Convection, stability and uniqueness for a fluid of third grade

Inr. i. Xon-Luwur Mrckanrcs. Pnntcd I” Great Bntam.

CO20-7462,88 S3.00 + 0.M) Pergamon Press plc

Vol. 23. No. 5,6. P&Y.377-384. I988

CONVECTION, STABILITY AND UNIQUENESS A FLUID OF THIRD GRADE

FOR

F. FRANCHI Dipartimento di Matematica, Universita di Bologna, Piavla di Porta S. Donato 5.40127 Bologna, Italy

and B. STRAUGHAN Department of Mathematics, University Gardens, Glasgow G12 SQW, U.K. (Receiaed 31 August

1987, receicedfor pubficution 3 May

1988)

Abstract-The equations for a fluid of third grade derived by Fosdick and Rajagopal are first studied on an exterior region in three-dimensional space. A uniqueness theorem and a pointwise continuous dependence theorem (on the initial data) are proved. The conditions at infinity are weak, certainly L2 integrability is not required. Then, the equations for the third grade fluid are adapted to the problem of thermal convection due to heating from below. It is shown that the linear instability problem reduces to that ofa second grade fluid. Interestingly, a study of non-linear stability for the same problem reveals that the constitutive inequalities obtained by Fosdick and Rajagopai play a very important role; there may be stronger asymptotic stability than for a second grade fluid, although in certain cases the stability may be much weaker.

1. INTRODUCTION

Models for incompressible, homogeneous viscoelastic fluids where the extra stress may be expressed as a function of the Rivlin-Ericksen tensors have been widely employed in the rheological literature. Of relevance to the present work is the study of thermodynamics, stability and instability for incompressible third grade fluids due to Fosdick and Rajagobal [l], this having been preceded by a similar study for incompressible second grade fluids by Dunn and Fosdick [23. Fosdick and Rajagopal [l] establish an asymptotic stability result from the proof of which uniqueness and continuous dependence results follow easily when the spatial domain is bounded. In this work we address similar questions when, however, the spatial domain is exterior to a compact three-dimensional set: we shall not require Lz-integrability of the velocity field nor strong decay of the pressure. We shall also investigate linear and non-linear stability for the problem of density variation driven convection due to heating from below: our findings for the non-linear problem are interesting and would appear to differ from the corresponding ones given by Straughan [3] for a fluid of second grade. By employing the Clausius-Duhem inequality and demanding that the free energy be a minimum in equilibrium, Fosdick and Rajagopal [l] have shown that the stress relation for an incompressible homogeneous fluid of third grade is T = -pZ + /iA + r,Az -t a,A2 + j!?(trA’)A, where A and A, are the first two Rivlin-Ericksen gradient L by

tensors, defined in terms of the velocity

A=L+Lr, A, = DA,0

(1)

(2)

+ AL + LTA.

(3)

In these expressions p is the viscosity, ctl and xt are normal stress coefficients, /I is another material coefficient, D/Dt denotes the material derivative and thermodynamics (see Fosdick and Rajagopal Cl]) requires the coefficients to satisfy the following inequalities: P 2 0,

dll20,

P 2 0,

IQ + a21 I &4cta).

(4)

We are here regarding the above coefficients as constants and are treating the third grade fluid model as an exact one. Whether the third grade fluid system serves as a model for any 377

378

F. FRANCHIand E. STRAUGHAN

real fluid wifi only be determined by comprehensive future mathematical and experimental analyses. However, we befieve that to regard such a model as exact certainly requires that it generate well-posed boundary-initial value problems and so it is not unreasonable to require p > 0, aI > 0, p > 0. The Navier-Stokes fluid is an exact fluid of grade 1 and no one can dispute its success with p > 0 even though thermodynamics only tells us that p 2 0. We should point out at this juncture that xl c 0 can even lead to global non-existence, [l, 2,4]; furthermore, recent experimental work points to the fact that al > O.*

2. UNIQUENESS

ON AN EXTERIOR

DOMAIN

The dynamical equations are given by pDui/Dt = phi + Tji.j,

(3

Yi,l = 0,

(6)

where the motion is assumed to be isochoric, and where u, p, b are the fluid velocity, (constant) density and external body force, and where T is the stress tensor given by (1). Standard indicial and vector notation is employed throughout. Further, equations (5), (6) are defined on a region R, exterior to a compact set in R 3, and on the time interval (0, TJ, for some T(5 00). To demonstrate uniqueness, we suppose Ui and OF are two solutions to (5), (6) on R x (0, r] which satisfy the same boundary conditions on i3C2(the bounded boundary of Q) and the same body force. Then, defining u = u* - v we find u satisfies, where p-l is absorbed in the remaining coefficients and pressure, Ui,f+ Uj*Ui.j+

UjUl

j

r:

-_i+~Aui+a,Aui.~+xl[c~A~,,-v,Aij,,]~j +alCv~jA,r+V,TiA,*,-Vo,.jAir-V*,iA,j].j + a2 CA,*, A$ -

AimAmj3.j

+ /3[(trA*‘>A;

- (trA2)Aij], j,

where A$ = Vifj + u$, p denotes the differencepressure, ui = 0

on R x (0, T],

and

on K? x [0, T],

UJX, 0) = 0,

(7)

XER.

