A stability criterion for an infinitely long Hall-Héroult cell

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

Journal of Computational and Applied Mathematics 71 (1996) 47-65

ELSEVIER

A stability criterion for an infinitely long Hall-H6roult cell P. Maillard, M.V. Romerio* Department of Mathematics, Swiss Federal Institute of Technology, CH 1015 Lausanne, Switzerland Received 12 September 1994; revised 8 August 1995

Abstract This study is concerned with the stability of a Hall-H6roult cell which is assumed to be infinitely long. The modelling used leans on the magnetohydrodynamic theory for incompressible fluids. The presence of longitudinal electrical current inside the cell are taken into account. The influences of these last ones, on the stability of the cell, are studied. The modelling is obtained through a linearization of Navier-Stokes and Maxwell equations around a stationary solution. The problem defined by this process is solved with the help of a method related to the conservation of the energy flux. It allows in particular to establish a stability criterion related to the intensity of the longitudinal electrical currents. Peculiar features of possible unstable modes are analytically obtained. Numercial results confirm the existence of such unstable modes.

Keywords: Magnetohydrodynamics; Stability; Aluminium production AMS classification." 76E25

1. Introduction

This paper is devoted to the stability problem of an infinitely long Hall-H6roult cell. Cells characterized by this particular geometry were already used in other papers on the one hand to compute steady solutions for systems, described in the frame of M H D theory for incompressible fluids [8] and on the other hand to derive some results about stability [3]. In the present paper the used mathematical modelling is leaning on a linearized version of the equations derived in [3]; it can however afford a constant longitudinal electrical current, the presence of which is usually supposed to generate possible instabilities. The system of partial differential equations resulting from the linearization procedure will be studied both with the tools used in [3] and by spectral techniques.

* Corresponding author. E-mail: [email protected]. 0377-0427/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0 3 7 7 - 0 4 2 7 ( 9 5 ) 0 0 2 1 7 - 0

48

P. Maillard, M.V. Romerio /Journal of Computational and Applied Mathematics 71 (1996) 47-65

As in [3], we will look for solutions which are translationally invariant (in the direction where the cell is infinite). The system we are dealing with consists in a steady solution and in a set of linear equations (obtained by the linearization procedure mentioned above) which describes the behaviour of the oscillations around the steady motion. Both are obtained in Section 2. Making use of methods developed in [3] we obtain in Section 3 a criterion showing the influence of longitudinal electrical currents on motion stability. In Section 4 the spectral problem associated with the linearized equations is studied. The time dependence of different fields being assumed of the form exp( - At) (2 e C), we mainly show that if Im 2 :~ 0 then Re 2 > 0; in other words, when oscillating the system is stable. In the last section some numerical calculations showing the dependence of eigenvalues on the amplitude of the longitudinal electrical current are presented.

2. The modelling The Hall-H6roult cell we are dealing with is schematically represented in Fig. 1. We assume that this last one is a cylinder the generatrices of which are parallel to the x-axis. Let us introduce some notations. ~21 and ~t'~2 are, respectively, the domains occupied by the aluminium and the electrolytic bath (their boundaries are denoted by ~ 1 and ~(22). 51 and ~-'~2 correspond, respectively, to the cathode and anode blocks (their boundaries are denoted by ~ 1 and t~2). ~2 = (~1 w t]2) °

(domains occupied by the fluids),

~ = (~1 u-~2) °

(electrodes),

A = (~u~) °

(cell).

The interfaces between the different domains are given by F = ~'~1 (") ¢~'~2,

S = S 1 uS2,

z~ = , ~ 1 U z ~ z -

The different fields entering the modellings are: velocity ~, pressure ,6, induction b, magnetic potential ~, electric field ~, electrical potential ~, current density .~ We denote by x = xex + yey + zez the elements of •3, with its usual euclidean structure. In agreement with [-3, 8] we impose on these fields some restrictions which are contained in the assumptions given below. (H1) Velocity, magnetic potential and current density as well as gradients of pressure and electrical potential are x-independent. (H2) Current densities outside the cell are given; they are x and t (time variable) independent. (H3) The interface F, between the fluids is x-independent; it is described by the relation z=h(y,t),

y e [0, L],

where L is the width of the cell. From (H3) we see that the domains f2~ and (22 as well as F are depending on h. We will from now on recall this dependence by writing Qi(h), j = 1, 2, and F(h).

P. Maillard, M.V. Romerio /Journal of Computational and Applied Mathematics 71 (1996) 47-65

~ / ~

x~

/~2

49

Freem~rfaceF(h)

Y

[

[

Electrolytibath c &'},2(h)~ Aluminium~"~l(h) ~

AnodeblockE 2 CathodeblockE l

1

Fig. 1. Schematic representation of a cross section through a Hall-H6roult cell.

