The flow of a viscoelastic fluid between two parallel plates with heat transfer

Int. J. Engng Sci. Vol. 29, No. 12, pp. 1545-1556, 1991 Printed in Great Britain. All rights reserved

0020-7225/91 $3.00 + 0.00 Copyright 0 1991 Pergamon Press plc

THE FLOW OF A VISCOELASTIC FLUID BETWEEN TWO PARALLEL PLATES WITH HEAT TRANSFER VICTOR Institute

TIGOIU

Bd. Pacii 2200, 79622 Bucharest, Romania

of Mathematics,

by E. SO&)

(Communicated

problem of the flow of a third grade fluid between two parallel plates with heat transfer is studied. We prove that von Karman type solutions are not admissible for a general third grade fluid, but it may be experienced by a particular subclass which we put in evidence. The existence, the uniqueness and the dependence on the little parameters a = RePr of the solution of the heat transfer problem are then analysed. Some numerical experiments concerning the first two approximations of the attached Taylor expansion of the solution of this problem are represented.

Abstract-The

1. INTRODUCTION

The problem of finding admissible solution (for some flows) for viscoelastic polynomial fluids has very few works dedicated. In the paper of Verma et al. [3], for instance, a particular problem for second grade fluids was investigated. We try to solve it for a third grade fluid. If such flows (as von Karman’s solutions describe) may be experienced by a third grade fluid, then we can imagine some physical experiments in order to simulate this flow and so we expect to have a real chance to determine supplementary relations (different from viscometric ones) between constitutive coefficients. In order to do this we analyse in the first part of this paper the existence of von Karman’s solutions for the proposed problem. We already consider the conditions such as the constitutive coefficients be constant and we denote Al = Vv + (Vv)’ and A,, = &_1 + A,_IL + LTA,_, (n = 2, 3, . . . ) “V” is the gradient operator and “0” represents the material derivative) the well known Rivlin-Ericksen tensors. The main conclusion of this part is that a general viscoelastic fluid of third grade (such as it was described in [l, 21) cannot experience a von Karman flow, but this is possible for a particular subclass. For this subclass a solution can be constructed by a power series technique (in Reynolds number). For such a fluid and such a solution of the mechanical problem we analyse in the second part the heat transfer problem. A heat transfer equation is deduced and first we observe that the field of the absolute temperature 8 does not directly depend on the constitutive coefficient /3*. An existence and uniqueness theorem for the solution of the bilocal attached problem is proved. The regularity of this solution is investigated as well as the dependence of the solution on the parameter a = Pr Re (where Pr is the Prandtl number and Re is the Reynolds number). We prove that the solution may be developed in a Taylor series in “a”, in the neighbourhood of a = 0. The first approximations are calculated. Conclusions regarding some numerical experiments on the temperature field are pointed out at the end of this paper. The author is grateful to Professor An Cihol Ho for many discussions and some calculus performed concerning the first part of this paper (during his doctoral stage at the University of Bucharest). I am grateful also to Dr E. Soo’s for useful remarks concerning the final form of this paper and to Dr I. R. Ionescu for the help in the performance of numerical calculi.

2.

THE

FLOW

PROBLEM

A viscoelastic fluid of third grade is given by [ 1,2] T(x, t) = -PI + Ml

+ al(&

- A?) + PIA3 + &(AIA2 + A,A,) + &(tr Af)A, 1545

(1)

1546

V. TIGOIU

where T represents Cauchy’s stress tensor, p(x, t) is the hydrostatic pressure, p, oi, pi (i = 1, 2, 3) are constant constitutive coefficients with known significances, I is the identity tensor and tr(*) is the trace operator. Following [l, 21 we have the constitutive restrictions

The fluid flows between two parallel plates. The upper one is supposed to be porous and the fluid passes through with constant vertical velocity.

v(-vy)I,=d = -v0j (in order to conserve the mass of the fluid between plates) where “d” is the distance between the two plates. The other one moves with the velocity

v(x, y)lv=O = (cx)i, “i” and “j” are the versors in the horizontal and respective vertical directions and “c” is a given constant. We remark that the origin is preserved at rest (see also Fig. 1). The two plates are kept at constant but different temperatures. Due to the fact that the constitutive coefI?cients are independent of the temperature the flow problem is independent of the thermic one. In this part of the paper we are interested in the following problem: does a general viscoelastic fluid of third grade accept a von K&man solution for the flow problem described above? For that, let u and u be the two components of the velocity field in the horizontal and vertical Cartesian axes. A solution of von KarmBn type is given by v = -cdf(q)

