CC BY
7
УДК 532.517
The Joint Motion of Two Binary Mixtures in a Flat Layer
Viktor K.Andreev*
Institute of Mathematics, Siberian Federal University, Svobodny 79, Krasnoyarsk, 660041,
Russia
Received 10.09.2008, received in revised form 20.10.2008, accepted 05.11.2008 The invariant solution of the equations of thermodiffusional motion is investigated.. This solution describes the motion of two immiscible incompressible binary mixtures with a common flat interface under the action of pressure gradient and thermocapillary forces. The stationary flow of such system is found. If the pressure gradient in one of the mixtures tends to zero sufficiently fast, then the motion of mixtures is slowed down by the viscous friction. On the other hand, if there exists a finite limit of pressure gradient when time tends to infinity, then the solution tends to the stationary state.
Keywords: flat layer, thermodiffusional motion, invariant solution.
1. Problem Statement
Consider the motion of two immiscible incompressible binary mixtures with a common interface. Suppose that Qj (j = 1, 2) are the domains occupied by the fluids with interface r, Uj (x,t) and pj (x, t) are the velocity vectors and pressures, respectively, and 0j (x, t) and Cj (x, t) are the deviations of temperatures and concentrations from their average values. The equations of thermodiffusion and motion in the absence of external forces (g = 0) have the form [1]
du 1 „ „ dd.i
= X- Aß- ;
(1.1)
+ p- Vp- = v-Au-; -¿ = X-Aß-
dC ■
= dj Acj + aj dj A9j ; div Uj = 0,
where pj is the average density, Vj is the kinematic viscosity, Xj is the thermal diffusivity, dj is the diffusion coefficient, aj is the thermal diffusion coefficient, and d/dt = d/dt + u • V.
Suppose that the coefficient of surface tension a on the interface depends on the temperature and concentration, a = a(0, c). For many mixtures, the linear law provides a good approximation of this dependence:
a(Q, c) = ao - œ1(9 - 0q) - (c - co), (1.2)
where œi > 0 is the temperature coefficient and œ2 is the concentration coefficient (usually œ2 < 0 since the surface tension increases with concentration). Let us now formulate the conditions on the interface r.
1. Equality of velocities:
Ui = U2, x G r. (1.3)
*e-mail: andr@icm.krasn.ru (c Siberian Federal University. All rights reserved
2. Kinematic condition:
u • n = Vn, x e Г. (1.4)
This condition follows from the assumption that Г is a moving material surface. Here n is the unit normal vector to Г directed from Qi to Q2, Vn is the velocity of interface displacement in the normal direction, and u is the velocity vector on Г, which is the same for both fluids due to (1.3).
3. Dynamic condition:
(P2 — P1)n = 2<rHn + Vra, x e Г. (1.5)
This condition expresses the balance of all forces acting on the surface (pressure, friction, surface tension, and thermocapillary forces). Here Pj = — pj + 2pjVjD(uj) are the stress tensors, D is the rate of strain tensor, H is the mean curvature of Г, and Vr = V — (n • V)n is the surface gradient.
4. Temperature continuity and concentration balance on the interface:
d1 = в2, c1 = Ac2, x e Г, (1.6)
where A is the Henry's law constant.
5. The equality of heat fluxes on the interface:
, дв 2 , Зв2 ° Г (1 7)
k2 — ki = ° x e Г, (1.7)
where kj are the thermal conductivities.
6. The equality of mass fluxes through the interface:
d (dc2 + дв2 \ d (dci + дв1 \ Г (18)
«2 + = di — + ai— , x e Г. (1.8)
\дп dn J \дп dn J
The domains ^i and П2 can be in contact not only with each other, but also with rigid walls that will be denoted by Xj. On these walls, the no-slip condition should be imposed
uj = aj^t^ x e X, (L9)
where aj (x, t) is the velocity of the wall Xj. In addition, we assume that the temperature on Xj satisfies the following conditions
ej = ew(x,t), x e Xj, (1.10)
with given functions в-j. It means that temperature is imposed on the wall. The condition of absence of mass flux through the walls X j is written as
дп + aj ддп = °, x e X. (1.11)
For completing the problem statement, the initial conditions should be added to relations (1.1)-(1.6):
uj(x, °) = u0j(x), ej(x, °) = e0j(x), Cj(x, °) = c0j(x), x e Qj. (1.12)
In what follows, we consider two-dimensional equations of motion for two binary mixtures with a flat interface in the absence of external forces. It can be shown [2] that this system admits a one-parameter subgroup of transformations corresponding to the generator
d , d „ d „ , . d ddX + A dj + B ^ + fj(t) dj,
where Aj, Bj are constants and fj (t) are functions of time. The invariant solution should be sought in the form
Uj = Uj (y,t), Vj = Vj (y,t), pj = Pj fj (t)x + Pj (y,t), Oj = Aj x + Tj (y,t), Cj = Bjx + Kj (y,t).
It follows from the continuity equation that Vj is a function of time only, Vj = Vj (t). Projecting the momentum equations on y axis, we find p-iPjy = Vjt(t). Further we assume that Vj(t) = 0 (otherwise the no-slip conditions on the walls are not satisfied). Then the invariant solution is written as
Uj = Uj (y,t), Vj = 0, pj = pj fj (t)x + Pj (t),
(1.13)
Oj = Aj x + Tj (y,t), Cj = Bjx + Kj (y,t).
Solution (1.13) can be interpreted as follows. Suppose that on the interface y = 0 between two mixtures the surface tension linearly depends on the temperature and concentration: a(O, c) = a0 — ffiiO — ie2c, where > 0 and are constants (see (1.2)). Initially, the first and second mixtures are at rest and occupy the layers —li < y < 0 and 0 < y <l2, respectively. At t = 0, the temperature field Oj = Ajx and concentration field Cj = Bjx are created instantly in the entire layers. The thermoconcentration effect and pressure gradients fj (t) induce the motion of mixtures. In this motion, the interface is represented by the plane y = 0 and the trajectories are straight lines parallel to x axis. The functions Uj,Tj, Kj can be called the perturbations of the quiescent state.
Substituting (1.13) in the governing equations and taking into account the conditions on the interface y = 0, we obtain the initial boundary value problem
ujt = vj ujyy + pj fj Tjt — Xj Tjyy Auj!
