)(y) dy = 0.
J 0 Jl
The second approximation leads to inhomogeneous equations within the domains of definition (0 < y < l at j = 1 and l < y < h at j = 2)
(2) J3
W • = ——
3УУ
(2)
1
+----
v3 v
3
^•3 + 3
(2) . (2) n
w3 + v3y = 0
(2)
1 (2w3->„3.. + j jj)
(1.28)
(2)
—-a4 + - j
x3
„(1^(1)
In boundary conditions (1.27) the following changes takes place (upper index “1” should be changed by index “2”)
3УУ
3УУ
a!2) (0)=0, a(2)(h)=0, 6(12)(0) = 0, b(\h) = 0,
k2«2У^(l) - kia{J(l) = ^a11)(l)w11)(l), (1.29)
М2У}(1) - k16(12y)(l) = ^6(11) (l)w[r)(l).
2. Solution of boundary value problems of the first and second approximation
Problem (1.26), (1.27) has solution wj.1)(y), aj.1^(y), f3(1):
(1)( ) = ^(1 - q)Afc(3y2/fc2 - 2qy/h)
W1 (У) 27M2[7 + M(1 - Y)] ’
(1), ч ^Ah(3y2/h2 - 2(2 + y)y/h + 1 + 2y)
w2 (y) = —
a11) (y)
2(1 - yWy + M(1 - Y)]
(a20 - a10) У .
T + a10j
[Y + k(1 - Y)] h
a21)(y) =
f (1) = h = -:
Y + k(1 - y) 3*v(1 - y)A
k(a,20 - aw)T + kaw + y(1 - k)a,20 h
f (1) =
J 2 = -
3^yA
Yhp2 [y + M(1 - y)] ’ 2 (1 - Y)hp2 [y + M(1 - Y)]
where k = k1 /k2, v = V1/V2, Y = l/h < 1, у = M1/M2,
1
(2.1)
A
Ya20 + k(1 - Y)a10 Y + k(1 - y)
Velocities v(1)(y) are found by integrating w(1)(y):
(1)
v(1)(y) = -
ж(1 - Y)Ah2 f y
2YM2 [y + M(1 - Y)] \h3
tSYAh?
Zy3 _ Yy2\
Vh3 h2 у
(1b 8 ________&YAh_________Гу2 . . 0 +
V2 (y) 2(1 - yWy + b(1 - y)] |_h3 (2 + Y) h2 +(1 + 2y) h Y
(2.2)
(2.3)
Functions bj1'l(y) are obtained also:
W) = -h
y2 + (a20 - aw) y3
“io h2 + 3[y + k(1 - 7)] h3
b(i)(y) =__________h________
2 (У) Y + k(1 - Y)
+ Cy + bio,
y2
Itta^opo,) V± + [kauJ + y(i - t)02o]£} + Diy + D2
(2.4)
where k = к1/к2, b20 - bi0
C=
+
h
h[Y + k(1 - y)] 3[y + к(1 - y)]2
x 2
D1 = kC -
D2 = l(1 - k)C +
к + y(k - 1) ^3k(1 - y)2 - Y2 2l(k - !)[y&20 + k(1 - Y)aio]
^ aio + k - y(k - 1^4y2 - 6y + 3^ a2oj,
(2.5)
Y + k(1 - y)
2l2 (k - 1) 3[Y + k(1 - Y)]
{[y + 3k(1 - Y)]aio + 4y&2o} + bio-
To calculate the second approximation, which is the solution of problem (1.28), (1.29), we introduce the notation
Fj (y) = 1
v?wfJ + (wji})
Hj(y) = X. (2wji)a(i) + -
(2.6)
It is clear that Fj (y) are polynomials of the fourth degree by y, а Hj (y) are polynomials of the third degree. Further calculations in comparison with the finding first approximation are rather long and therefore only main stages will be describes below.
Integration of equations for w(2), а(2 from the system (1.28) leads to representations
f(2) y2 Г у
wf\y) = m^y + m2 - j + (y - z)Fj(z) dz,
^j J l
/У
(y - z)Hj (z) dz
(2.7)
with constants mi, m2, ni, nj2, j = 1, 2. The constants fj2'1 are also unknown.
