CC BY
10
Journal of Siberian Federal University. Mathematics & Physics 2013, 6(2), 211—219
УДК 517.9
^mbined Motion of Three Viscous Heat-conducting Liquids in a Flat Layer
Elena N. Lemeshkova*
Institute of Computational Modelling, SB RAS, Akademgorodok, 50/44, Krasnoyarsk, 660036
Russia
Received 17.11.2012, received in revised form 16.12.2012, accepted 09.01.2013 The joint unidirectional motion of three viscous liquids under the influence of thermocapillarity forces and pressure difference has been researched. An exact stationary solution of the problem has been found. The solution of the non — stationary problem has been obtained in the form of final analytical formulas in the image using the method of Laplace transformation. By the numerical inversion of Laplace transformation the evolution of the velocity fields and of the temperature perturbation to the stationary regime for specific liquids has been obtained.
Keywords: boundary value problem, Laplace transformation, thermocapillarity.
1. Statement of the problem
Assume that there are three layers of viscous incompressible liquids with the thickness of h,l2 and l3 — l2, with the interfaces y = 0,y = l2, and solid walls y = —li,l3. Motions in the layers are described by the system of viscous heat — conducting liquids equations in the absence of external forces (j = 1, 2, 3)
duj 1
+--vpj = Vj Auj, dtvuj = 0,
dt pj
d&j ~dt
XjAOj, (1.1)
where uj,pj is the vector of velocity and pressure; Qj is the deviation from the average temperature value; pj is the density; Vj is the kinematic viscosity; Xj — thermal diffusivity, d/dt = d/dt + Uj • V. We suppose that the motion is unidirectional as
Uj = (Uj (y,t), 0, 0).
Then the pressure in each liquid can be represented as pj = pj fj (t)x + a.j (t) with the arbitrary fj ,aj, and temperature — Qj = -Ajx + Tj (y,t) with the constants Aj. Assume that the coefficient of the surface tension a on the interface depends on the temperature linearly: aj (Q) = ajj — (Qj — Q0), ,a0, Q0 = const > 0, j = 1, 2. After the substitution into equations (1.1) the functions Uj(y,t),Tj(y,t) satisfy the equations
* lena_lemeshkova@mail.ru © Siberian Federal University. All rights reserved
Ujt = VjUjyy — (i), (1.2)
Tjt = Xj Tjyy + Aj Uj. (L3)
The conditions of continuity of the velocities and temperatures on the interfaces (in the general view the conditions on the interface are shown in [1]) and give equalities
ui(0,t)= u2(0,t), u2(Z2,t) = U3(l2,t), (1.4)
Ti(0,t) = T2(0,t), T2(l2,t) = T3(l2,t). (1.5)
Moreover the heat fluxes are equal to
klTiy(0,t) = k2T2y(0,t), k2T2y(l2,t) = k3T3y(l2,t), (1.6)
and there are jumps of tangential stress
M2U2y(0,t) - MiUly(0, t) = Affil, ^3U3y(l2, t) - M2U2y(¿2,i) = A^2, (1.7)
where kj are the heat conductivity coefficients, ^j = Vjpj are the dynamic viscosities. In equation (1.3) and boundary condition (1.7) A = A1 = A2 = A3 (it is a consequence of the equality of the temperature at y = 0 and y = ¿2, see (1.5)). The conditions for normal stresses are reduced to pressure equality in liquids and the kinematic conditions at y = 0, y = l2 are satisfied identically. Since the walls y = —Z1; y = l3 are solid, then the conditions of sticking can be written as
u1 (—Z1,t) = 0, U3(l3,t)=0. (1.8)
It is believed that the temperature gradient is constant that is
T1(—l1,t) =0, T3(l3,t) =0. (1.9)
It is assumed that motion arises under the influence of thermo capillarity forces and the pressure difference from state of rest so
Uj (y, 0) = 0, (1.10)
Tj (y, 0)=0. (1.11)
The equations (1.2)-(1.11) form two logically current tasks for the velocities Uj and the temperature perturbations Tj.
Remark 1. The considered solution of equations (1.1) is invariant relatively of to a one-parameter sub-group of continuous transformation corresponding to the operator d/dx + pf (t)d/dp — Ad/d©.
2. The solution of stationary problem
Suppose that velocity, pressure and temperature do not depend on time — stationary flow then the initial conditions (1.10), (1.11) are not stated. Therefore Uj = U0(y), Tj = T?(y), fj = f = const and equations (1.2), (1.3) take the form U0yy = fj/vj, T.?yy = — Au0/xj, j = 1,2,3, it follows that
f
A, j
«0 = jy2 + j + j j = — (jy4 + jy3 + jy2) + c3y + c4.
2v.