Kjt — dj Kjyy + aj dj Tjyy — Bj uj at -h <y< 0 (j — 1), 0 <y<l2 (j — 2);
ui(0,t) — u2(0, t), T1(0,t) — T2(0,t), K1(0,t) — XK2 (0, t);
kiTiy (0,t) — k2T2y (0, t);
di(Kiy (0,t) + aiTiy (0,t)) — d2(K2y (0,t) + «2 T2y (0, t)); P2V2u2y(0, t) - piviuiy(0, t) — -ffiiA - «2ßi = H; uj (y, 0) —0, Tj (y, 0)—0, Kj (y, 0)—0.
(1.14)
1.15)
1.16)
1.17)
1.18) 1.19)
In the second equation (1.14), A = Ai = A2 (it follows from the equality of temperatures at y = 0). In the boundary condition (1.15), A = const is the Henry's law constant, so Bi = XB2. In addition, Vj, Xj, dj, aj, kj are positive constants that characterize the physical properties of the
mixtures. The above relations should be supplemented by conditions on the rigid walls y = —li and y = I2. These are the no-slip condition
ui(-Zi,i) = 0, U2(Z2,t) = 0,
(1.20)
condition of absence of temperature perturbations
Ti(-Zi,t) = 0, T2(Z2,t)=0,
(1.21)
and condition of absence of diffusive fluxes
dK1 dT1 dy dy
y=-ii
dK2 dT2 —--+
dy dy
y = l2
(1.22)
It can be seen that equations (1.14)-(1.22) form three problems for functions («i, «2), (Ti, T2), and (Ki, K2). These problems can be solved successively. Since the problem for the velocity field is linear, it can be decomposed into inhomogeneous problem with fj (t) = 0 and zero boundary condition (1.18) and homogeneous problem with fj (t) = 0 and non-zero boundary condition (1.18), i.e. H = 0.
Remark 1. Since pi = p2 at y = 0 for all x, it follows from the dynamic condition on the interface that [1]
Pifi(t)= P2f2(t), Pi(t)= P2(t). (1.23)
0
0
2. Determination of the Velocity Field Under Given Pressure Gradient
Taking into account the above considerations, let us first consider the problem of determining the velocity field only under instantly imposed pressure gradient in one of the layers. In this case, we have the following adjoint linear initial boundary value problem (f (t) = fi(t))
uit = Viuiyy + f (t), -Zi < y < 0; (2.1)
ui(-Zi, t) = 0; (2.2)
U2t = V2U2yy + — f(t), 0 <y<Z2; P2 (2.3)
U2 (Z2, t) = 0; (2.4)
ui(0,t) = U2(0,t), ^iuiy(0,t)= ^2U2y(0,t), t > 0; (2.5)
ui(y, 0)=0, -Zi < y < 0, U2(y, 0)=0, 0 < y < Z2. (2.6)
Relations (2.2) and (2.4) represent the no-slip conditions on the fixed rigid walls, while equations (2.5) express the equality of velocities and shear stresses on the interface [5, p. 268]. In addition, vi,2 = Mi,2/pi,2, where are the dynamical viscosities.
Remark 2. Without loss of generality, we can assume that Pi(t) = P2(t) = 0 in (1.23) since these functions do not influence the motion of mixtures.
A priori estimates. Let us derive some a priori estimates for the solution of problem (2.1)-(2.6). First, we multiply equation (2.1) by giwi(y,t) (equation (2.3) by ^2w2(y,t)) and integrate it with respect to y between —¿i and zero (between zero and ¿2). Summing up the obtained relations and using boundary conditions (2.2), (2.4), and (2.5), we find
0 ¡2 "
dE(t)
dt
where
+ vi J n\y dy + wj U22y dy = pif (t)(/ Ui dy + J U2 dy^ , (2.7)
12
E(t) = 2 pi J Ui(y,t) dy + 2 P2 ^ U2(y,t) (2.8)
J'2V
-h 0 is the total energy of two layers.
The uniqueness of solution for problem (2.1)-(2.6) follows from (2.7). It can be seen that if f (t) =0, then ui(y,t) = u2(y,t) = 0.
Relation (2.7) allows us to determine the asymptotic behaviour of solution when t ^ to under some restrictive assumptions on the function f (t). Indeed, owing to conditions (2.2) and (2.4), the Friedrichs inequalities hold for ui(y,t) and w2(y, t):
0 2 0 ^ ¿2 £2 /u1(y,t) dy < (y,t) dy, J u2(y,t) dy < ^ Ju2y (y,t) dy. (2.9)
lil
Using inequalities (2.9) and the Cauchy-Bunyakovski-Schwarz inequality, we find from (2.7) (since yfâ + Vb < \/2(a + b), a > 0, b > 0)
^ + 4JE(t) < 2^i |f (t)|VE(i), (2.10)
dt
where J = min(1-2v1, ¿-2v2) and ¿1 = p1 max^¿i/pi, -^¿2/^2). Taking into account that E(0) = 0, and according to (2.8) and initial conditions (2.6), we obtain from (2.10)
t 2
Hence, if the integral
E(t) < ¿2(/ If (t)|e2Ät dt) e-4Ät. (2.11)
0
cc
J If (t)|e2Ät dt = VC! > 0, (2.12)
converges, then it follows from (2.11) that
E(t) < ¿2Cie-45i (2.13)
for all t ^ 0. Therefore, L2—norms of functions ui(y,t) and u2(y,t) tend to zero as t ^ to exponentially and uniformly with respect to y G (—/2, 0) and y G (0,/2) provided that (2.12) is satisfied. To derive the estimate for |«j (y, t)|, it is necessary to estimate the integrals
0 ¡2
j uiy ^ j u2y dy.
-i, 0
0
Let u(y,t) be a solution of the equation ut = vuyy + F(y,t), y G [a,b]. Then the following identity holds:
t b b t b b t b j j (u2 + v2u2yy) dydt + v j u2 dy = 2v j (utuy) dt+v j u0y dy+ j j F2(y,t) dydt, (2.14)
0 a a 0 a 0 a
where u0(y) = u(y, 0). Identity (2.14) follows from the equalities t b t b
(ut — vuyy)2 dydt = J J F2(y,t) dydt, utuyy = — (ut%) — 1 — (uj;).