Taking into account the sticking to the walls y = 0 (j = 1) и y = h (j = 2) the functions v(2) (y) are found from the equations of mass conservation in layers:
ГУ rn
v(2)(y) = - w(2)(z) dz, v(2)(y) = - w(2)(z) dz-
Jo Jy
The following integral equalities are valid
pl rh
/ w(2)(z) dz = 0, / w(2)(z) dz = 0
Jo Jl
(2.8)
(2.9)
since v(2)(l) = 0, v2A>(l) =0 on the interface. Thus, there are ten boundary conditions to determine ten constants mj, nj, fj2), i,j = 1, 2: (2.9) and
w(2)(0) = 0, w(2)(h)=0, a(2)(0)=0, a22)(h) = 0,
,(2)
j
w(2)(/) = w22\i), «i2)(i) = a2)(/),
(2.10)
М2W22y(l) - Miwiy (/) = -2жа\л>(/), k2«2y(/) - kia\y(/) = (/)wii;(/).
Substitution of representations (2.7) in conditions (2.9), (2.10) allows one to find the above-mentioned constants uniquely
,(2) /
,(2)
(2),
,(2)/
/О /*0
zFi(z) dz, n2 = zHi(z) dz, nj; =
i = k2Di - (h - /)D2 k2h[j + k(1 - y)j ’
Di = f zHi(z) dz -I (h - z)H (z) dz, D2 = J и Л
n2 =
Y(1 - Y)^2Ah[Ya2o + k(1 - Y)aio]
H 2M2[Y + M(1 - Y)][Y + k(1 - Y)] ’
kiDi + /D2 2 f/ \ 7 kiDi + /D2
,-h
, n2 = -J (h - z)H2(z) dz -
k2h[Y + k(1 - y)] л
i _ M2D3 - (h - /)D4
k2 [Y + k(1 - y)]
A f(2)
2v2
F2h[Y + M(1 - Y)] 2^ cl
■ /f +(
o
h2 (1 Y2 ) l h
D3 = 2- /Г; + l2~7J /22) + ^ zFi(z) dz -J (h - z)F2(z) dz,
D4 = -2ж(п\/ + n2i) + /p2/2(2) - /pi/i(2),
M4D3 + /D4 F2h[Y + M(1 - Y)]
m2
_hl
2v2
h
(h - z)F2(z) dz -
P4D3 + /D4 F2[Y + M(1 - Y)]
/(2) = x {(Y - 1)[4Y + M(1 - Y)]Ki - 3v(1 - Y)2k2} , v = —
1
vi
Д
V2
/2(2) = X {3PYKi + Y[Y + 4M(1 - Y)]K2} , P = — ,
P2
Д = 4Y(1 - Y)[Y + M(1 - Y)]2
(2.11)
Ki
K2
12vi^(y - 1)(ni/ + ni) 12vi[y + m(1 - Y)] t fv
l^2h h/2
6vi [y + 2^(1 - y)] ^l
o
+
h/
[ [ (У - z)Fi (z) dz dy+
ol
J zFi(z) dz +h~ (h - z)F2(z) dz,
12^Y(ni/ + n2) 6^v2
P2h
l
zF1(z) dz -o
x J (h - z)F2(z) dz +
h2
h
6v2 [2y + m(1 - Y)]
(1 - Y)h2
12v2 [Y + M(1 - Y)] fh Г (1 - y)2h3 Ji Ji
ЛУ
(У - z)F2(z) dzdy.