(3)

U = CXf’(?j) where y = dg. A straightforward motion

but very long calculus gives us the following system for the equations of

j3$‘$f”Zf”+ c; {C4flR+ Q11C(f”2+ yp-f’“)

pC2X[f’2-,,] = q/3,+

-+ &c2(28f’fff2 - 4#‘7” + 8f’2f”‘) +

t t

t

+

4

+ t

t

8f’2f”‘)} - g

+ 4 i

y -///////f/////f/

-0

+

Y

Cd

i ////////f///J

~,~~(%tf’f”~

-

Fig. 1. Geometric representation of the Aow domain.

c

X

1547

(4)

The form of the equations (4) demands with necessity, after some calculi, to suppose that the hydrostatic pressure has the following form El;P = -Pt(ri> +$A??~ Then, multiplying the equation (41, by d*/c,ux and the equation (4)2 by d/PC we find f’” = Re( fr2 - y)

- 1’0~+ f13)$f”2r

_ fi, Re’(9f’f”

+

- C, Re(f”’ -I-2f ‘f’” - #I”“)

4f‘*f” - 13Ff"'-ff'f"" + f"f",- j&Re’(28~‘~~’

- 4rf”’ + 8f”f”‘) - fi3 Re’(24f”‘f’ + 8ff2f) f” = {--Reff’

+ 9r’)

-t 2p2

+ a1 Re(5f’f” + p’) C & Re*(4~~‘~ + /!12Re’(-48.‘2f”

+ lS$‘f’” + 3f2f Iv

- f@“* - Sff’f”‘) - Xl& ReZf ‘T”} f pi

-t $ {2&r Ref’%” + 68, Re”(f”” -5f ‘f”f” - r2 f p2 Re2(4p3 - 6f'f"f"' - 2p* - Z#““‘)

-pfJv)

+ & Re2(2f”3 - Sf’f’%‘“) - pi}

where Re = cpd2/p is the Reynolds number, iil, fir, f12, & are the dimensionless coefficients corresponding to lyl, p,, /12, &. Then we identify the coefficients of (x/d)’ and (x/dj2 and we obtain a system of four differential equations for three unknown f~~ct~o~s f, p,, pz f’” = Re(f’* -A”‘) - &r Re(f’” + 2f’f”’ -Jsr”> - @, Re”($lf’f rr2 + 4j”f’” - 13yr

-@‘j-‘” +f’f”)

- S/$ Re2(3f”2f’ +f’“f”)

- 4jj2 Rez(7f”“f’ -A”“f”f”’+ 2f’2f”“) + 2p2

(j$ -t &f”‘LfVP= 0 f”= -Reff’

+ 2E1Re(Sf’f” +r)

+ 38, Re’(l5f”f”

+8~~Re2(-4f’2f”-~Z-~‘f.r)-4a~3Re2f’2f”+p~ pi = 2g1 Re f’“f + 68, Re2(y3 + f ‘f”f m- r2

+ 2fizRe2(2r3 - 3f’f”f”’ -#“” -rf”)

+

5#‘f” f f 7” + 3~$‘*)

(6’)

- #“jFJv) + 2f13Re”(f’” - 4f7-‘%“‘)

We easily observe that any general solution of the equation ($3 4 &)flr2f,” = 0

(7) cannot satisfy the other equations. Indeed, we see that the equation dJdy(f"j3 = 0 has no solution if we consider any bilocal problem obtained from the described mechanical problem [for instance f(O) =f’(l) = 0, f’(O) = 1, f(l) = vn/cd]. So the system may accept a solution if and only if a2 -I-& = 0. As we shall see, this is a very strong condition. Indeed, if we consider also the constitutive restrictions (2),,d we have !%=Q

i%+A=Q

(8) Thus we conclude that a general viscoelastic fluid of third grade cannot accept von K&&n’s solutions for any problem of the kind mentioned above.