0 a 0 a
Let us first put u = ui, a = —li, b = 0, v = Vi, F = f (t) in relation (2.14) and multiply it by pi. Then we take u = u2, a = 0, b = I2, v = V2, F = gig-if (t) and multiply the same relation by P2. Summing up the results, we obtain another integral identity for problem (2.1)-(2.6):
t 0 t h
Pi J J (uit + v2u2yy) dydt + P2 J J (u2t + V2u2yy) dydt+
0 -01 i2 0 0 t (2.15)
J u2iy dy + ^2 j u\y dy = pi(li + I2) J f2(t) dt.
In the derivation of (2.15), initial conditions (2.2), (2.4), and (2.5) and boundary conditions (2.6) were taken into account. Consequently, for all t ^ 0
h
f u2y dy < E1(t) , fu2y dy < E1(t) , (2.16)
7 y mi 7 y M2
-i 1 0
where E2(t) is the right-hand side of (2.15). Therefore, if
Jf 2(t) dt = C2 > 0 (2.17)
in addition to (2.12), then the following uniform estimates with respect to y (y G (—12, 0) and y G (0, I2)) hold:
12C1C3\ 1/2
u(y,t)K K26i j;) e , (2.18)
where C3 = pi (li + l2)C2, j = 1, 2. These estimates are obtained with the help of
y 1 2
u2(y, t) = 2 J ui(y,t)uiy(y,t) dy, u2(y,t) = —2J u2(y,t)u2y(y,t) dy, - 11 y
and with the help of inequalities (2.7), (2.16), (2.17), and the Cauchy-Bunyakovski-Schwarz inequality.
Remark 3. It can be shown that if condition (2.12) is satisfied, then relation (2.17) also holds true.
0
We have proved
Theorem 1. The solution of problem (2.1)-(2.6) tends to zero as t ^ to subject to condition (2.12). The rate of convergence satisfies estimates (2.18) that are uniform in the intervals (—li, 0) and (0, l2).
In other words, if the pressure gradient in one of the mixtures tends to zero sufficiently fast, then the motion of mixtures is slowed down by the viscous friction according to inequalities (2.18).
Solution in Laplace representation. To obtain more detailed information on the behaviour of uj(y,t), let us apply the Laplace transform to problem (2.1)—(2.6):
CO
uj(y,p) = J e-ptuj(y,t) dt (j = 1, 2) (2.19)
0
(the conditions for the applicability of formula (2.19) can be found, for example, in [6, p. 494]). As a result, we obtain a boundary-value problem for representations uj (y,p) :
u'i — — ui = — ® (—li <y< 0); (2.20)
Vi Vi
ui(—li,p) = 0; (2.21)
uy — p u = — ^ f (p) (0 <y< l2); (2.22)
V2 Q2V2
u2(l2,p) = 0; (2.23)
ui(0,p) = u2(0,p); (2.24)
Miu2(0,p) = ¡J,2u2(0,p), (2.25)
where the prime denotes differentiation with respect to y. After some calculations, we obtain from (2.20)-(2.25)
- ( ) f (P) ui(y,p) — --
pW (p)
P - (P - 1) ch\l — l2 V2
sh J — (y + li)-V Vi
(2.26)
- shVViy + shvVilVchVV2l2 + MchvViy-chvVili)shVVpl2y;
- t s f(P) \ M
u2(y,p) — - pw—){ TV
1 + (p - 1)chA/pli V1
shj — (l2 - y)+ V2
(2.27)
+ sh^lVp y — sh y VP ch y VP li + p^ ch y -p y — ch y -p sh y -p li
Here f (p) is the representation of f (t), p = pi/p2, and W (p)= shJ^pl2 chJV-li^-Jv + cthy^2 thJ—^li) , j = ji/j2, v = Vi/V2. (2.28)
The originals «j (y,t) (j = 1, 2) are reconstructed by the formula
U (y,t) = 2ni / ePtUj (y,P) dP.
(2.29)
Suppose that lim f (t) = f0 = const exists, then lim pf (p) = f0 [6, p. 521]. Of course,
t—p—>0
in this case the function f (t) does not satisfy condition (2.12). Let us calculate lim pUj (y,p)
p— 0
according to (2.26) and (2.27). Simple but cumbersome calculations with the use of asymptotic representations sh x — x + x3/6, ch x — 1 + x2/2 as x ^ 0 show that
lim pui(y,p) =
2 + M - l2 /y\ + +1)
h) + -(m + -) \ij + -(m + -) _
= u0(y);
(2.30)
lim pu2 (y, p) = -2f0M
p^0
2vi
2 + m-_ZY yA + + 1)
h ) M + I \h J M + -
= u0(y),
(2.31)
where the relation gv = p was employed. It can be easily checked that the right-hand sides of (2.30) and (2.31) represent the exact stationary solution of problem (2.1)-(2.6), where f (t) should be replaced by f0. So, the solution of problem (2.1)-(2.6) approaches the stationary regime w°(y), «2(y) as t ^ to.
Solution for semi-bounded layers. To construct this solution, we consider the case when /i and /2 tend to infinity in formulae (2.26), (2.27). Taking into account that relation (2.28) when /i, /2 ^ to becomes
W <p> - (1+Tv)(S <■+v£ '2
and denoting the limits of Uj(y,p, /i, /2) by Uj(y,p), after some calculations we find
f/i(y,p) =
f (p)
1 , yv (g -1) A Pp
1 +-:—exp , / — y
M + V v V V vi
(2.32)
U2(y,p) =
f (p)
M(g - 1)
g--:—r^ exp I -\l — y
M + V v V V V2
(2.33)
It can be easily checked that U, U2 satisfy problem (2.20), (2.22), (2.24), (2.25) (we recall that y < 0 in (2.32) and y > 0 in (2.33)).