As usual in m a g n e t o h y d r o d y n a m i c theory (MHD) of incompressible fluids we assume that different fields are governed by the following equations pOtu +

p(~, V)ti = rl Aft - V(~ + pgz) +fAb

div ~ = 0 -

in ~'~l(Tl)w~t~2(h),

(2.3)

in O l ( h ) u Q 2 ( h ) w ~ u A c,

#,h = (~, V(z

- h))

(2.1) (2.2)

in ~l(~-/)k.J~E(~'/)k.J-,~k-J/l c,

A~ = Pof

divff = 0

in f21(h)uf22(/Tz),

(2.4)

on F(h'),

(2.5)

in ~r21(h)k.J~r~2(~-/)k.J,--~k.)A¢,

(2.6)

where = curl ~

J'" =

a ( - g ~ - ~ta + a ^ b)

in f21(h)wO2(h)

~ ( - Vq~ - O,~i)

in ~,

jo

in ,~c

(2.7)

(jo

given),

p, q and a are, respectively, the density, the viscosity and the electric conductivity (both having constant values in different domains where they are defined) whereas g is the gravity constant and Pc is the magnetic susceptibility of the vacuum. In order to describe the conditions satisfied by the fields on the different boundaries and interfaces we denote by {S}~, the "jump" of a generic field S on the interface e~. F o r ~o = F(h) we have for instance {S},o = S I~,t~) - S [a~ts). We now give these conditions (li represents the outward unit normal on the corresponding domains). li = 0

on 00,

{ti}r(~) =0,

(2.8)

3 {( --P(~ij + ?](OiUj + OjUi))~lj}r(~} = 0 , j=l

i = 1, 2, 3,

(2.9)

50

P. Maillard, M V. Romerio/Journal of Computational and Applied Mathematics 71 (1996) 47-65

ti = O(ln(y 2 + z2))

for [(y,z)[ ~ oo,

{~i},o =0,

{b/x ti},~ = 0

{~}o, = 0

for e~ = S

( f ti) = t 0 (jo, ti) {(.L ri)},o = 0

for ~o = O A w S and F(h), and ~o = F(h),

on ~ A \ Z , on Z,

(2.11) (2.12) (2.13)

for (n = S and F(h),

f ~ h(y, t) dy =0,

(2.10)

t/> 0.

(2.14) (2.15)

Remark. The modelling presented here is very similar to the one used in [3, 8] but for assumption (H2), which is mathematically expressed by condition (2.7). With (H 1), we moreover have that/~ and q~ can be written in the form

~(x, y, z, t) = p0(y, z, t) + ~(t)x, $(x, y, z, t) = q~o(y, z, t) +/~(t)x,

(2.16)

where c7 and/~ are arbitrary functions of time. Being interested in a linearization around a steady solution we introduce the following representation of a generic vector field g (where g represents any unknown of the system). g(x, t) = s(x) + S(x, t).

(2.17)

The same type of decomposition is applied to a scalar field. For a vector field we moreover decompose s and S in the following way. s(x) = s*(x) + s(x)ex

and

S(x, t) = S*(x, t) + S(x, t)ex,

(2.18)

where (s*, ex) = 0 and (S*, ex) =0. We thus decompose each vector field into a vector along ex and another one perpendicular to ex. 2.1. Steady states The steady solutions associated to the evolutionary problem (2.1)-(2.15) are obtained by using techniques developed in [8] (see also [5] for more details). They are characterized in the following proposition.

Proposition 2.1. Any solution of the steady problem satisfies (a) u* = 0 in (2; (b) j = - a f l i n A ; (c) Po = - pgz - (1/2po)b 2 - afla + 0 in Y2, with 0 = Oi in Y2i(h), i : 1, 2, where Oi and 02 are two real constants such that

P. Maillard, M.V. Romerio/Journal of Computational and AppliedMathematics 71 (1996)47-65

(d)

#°If~2n° ln((Y-

a(y,z)-

51

¢)2 + (z-~l)2)l/2J°(~'tl)d~dqJ

°EfA

1

ln((y - ~)2 + (z - r/)2)1/2 ( - aft) de dq + 03, (y, z) e ~2,

where

03

is a real constant; moreover, a ~ C1(~2); {a}r(h)fl ( a ( y ' h ( Y ) ) - - l f ~ ) {P}r(h) 9 -~ a(~, h(~)) d( , y e [0, L];

(e)

h(y) -

(f)

{~3z(Po + ~pol b2)}r(h) = { - - p g - - a f l O z a } r ( h ) o n F ( h ) .

Let us point out that in Proposition 2.1 the longitudinal currentj is proportional to fl, which thus appears as the main parameter of our study. When fl =0 we obtain a unique solution for which h =0 i.e. a flat interface. When fl 4:0 the shape of the interface is obtained by solving the problem corresponding to (d) and (e), of Proposition 2.1, for instance with the help of a fixed-point method. We add the following assumption which is clearly valid for fl = 0. (H4) There exists a positive number flo such that the steady problem admits a unique solution satisfying (it is implicitely assumed that Pl > P2) 0 < Mo < { - P9 - aflOza}r(h) < M1

on r(h),

V l/~I < %.