V. TIGOIU

1548

For the particuiar subclass of third grade fluids described by (1) and (8) we see that, by simple manipulations, we can obtain from the system (6’) a differential equation for f and two other equations for p1 and p2 f~=Re(f'f~-~)~~~Re~v-f~zv) p;=f"+ Reff’-

a1 Re(5f ‘r + #“‘) + 8& Re’( f 12r+ #"' + flp'f "')

We restrict our attention on the equation for f f'"=Re(f'f"-r)+

&Rewv

-f'f'")

(9)

We remark immediately that the velocity field will not depend on j& (thus, in a sense, it is an universal field). The pressure field will depend on /3* and the Cauchy’s tensor, in each fluid particle, will depend also on &. The data for our problem are f(O)=f'(l)=O,

f'(O)= 1,

We seek the solution of the problem (8)-(10)

f(l)=volcd

(10)

in the form (Re << 1)

A sequence of linear problems is obtained which solutions are poIynomials in n E [O, I] and depend on 9, al and the boundary data. That is fy= 0;

fY= -Gfd"-f;f13

f~v=f~f~+f~f~-~~~

~~(~f~_f~f~")

(12)

with the conditions

f;(o) = 1, ~(o)=fi(l)=f;(o)=ff(l)=o,

MO) =fXl> =o>

fo(l) = v,/cd i = 1,2, . . . , II,

...

(13)

Finally the solutions are of the form u = -cd{_&&) + Reft(r)

+ Re*f,(r7)) + o(Re3)

u= cx{fXrl)+Refi(rl) +Re'fi(rl)) +O(Re3) where J(q) are all polynomials in 9. In conclusion, a general viscoelastic fluid of third grade cannot experience flows. A particular viscoelastic fluid of third grade given by T=-pi+

~A,-~*(A~-A~)~~~(A*A~+ A1A2- A,tr(Az))

(14) von Kairm&n type (‘15)

can experience such flows (at least in the approximation described above). We observe also that even this particular class of fluids is yet more general than that which was obtained by Fosdick and Rajagopal 141. We remark that in the case &-+ 0 we formally obtain from (15) the constitutive equation for a second grade fluid and due to the form of the equation (9) we have the solution obtained in [31. It is necessary also to remark that in the case described above the function F(n) = (5/2)uf'+(1/4)a"f (w here n = Re) is at least in the first two approximations of f, strictly positive for rj E [0, 11, wRen a <<1 and vo/cd is greater then a suitable constant. This should be of interest in the second part of this paper. We see that the shear stresses are independent of the constitutive coefficient &. But if we evaluate the difference between the normal stresses (in fixed x points) on the two plates we

Tile flow

lSpTn*n=

-

of ii viscoelasticfluid

-.4/+~+$Ap,

)

1549

-tAT,n~n

where we denote Ah = h(x, 0) - h(x, 1). Thus employing the equations (6”),,X we conclude after some straightforward calculi and keeping in mind that n is the normal versor to the plates that bTn*n

+ T&)= -3f’ - i Ref* + ci, Re(3ff’ c 2f”)

c-$-p

-ir

-16~,Re2~‘f”+p,,+~

+ 5 Re(f’” -p)

+ i 8, Re(f”” - 2f’f’” +flfV)),

Thus we see that the difference ATn * n between two points on the two plates [due to boundary conditions (lo)] will be independent on f12. It will represent a ~ompat~b~Iity condition for such flows. If the normal pressure lapse may be measured in two interior point with x = 0 then we have

which is significant starting with the third approximation

3. HEAT

TRANSFER

in Re.

IN A THIRD

GRADE

For a viscoelastic fluid of third order we have obtained restrictions (from thermodynamical point of view)

FLUID

[1,2] the following constitutive

%+I, A,, 0) = &L Aa 0) + WC%,AZ,@)

Cl@

where 4 is the specific entropy, B the specific internal energy, $ the specific free energy and 6 the temperature field. We suppose as in [lt 21 that the specific heat at constant volume ct(Al, AZ, 8) = -i3&,e (A,, A,, 6) is positive on eq~j~ib~urn states (c,(O,O, 6) > 0). On the other hand 4 is a linear function in & (see [15 21). p&L,