Using the properties of inverse Laplace transform [6, p. 506, p. 510], we reconstruct the originals
t
TV (g — 1)
where
Ui(y,t) = y f (t )
1+
Erf -
U2(y,t) = y f(t)
0
g-
M + Vv V vi(t - t )
M(g -1) Erf ( , y M + a/v W v2(t - t )
dT;
dT,
(2.34)
(2.35)
Erf z = 1 — erf .
erf z = —==
2 f -z2 ,
e z dz.
a—ioo
p
p
t
z
Formulae (2.34) and (2.35) provide the solution of problem (2.1), (2.3), (2.5), (2.6) in semi-bounded layers. Suppose that
fi
f <"—-
(2.36)
with the constant fi. Then, after some calculations, we find from (2.34) and (2.35) (formulae (2.32) and (2.33) can also be used):
Ui(y,t) — 2fiVt{ 1 +
I M + VV
U2(y,t) — 2fiTt (q - ^^ I M + VV
exp (-if)+2 ^ /exp (-T )d1
— tt tt
exP ' -t") - 2 j exP (-J J di
«2
(2.37)
(2.38)
where £j = y/^/Vjt is the similarity variable. In other words, if the pressure gradient is given by
(2.36), then the solution of problem (2.1), (2.3), (2.5), (2.6) is self-similar and given by formulae
(2.37) and (2.38). It is not surprising since only in this case equations (2.1) and (2.3) are invariant under the group of dilatations u' = au, y' = ay, t' = a2t with parameter a.
From (2.37) and (2.38), we find the asymptotic behaviour of velocities as t ^ to ^ 0) at any finite y, |y| ^ M = const
Uj(y,t) = fi(j+gf) St [1 + O(i)].
J + V V
On the other hand, when t is fixed and |y| ^ to (£i ^ —to, £2 ^ +to), one obtains from (2.37) and (2.38)
Ui(y,t) = 2fiV~t [1 + O (exp (—£2/4))] , U2(y,t) = 2figV~t [1 + O (exp (—£2/4))] . In the derivation of these relations, the results of asymptotic behaviour of integrals of the type
tt
F(z) — J f (i) exp[-S(i)] di
as z ^ to were used [7, p. 58].
On determining the pressure gradient. Often the volume flow rate through the layers is specified instead of the pressure gradient:
0 12
(2.39)
Qi(t) — J ui(y,t) dy, Q2(t) — y u2(y, t) dy.
-h
For example, suppose that (—12, 0) is the layer of water and (0, l2) is that of oil. The flow rate of oil Q2(t) is given. Applying the Laplace transform (2.19) to relations (2.39) and using formulae (2.26), (2.27), we find
Qi(p) — -
f (p)
pW(p) p
VM ch J pli - 1
Q - (q - 2) ch J — l2
+
+ —^ </— shj — li sh w — l2 - lA sh < / — li chj — l2 + —^ chj — li sh < / — l2
VH p V Vi V V2 V V Vi V V2 a/v V Vi V V2 ,
(2.40)
Q2M"fj ^ /f Gh '2 - !
1 + (2ß-1) ch JV^Zi
+£4/— shA/ — Z2 shA/ — Zi - ^ ( —^ sh w — Z2 ch,/ — Zi + ch,/ — Z2 sh w — Zi
p V V2 V vi Vvv V V2 V vi V V2 V vi
(2.41)
One can determine f (p) from (2.41) and reconstruct f (t) according to formula (2.29). The flow rate of the first liquid (water) is determined from (2.40) and (2.29).
It is interesting to calculate the flow rate for stationary flows (2.30) and (2.31). In this case,
0 ¡2
Qi = /ui(y) dy = 12f + 1) (4^ + 3M +12), Q2 = / u0(y) dy = ¿) (m + 41+ 312).
-¡1 0
The ratio between flow rates
Q| = m (m + 41 + 312) Q? = 12 (4m1 + 3m + 12)
strongly depends on the thickness of the layers. For example, if we take 1 = 0.25 (12 = 41?), then for water and oil with m = 0.312 we find Q°/Q? « 5.71, while for 1 = 0.5 (12 = 21?) we have
Q2/Q0 « 2.11.
3. Determination of Velocity Perturbations Induced by Thermocapillary Forces
In this case, the initial boundary value problem is written as
uit = viuiyy, —1? < y < 0; (3.1)
ui(- 1?,t)=0; (3.2)
«2t = V2«2yy, 0 <y<12; (3.3)
U2(M)=0; (3.4)
M?(0,t)= u2(0,t), M2u2y(0,t) - p?«,^(0,t) = H, t > 0; (3.5)
ui(y, 0)=0, -1? < y < 0, u2(y, 0)=0, 0 <y<12. (3.6)
Remark 4. There is a discontinuity in condition (3.5) at the initial moment of time since its left-hand side is zero at t = 0 according to (3.6) but H = 0.
Problem (3.1)-(3.6) has a stationary solution (Couette flow in layers)
r0 _ J, , „0 _ J, y
where
u! = M 1 + f ) , U0 = a ( 1 - ) , (3.7)
H/ /
a =--—, H = —(ffiiA + ®2Bi), / = / . (3.8)
M2(M + /) /2
The application of Laplace transform (2.19) to problem (3.1)-(3.6) leads to the boundary-
value problem
< — — Ui =0, —/i < y < 0; (3.9) 1v1
ui(—li,p) = 0; (3.10)
u2' — — u2 =0, 0 <y <l2; (3.11) -2
u2(l2,p) = 0; (3.12)
ui(0,p) = u2(0,p); (3.13) —
J2W2 (0,p) — Jiui(0,p) = — , (3.14)
p
where the prime denotes differentiation with respect to y. The solution of problem (3.9)-(3.14) can be easily obtained
ui(y,p) =--^— ,- - sh, (li + y), —li <y< 0; (3.15)
J2\fp Wi(p) chyjp V -ili V Vi
Jv2 H thJpv-1 li rp-
Mv,p) =----shJ p (I2 - y), 0 <y<l2, (3.16)
Wi(p) shy/pv-112 V V2
where
Wi(p) = = + th,/^li cth J -— I2. (3.17)
1 2
From (3.15)-(3.17), one can find the limits
lim puj (y,p) — u0 (y)
with the functions u0(y) from (3.7) and (3.8) as it should be. The flow rates are given by
O 12
Q0 = / u2(y) dy = a2, q2 = / u0(y) dy = ^ , (3.18)
-i 1 0
and their ratio is Q2/Q0 = 1//.