(2.19)

2.2. The linearized problem

Taking (2.17) and (2.18) into account we now exhibit the equations and conditions obtained when performing a linearization of the relations (2.1)-(2.15) (see [5] for more details). Keeping in mind the peculiar properties of steady solutions, especially the fact that the transverse velocity field is vanishing, it can be shown that, under the assumptions (HI) and (H2), the problem is partially decoupled into two parts. The first one, which is the only one considered here, consists in finding U*, Po, A, H and fl satisfying: p~3,U* = qAU* - VPm + (jVA + JVa)

div U* =0

in

-- AA = #oJ

in ~l(h)LyQ2(h),

(2.20)

~x(h)u~2(h),

(2.21)

in (21(h)wf22(h)w~uA c,

(2.22)

on V(h),

(2.23)

in f2x(h)~22(h),

(2.24)

O,H = (U*, V(z - h))

where 1

P,,, = Po + - - bB

/~o

P. Maillard, M.V. Romerio/Journal of Computational and Applied Mathematics 71 (1996) 47-65

52

with B = (curl A *, ex) (B is the x-component of the induction magnetic field).

cr(- fl - O,A - (U*, Va)) in Ol(h)uQ2(h), J=

a(-fl-0,A)

in ~,

0

in ~c,

(2.25)

with the following boundary and interface conditions: U* = 0

on ~?f2,

{U*}r(h~=0,

(2.26)

3

{ ( - P o + (Ozpo)H)6ij + rl(c~iU* + (?jU*)nj}r(h) =0,

i=2,3,

(2.27)

j=2

A = O(ln(y 2 + z2)) {A},o = 0

for I(y,z)l--, oo,

for co = O A u S

{c3A/~n }OA~S=0,

and

(2.28)

e) = F(h),

{c3A/an}r(h) = --/to {O'}r(h)/~

(2.29) H

IV(z

h)['

{B + (c3zb)H}r(h) =0, (h) lV(z Z h) l da =0' fA J dX =O.

(2.30) (2.31) (2.32) (2.33)

The stability problem we are interest in will now rest on a study of Eqs. (2.20)-(2.33). The second part of this system consists in equations which are similar to the above ones. They contain the unknowns A* and U. The field B, which derives from A*, (2.24), appears in (2.31). However, as it will be seen later, this last one never enters the different proposition concerning the stability. The derivation of the set of equations for A * and U is thus unnecessary for our purpose. Uniqueness and existence of solutions for the linearized system derived for (2.1)-(2.7), in its general form, has been studied in some previous papers, e.g. [2] and publications therein.

3. A stability result Making use of an energy conservation, similar to the one used in [3], for the linearized problem, we first derive a result concerning the asymptotic stability of the system.

3.1. An equation for the energy Proposition 3.1. The unknowns of the linearized problem satisfy the following relation:

~3 (c~,U* + c~,U*)2 dx d(fol~ p] v,12dx ) + f,1~ i,j=2

P. Maillard,M.V. Romerio/Journal of Computationaland AppliedMathematics71 (1996)47-65 +

afl(VA, U*) dx +

+~

JOtA dx +

O"

~h) 2 {- PO - aflO~a}r~h)[V(z-

53

dx

h)l do- =0,

Vt ~> 0.

Proof. Integrating (2.20) against U* on f21 (h) and O2(h) (which are time-independent domains), we get, with (2.21):

P u , 12 d x - fQ Z3 ,u*~j(tju* + o,u*)dx + f,~ (vP,,, u*)dx dtd f 51 i,j=2

-- fo [j(VA, U*) + J(Va, U*)]

dx =0,

Vt ~> 0.

(3.1)

Let us consider separately some of the terms entering expression (3.1). Making use of Gauss theorem and taking (2.26), (2.27) and (2.23) into account we write the second integral of (3.1) in the form

,.~..~

i ,.j=~Z (~,u* + oju*) 2 d~ +

fr

(h)

{Po +

(3zpo)H}rth)IV (zOtH - hll do'.

(3.2)

Similarly, with (2.24), (2.21), (2.26), (2.27) and Gauss theorem again the third term of (3.1) reads

fo(VPm, U*)dx= fr ~h){ or with the continuity of b on

P°-I

d_t-H__h)l da, ~--oobB~Jmh)lV(z

F(h) and

(2.31) this leads to

fQ(VPm, U,)dx : f F {__ po _. 1 O O z b 2 ~ th)

(3.3)

2po

~,O

Jrth) lV(z Z

h) l

do'.

(3.4)

Finally, with (2.25) and (2.33) one can show that

- foJ(Va, U*) dX = fA (JOtA + l jz) dx.