t, 0) = 2&L’ - k f p&L, 0) = ;&Al

=A, + pd(A,,

@),

(171

where L=Vv and L’ is the symmetric part of L, We shal1 determine the free energy for the particular class of third grade fiuids described in (15). We introduce (17) in Clausius-~~hern inequality written on isothermal processes and it results that the inequality (

~~~,(A~~ @- ; @IAB- &A;) - & -+A:~0

(181

must hold whatever be AI, AI, Az. Then, with necessity, it results that ~(AI,

@=&AI

l

A,+;&A:&+P~(B)

(19)

and

(20)

1550

V. TIGOIU

Supposing that the internal energy is a measure (at least on equilibrium processes) of the heating of the body, and that the specific heat coefficient is independent of 8 (i.e. is a material constant) we have E(0, e) = c,e

(21)

egye) -g(e) + c,e = 0

(22)

and employing (20)

Solving this equation we obtain ,os(x, t) = ; a1A1 - Al + ; j&A;. Ai + pc,e

(23)

We remark (see [2] Chap. 1.E) that for the general case of a third grade fluid the free energy shall be given by ,ow(x, t) = ; /!?i&. Al + ;&A:

- Al + ; alAl . AI - /xl0 In 8 + p&,0,

(24)

where K0 is a constant. On the other hand (see [2] Chap. 1.C) in the neighbourhood of equilibrium states the Fourier formula is a good approximation for the heat flux and then the equation for the heat transfer is given by pc,8 - div(k grad 6) + i pi& . & - i /3,[A,L i LTAl + AIL + LrA2] . A, -;

(i%- /%)(AI - &I2 - ; CL& - A,) = 0 (25)

where k is the thermal conductivity coefficient. In the particular case considered above (the first part of this paper) supposing that the coefficient of thermal conductivity is a material constant the equation (25) can be written in a much simpler form p+kAt+(A,.AJ=O

(26)

We suppose that the temperature variation between the two plates is so as the constitutive coefficients remain constant during the experience. Then we obtain the following stationary problem -pc,VoVB

+ kA8 + ~c’(4f’~ + $ffJ2) = 0

(27)

and the boundary conditions (28)

0(x, Y)L=~ = e0 0(x, Y)ly=+ = e1 where &, 8, are the temperatures of the plates. We seek the temperature field in the form

(29)

e-eO=M(~(rO+$~ol$

where f and v are obtained in the Chap. 2 and M = pc/pcl ([Ml = C” Celsius grades). We introduce the solution v, and (29) in (27) and equating the coefficients of the powers of x/d we obtain v”(rl) + Re Prf(rl)q(rl)

=0

(30)

+ 2tP(rl) + 4 Re Pr_Y2(n) = 0

(30’)

- 2 Re Prf’(n)q(rI)

@‘(rl) + Re Prf(q)rp’(rl)

+ Re PrY2(n)

1.551

The flow of a viscoelastic Ruid

and the following bilocal problem V(0) = V(l) = 0 cp@) = 0, q(l) = (0, - ~“)/~ We shall analyse first the existence (3O)-(31). To do this let denote

and the uniqueness

r(rl) = ex&

e~‘f(sI

of the solution

(31) (31”) of the problem

+,j

g(r)) = W(rl)r(q) F(q) = (2af’z -t Y’)l’Z G(q) = af”*r

(32)

With these notations the problem may be written - (8’)” + FC17)g= C(ll) a(0) = 0 gfl) = 0

(33)

where F is at least a continuous function. We denote with 5? the operator of the problem (33) and having in mind that the optimal Friedrichs constant on [O, l] is l/n* we obtain 6% g) = l* 8” drt + i’ ~(~~g2~~) drJ 3 s’i’ Let N be such that Ifl, If’] sN.

8 & + 1’ ~(~~g~t~~ drl 0

(34)

Thus it results that

An elementary calculus shows (see the remark at the end of the first part) that the corr~s~n~ng expressions for F, obtained with the first two approximations of f(f, and fi) are non-negative on [0, l] and that there exists an N > 0 independent of a. Then let be C=IF* - $aN. The condition C >O gives us a << (2n2)/5N which is perfectly realizable in our problem (where we have supposed that a << 1). Finally it results that 9 is coercive. With this we can say that there exists an unique solution of the problem (33) and this one is the same as the classical solution of this problem. Consequently g E C’((0, 1)) rl C1([O, 11) and due to the notations (32), I&has the same properties. For the ~rob~~rn (30)-~~), denoting by h(n) iifz-2$(q)

