The motion of a binary mixture and viscous liquid in a circular pipe under the action of an unsteady pressure gradient Текст научной статьи по специальности «Математика»

УДК 517.994

The Motion of a Binary Mixture and Viscous Liquid in a Circular Pipe under the Action of an Unsteady Pressure Gradient

Viktor K. Andreev*

Institute of Computational Modelling SB RAS, Akademgorodok 50/44, Krasnoyarsk, 660036, Institute of Mathematics, Siberian Federal University, Svobodny 79, Krasnoyarsk, 660041,

Russia

Alexander P. ChupakhiU

Institute of Hydrodynamics SB RAS Lavrentyev 15, Novosibirsk, 630090,

Russia

Received 10.01.2010, received in revised form 10.02.2010, accepted 20.03.2010 We study an invariant solution of the equations of thermodiffusion motion and of a viscous heatconducting fluid, which is treated as an unidirectional motion in a circular pipe with a common interface under the action of an unsteady pressure gradient. A priori estimates of the velocity, temperature and concentration are obtained. The steady state is determined and it is shown (under some conditions on the pressure gradient) that, at larger times, this state is the limiting one. On the other hand, if the pressure gradient of binary mixture tends to zero sufficiently fast, then the motion in pipe is slowed down by the viscous friction.

Keywords: thermodiffusion, invariant solution, interface, a priori estimates, Laplace transformation.

1. Basic Equations and Boundary Conditions

The governing equations of thermodiffusion motion (in cylindrical coordinates r, p, z) in the absence of external forces have the form

v v2 1 ( 2 u

ut + uur + pr + V ^ 2 v<^ 2

v

uv

1

vt + uvr + - v^ + wvz +----=------+ v Av----------2 u^----2 ,

r

r

pr

2

v

r

v 1 u 1

Wt + uwr + - w<o + WWz = — Pz + vAw, ur +-+ - v^ + Wz =0,

r p r r

vv

0t + uOr +— 9^ + wOz = xA 9, ct + ucr +— c^ + wcz = dAc + adA9,

(1.1)

where u, v, w are the velocity components; p is the pressure; 9 is the temperature; c is the concentration; p is the density, v is the kinematic viscosity, x is the thermal diffusivity, d is the diffusion coefficient (p > 0, v > 0, x > 0, d > 0, a are constants); A = d2/dr2 + r~18/dr +

*andr@icm.krasn.ru ichupakhin@hydro.nsc.ru © Siberian Federal University. All rights reserved

r 2д/дф2 + d2/dz2 is the Laplace operator. The system (1.1) admits a two-parameter subgroup of transformations corresponding to the generators [1]

д/дф, d/dz + Лд/дв + Bd/dc — pf (t)d/dp,

where Л, B are constants and f (t) £ is an arbitrary function. The invariant solution has the form

u = 0, v = 0, w = w(r,t) p = —pf(t)z + D(t), в = Az + T(r,t), c = Bz + K(r,t).

(1.2)

The solution (1.2) can be interpreted as follows. Consider the motion of an immiscible viscous heat conducting liquid and a binary mixture on the interface r = a. Initially, the mixture and the liquid are at rest and occupy the cylindrical domains 0 ^ r < a and a < r < b, respectively.

Aj z and concentration field c

Bz in the mixture

At time t = 0, the temperature field 0j are created instantly in the whole domains. The pressure gradients fj (t) induce the motion of mixture and liquid. In this motion, the interface is represented by the cylindrical surface r = a and the trajectories are straight lines parallel to z axis, see Fig. 1.

Thus the solution (1.2) can be written as follows (j = 1, 2):

wj = wj (r,t), pj = —pj fj (t)z + Dj (t), ej = Aj z + Tj (r,t), ci = Biz + K (r,t). (1.3)

The functions Wj, Tj, K can be considered as perturbations of the quiescent state.

On the interface r = a, the following conditions are imposed [2]:

wi(a,t) = W2(a,t), Ti(a,t) = T2(a,t), Ki(a,t) = \K2(a,t),

0;

д^^,, t) дT2(a,t)

ki---r----- = K2~

дK (a,t) д^^В)

+ a1-

дт- дт- ’ дт- дт-

Ai = Л2 = Л;

f2 (t) = pfi(t), D2 (t) = Di(t), p = Pi/p2;

f^2W2r (a,t ^1 w1r (a,^,') °?

where A is the Henry’s law constant, p,j = Vj/pj, kj are the thermal conductivities, kj On the rigid wall r = b,

W2(b,t)=0, T2(b,t) = 0.

(1.4)

(1.5)

(1.6)

(1.7) const.