(3.5)

Substituting (3.2)-(3.5) into (3.1) and taking Proposition 2. l(f) into account leads to the announced reuslt. []

3.2. Some preliminary lemmata The study of the differential equation obtained in Proposition 3.1 will require some estimates of different terms. These estimates will now be stated in a sequence of lemmata.

54

P. Maillard, M.V. Romerio /Journal o f Computational and Applied Mathematics 71 (1996) 47-65

Lemma 3.2. There exists a positive constant c1, independent of t and fl such that ff2 -~ q Z3 i,j=2

(~,U*+8~U*) 2dx>~c, f -~plU l "12 dx.

Proof. See [3] or use Korn inequality (cf. [7]).

[]

Lemma 3.3. There exists a positive constant c2, independent of t and fl such that

d fr 1 H2 do" dt the-2{ - P9 -- o"flt3za}rth) IV(z - h)l

~
~ Z (~,uj* + ~iuj*) 2 i,j=2

x

(fr

1 ~h)2 { - pg -- o"flOza}r(h)

IV(zH2- h)l do")X~l/2.

Proof. Taking the time derivative of the 1.h.s of the inequality (H being the sole time-dependent function) we get, with Cauchy-Schwarz inequality and (2.23) the existence of a constant d2 such that d Ir

HE

1

~. (h)-2 { - P9 -- o"flOza}r(h)IV(z - h)l

<~d2(f

(U*,n)2[V(z-h)ldo") '/2

(;1

F(h)

x

do"

(h) -5

H 2

{ - pg

-

o"flaza}~h~IV(z h)l

,/",~1/2 do"]

.

(3.6)

Applying a "trace result" (see for example [-2]) we can exhibit a positive constant d3 such that IU*12do.-~
~p

[U*12+

(h)

~, ( 0 i g t ) 2 dx.

(3.7)

i, j = 2

Noticing that

fr

(h)

(U*,n)2lV(z-h)[do. ~d4 fr

(h)

]U*12 do.,

we get the announced estimate by using Lemma 3.2.

(3.8) []

Lemma 3.4. Let A and .4 be two fields defined by

A(x,t)-

2rcP°f A l n l x - x ' l J ( x " t ) d x "

ft.(x, t) -

#o fr

(h)

H(x', t)h)l da(x'), ln(lx --x' [) {a}r(h)fl ]V(z

then A(x, t) = A(x, t) + A(x, t) satisfies Eq. (2.22) with conditions (2.29)-(2.31).

P. Maillard,M.K Romerio/Journal of Computationaland AppliedMathematics 71 (1996)47-65 Proof. Follows from classical potential theory (see [4]) or from a direct derivation.

55

[]

Lemma 3.5. For all time t >~O, ft satisfies

A=O((yE + z2)-I/2),

(a)

[7A=O((yE+ z2)-I)

forl(y,z)[~oc;

[2dx);

d

(c) there exists a positive constant c3, independent of t and fl, such that

IVAI2 dx.

! j2 dx >/ c3

Proof. (a) Is a consequence of (2.23) and Lemma 3.4. (b) From Lemma 3.4 we get (since J = 0 in ~c)

fAJ(X, t)~,A(x, t)dx -

Po II lnlx - x'lJ(x, t)O,J(x, t)dx' dx 2rt .).JA×A

d (l fo J(x,t)A(x,t)dx).

dt

(3.9)

2

From the expression of A given in Lemma 3.4 we draw that AA = - Po = / i o so that one easily gets from Green's formula and (a)

-1f2

JA dx -

2polf.~

AAAdx=--of

~ ,VAl2dx.

(3.10)

Substituting (3.10) into (3.9) yields (b). (c) Applying Cauchy-Schwarz inequality to (3.10) we get (with d = 0 in/lC)

1 fR [VA[2 dx ~'~ 1 (fA 2

2P0

j2 dx)I/2(fA A 2 dx) 1/2, Vt ) 0.

Proceeding in a similar fashion and starting from the expression of A given in Lemma 3.4 we obtain the existence of a constant ds independent of x, t and fl such that

A2(x,t)<<.dS fAJz(x',t)dx',

x~A,

or after an integration

(fAA2dx) 1/2 <~d6(fAJ2dx) 1/2' for a suitable constant

d 6. Introducing

(3.11) (3.11) in (3.10) allows us to conclude.

We now derive estimates for ,4; these are given in the following lemma.

[]

P. Maillard, M.V. Romerio /Journal of Computational and AppliedMathematics 71 (1996)47-65

56

Lemma 3.6. (a) For I(y,z)l

~ oo, the behaviour

2 = o ( ( y ~ + z2) -'/~),

of.4 is given by

v A = O ( ( y ~ + z~) -');

d foafl(va,v,)ax- dt(2@of lVAlZd x ) ;

(b)

(c) there exists a positive constant c4, t and fl independent, such that d

1

~/(~of~ Iv/i[ 2dx)

<<-c4lfll(fR-plU*12dx)'/2( ll7Al2dx)l/2; pof. 1

1

(d) there exists a positive constant c5, t and fl independent, such that 1 2/~o

~

117AI2 dx ~

C5f 2

(h)

-~ { -- pg -- o.fl~a}r(h)IV(z

--

h)[ do'.