A priori estimates. Let us introduce new functions
Wj(y,t)= u0(y) - Uj (y,t). (3.19)
Then Wj (y,t) satisfy the problem
wit = viwiyy, -/1 <y < 0; (3.20)
W2t = V2W2yy, 0 < y < /2; (3.21)
wi(0,t) = W2(0,t), M2W2y(0,t) - Miwiy(0,t) = 0; (3.22)
w1(-/1,t)=0, w2(/2,t) = 0; (3.23)
wi(y, 0)= u?(y), W2(y, 0)= u2(y). (3.24)
Note that now the initial conditions are non-zero, and the second boundary condition in (3.22) is satisfied for any t > 0 (at t = 0, its right-hand side equals to H).
Let us multiply equation (3.20) by piwi and integrate it with respect to y between —/i and 0:
0 .n 0
2
9 1 [2 d 0 [2 d
dt 2 PM Wi dy = viwiwiy - dy.
J _ li J
—1 1 -ii
Similarly,
l2 ; l2
9 1 [ 2 d l2 [ 2 d
dt 2 p^ W2 dy = p2W2W2y - p2 y W2y dy.
Summing up these equalities and using boundary conditions (3.22) and (3.23), we obtain
0 ¡2 f 0, t > 0;
f + M ? y w2y dy + M2 J dy = f H21 ? (3.25)
-¡1 0 I M2(M + 1), t = 0,
where the 'kinetic' energy of layers is given by
0 I2
E(t) = 2 p 1 f w2 dy + 1 p2 y w2 dy. (3.26)
The Friedrichs inequalities (2.9) hold for wj due to boundary conditions (3.23). Then from (3.25) we derive the inequality (J = min(/-2vi, 1-2v2))
dE
— + 4JE < h(t), (3.27)
where h(t) is the right-hand side of (3.25). Integration of (3.27) with initial conditions (3.24) leads to
E(t) < E(0)e-4Äi, (3.28)
where
0 '2 2 E(0) = 1 p 1 j w2(y, 0) dy +2 p2 j w2(y, 0) dy = a;2 (p iZ 1 + p2Z2). (3.29)
-ii 0
due to boundary conditions (3.7) and (3.8).
Remark 5. In the derivation of relation (3.28), the Gronuoll inequality was used [8, p. 183]. It is applicable since h(t) is a summable function and integral of it is equal to zero.
Hence,
0 ¡2
f w2 dy < 2E(0) e-45t, i w2 dy < 2E(0) e-45t. (3.30)
7 Pi J P2
-¡1 0
imate the L2—n
obtain
To estimate the L2—norms of wjy, we again apply identity (2.14). Then, instead of (2.15) we
t 0 t i2 0
pi JI (w2t + ^yy) dydt + p2 //(w2t + v^yy) dydt + vi j w2y dy
0 -ii 00 -ii
'2 0 '2
+P2 j wly dy = pi j (u?y )2 dy + P2 y (ui°y )2 dy = a2 ^P- + P^ = Di.
i
It follows that
w\y dy < — , i wly dy < — . (3.32)
y Mi J y M2
0
Now from (3.30)—(3.32) and the Cauchy-Bunyakovski-Schwarz inequality, we derive the a priori estimates _
|wj(y,t)| < e"2it, (3.33)
V Pj H
where E(0) and Di are given by formulae (3.29) and (3.31), respectively. Returning to substitution (3.19), we obtain the following result.
Theorem 2. The solution of initial boundary value problem (3.1)-(3.6) is unique and approaches the stationary state (3.7) as t ^ to. The rate of convergence is estimated by
lUj (y,t) - u0(y)| < e"2it (3.34)
j V pj Pj
with constants E(0) and D\ from (3.29) and (3.31), respectively.
According to (3.34), the solution of initial boundary value problem (3.1)—(3.6) converges exponentially to the stationary solution.
4. Evolution of Temperature Perturbations
In this case, the initial boundary value problem has the form
Tit = XiTiyy - Aui, -Zi < y < 0;
T2
2t
Ti(-M) = 0;
X2T2yy - Au2, 0 <y<l2;
T2(l2 ,t) =0;
Ti(0,t) = T2(0,t), kiTiy (0, t) = k2T2y (0,t); Ti(y, 0)=0, T2(y, 0)=0.
(4.1)
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
Note that boundary conditions (4.5) are identically satisfied at t = 0 as well.
Problem (4.1)-(4.6) exactly coincides with problem (2.1)-(2.6), where one should replace f (t)
by -Aui(y,t), pip— f (t) by -Au2(y,t), Vj by Xj, and pj by kj. Note that Xj = kj/Pjcoj, where coj are the specific heats of the mixtures. Let us multiply equation (4.1) by picoiTi (equation (4.3) by P2C02T2), integrate it with respect to y between —Zi and 0 (between 0 and Z2), and sum up the results. Similarly to (2.7), we find
0 12 0 1 2 dEt2 + ki J T2y dy + k2 J T2y dy = -A picoi J uiTi dy + p2C02 J U2T2
dy
(4.7)
-i 1
-i 1
where
0 ¡2 E2(t) = 1 PiC0^ T2 dy +2 P2C02 y T22 dy. (4.8)
-¡1 0
The velocity field induced by the pressure gradient only. In this case, estimate (2.13) holds. It follows that
0
f 2 , 2J2Cie-4dt [ 2 , 2J2Cie-4dt , N
u2 dy < -, u2 dy < -. (4.9)
7 Pi J P2
-¡1 0
The functions Tj(y,t) satisfy the Friedrichs inequalities (2.9). So, from (4.7) we obtain the inequality similar to (2.10):
dE2 + 4^2Ei (t) < 2J3^E2(t) e-2<5t,
where J2 = min(/-2xi, 1-2X2) and J3 = %/2 |A| Jiv^Ci max(yc0i, yc02). It follows that
i _¿1_ (V2-»- e-2^2i)2 J2 = J;
E2(t) < 4(J2 - J)2 ^ e j , J2 = J; (4.10)
[ J32^e-4^, J2 = J.
In the derivation of estimate (4.10), we take into account that E2(0) = 0 due to (4.8) and initial conditions (4.6).