(1.8)

The boundedness conditions at r = 0 are given by

|wi(0,t)| < те, |Ti(0,t)| < те, |K(0,t)| < те.

The initial conditions have the form

Wj(r, 0) = 0, Tj(r, 0) = 0, K(r, 0) = 0.

(1.9)

(1.10)

Substituting (1.3) into the governing equations (1.1) we obtain the system of parabolic equations with the unknowns Wj (r, t), Tj (r,t), K(r,t) (j = 1, 2):

wjt fj(t) + vj ^wjrr + r wjr^ ;

Tjt = Xj \ Tjrr + r Tjr j Awj;

Kt = di ( Krr +— Kr j + aidi fTirr +— Tir ) — Biwi.

(1.11)

(1.12)

(1.13)

For j = 1, r £ [0, a), and for j = 2, r £ (a, b). Thus we have the linear conjugate initialboundary problem (1.11)—(1.13), (1.4)—(1.10).

2. Stationary Solution

We find the steady-state solution of the problem (1.4)—(1.13) (the initial data (1.4) are ignored in this case). All unknowns do not depend on time, so w°(r), Tj0(r), K0(r), fi(t) = /° = const, Di(t) = D° = const. Such a solution has the form

л

4vi

(b2 — a2)p, + a2 ( 1------2

w0

/i°b2M л rA /

TVT l1 — b? , M = Mi/M2,

T 0

Ar2/° 16x ivi

a2 + p,(b2 — a2) —4

T0 = ArVl° I b2 r

16X2V2 C — 0+ 1П r + C“’ K

where Ci,..., C4 are well-known constants.

a iAr2/° 16xivi

+ Cb

a2 + p,(b2 — a2) —4

(2.1)

+ Cp

Remark 1. Stationary distribution of concentration is only possible in the absence of concentration gradient in the direction of motion at the initial moment of time. When B = 0, the concentration is always non-.stationary.

3. A Priori Estimates of Velocities

It can be seen that equations (1.11)—(1.13) form three problems for functions (w i, W2), (Ti, T2) and (K). These adjoint problems can be solved successively. We have the following adjoint linear initial boundary value problem first,

W 11 = vi

(w irr A^w irj + /i(t), 0 < r < a;

(3.1)

0

w

1

2

W2t = V2 \ W2rr + - W2 rj + pfi(t), a <r <b;

wi (r, 0) = 0, W2(r, 0)=0;

|wi(0,t)| < to;

W2 (b, t) = 0;

w i(a,t) = W2(a, t), P2W2r (a, t) — ^iwir (a,t) = 0.

There exists the energetic identity

a b a b

д E + pi J rW2r dr + p2 jrw\r. dr = pi fi(t)^J -Wi dr + J-W2 dr^j ,

where

E (t) = 2

a b

pij-w2 dr+«/ -w2 dr

The uniqueness of solution for problem (3.1)—(3.6) follows from (3.7).

The right-hand side of (3.7) admits the estimate

(j rdr^ ! (j rw 2 dr^f + (J rdr^1 (J rw\ dr^1 < Ci^EJt),

0 0 a a

^ a Vb2 — a2 \

Ci = y/2

max

^\[РГ \ГР2 )

Moreover, there is another identity for the problem (3.1)—(3.6)

t b

t a

p1r

00

2 2 I 1

Wit + Vi ( Wirr + - Wir

drdt + p2 j J r

0a

w\t + v| ( W2rr + r W2r

b t

+p ij rw2lr dr + P2 j rw\r dr = pr [a2 + p(b2 — a2)] J f2(t) dt.

Therefore, if

then the following estimates hold

СЮ

J f2(t) dt = C22,

rw2r dr ^

[a2 + p(b2 — a?)]C^2

rw\r dr ^

p i [a2 + p(b2 — a2)]Cf 2P2

2

a

b

a

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

(3.7)

(3.8)

(3.9)

(3.10)

drdt+

(3.11)

(3.12)

(3.13)

Lemma 1. The following inequality holds:

a b

rwf dr + rw2 dr ^ Mq I pi / rw(r dr + p2 rw^r dr

(3.14)

where Mq is the solution of the variational problem

Mq = sup

vi,V2EV

f rv2 dr + f rv2 dr

0 a

a b

pi / rv2lr dr + p2 / rv?;r dr

0 a

Here V C W2l(r;0, a) x W2i(r; a,b), and conditions (3.3)-(3.6) for vi, v2 are satisfied. The solution of Euler equations for (3.15) is given by [3]

vi = CiJq( 1 r\ V2 = C2Jq( 1 Л + C3Yof 1 r

VMoMi

V VM0P2

V M0P2

(3.15)

(3.16)

Let us denote ai = b/a, a2 = vVTV, x = a/yfpiMQ, then Mq (more precisely x) is the root of equation

Jo(x)[Ji(a2x)Yo(aia2x) - Jo(aia2x)Yi(a2x)] +

(3.17)

+a2 Ji(x)[Jo(aia2x)Yo(a2x) - Jq(a2x)Yo(aia2x)] = 0,

where Jk, Yk (k = 0,1) are the Bessel functions.