Proof. (a) Follows from Lemma 3.4, and (2.32). (b) From (2.21), Gauss theorem (2.26) and (2.29), the 1.h.s. of (b) reads

fa o.fi(VA, U*) dx = -- fr(h) {a}r(h, flA(U*, n) da or with (2.23)

foo.a(v..V*)dx= - frth)

OtH

IV(z - h) l

do'.

(3.12)

From the expression o f / ] given in Lemma 3.4 we have AA = 0 and since A is continuous across F(h) we can substitute A to A in (2.30). Integrating this equation against A we then get, with Green theorem, that

1 2

fr (h) {o.}rth)flAiv(z H- h)l do.=~ l f~2 [V/il 2 dx.

(3.13)

Combining (3.13) and (3.12) yields (b). (c) One applies Cauchy-Schwarz to the 1.h.s. of (b). (d) From (3.13) we get with Cauchy-Schwarz again that

l fo I yA 12dx ~< dv[31 (fr

2po

(h)

"42do.)'2(fr (h)

IV(zH2 Z-h)l z do') 1/2

(3.14)

where d7 is a positive constant independent of t and ft. Applying Cauchy-Schwarz once more to the r.h.s, of the expression defining A (Lemma 3.4) we finally obtain

nZ(x ', t) da(x'). ~2(x, t) <. dsfl 2 ~ dr ~,) Ie(z - h)l 2

(3.15)

57

P. Maillard, M.V. Romerio/Journal of Computational and Applied Mathematics 71 (1996) 47-65

Integrating this inequality on F(h) leads to ( 4 2 do') 1/2 ~< d9lfll

~h)

~h)[V(z -- h) 12

do')

(3.16)

where d 9 is a positive constant independent offl and t. The result then follows by substituting (3.16) into (3.14) and in taking assumption (H4) into account. [] In order to rewrite the curvilinear integral on F(h), two coupling terms still have to be considered. Lemma 3.7. (a) There exists a positive constant c6, independent of t and fl, such that

1 12 dx)1/2. <<.c6, ,(2 of 2lVAl2ctx)l/2 (f splU*

;

(b) There exists a positive constant c7, independent of t and fl, such that

fAJ ,2 dx


1j2dx )1/2(;Q~ E3 G

i,j=2

2dx)1/2"

Proof. (a) is a formulation of Cauchy-Schwarz inequality applied to the 1.h.s. (b) By Cauchy-Schwarz inequality we have that

fAJt~tA dx <. ( ; a j 2 dx)l/2 (fA(atA)2 dx)l/2.

(3.17)

Deriving ~] (defined in Lemma 3.4) with respect to t and taking (2.23) into account we obtain

A(x,t) =

#o fr l n l x - x ' [ {~r}rtn)fl(U*(x,t),n)d~(x'). 2rt th)

With Cauchy-Schwarz inequality applied once more to the r.h.s, we get, after differentiating with respect to t (0,.4(x, t)) 2 ~< dlofl 2 ~

dr (h)

(U*, n) 2 do',

where dlo is a positive constant independent of x ~ A, t and ft. Integrating this last relation on A and making use of Lemma 3.2 we get

(OtA)Z dx

< daIlfll

~ ~ (c3iU* + OjU,*)Z dx i,j=2

Substituting (3.18) into (3.17) leads easily to the announced result.

(3.18) []

P. Maillard, M.V. Romerio/Journal of Computational and Applied Mathematics 71 (1996) 47-65

58

3.3. A stability criterion With these preliminary results we are now in a position to establish the main result of this paper. Proposition 3.8. There exists a positive constant fl such that if lfll < fl then

,~o ~pl limEfo,

dx +y~ ° siva dx

l

=0.

Proof. We introduce some notations.

X , ( t ) = f ~l p [ U * i2 dx,

x:(t) = ~o

~ IVA

d~,

X(t) = X l ( t ) + X2(t), Y1 (t) =

r/

( 6 U * + ~jUi*) 2 dx, i,j=2

r(t) = r , ( t ) +

Y2(t) = f A 1a j 2 dx,

r2(t), (3.19)

Wl(t)

f~ ~fl(vA, u *) dx,

W2(t) ---- fA JOtA dx,

W(t) = Wl(t) + W2(t), /l(t)

= ~o

~ IVAI2 dx,

Z2(t) =

fr

1 H 2 do, (h) ~{ - pg - aflOza}r{h)iV(z h)l

Z(t) = Z2(t) -- Z,(t). We note that with assumption (H4) we are already insured that

131 < 13o.

(3.20)

We thus have that Z2(t) ~> O.