The estimates of integrals
0 ¡2
y Tly ^ y T2y dy
-l1 0
are obtained from identity (2.14), where one should replace vj by Xj, Uj by Tj, and Fj by -Auj. By analogy with (2.15) we can obtain the following identity
t 0 t ¡2
Pic0i/ / (T" + X2Tlyy) dydt + P2c0^ y y (T22t + X2TU) dydt+
0 -li 00 (4.11)
0 ¡2 t 0 t ¡2 V ^
T2y dy+k2 JT^y dy = A2 Pic0 i y y ui dydt+P2C0^ y«2 dydt .
-¡1 0 0 -¡1 00
With the help of inequalities (4.9), it follows from (4.11) that
0 ¡2
0 If. dy < , i It dy S , (4.12)
-¡1
where
x A2J2Ci . 04 = —^— (C0i + C02). 2J
Since
y ¡2
T2(y,t) = ^ Ti(y,t)Tiy (y,t) dy, T22(y,t) = -2 j T2(y,t)T2y (y,t) dy,
we obtain the following estimates from (4.10), (4.12) and (4.8):
T22 < 2
T2 dy
1
2 /2
1
T2y dy) < 2
2/2 2Ö4E2 (t)
k ip 2C0 2
|Ti(y,t)| < 2,
I2S4 E2(t)
k iP 2 Co 2
/2
Similarly,
|T=<yt)K (2/^)
/2
(4.13)
(4.14)
Therefore, in this case the temperature perturbations decay exponentially with time (as e St for S < S2 and as e—&2t for S > S2).
The application of Laplace transform to (4.1)-(4.6) leads to the following boundary value problem for representations
f„ P f Au2 (y,P) , ^
T 1--T 2 =-, —l 2 < y < 0;
X 2 X 2
T" — T2
2 X2
P f AU2(y,p)
X2
, 0 <y < I2;
T2 (0, p) = T2 (0,p), kT[ (0,p) = T2 (0,p); T2(—12,p) = 0; T2 (I2, p) =0,
(4.15)
(4.16)
(4.17)
(4.18)
(4.19)
where k = k i/k2 and the prime denotes differentiation with respect to y. The solution of problem (4.15), (4.16) can be written as
T2 (y, p) = L 2 sh, j-^y + L2 ch< P^y +--A- f U 2 (z,p) sh
X X -
X pX - 1
— (y — z) X
dz; (4.20)
I pp I pp A
T2 (y, p) = L3 shJ — y + L4 chJ— y +
X2 X2 -
u2(z, p) sh
X2 pX2 0
— (y — z) X2
dz. (4.21)
From (4.17)-(4.19) we have the system of algebraic equations for Li(p), i = 1,4:
L2
X
0
— U2(z,p) shw — z dz = L4
x/pXr1-1 v X 2
0
kw — L2 +--U2 (z,p) ch,/ — z dz = ,/ — L3,
X X X X2
-i 1
— sh, / — L 2 + ch, / — 12L2 = 0,
0
0
or
y
A '2
shw — Z2L3 + ch</ — Z2L4 +--, U2(z,p) sh
V X2 v X2 . L., -1
X2\/ PX2 0
- (Z2 - z)
X2
dz = 0.
It follows that
Li = —1(p), , 2(p) , L2 = Li th , /-^Zi, W2(p) V xi
L3 = —= Li - —1, aA
L4 = L2
Xi
0
— Ui(z,p) sh</—zdz,
\zpxt1 -ii V X1
(4.22)
where the following notations are used
—1(p) =--i/X2 [ U1(z,p) ch, /— z dz,
Xi V P J V Xi
-li
A cthJpX-1 Z2 0 rp~
—2(p) =--, — Ui (z,p) sh</— zdz+
X^/PXr1 VXi
XU PXi
(4.23)
+
A
X2\/ PX2 1 sh \/PX2 1 Z2 0
U2 (z, p) sh
^ (Z2 - z)
X2
dz,
W2(p) = VL + th , f^h cth , /^12. vx V x? V X2
Let us find the stationary solution of problem (4.1)-(4.5) (boundary conditions (4.6) are not
taken into account here). We have the following problem for functions T^y) and T20(y):
4
Ti0yy = — u0(y), -Zi <y< 0;
X1 A
T0yy = — u2(y), 0 < y < Z2;
X2
Ti0(—/i) = 0, T20(/2) = 0; T0(0)= T20(0), kT* (0)= T20y (0), k = ki/k2. If we substitute the functions w?(y), u0(y) from (2.30) and (2.31) to the right-hand (4.24)-(4.27), then integration of (4.24)-(4.27) and further simplification lead to
'•4 (p — /2)y3 , p(/ +1)y2
Ti0(y) = —^ 2Xi vi
—Z2f0p
y
+
+
12Z2 6Z1Z(p + Z) 2Z(p + Z) _
T20(y)
2X2 v?
where the constant a1, a2 are given by
A1?/0
+
(p - Z2)y3 , Z(Z + 1)y
+
+ aiy + a2,
+ kaiy + a2,
(4.24)
(4.25)
(4.26)
(4.27) sides of
(4.28)
ai
a2 =
24Xivi(p + Z)(k + Z) —ZiZ3 f0
24Xivi(p + Z)(k + Z)
12Z§ 6Z2(p + Z) 2(p + Z) _
[Z3(5pZ + 4p + Z2) - Xp(p + 4Z2 + 5Z)], [kZ2(5pZ + 4p + Z2) + Xp(p + 4Z2 + 5Z)].