If xq = xo(aia2) is the minimal positive root of equation (3.17), then

Mq = ---2 •

pixQ

(3.18)

If a2 = 0, then xQ is the first positive root of function JQ(x). If a2 = 1, then xQ = 2.40482/ai, i. e., the hyperbola.

Using the inequality (3.14), we obtain

dr + p2 J a rw)r dr ^ 2dE, (3.19)

1 • / —— mm Mq \ - ,“ ) • vPi P2 / (3.20)

dE

Therefore, it follows from (3.7)-(3.10) that-+ 2SE ^ Cipi|/i(t)k/E, and

dt

E < CM

t 2

J |/i(r)|e5r ddj e-2St.

(3.21)

If

СЮ

j |/i(t)|e^T dT = C3,

(3.22)

b

a

2

b

a

2

r

d

then

a

/ rW dr ^

piC12C|e-2'5t

2

0

Remark 2. Boundedness of the integral (3.22) Now, we have

b

j rw2 dr ^

p\Cl Cfe"2<5t 2P2

(3.23)

a

implies the boundedness of the integral (3.12).

w‘2(r,t)

b

j(w2)r dr

r

2

< -a

b

rw“2

dr

1/2

a

b

rw22r dr

a

1/2

< С4е-У

(3.24)

C C C C I P [a2 + P(b2 - a2)]

у М2 a

or

|w2(r,t)| e-st/2 (3.25)

for all r £ [a, b].

Unfortunately, similar arguments are not applicable for the estimate of |wi(r, t)|. We have to find another way. From the boundary condition (3.6) and (3.25) we obtain

|wi(a,t)| — |w2(a,t)| < \JC4 e-St/2. (3.26)

The initial value problem

wit — vi ( wirr + r wiM + fi(t), 0 < r < a;

wi (r, 0) — 0, |wi (0, t)| < ж, wi(r,t)l — wi(a,t),

has the solution

, ,n 2vi MnJo(Mnr/a)

wi(r't) — ^ § Ji(Mn)

!wAa’r )e-,,,4<'-T )/a2 dT+

+ 2 J0(Mnr/a)

a n—i MnJi(Mn)

t

J fi(T)e-VlMn<t-T)/a2 dr.

After some calculations we obtain the estimate

-5t/2

I / С1/ V J " >

И(М)| < e-,1vi./a2, 2M2vi/a2,

t

(3.27)

(3.28)

where Mi is the first root of the function Jo(m) — 0. Thus, the following theorem is proved

Theorem 3.1. The solution of the problem (3.1)-(3.6) tends to zero as t ^ ж subject to condition (3.22), and the estimates (3.25), (3.28) hold.

In other words, if the pressure gradient in the mixture tends to zero sufficiently fast, the motion of mixture and fluid is slowed down by the viscous friction according to inequalities (3.27), (3.28).

4. Solution in the Laplace Representation

Let us apply the Laplace transform to problem (3.1)—(3.6)

СЮ

,(r,ri = /

(4.1)

As a result, we obtain a boundary-value problem for the ordinary differential equations:

- , 1 h P ~ fi(p) 0 < < ~ + 1 - P ~ h(p) < < b

w\rr +— wir-------wi =---------, 0 < r < a; w^rr H— w-2r-----г-2 =--------, a < r < b;

r V1 Vi r V2 V2

w\(a,p) = W2(a,p), W2(b,p) = 0; М2 w-2 r (a,p) = Ml w-ir (a,p),

with the exact solution

whi = CiI0[J — r\ +

f i(p)

(4.2)

(4.3)

(4.4)

(4.5)

, v ,, , (4-6)

V2 V2 p

where Ci, C2, C3 are well-known constants.

Suppose that the lim fi(t) = f0 = const exists. Of course, in this case the function fi (t) does

t——

not satisfy condition (3.22). Simple, but cumbersome calculations with the use of asymptotic representations for Bessel’s function show that lim Wj(r,t) = lim pwhj(r,p) = wj(r), where

t—ж p—0 j

wj0(r) is the stationary solution (2.1).