(3.21)

Let us first write some of the results obtained in the previous lemmata (Lemmata 3.2, 3.3 and 3.5-3.7) with the above notations. They read Yl(t) ~> Cl Xl (t),

(3.22)

Y2(t) >1 c3Xz(t),

IZi(t)l < c41Bl(X~(t)) ~/2 (Za(t)) m,

I Z i ( t ) l ~< c2(Yl(t)) 1/2 (Z2(t)) 1/2,

Z~(t) < c ~ Z ~ ( t ) , I W,(t)l ~< c61~l(Xl(t)) ~/2 (X2(t)) x/2,

(3.23) (3.24)

W2(t)l ~ c71 fll (Yl(t)) 1/2 (Y2(t)) 1/2.

(3.25)

P. Maillard, M.V. Romerio /Journal of Computational and Applied Mathematics 71 (1996) 47-65

59

Relations (3.22) and (3.25) insure the existence of two positive constants c8 and c9, fl and t independent, such that

Y(t) >~csX(t), W ( t ) [ ~ c 9 Jill

(3.26)

Y(t).

(3.27)

With the decomposition of A, as a sum of A and .3, given in L e m m a 3.4 and with the relation (3.19) as well as L e m m a t a 3.4(b) and 3.5(b), the result of Proposition 3.1 reads

X'(t) + Y(t) + W(t) + Z'(t) =0.

(3.28)

With the help of (3.28) this leads to

X'(t) + Y(t) + Z'(t) <~c9[fl[ Y(t).

(3.29)

Integrating (3.29) on the interval (0, t) we get

x(t) +

(1 - c91fll) fl v(v)

+ Z(t) <. X(O) + Z(O).

Let us now insure that 1 - c9 I/~1 > 0 and that such that /3~ - (1 -

(3.30)

Z(t) >>,0 for all t ~> 0. Let us give an m e (0, 1) and fl~

m)/c9.

(3.31)

We then choose fl such that I/~l < / ~ .

(3.32)

F r o m (3.19) and (3.24) we deduce that

Z(t) >~(1 - csfl2)Z2(t),

(3.33)

so that, Z : being /> 0, we can insure that

Z(t) is non-negative as soon as (1 -c5fl 2) > m, i.e., for

( l - - m~ 1/2

Jill • f12

- ,It where f12 = \ - C5

(3.34)

•

Finally (3.33) and (3.34) yield

Z(t) >>,mZ2(t) ~ O.

(3.35)

With restrictions (3.32) and (3.34) and taking into account the non-negativity of X and Y, we deduce from (3.30) the existence of a constant dl > 0 such that

(a) X(t) <~dl ;

(b)

fl Y(z) dr ~< d~ ;

(c) 0 ~< Z(t)

<~d~.

(3.36)

Z(t) is thus b o u n d e d and, from the above results, ZI and Z2 are themselves bounded. Taking (3.36) into account we get from (3.23) and (3.26) the existence of constants d2, d3 and d4 such that

f

' X(r) dr ~< de,

0

IZ~(t)l ~< d31fl[,

IZ~(t)l ~< 2 d 4 x / - ~ .

(3.37)

P. Maillard, M. V. Romerio /Journal of Computational and Applied Mathematics 71 (1996) 47-65

60

With (3.37), we draw from (3.29) that X'(t) + ( 1 - c91/~1) Y(t) - 2 d 4 x / - ~ ~< d31/~]. F r o m our previous assumptions on/~ we have the relation 1 > (1 - c9 [/71) = d 2 > m, so that our last relation takes the form

( Let us n o w define /~ = min(/~o,/~1,/32).

(3.39)

]/~l being b o u n d e d by/7, we deduce from (3.38) the existence of a time-independent constant M such that X'(t) <, M.

(3.40)

With the arguments of [-3] the inequalities (3.36), (3.37) and (3.40) allows us to conclude.

4. Characterization of possible instabilities O u r purpose being to describe the behaviour of possible instabilities of our system, we study the spectral problem associated to the linearized problem defined by (2.20)-(2.33). F o r each u n k n o w n S of the problem we assume a time dependence of the form S(x, t) = Re(S(x)e-~t),

t ~> 0,

2 ~ C.

(4.1)

Stable situations are thus characterized by Re(2) ~> 0. The spectral problem consists in finding non-trivial solutions of the system -2pU*

= 17AU* - [7Pro + ( j V A + Jga)

divU* =0 - AA = PoJ

in ~21(h)wf22(h),

in ~21(h)w~22(h),

(4.3)

in Q l ( h ) u Q z ( h ) w F , u A c,

- 2H = (U*, g(z -

h))

(4.2)

on F ( h ) ,

(4.4) (4.5)

where 1 Pm= Po + - - b B , I~o J =

(4.6)

o'(2A - fl - (U*, ga))

in ~l(h)w~22(h),

o'(2A - fl)

in

0

in ~c.