- 4 Re Prf”(rl),

(35)

we have an equivalent system of first order differential equations

(36)

W(q)= -exp(-a[f(s)ds)#O whatever be t] and then we construct the solution

(37)

1552

V. TIGOIU

where the constants Ci, C2 are determined from (36)3,4. The uniqueness of the problem (30’)-(31’) is evident with the remarks concerning the problem (30)-(31). In what follows we will analyse the behaviour of the solution [for instance for the problem (30)-(31)] as a function of the data, of the parameter “a” and the regularity of the solution. In fact we produce some apriori estimates of the L2 norm of the solution and of his derivatives. Let the problem be ~“+&‘-2uf’lJJ+uf”2=0 V(O) = w(1) = 0

(38)

We multiply with I/J and integrate over [0, 11. After some simple calculi we obtain (39) Then employing the above mentioned

notations and Schwartz’s lemma we have

Ilw’ll:+v

llv412,~~~ llf”211LZ IllvllL~

(40)

Then with Friedrichs inequality we obtain finally

IIVIILJ6

u5

Ilfn211LZ

(41)

n2--UN 2 (We have in mind all the time that f is sufficiently smooth and N is independent On the other hand we reexamine the inequality (40) and easily observe that

IIV’IIL?c

u5

lIf”zIIL~

of a.)

(42)

n2--UN 2 From the equation (38), we obtain that $J” E L'([O, 11) and then ly)Q= $(f’v”

+

4f Q@-

4f tfrr2 + f”4)

4ff’q’rj) + 2j72qr -

(43)

which means that t/Y E L2([0, 11) and finally we obtain

IIV’IIL~~aM2

llf”211tZ

(44)

where u(niV’ M*(a)

=

2N)

+ 5

+l-,l

when

a-*0.

n2--UN 2 We observe that due to the equation 7#E Cm((O, 1)) and

(38), and the regularity

IIv(“‘IILz~ uM,,

of coefficients it results that (45)

whatever be n = 1,2, . . . , and where M,,depend on “a” and the maximum of the absolute value of coefficients on [0,11.Since the solution of the problem (38) is V = I#(% a) from the apriori estimates obtained above we see that lim ~~I/J(“)~~~~ =0 O-+0

The flow of a viscoelastic

1553

fluid

whatever be n. On the other hand for a = 0 the problem (38) has the solution V(rl, 0) = 0 and then the solution 9 = ly(q, a) is continuous also are his derivatives. Indeed

(46)

as a function depending on “a” for u = 0 and

lim II@‘“‘(*,a) - ?#J(“)(* , O)\ltz = 0 u-0 Let now a’ E (0, t), z << 1. Let {a,}, be a sequence a,, E solution of the problem (38) corresponding to the value a’ sequence of solutions which correspond to {a,},. We substract (we suppose that f and his derrivatives are independent on a,) for a’. Denoting $ = q, - I&, we have the problem $“+ alfi31’

- 2a’f’+

+ (a’

- a,){fvA

-

(47)

(0,

t), a, -+a’. Let v, be the of the parameter and {?&}, a the equation (38) written for a, from the same equation written

2f’fllCln +f”*} = 0

l&O) = l&l) = 0

(48)

We multiply (48) with V& integrate over [O, 11, employ the data (4Q3 $ E C2((0, 1)) n C’([O, 11) an d we obtain after some calculi

and the fact that

If we employ now Schwartz’s and Friedrichs inequalities we obtain (J+QV)

tlljll&s

Id - %I (N’ llw:IIL~ IIWIlL4N

II%llt2 lliQlL~f Hf211L~IlhA

cm

NOWif we observe that a’ C 2z2/5N and we employ the estimations (41) and (42) we have /Jlyjl#Q.+.~,l {

~R(~~‘-z~~ X2_;o’N

+I}

.:;:,*

@lJ

and thus

We conclude that q(q, a) is a continuous function of a whatever be 11E [0, z) and then employing Weierstrass theorem we see that v may be approximated with a sequence of polynomials PG(rr) which is convergent to II, at each fixed q E [O, 11. That is for E >O there exists nE so that

whatever be n > n,, We are interested now in the study of the regularity of the solution in a = 0 as a function of %“~ For this, we folly derive the eq~tion (38) and obtain the problem ~~-afiy~-ZQ~‘lv,Sf~‘-2f’~+~2=0 R?(O) = 11100)= 0

041

where Ill?) ~~~~,

a).

The same arguments as for the problem (38) lead us to the conclusion that exists and is unique the solution VUof the problem (54) and &Jb, a) E C’((O, 1)) n C’([O, 11).