So, the values nj (i,j = 1,2), m2 are calculated from the first approximation and the input data. Then /(1\ j = 1,2 are found; with their help we determine the constants D3, D4, and
hence, m^ m2, ml. Functions wj )(y), aj )(y) are given by formula (2.7), and v( ) are given by equalities (2.8). It should be noted that the functions wj1'1, vj1'1 are proportional to ж, and wj2\ vj2^ to ж2.
i
i
m
2
3. Analysis of the influence of internal energy on the example of liquids layers of equal thickness
Suppose for simplicity that 7 = l/h =1/2 and upper wall y = h is under the influence of a constant temperature, i. e. a20 = 0. We introduce the notation of the Marangoni number in a convenient form for computation (aio = 0)
M = kma 10 h3 k = kl = Ml
Xi+ 1)(m + 1) ’ k2 ' M м2 '
(3.1)
In this case, the characteristic temperature is 9* = a10h?. Then, in the dimensionless form, the formulas of the first approximation (2.1), (2.3) (recall that bj1 from (2.4) does not affect convection) are
wfV)
„(i)
(y)h
Xi
(W
-M(3£2 - £),
Am(() = OlM = 1
a10
V( 1 V) =
v(1) (y)h Xi
=M
2£
k + 1 ’ (£ -1£2)
F (i)=Ah
2 = 6P1M
X2
at 0 < £ = y/h < 1/2, Pi = v1/x1 is Prandtl number of the first liquid;
W21)(£) = = -m(3£2 - 5£ + 2),
X1
(3.2)
A(i)(£) = a2 )(y) = 2k(1 - £)
A2 (£) = „ L 1 1 ,
k+1
V(1)(£) =
V2)(y)h
Xi
= У£3 - 2£2 + 2£ - i)
(i) _ f^h4 _ 6P2M
F2(1) =
Xi
X
(3.3)
at 1/2 < £ = y/h < 1, P2 = v2/x2 is the Prandtl number of the second liquid, and x = X1/X2 is the ratio of thermal diffusivity coefficients.
To calculate the second approximation, we give the form of the functions Fj, Hj. The integrals of these functions are included in the representations for w22\ v22 and all the constants. Denoting for brevity
B
;saioh
M2(k + 1)(m + 1)
we get
Fi(£) = ^ (V - 2£3 +
Hi(£) = -\- 5£3
Xi(k + 1) L
+ ^3k + ^ £2 - (k + 1)£
(3.4)
(3.5)
1
at 0 < £ < 1/2;
F2(0 = B (V - 10£3 + 225 £2 - 7£ +0 ,
нт=-A+1) (- 5£3+f£2 -12£+D
(3.6)
at 1/2 < £ < 1. Taking into account the first part of the formulas (2.11) and functions Hi(£), H2(£) from (3.5), (3.6) we find
aioM
(
— 2k +1
,i _
1 4(k + 1)2hV 24
kaioM
+ kMo
)•
2 aio M(2k — 1)
П1 = 192(k + 1)
n2
4(k + 1)2h
n2 =
kaioM
'2 = 64(k + 1) with dimensionless constant
X ■
(3^^ — Mci) •
(
16
k + 1
24
3kx — 2k +1
24
(3.7)
—M0
)
M0
$Xi
k2h
(3.8)
Therefore
42)(£) =
a?)(y)
ai0
M
/2 (
X 4 Lk + 1 v
3kx — 2k + 1 , _ \ 1
k + 1 V 24 0/ 8
+
k + 1 4
(k + 1) a3 (2k + 3) M , 1 л5
£+
A22)(£)
42)ы
aio
3
kM k + 1
£3 —
°)
£4 + 2 £5} • 0 < £
X — 32+32£—2£2+4£3—4£4+2£5)+
4
1
< - • 2
(3.9)
1 (3kx — 2k +1 \
+ —------ —-------— + kM0 (£ — 1)
4(k + 1)V 24 0/V _
- < £ < 1. 2
Similarly, is calculated wj2)(£). The final form of the functions Aj(£) = Aj4£) + Aj2^(£) and Wj (£)= Wj(1)(£) + w(2) (£) are
Ai(£) =1 —
2£
M
+ k+1\
1
+
k + 1 k + 1 t 32(k + 1)
(k + 1) a3 (2k + 3) M , 1 a5
3 (3kx — 2k + 1) — 1
£+
3
£3 —
4
£4+2 £5j+
k1
—-^Mi£^ 0 < £ •
4(k +1)2 ^ 2
A2(£) =
2k
k + 14~ ^' ' k + 1
3kx - 2k + 1
(1—£)+k+i{ x(—2.+S £—2£2+4£3—4£4+1 £5)+
+
Mi =
96(k + 1)
ke2aioh2
(£ — 1)} —
7 47
32 + 32 " 2
k1 i(kWMi(£ — i). 2 <£ < i.
kE
(3.10)
M2k2(M + 1)(k +1) (m + 1)(k + 1)
W1®
W&)
wi(y)h?