(4.29)
0
2
4
2
y
It can be shown that lim Tj(y,t) = To(y), i.e. the temperature perturbations in the layers
approach the stationary state with time if lim f (t) = f0. To prove this, it is sufficient to
t—
calculate the limits lim pTj (y,p). As an example, let us consider the case j = 1. First, we recast
p—o
expression (4.20) with the help of (4.22)
Gj(
W2(p) ch^pX-112 VX2
y (4.30)
+--[ U2(z,p) sh (y — z)
T2(y,p)= G2(p) — sh ^/Z (y + l2) +
/ / dz.
xi\J Xi -p —h IVX1 '
Second, we substitute Uj(y,p) from (2.26) and (2.27) into (4.22), (4.23), and (4.30) and obtain a cumbersome expression for Ti(y,p), which is not presented here. However, there is an easier
way of calculating the limit limpTi(y,p) from (4.30) and the limits limpUj(y,p) = ul°(y) given
p— o p— o j
by formulae (2.30) and (2.31). As p ^ 0 (shx ~ x, chx ~ 1, x ^ 0), it follows from (4.23) that
o
W2 (p) ~ , pGi(p)---f u?(z) dz,
VX XiJ X—ip —Jh
pG2 (p)
0 i2 — j U^(z)zdz + xj U^(z)(l2 — z) dz
X2l2\JpX-
-h o
The integrals in the right-hand sides can be easily calculated with the help of (2.30) and (2.31):
o
/u?(z) dz =12^+) (4p + 3P + Z2),
-h o
jui(z)zdz = -24^+) (3p + 2P + Z2),
-h
'ju°2 W * = 1^+) (P + 3'2 + 4'>.
o
Therefore,
lim pG2(p) — pG^) sh ^ (y + l2) =
W2(p) ch^pX-2 l2 V X2
A/0l2f [kl2(8pl + 6p + 2l2) + lf(3pl + 2m + l2) — px(p + 4l2 + 5l)]
(4.31)
(y + l2).
24viXi(p + Z)(k + Z)
The second term in the right-hand side of (4.30) multiplied by p has the following limit as p ^ 0
y
A f o( )( ) d AfoZU y4 , (p - Z2)y3 + P(Z +1)y2 +
XiJ ui(z)(y - z) dz = 2ViX r 12! + 6Ti-P+) + HP+T +
-li (4.32)
Zi(8pZ + 6p + 2Z2)y + Z2(3pZ + 2p + Z2)'
+-
12l(p +1)
l
Summing up (4.31) and (4.32) gives precisely formula (4.28) for T0(y). It can be shown similarly that limpT2(y,p)= T0(y).
p^O
Determination of temperature perturbation induced by thermoconcentration forces. Let us first find the stationary solution of problem (4.1)-(4.5) with the functions ui(y), u2(y) from (3.7) and (3.8) in the right-hand sides of equations (4.1) and (4.3). In this case, the functions T0(y) satisfy the boundary value problem (4.24)-(4.27). The integration gives
TOO(y) _ — (6y3 + + aiy + 02, T0(y) _ 0A( - + + kaiy + a2, (4.33) Xi \ 6/1 2 ) X2\ 6/2 2 )
_ aAh(/2 - X) _ _ aAhhik/ + x) 01 _ 3X1 (k + /) , 02 _ 3xi(k + /) ■
Here the functions T0(y) are expressed by third-degree polynomials in y in contrast to (4.28). As in the previous paragraph, in this case it can be shown with the help of (4.20)-(4.23) and (3.15)-(3.17) that limpTj(y,p) _ T0(y). So, the temperature perturbation approaches the stationary
p^O j
regime with time.
5. Evolution of Concentration Perturbations in the Layers
The initial boundary value problem for concentration perturbations has the form
Kit _ diKiyy + ^ Tit + (^ - XB2) ui; (5.1)
Xi V Xi J
K2t _ d2K2yy + ^ T2t + (a2dA - B2) U2; (5.2)
X2 V X2 /
Ki(0, t) _ XK2(0, t), d(Kiy(0, t) + aiTiy(0, t)) _ K2y(0, t) + a2T2y(0, t); (5.3)
Kiy (-/i,t) + aiTiy (-/i,t)_0, K2y (/2,t)+ a.2T2y (M)_0; (5.4)
Ki(y, 0)_0, K2(y, 0)_0■ (5.5)
Equations (5.1) and (5.2) are satisfied for -/i < y < 0 and 0 < y < /2, respectively. The term Tjyy was replaced from the second equation (1.14). In addition, Bi _ XB2. So, (5.1) and (5.2) are inhomogeneous parabolic equations with known right-hand sides (see sections 2-4). In boundary condition (5.3), d _ di/d2.
Stationary distribution of concentrations. To find this distribution, we assume that Kjt _ 0 and Tjt _ 0. Then one obtains the following boundary value problem instead of (5.1)-
(5.4):
K0yy _ (^B -a^)ui(y), -/i <y<0; (5.6)
KOyy _ (B2 - ^) u O(yh 0 < y < /2; (5.7)
K0 _ XK2(0), d(Kiy(0) + aiT0y(0)) _ K0y(0) + a2T0y(0); (5.8)
K0y(-/i) + aiT0y(-/i) _ 0, K0y(/2) + a2T0y(/2) _ 0, (5.9)
where the functions u°(y), T0(y) are given by formulae (2.30), (2.31) [ (3.7), (3.8)) ], (4.28), (4.29) [ (4.33) ]. The choice of particular functions depends on the factor that induces the motion
of mixtures, i. e., the pressure gradient or thermoconcentration forces. In the former case, we substitute u0(y) from (2.30) into (5.6) and u0(y) from (2.31) into (5.7). With the help of the first condition in (5.8), integration leads to
^o^ ¿i/o/ AB2 a\A
y4 + (m - l2)y3 + m(1 +l)y2
1212 6l1l(M + l) 2l(p + l) _
7^0/ ^ l<2/0M fB2 «2AA
K2(y)="¿V" U
y^ (m - l2)y3 + l(l + 1)y2 12l§ + 612(m + l) + 2(m + l)
+ biy+A62, -li <y < 0;
(5.10)
+ b3y + b2, 0 <y <l2.
Z2
The constants bi, 63 are found from the boundary conditions on the walls (5.9):
bi = -iTiy<-''> +If) (TT - lA)(4P' + 3p + 63 = ^ №) - f) (B - )(3'2 + 4' + p)
The second condition (5.8) on the interface provides the following relation
dbi - 63 = (k«2 - d«i)ai, (5.11)
where ai is a constant from (4.29). Since it follows from (4.28) that
T?y,'» = - 12X,!Afo + 0 (4p + 3p + (2)'
T?y «2) = k«i + 12Xfny(3(2 + 4Z +
condition (5.11) is satisfied if and only if B2 = 0. Therefore, the stationary distribution of concentrations is possible only in the absence of their gradients in the direction of motion at the initial moment of time. When B2 = 0, the distribution is always non-stationary. So, if B2 = 0, then we have in (5.10)
bi = -aiai, 63 = -a.2kai. (5.12)
The constant b2 remains arbitrary and without loss of generality it can be assumed to be zero since adding constant concentrations Xb2 and b2 to K? and K?, respectively, does not change the problem for K? (y).