W2 = C2I0I.I + CsKolJ + f2(p)

5. On Determination of Flow Rate or Pressure Gradient (Inverse Problem)

The volume flow rate through the layers is often specified instead of the pressure gradient:

a о

Qi(t)=2 к j rwi(r,t) dr, Q2(t) = 2 к j rw2(r,t) dr.

(5.1)

For the stationary solution, we have

Qo Ka4f0 Qi = 4v i

1 b2

2V 02 - L M

m = m i/M2;

The ratio of flow rates is

Q2 = (b2 - a2)2.

Q0 = 4 [1/2 + M (b2/g2 - 1)] Q2 = M (b2/a2 - 1)2

For example, if b/a = 1.1, m = 0.1, then Q° « 500 Q°.

(5.2)

(5.3)

w

Assume that Qi(t) is known, therefore Qi(p) can be found as

fi(p)ai

Qi(p) = 2п I rwi(r,p) dr = 2n

2p + p4v Via

(5.4)

From this equation, we can find fi(p) and hence

l + itt

1

fi(t)=2ni ePtfi(p) dp■

l-i

Thus, we can find the unknowns wi(r,t), wi(r,t), fi(t) and the inverse problem is solved.

6. Determination of Temperatures

In this case, the initial boundary value problem has the form

Tit = xi ^ Tirr + 1 Tir) - Awi(r,t), 0 <r < a;

Tit = xifrirr + 11 Ti^ - Awi(r,t), a < r < b;

Ti(a,t) = Ti (a,t), kiTir (a,t) = kiTir (a,t); Ti(b,t)=0, ITi(0,t)l < ж;

Ti (r, 0)=0, Ti (r, 0)=0.

(6.1)

(6.2)

(6.3)

(6.4)

(6.5)

Problem (6.1)—(6.5) exactly coincides with the problem (3.1)—(3.6) for Wj (r,t), where one should replace Wj ^ Tj with fj ^ -Awj, and Vj ^ Xj with pj ^ kj. Note that kj = Xj Pj cj, where cj are the specific heats of the mixture and liquid.

Similarly to (3.19), (3.22), we find the estimates

ti/ie-st/i,

\Ti(rdt)\ ^ NA 1 e-5lt|i/i

^i = s,

Si = s,

|Si - S|i/i

s- = Mmin{t-A |ri(r-()Kwl^1/2e-^si>0

Si = Si(S,Si,xip\/ai,vip\/a2), pi « 2.408.

(6.6)

(6.7)

(6.8)

The application of the Laplace transform to (6.1)—(6.5) leads to the following boundary value problem for representations

Tirr 1 + TT1r - — Ti A = — wi, 0 < r < a; (6.9)

r x1 Xi

'Tirr 1 + Tir — — Ъ A = — wi, a < r < b; (6.10)

r Xi Xi

Ti(b,p) = 0, ITi(0,p)l < ж; (6.11)

Ti(a,p) = T2(a,p), kT\r(a,p) = T2r(a,p), k = A .

k2

(6.12)

The exact solution of the problem (6.9)-(6.12) can be written as

T ( ) D r ^ [V ) Afi(p) ACi

Ti(r,p) = DiI0\J — r--------2----------j—------—

VV xi / p2 pxi(1/xi- i/vi)

Io L/ —r

T2 C,p) = Wof, /J r) + D3K„( r) - ^

(6.13)

AC2

. I t A°3—/—1 Ko(\ 1^ r V2 J px2(1/X2 - 1/V2) VV V2

-IoL/-r I -

pX2 (1/X2 - 1/V2)

when vi = xi, V2 = X2, i- e. Prandtle numbers are not equal to one. If vi = xi, V2 = X2, then

i’,(r,p)=djoLX- r -

Afi(p) AC, r

+xmx I4v vir

T2(r,p) = D2 * | 2 /'Vf r) + DK„ ( Д r) - fl +

AC2r .Ii(./ Hr I -

AC3r K / /_P r 1

2^p/v2 x WV VW 2^p/v2 x2 VV v2

(6.14)

It can be shown that

lim pTj (r,p) = TO(r),

p——O

if lim pfi(p) = f°, where T°(r) is the stationary solution from (2.1).

p—O j

7. Determination of Concentration

The initial boundary value problem for concentration perturbations has the form (B = 0)

(7.1)

Kt — di ( Krr +— kA + aqdi | Tirr +— Tir

rr

K (r, 0) = 0;

Kr (a, t) + aiTir (a, t) = 0;

|K(0,t)| < to.