XiM W2(y)h2
-U + « - 2 F1? + mA + M (-0t6 - -0t5 + Ь?), 0 < {
1
1
^ 2 ’
XM = -3t2 + 5t - 2 + 2 F2(1 - t2) + m2(t - -)+
i VM (1 t6 - 1 t5 + 25 t4 - 7 t3 + 3 p2_9) I < t < 1
+ Pi l 10 t 2 t +24 t 6 t + 4 t 40) ’ 2 ^ t
(3.11)
Fi
M
(I + 5^ + \70 224
(у + 1)РД 70 224 1120 4(k + 1)
9V ^ + 1)Pi (3x + 2k - 1)) - 3kM
4(k + 1) ’
i
F2
M
70P1
(v
1) + Fi , mi
1 M 5 893 vM
— Fi —-------- m2 = — F2 —----------
6 1 1120P1 ’ 2 6 2 3360 Pi ’
V'«>=XM=- Г W-(t) *• V2(t)=XM=- /X©*■
In the formulas the terms including M1 show the contribution of influence of interfacial internal energy on the functions Aj(t) and profiles of the longitudinal velocities. We emphasize important feature of the formation of Marangoni finite stresses through increments of the interfacial internal energy [1]. It does not require the inflow of energy into the system from outside in a thermal or chemical form. Such stresses can also be formed in the isothermal state of the interface.
The profiles of the functions Wj (t) and the vertical velocities Vj(t) are shown in Fig. 2, 3. Under the influence of the parameter, the velocity profiles deform, but this deformation is not significant: max \Vj(t, M, M1) - Vj(t, M, 0)| ~ 0,1. Of course, this smallness is due to the smallness of the Marangoni number. It is of interest to study the general nonlinear problem even for the isothermal case of the interphase boundary.
0.15----------------------------------------------------
§
Fig. 2. Profiles of the functions Wj(t) at a1o > 0 for model liquids
This research was supported by the Russian Foundation for Basic Research (17-01-00229).
0.02
'°'°20 0.25 0.5 0.75 1
§
Fig. 3. Profiles of the vertical velocities Vj(£) at aw > 0
References
[1] V.K. Andreev, V.V. Pukhnachev, Invariant solutions of thermocapillary motion equations, Numerical methods of continuum mechanics, Novosibirsk, 14(1983), n.5, 3-23 (in Russian).
[2] V.K. Andreev, V.E. Zahvataev, E.A. Ryabitskii, Thermocapillary Instability, Nauka, Novosibirsk, 2000 (in Russian).
[3] F.E. Torres, E. Helborzheimer, Temperature gradients and drag effects produced by convection of interfacial internal energy around bubbles, Phys. Fluids A, 5(1993), no. 3, 537-549.
О влиянии внутренней энергии границы раздела на термокапиллярное течение
Виктор К. Андреев
Институт вычислительного моделирования СО РАН Академгородок, 50/44, Красноярск, 660036 Институт математики и фундаментальной информатики Сибирский федеральный университет Свободный, 79, Красноярск, 660041
Россия
Изучается двумерное ползущее движение двух несмешивающихся вязких теплопроводных жидкостей, на границе раздела которых поверхностное натяжение линейно зависит от температуры. На твердых стенках температура имеет экстремальные значения, что хорошо согласуется с полем скоростей типа Хименца. При малых числах Марангони найдено точное решение возникающей обратной краевой задачи и дама оценка степени влияния внутренней энергии границы раздела на стационарное течение.
Ключевые слова: граница раздела, термокапиллярность, внутренняя энергия границы раздела, обратная задача.