In the case when the velocity field is determined from (3.7) and (3.8) and the perturbation of temperatures are found from (4.33), integration of equations (5.6) and (5.7) gives
k? <"=(f - ^M £++iy+-2,
(5.13)
K?<"= (t - OAM - 62 + £) + hi + b2. 0 <y<„.
where one should again put B2 = 0. The constants bi and b3 are given by (5.12), where ai is a constant from (4.33).
So, one should put B2 _ 0 in equations (5.1) and (5.2) when studying the behaviour of solution for the problem (5.1)-(5.5) as t ^ to.
Solution in Laplace representation. Let us find the solution of problem (5.1)-(5.5) using the Laplace transform. Taking into account zero initial data for Kj and Tj, we obtain the following boundary problem for representations Kj(y,p):
P h a1 h , f AB2
Ki - — Kl =--pli + —
di xi \ -1
a1A\
- )ui;
Xi J
- „ P - a.2 - (B2
K2 - ~T K2 =--PT2 + -1---lu2;
d2 X2 \-2 X2
Ki(-li,p) = -aiTi (-li ,p), K2 (I2, p) = -a2T2 (l2,p).
Note that the right-hand side of (5.16) is equal to
(5.14)
(5.15)
K1(0,p) = AK2(0,p), dK[(0,p) - K2(0,p) = a2T2(0,p) - aidp(0,p); (5.16)
(5.17)
a2T2(0,p) - a1—Ti(0,p) = (ka2 - aid)Ti(0,p).
(5.18)
due to (4.17). The solution of problem (5.14)-(5.17) is written as
I— y
— I p I p d- f
K- (y,p) = D- shj— y + D2 chj — y + W — h-(z,p) sh yd-i y di \ p J
-h
'"T (y - z) di
dz;
(5.19)
/1 — f — y + D4 chj —y +* — h2(z,p) sh d2 d2 p
'"T (y - z) d2
dz,
(5.20)
where
ai ~ f\B2 a1AV U a2 h , (B2 a2AY
hi = - — pli + ( —j---— lui, h2 = - — pl 2 + ( —---)U2.
xi
di
xi
x2
d2
x2
(5.21)
After substituting (5.19) and (5.20) into boundary conditions (5.16) and (5.17), we find Dj(p) (j = 1, 2, 3, 4) with the help of (5.18)
Di(p) =
A L () G () [-2 T2(l2,p) ai [di th l2 ( )
G3(p) - G4(p) - a2i--j=---- \--\- T2(-li,p)
W3 (p)
ch y pd2 l2
sh ypd- li
D2(p) = J d-i aiTi(-i,p) + Di(p) cth, /Jli, V p sh,J—d-1 li V di
D3(p) = VdDi(p) - G3(p), d = —i/—2,
I— 0 _
D4(p) = ^ - AJj j hi(z,p)shV—izdz, v -h
(5.22)
G3M = --\ — [ hi(z,p) chj -p z dz + 4 — (kai - ai—)Ti (0,p), p di p i
-h
0
j--- o _
1 / -i P f P
G4M = - — thw — l2 hi (z,p) shw — z dz+
A V P V a,2 J V di
+
1
ch 1/ pd2 l2
'-i/„ V P
- l2
— h2(z,p)sh P
i «2- z>
dz,
W3(p) = AVd + cth Jf Zi th J-p Z2.
\ di V d2
Using formulae (5.19)-(5.22), it can be shown that limpKj(y,p) = K?(y) when B2 = 0, where
p— o j
Kj? are given by (5.10) or (5.13). It is done in the same way as in section 4.
On a priori estimate of concentration perturbations. Let us write equations (5.1) and (5.2) in the form
Kit = diKiyy + aidiTiyy - AB2U,, -Zi <y< 0; (5.23)
K
2t
d2K2yy + Q-2d2T2yy - B2U2, 0 <y<l2.
(5.24)
We integrate these equations with respect to y, taking into account the second boundary condition (5.3) and conditions (5.4) and (5.5). As a result,
o I2
J Ki dy + J K2 dy = -B2
-h o
One can only deduce from this relation that
to t I2
A J j ui dydt + j J U2 dydt
. 0 -ii 00
0 I2
Ki dy + K2 dy —1 0
is bounded for t ^ 0. In particular, this expression is zero when B2 =0 (note that Kj (y,t) can have arbitrary signs since they represent the concentration perturbations).
On the other hand, multiplying (5.23) and (5.24) by Ki and K2, respectively, and integrating again with respect to y, we obtain the integral identity
012 0
-E- + di J Ky dy + d2 J K22y dy = -aidi J KiyTiy dy-
-ii 0 -h
-i 1 12
-a2d2jK2yT2ydy-b2 (ajuikidy+jU2K2dy),
(5.25)
where
0 12
E(t)=1 j k2 dy+2| K22 dy.
-i 1 0
(5.26)
It can be easily deduced from these relations that K12 dy and K22 dy are bounded for any
-11 o
finite t when B2 = 0. It can be done with the help of elementary inequality ab ^ ea2/2 + b2/2e,
0
2
Ve > 0. Here it is difficult to obtain an inequality of type (2.10) or (4.10) from (5.25) and (5.26). The point is that the Friedrichs inequalities (2.9) does not hold for the functions Kj(y,t).
0 ¡2
However, they are satisfied if the mean values J K\(y,t) dy = 0 and J K2(y,t) dy = 0. It follows
-l i 0
from a more general Poincare inequality
b b 2 b J f 2(x) dx J f (x) dxj + 2(b - a)2 J f /2(x) dx.
a a a
However, the mean values are non-zero here. Therefore, this procedure does not allow us to determine the rate of convergence of Kj (y,t) to zero when condition (2.12) is satisfied.
This work was supported by the Russian Foundation for Basic Research (Grant Nr. 08-0100762).
References
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