(4.7)

The boundaries and interface conditions are U* = 0

on c3f2,

{U*}r~hl=0,

(4.8) (4.9)

P. Maillard, M. V. Romerio /Journal of Computational and Applied Mathematics 71 (1996) 47-65

61

3

Y~ {(-

Po + a z p o . I - I ) 6 . + , ( ~ , u * + ~jU*)nAF(h) =0,

i = 2,3,

(4.10)

j=2

A = O(lnlxl) {A}o, = 0 0A

-~n t~AwS

when [(y,z)[ ~ 0%

for o) = =0,

(4.11)

OAuS and e) = F(h),

N

g(h)

(4.12) (4.13)

= - ~0{~}F(h)fl IV(z - h)t'

{B + ~b'H}r~h) =0,

(4.14)

fr (h) IV(z H- h) l da =0,

(4.15) (4.16)

A J dX -=O.

Mimicking the proof of Proposition 3.1 we get

Proposition 4.1. The unknowns of the spectral problem satisfy the relation -2

plU*12dx+

I~iU? At- ~jU/*I 2 HX

~ i,j=2

+ fo afl(VA, YY*)dx-X fA J A d x + ;A -IjI2 dx 1 (7 - 2fr (h) {-- pg - afl~Oza}r(h)[V (zIHI2 -- h) ] d a

= O.

Let us give a more precise form to some of the terms of the preceeding relation.

Lemma 4.2. (a)

f. afl(VA,CZ*)dx=(--~)(--fr(h{a}r(h)fllV(z_h)l

(b)

f,

A/-/

lfo,

JA dx = ~oo

leA

2

dx -

{0"}r(h)fl (h)

da);

.,ill IV (z -- h) I

Proof. This can be obtained in a wa~ similar to the one used in the proof of parts Lemmata 3.5(b) and 3.6(b). [] Our main result can now be stated as follows.

Proposition 4.3. Every eigenvalue 2 of the spectral problem, with U* ~ 0, satisfies (a) If H = 0 then Re(2) > 0; (b) I f l m ( 2 ) ¢ 0 then Re(2) > 0.

P. Maillard, M.V. Romerio /Journal of Computational and Applied Mathematics 71 (1996) 47-65

62

Proofi We introduce some notations

X1 = f plU*[Zdx > O, X2=--

w

#o

=

(h)

2

IVAI2 d x > / 0 ,

Y1 = Y2 =

fi 2 R e ( A H ) d a , --

-2 i, j= 2

(OiU? -~- ~ j U ~ ) 2 (ix >/0,

-[J a

dx>/0,

{ -pg-aflOza}r(h)lV-~--h) I dm

Z =

[]

(4.17)

(h)

Let us make two remarks. • W and Z are both real; • U* ~ 0 which implies X1 > 0 insures us that Y1 > 0 (see Lemma 3.2). Taking Lemma 4.2 into account Proposition 4.1 takes the form 2X1 + ) , ( X 2 W + Z) - (Y1 + Y2) =0. This yields for the real and imaginary parts the following: (a) Re(2) (S~ + X2 - W + Z) - (Y1 + Y2) =0;

(b) Im(2) (X~ - X2 + W - Z) =0.

(4.18)

(i) When H = 0 we draw from (4.17) that W = 0 and Z = 0 so that, with (4.18(a)), Re(2) > 0; (ii) When Im(2) # 0 we draw from (4.18(b))

-W +Z=X~-X2, so that (4.18(a)) yields Re(2) = (Yx + Y2)/2Xx > O. These results give us two important features of the spectrum: • the first part tells us that instabilities cannot appear for motions characterized by a steady interface; • the second part shows that a mode which has an oscillating motion is always stable.

5. Some numerical computations The theoretical results derived in Section 4 do not give us information about the existence of unstable modes. The purpose of this section is to present some results obtained by a numerical approximation of the spectral problem defined by (4.2)-(4.16). The computation, for pressure and velocity, are based on a finite element method using Q1 - P o elements with It penalization of the pressure. The magnetic potential is determined by the same type of method, the boundary conditions being obtained by a Green formula. We consider the modelling given in Fig. 2 corresponding to a cell which is 3 m wide. The thicknesses of the aluminium, electrolytic bath, cathode and anode blocks are, respectively, 0.2, 0.06, 0.2 and 0.24 m. Note that the four bus bars, which are parallel to the x-axis, simulate the usual bus bars arrangement carrying the electrical current into the cell. We assume that the steady interface is fiat.

P. Maillard, M.V. Romerio /Journal of Computational and Applied Mathematics 71 (1996) 47-65

6)

6)

100'000 A

I00'000 A

63

Free interface Y'(h)

, oo,ooo

i iiiiiiiiiiiiiii i iiiiiiiiii!iiiiiiiiiiiiiiiiiioii!iiii}iiiiiiiiiiiiilliiiiiiiiiiii!ii Iiiiiiiii!iiiiiZiiiiiJiiiiiiiiiiiiiiiiiiiiiiiiii " iiii!i!iiii , oo,ooo

Electrolytic bath .Q2(h)

~

Anode block ~2

Aluminium [21(h)

~

Cathode block '~1

Fig. 2. Cross section of the cell used for numerical computations.