It can be easily verified that

for all t > 0.

a

rK( r, t) dr = 0

(7.2)

(7.3)

(7.4)

(7.5)

Lemma 2 ([3]). Assume that the function g(r) £ C[0, a], a > 0, gr £ L2(r; 0, a) and f rg(r) dr

0, then the Friedrichs inequality holds

a 2 a

j rg2(r) dr ^ — j rg)2 (r) dr.

2

p

Remark 3. Instead of 1/4 we can take 1/xQ, where xq к 3.8317 is the first positive root of the Bessel function Ji(x). In fact, it is enough to solve the variational problem

a

f rw2 dr

Mq = sup ^---------,

w=0 f rw2 dr

0

with |w(0)| < ж and

a

j rw(r) dr = 0.

0

Taking into account the inequalities (3.25), (3.28), (6.6), (6.7) and (7.6), we obtain the estimates

a

j rK2 (r,t) dr ^ N3ti/2e-Sat, (7.7)

0

|K(r,t)| ^ N4ti/4e s'it/2, where S3 > 0 is the function of Si, S, xidl/a2, vidi/a2.

The problem for the Laplace representation K(r, p) has the form

K" + 1 K' - PK = -Pdf - — Wi = F(r,p), 0 <r<a; (7.8)

r di Xi Xi

K'(a,p) + ai Ti(a,p) = 0; |K(0,p)| < ж.

The exact solution of this problem can be written as

K (r, p) = LiIo(JpA +

r

+ J yF(y,p)

IoU i r)KoU % У) - IoU ^ y)KoU J r

dy-,

l i = -

d

1 -i

yF (y,p)

^ 1% J K0\ \ di y ) +

^ Idt y)K\\j di a

dy + aid — Ti(a,p) У

p

Again, if limpfi(p) = lim fi(t) = f0, then

p——0 t—

lim piK (r,p) = K 0(r),

p—0

a

(7.9)

(7.10)

(7.11)

(7.12)

where K0(r) is the stationary concentration from (2.1).

Conclusions

The invariant solution of the equations of thermodiffusion motion is investigated. This solution describes the unsteady motion of an immiscible binary mixture and viscous liquid (lubricant) with a common cylindrical interface under the action of pressure gradient. It was shown that this problem can be reduced to the linear conjugate initial-boundary-value problem for parabolic equations. A priori estimates of the velocity, temperature and concentration are obtained. If the mixture pressure gradient tends to zero rapidly enough, then the motion is retarded by viscous friction. The stationary solution of this problem is found and if the mixture pressure gradient has a finite limit when time tends to infinity, then the solution tends to the stationary state. This statement is proved by using the properties of the Laplace transformation. Thus there are three possibilities of the solution behavior. In the first place, all unknowns tend to zero as time increases if the condition (3.22) is satisfied. Secondly, all unknowns tend to a stationary state as time increases if the there exists a finite limit lim f2(t). And the last case, the problem

t—— ^

considered has unsteady solution at any time, see the formulae (4.5), (4.6), (6.13), (6.14), (7.11), (7.12). There has been previous work on planar layers [4].

This work was supported by Siberian Branch of Russian Academy of Sciences on integrated project No. 65.

References

[1] V.K.Andreev, I.I.Ryzhkov, Group classification and exact solutions of thermodiffusion equations, J. Diff. Equations, 4(2005), no. 4, 508-517.

[2] V.K.Andreev, V.E.Zakhvataev, E.A.Ryabitskii, Thermocapillary Instability, Novosibirsk, Nauka, 2000 (in Russian).

[3] V.K.Andreev, On Inequalities of the Friedrichs Type for Combined Domains, J. Siberian Federal University., Mathematics & Physics, 2(2009), no. 2, 146-157 (in Russian).

[4] V.K.Andreev, The Joint Motion of Two Binary Mixtures in a Flat Layer, J. Siberian Federal University., Mathematics & Physics, 1(2008), no. 4, 349-370.

Движение бинарной смеси и вязкой жидкости в цилиндрической трубе под действием нестационарного градиента давления

Виктор К. Андреев Александр П. Чупахин

Исследовано инвариантное решение уравнений термодиффузионного движения и вязкой теплопроводной жидкости, которое представляет собой однонаправленное движение в цилиндрической трубе с общей границей 'раздела. Получены априорные оценки скорости, температуры и концентрации. Найдено стационарное состояние и показано (при некоторых условиях на градиент давления), что это состояние является предельным. С другой стороны, если градиент давления бинарной смеси стремится к нулю достаточно быстро, то движение в трубе останавливается за счет вязкого трения.

Ключевые слова: термодиффузия, инвариантное решение, поверхность раздела.