We repeat the arguments leading to the estimates for V&and his derivatives the following estimates for the derivatives of v

where M(a)-, 1 On the other substructing the employ also the

when

a-,

q(“’ and obtain

0.

hand we consider the problem (54) for II = 0, denoting tj = q, - qtltoand two corresponding problems we obtain finally the foliowing two estimates (we estimates (41), (42) and (55))

which lead to

(57) That is, + is derivable in a = II, by respect with “a”, and his derivative is a continuous function, Now it is easy to prove by induction that li, is indefinite derivable in a = 0, and that ,,~ljp...o,,~l s n!

5 ((CI‘N+ Q2 + 1)(&V + l)n-3,

where

,2-L n:

Jr2’

Keeping in mind (58) we see that alI the derivatives consider

of +# are nun-zero functions,

then we

where a* E (0, z). For this series we observe that the radius of convergence (a st~aigh~o~ard but lang calculus using well known theorems on series convergence) is R = l/(cuN f 1) and consequently we have

where y1+ *.

\\.\ \\ l.k 1

1555

The flow of a viscoelastic fluid

“,

‘.‘,

\\

* \ \ \

‘\.‘\

‘i,

\’

‘\

‘5 \

x=1.2

x=0.7

*\ \\

‘\\\

x12.0

x=2.7

11

Fig. 2. Relative variation of the first two approximations of the temperature for u,lcd = 0.5, (0, - 0,)/M = -0.4. (1) a = lo-‘, 8”) continuous line, f?‘) - - - - ; (2) a = lo-*, 19”’ continuous line, #‘) _ . _. _.,

In conclusion the function I+!J(v,a) is developable in a Taylor series in the neighbourhood of a = 0, that is the equality (59) holds for each a E (0, rO) where to = min{ t, R} and: whatever be E > 0 there exists nE > 0 such as whatever be n > nE we have

(61) for each a E (0, to). Some simple calculus led us to evaluate the first two approximations temperature field (nondimensional) (8 - &J/M, namely 6 - 60

of the relative

co) 61 - 80

( M > =Mv=vo(tl)

iP(yl,x) = and respectively

P(q, x) =

(yy’) =qo(rl)+ (rpl(rl) +f m)u

A set of numerical experiments are presented in the Figs 2-4. We remark that due to the lower plate movement at great distances the fluid is much more heated. The temperature variation is linear throughout in the neighbourhood of this plate (due to grate internal dissipations). A small variation of the temperature is to be remarked in the vicinity of the upper plate due to the continuous advection of mass fluid [all these remarks concern the approximation of first order give by the formula (63)]. If the difference between the temperature of the two plates is smaller we see that the difference between the zero and first order approximations is more important than in the case where we have a large difference between these temperatures. So we may consider that for large differences between the

x-o.7

x11.2

x-2.0

x-2.7

rl

Fig. 3. Relative variation of the first two approximations of the temperature for (19, - 0,)/M = -0.4. The rest of the notations are the same as in Fig. 2.

v,/cd

=

1,

1556

V. TIGOIU

0

x-o.7

x11.2

x-2.0

x-2.7

1

Fig. 4. Relative variation of the first two approximations of the temperature for u,/cd = 1, (0, - f&,)/M = -4. The rest of the notations are the same as in Fig. 2.

of the two plates the approximation of the first order is sufficiently good one in a of the order of O(d3) in the “x” direction. Finally we see that if we repeat the calculus with a = l/100 the difference between zero and first order approximations is much smaller then in the case when a = l/10 and this is practically the same whatever be the difference between the temperatures. temperatures region

REFERENCES [l] V. TIGOIU, Prepr. Ser. Math. 69. INCREST, Bucharest (1984). [2] V. TIGOIU, Studii si Cercetari Mat. N(4), 279-348 (1987). [3] P. D. VERMA, R. SINGH and P. R. SHARMA, Znd. 1. Tech. 24,557-564 (1986). [4] R. L. FOSDICK and K. R. RAJAGOPAL, Proc. R. Sot. London A 339, 351-377 (1980). (Received

12 February

1991; accepted 25April

1991)