Table 1 Values of the physical parameters entering the computations Density p (kg m-3)

Viscosity q (Pa s)

Electric conductivity (7 ( 0 - 3 m -3)

Aluminium 01(h) Electrolytic bath O2(h) Cathode block ~x and anode block "~2

2.27 × 103 2.13 × 103

4.5 × 10 2 1.25 x 10 -1

--

--

3.33 x 106 6.66 x 102 2.0 x 104

For the results presented here, we take the physical values given in Table 1; the gravity constant g respectively the magnetic susceptibility of the vacuum #0 is 1 0 m s -2 resp. 4 m x 10 - 7 gs A - I m-1. This leads to results given in Fig. 3. This last one represents the eigenvalue (as well as its complex conjugate) for which the real part is the smallest for various values of the parameter/~. We remark that the system is becoming unstable as soon as/~ is larger than 2.69 x 10- 2 V m - 1 In agreement with Proposition 4.3 these unstable modes are non-oscillating ones.

6. C o n c l u s i o n s

In order to understand the presence of possible instabilities in Hall-H6roult cells, several modellings have been derived by different authors. Two main classes of such modellings can be distinguished in the literature. In the first one the cell is assumed to be infinite in horizontal direction and instabilities are generated by sufficiently large horizontal electrical currents (see for instance [6]).

64

P. Maillard, M.V. Romerio /Journal of Computational and Applied Mathematics 71 (1996) 47-65 0.2

0.15

0.i

0 05

b~

0

-,-"1

H H

-0.05

-0.i

"',. -0.15

-0.2 -0.01

0

i 0.01

, 0.02 Real part

0.03

0.04

0.05

Fig. 3. First eigenvalue for different values of the parameter ft. (Vq) correspond to values of fl varying between 0 and 2.5 x 10 -2 (steps of 0.5 × 10-2); (O) correspond to values of fl varying between 2.5 x 10-2 and 2.7 × 10-2 (steps of 0.05 × 10-2). In the second class the geometry of the cell corresponds to a finite domain of ~3 and, in this case, possible instabilities are produced by perturbations introduced in the purely hydrodynamic frequency spectrum by the electromagnetic force field (see [1]). In this situation instabilities are thus strongly depending on the aspect ratio of the horizontal dimensions of the cell. The interest of the present modelling rests firstly on the fact that it is infinitely long but still has a finite width and secondly on the possibility to start a linearization procedure, defining a spectral problem, from a steady solution which can be explicitely obtained. It is thus important to show that this result remains valid when the cell affords a certain a m o u n t of longitudinal electrical current. With spectral techniques moreover it is interesting to characterize possible unstable modes. In fact we show that unstable modes of oscillation are purely exponentially increasing modes thus non-oscillating ones. The existence of such unstable modes is here numerically established.

References

[1] V. Bojarevics and M.V. Romerio, Long waves instabilities of liquid metal-electrolyte interface in aluminium electrolysis cells: a generalization of Sele's criterion, Eur. J. Mech. B Fluids 13 (1994) 33-56• [2] J. Descloux, M. Flueck and M.V. Romerio, Spectral aspects of an industrial problem, in: E. Sanchez-Palencia, Ed., Spectral Analysis of Complex Structures, Travaux en Cours (Hermann, Paris, 1995) 17-33.

P. Maillard, M.V. Romerio/Journal of Computational and Applied Mathematics 71 (1996) 47-65

65

[31 J. Descloux, Y. Jaccard and M.V. Romerio, A bidimensional stability result for aluminium electrolytic cells, J. Comput. Appl. Math. 38 (1991) 77-85. [4"1 N. Giinter, Die Potentialtheorie und ihre Anwendung auf Grundaufgaben der mathematischen Physik (Teubner, Leipzig, 1957). [5"1 P. Maillard, Sur la stabilit6 magn6tohydrodynamique d'une cellule de H6roult de longueur infinie, Th~se EPFL No. 1'102, Lausanne, 1993. [6] R.J. Moreau and D. Ziegler, Stability of aluminium reduction cells, in: Light Metals AIME Meeting (1986) 359-364. [7] P.A. Raviart and J.M. Thomas, Introduction ~ rAnalyse Numbrique des F,quations aux Dbrivbes Partielles, Collection Math~matiques Appliqu6es pour la Ma~trise (Masson, Paris, 1983). [8] M.V. Romerio and M.-A. Secr6tan, Magnetohydrodynamic equilibrium in aluminium electrolytic cells, Comput. Phys. Rep. 3 (6) (1986) 327-360.