Xj 24Vj
(2.1)
Constants cl, c2, c3 h c4 are determined from boundary conditions (1.4)-(1.9) and after some calculations find a representation for velocities in the dimensionless form
u?(0 = N(l—2e2 + l—B(e + 1) - l?) + «i(£ + 1), -1 < e < 0,
Uo(£) = N (h Vi£2 + /"iB(^i e +1) - 1 ) + a>2£ + «1, 0 < e < Z^/Zi,
u3(e) = nmim2(ii2e2 + B(fie -1) -1) + «3(e - -), z 2/Zi < e < 1/ Zi,
1
Ti
(2.2)
and temperature
T0(e) = N ( 12 e4 + ^ e3 + e2 +| (e + 1) - ^ + 52)- ai(e3 + 3e2 - 2) + ^(e+1), -1 < e < 0, 6 02
22 T0(e) = n ( ^ e4 + Xh^e3 + X-ili(B- li) e2 + | tee+1) - ^ + 52-)- Xi(«2e3 + 3aie2) + hf (kie+1) + ai, 0 < e < h/h,
6 02 3
2
T0(e) = N (^-2 X2( ^ e4 + ^e3 - ^ e2) + b3e+^hX (5 + B - h3 )-
(2.3)
12
6
2
3l 1
4
li
X 2 «3^3 3e2
~(e -X
kik2b2 l2, X2«3 / I2 ^ _,- , , l 2«2 nnn k2b2
~ + n(2 - 1) - Xik2(ai + w^^T
1 /X2l2a3/ l2 . l 2«2 \ , X 2«3 \
- ^(2 - 1) - X ik2l2(ai + w) + it)j
where e = y/li, li = li/l3, ¿2 = l2/l3, Mi = M1/M2, M—2 = M2/M3, ki = ki/k2, k2 = k2/k3, Xi = Xi/x2, X2 = Xi/X3, Mi; M2 are Marangoni numbers, N = j^lg^v2 is the dimensionless pressure gradient. As the characteristic velocities and temperature perturbations the relations vi/li and Ali v1/x1 are selected, respectively. Therefore
Mi = Aœil2 m2 = Aœ212 ^ = -MiM2(1 - -2) - Mil 22 + li2
vi M2 ' viM^ Mi/-2(1 - l 2) + Mil 2 + l 1
0 i = M—1M2(l2 - 1) - Mil2 - l 1,02 = kik2(1 - l 2) + k— l 2 + l i,
a i
1
01
M2
«3 = --T" 0i
(l2 + M—2 - M2l2)Mi + (1 - l— )M2M2) TiMi + (l"i + M-il2)M2
, a2 = -
liMi + MiM-2(l2 - 1)M2
0
1
2
bi = ^^(1 - f2)(f2(I + §)■+ B + 1) + ^(Г2 - 3 + §) + h(B - 1\))+ _ _ _ _ _ _ -2 - - -2 - _ -2 _ - -+^ (b+4)+^ (3 (2+в) - (b+1))+l^(B - 51) - ^ (Ц2 (I+B)+ 2
+ (B - r)), li
b2 = (h - 1)(1- - 1) - (ai + h02 )(12 - 1) - + (ai + %)_
2 Ц2 2 )K2 > 11 2li > 31\2 2li 1 3li ;
2
X 2l2 аз (l 2 1) ail i
- (з -1) - T•
From the representations of solutions (2.2), (2.3) it is seen that the influence of pressure gradient and thermocapillarity forces is independent of each other. This is a consequence of the problem linearity (1.2)—(1.9).
3. The solution of the non-stationary problem using method of Laplace transformation
Apply the Laplace transformation to the problem (1.2)-(1.9). Taking into account initial conditions (1.10), (1.11), obtain in the Laplace presentation equations for velocities Uj(y,p) and the temperature perturbations T j (y,p)
PUj(y,p) = vjUjyy(y,p) - Fj(p), pTj(y,p) = XjTjpp(y,P) + AU(y,p). (3.1)
Added to (3.1) are the converted conditions (1.4)—(1.9)
M2U2y(0,p) - MiUiy(0,p) = A^i/p, (3.2)
М3U3y (l2,p) - V2U2y (l2,p) = AiB2/p, (3.3)
Ui(0,p) = U2(0,p), U2 (l2,p) = U3(l2,p), (3.4)
Ti(0,p) = T2(0,p), T2(l2,p) = T3(l2,p), (3.5)
Ui(-li,p)=0, U3(l3,p)=0, (3.6)
Ti(-li,p)=0, T3(l3,p)=0, (3.7)
kiTiy(-li,p) = k2T2y(-li,p), k2T2y(0,p) = k3T3y(0,p). (3.8) The general solution of first equation (3.1), j = 1,2,3 of the form
Uj = Ci sh ./J1 (y + li) + Cj2 ch ^ f^(y + li) - Fj, (3.9)
of the second one
where
Vj V Vj P
Tj(y,p) = C j sh /XLy + Cjj ch. /—y + Tjr, (3.10)
V xj V xj
Tjr = —-- i Uj(z,p) sh. /— (z — y)dz
sjvxj.) v Xj
is the particular solution.
The constants (i, C2, Cj, Cj are defined from boundary conditions (3.2)-(3.8) and one obtains (p = pl2/v1)
Ui(e,p) =
U2(e,p)
NFi(p) p
NFi(p) P
ci s^vP(e+1) + c^vP(e + 1) -1
+
C1 s^vPe + C2 c^vPe
ci sh v/—ip(e +1) + C22 ch v—iP (e+1) - p 1
+
+
U3(e,P)
(72 s^v/vipe+c| ch v—ipe NFi(p)
Ci sh V—i—p(e + 1) + C3 ch a/VI^p(e +1) - P2
+
+
(73 s^V—1—2p£ + C3 ch a/—T—2pe
(3.11)
«1 = sh v/—Lp(Z2/Z 1) + 1 + -7= tanh a/VT—2p(1 - Z 2)/Z 1 ch i/hip(-2/Z 1 + 1),
V—2
a2 = ch ——ip-2/-i) + 1 + —^ tanh ——1 —2p(1 - -2)/-i sh Vhip(-2/-i + 1),
V—2
M1M2 , Mi M1M2 , . , ^ Mi , Mi ,
_ _ .=---Z Z + — - —, h2 = 1 - ch a/p - —, b3 =--== sh Vp ,
—1—2 c^—1—2p (1 - - 2)/Zip —1 —1—2 —1 V—1
A = «2
Mi
ch — sh \J—ip - sh — ch a/—ip
+ «i
sh —p sh ——ip--ch —p ch ——Tp
V—1
C1 = 1 Ci = A
C21 = x 2A
C22 = x 2A
C31 =
-hi - («2h2 + «ib3) ch—^ + («2b3 + «ib2) sh a/—p
-hi(-sh —psh —hip + —^ ch —pch ——ip) + «2^3 sh —p - b2 —^ ch —p)
V —1 V —1
bi( —^ ch —sh —Tp - sh —p ch —-Tp) - «i(b3 sh — - h2 sh —p)
—1 —1
M2
C21 ch—Tp(1+Z-2/ Z-i)+C22 sh—Tp(1+Z-2/Z-i)
(3.12)
ch^—1—2p(1 + 1/z 1 )-p2 sh^—1—2p(1 + -2/ Zi)
ch^—1—2p(1-l2 )/li
CI
P2 sh —^ ( 1 + -2 / -i ) -
C21 ch—(1+Z 2/Z 1) +C2 sh—^ (1+Z 2/Z 1)
sh^—1—2p (1 + 1/ Z 1 ))
ch,/—1—2p(1 -l2)/li
q = s^a/—1—2pl2/li - ch V—1—2pl2/Z 1 tan^ V-T—2p/li,
qi = c^a/—1—2pl2/li - s^a/—1—2pl2/li tanh a/—1—2p/li,
h
1
< =
1
p V-i^p
9^2M2 - qiMi^/v2sYi^iphlh + q^Mi ch^/jJipl2/li
_ 9i (—1 sh^/iipl21 l i + ch^//pl2 / / i tanh ^/p)
qM 1M2
C|
¿72 = < tanh Vp, <3
Mi
Ci +
M
a//1 pv^ip'
<32
~ ¿71 M 1 - - - -
<3 = — (—= s^v//ipl21 l 1 +C^A/li]ll2| l 1 tanh i/p ) + _ ^^
q vvi qpVvip
ch^/lp^1 l i - —22 sh^/p 2I1 i tan^v/p = — <3 tanh a/Vi V2p|l1, M3
sh^/lp 2I1 i.
Because of the complicated expressions the temperature perturbations in the Laplace representation are not given here.
Using equalities (3.10)-(3.12) and performing the calculations which are long enough one can prove limiting equalities limp^0 pTTj (y,p) = Tj0(y) and limp^0 pUj (y,p) = u0(y), when T0(y),u°(y) is the stationary distribution from (2.2), (2.3). Let us apply the numerical method of inversion of Laplace transformation to the obtained formulas (3.11), (3.12). The graphs only for the velocities are given because it has a real physical meanings.
Figs. 1, 2, 3 present the profiles of dimensionless velocities for the system of silicon = 956kg|m3, = 10.2 • 10-6m2|s, Mi^ = 9.71 • 10-3kg|(m • s), k3 = 0.133 kg • m|(s3 • K), x = 0.0675 • 10-6m2|s, ^ = 6.4 • 10-5kg|(s2 • K)) - water (p2 = 998 kg|m3, v2 = 1.004 • 10-6m2|s, M2 = 1.002 • 10-3kg|(m • s), k2 = 0.597 kg • m|(s3 • K), x2 = 0.143 • 10-6m2|s, = 15.14 • 10-5kg|(s2 • K)) - air (p3 = 1.205 kg|m3, v3 = 15.11 • 10-6m2|s, m3 = 0.018 • 10-3kg|(m • s),
k3 = 0.0257 kg • m|(s3 • k)----21 • 10-6m2
the dimensionless time t
in the first and third layers. The dimensional time at t = — the dimensionless pressure gradient in the first liquid.
K), x3 = 21 • 10 6m2|s) at 20°C. It is seen that with increased of = v3t|l2 the solution reaches a steady state, this being the fastest
10 is t = 1s, /(t) = /i(t)|/0 Fig. 1 illustrates the case when
x
Fig. 1. Velocity profiles in the layers at l ]_ = 10 3 m; l2 = 1.5 • 10 3 m; l3 = 2 • 10 3 m; N = 0.0001; M3 = 2; M2 = 3; /(t) = 1 + e-Tcost; curve 1: t = 1; curve 2: t = 3; stationary decision (- - -)
|N| << |aj| then thermocapillarity forces are predominating and we have almost linear profiles of the velocities — the Couette flow. Fig. 2 presents the case when |N| >> |aj| then the pressure
Fig. 2. Velocity profiles in the layers at l1 = 10 3 m; l2 = 1.5 • 10 3 m; l3 = 2 • 10 3 m; N = 10; M1 = 2; M2 = 3; f (t) = 1 + e -Tcost; curve 1: t =1; curve 2: t = 3; stationary decision (---)
gradients in layers become the main ones and the profiles are parabolic — the Poiseuille flow. Fig. 3 shows the case of roughly equal contributions to the mechanism of the flow of the above factors. At f (t) = smr the solution will not converge to a stationary one because the limit f (t) at t ^ to does not exist. In Fig. 4 curves 1, 2 correspond to the positive pressure gradient and curves 3, 4 to the negative one that is the motion is reversed and the process is repeated in t = 2n.
Fig. 3. Velocity profiles in the layers at li = 10 3 m; 12 = 1.5 • 10 3 m; 13 = 2 • 10 3 m; N = 1; M1 = 2; M2 = 3; f (т) = 1 + e -T cos т; curve 1: т = 1; curve 2: т = 3; stationary decision (---)
If the dimensionless pressure gradient is
*
ft Л ) 0 < т < т*;
f (т)H т ^
1 — 5eT T, т ^ т*.
then liquids will move in the positive direction at first, in Fig. 5, curves 1,2, and at t = t* the pressure gradient changes its sign and the reverse flow occurs, curve 3. With the time increase of time the motion will reach the stationary state, curves 4, 5.
Fig. 4. Velocity profiles in the layers at l 3 = 10-3 m; l2 = 1.5 • 10-3 m; l3 = 2 • 10-3 m; N = 1; M3 = 2; M2 = 3; /(t) = sin t; curve 1: t = 2; curve 2: t = 3; curve 3: t = 4; curve 4: t = 5; stationary decision (- - -)
Fig. 5. Velocity profiles in the layers at l 3 = 10-3 m; l2 = 1.5 • 10-3 m; l3 = 2 • 10-3 m; N = 1; M3 = 2; M2 = 3; curve 1: t =1; curve 2: t = 2; curve 3: t = 3; curve 4: t = 4; curve 5: t = 7; t* = 2; stationary decision (---)
Author thanks Professor V.K.Andreev for the statement of problem and some comments during the work.
The work was supported by the Russian Foundation for Basic Research 11-01-00283 and integration project SB RAS 38.
References
[1] V.K.Andreev, V.E.Zahvataev, E.A.Ryabitsky, Thermocapillary instability, Novosibirsk, Nauka, 2000, 31 (in Russian).
Комбинированное движение трёх вязких теплопроводных жидкостей в плоском слое
Елена Н. Лемешкова
Исследовано совместное однонаправленное движение трёх вязких жидкостей под действием термокапиллярных сил и перепада давления. Найдено точное стационарное 'решение задачи. Решение нестационарной задачи получено в виде конечных аналитических формул методом преобразования Лапласа в изображениях. Путём численного обращения преобразования Лапласа получена эволюция полей скоростей и возмущений температур к стационарному режиму для конкретных жидких сред.
Ключевые слова: краевая задача, преобразование Лапласа, термокапиллярность.