(8) (9)

We are now in a position to state our uniqueness theorem. Before that, however, it is convenient to state the class of solutions in which we are interested. A classical solution to (7H9) is of class K if it satisfies conditions (tO)-(13) below: 10,

IVVL Iv1
IVu*l,

jAvlrM
(10)

I”I,jk19

I”ifjkl I lvrk*

(11)

for constants M, k > 0, where I = (XiXi)“*. p = O(Iogr”/r”2),

(12)

for some GE [0, +). sup N Y2 tero. r1

as 7 3 0,

for some 7’ c co and some E > 0, where ( . ) denotes the integral over Q, and where g is defined by g = exp (- mrY), (m 2 41, (14) with y (> 0) to be assigned. Theorem 1. A solution to (7)-(g) of class K on S2 x [0, TJ necessarily is such that u - 0.

l

We are very grateful to Professor K. R. Rajagopal for drawing our attention to this fact.

379

Fluid of third grade

Proof: to prove the theorem we employ a variant of the method of Rionero and Galdi [S]. Define now (15) F(t) = ~((C71~12>+%
Next, multiply (7) by gUi and integrate by parts to find

dFld~=~(g,~~~l~12)-(~~i~j~i,j) + (g,iUiP)-P
-~~-~~(bui),jC~~j~ + $iGj + k,jAir + ur,iArjl) -crt((gui),jCA~a,j+ai,A,jl) -B((gui),jC(trA*2)aij+ AZAijasr+aAsrAijI) (16)

+ ~1(g.jUi.jU,,t)+ al
where Uij = Ui,j + Uj.i. [TO arrive at (16) we consider F defined on the intersection of Q and a large sphere, radius R, any integrations by parts are carried out and then R is allowed to tend to infinity. Thanks to the weight the boundary terms disappear. This procedure will be understood in what now follows.] We now denote the terms on the right hand side of (16) by I, - I, 1. Thanks to conditions (10) and (11) and the forms (14) and (1.5) for g and 1; the terms I,, I,, I,, I,, IT-I9 are immediately bounded by (17) c,F, for a computable constant cl. To handle I, we use several integrations by parts and the fact that the velocity fields are solenoidal to rewrite it as a1--‘I 6

=

!f(g,m”~Ivu12>

+


+

(g.iv2Ui.jUj.m)

+

+

(g,mUi,jUmAij

> -

+

WZ,iUi,jUj,m) (g.jUi,mumAij

(g.jUi.tnu~aij) + > -

(g.jUi.mumAij

> +


> -


>

(gui,mUmAUi)*

In this form, with the bounds (10) it is easy to see that I, sc,F,

(18)

for another computable constant c2. To treat the pressure term I,, the analysis follows exactly that of Rionero and Galdi [5, p. 2991 who demonstrate that 13 ,< c,(glu12)+c:y~+c~y1-~6,

(19)

for computable constants cj - c5, independent of y. Finally, using the arithmetic-geometric mean inequality terms I,, and I, 1 are estimated as: I,,+!,, Ic~F+w2
is then completed

F.

(21)

as in [S], allowing y +O to obtain

a

Notes

(1) For a Navier-Stokes fluid the pressure conditions have recently been considerably weakened by Galdi and Maremonti [63 and for the backward in time problem by Galdi and Straughan [7]: those papers obtain a Poisson equation for the difference pressure field and use the theory of singular integrals. Whether these techniques are extendable to the present problem remains to be seen. (2) Any attempt to obtain a uniqueness theorem for a weak solution will need to take account of an extra piece which wilLappear in (6) due to the term a,Uj,irj which has no counterpart in (7). (3) We believe conditions (10)-(13) are not unnecessarily strong restrictions on the velocity field. This may be judged if one compares the non-linearity in (7) and that of a Navier-Stokes fluid with conditions (10)-(13) and those required by Rionero and Galdi [S]:

380

F.

FRANCHI

and B. STRAUGHAN

the major limitation is (13), but we observe that this is consistent with the behaviour Iu,~ = 0(logrb/r1’2), r+ co, similar to (12) for the pressure.

3. CONTINUOUS

DEPENDENCE

We now state a theorem of continuous dependence on the initial data. Only brief details of the proof are sketched. For technical reasons we here restrict attention to the weight -#r , 9 =e

!z > 0,

(22)

as we wish to establish Holder continuous dependence on the initial data; the weight used in the proof of Theorem 1 leads only to a weaker logarithmic continuous dependence, cf. the discussion of Galdi and Rionero [8] for a Navier-Stokes fluid. The class of solutions under consideration in this Section is now defined. A classical solution to (7x9) is of class M if it satisfies: Conditions (10) and (11); r-co,

p =: G[r-(++J)],

62-O;

Condition (13) holds with y replaced by a and with g as in (22).

(23) (24) (25)

Theorem 2. Suppose al solution to (7), (8) of class M is such that the initial velocity difference u,, satisfies IUOI? IV&l 0). Then u satisfies the estimate XER,

IUII Qd,

rc(O, Tl,

(27)

for computable constants K, 6( > 0). Sketch of the proof: with g as in (22) and F as in Theorem 1 we proceed as in the proof of that theorem to now obtain dF/dt s b, F + b, d, for some Bs(O, 1) and computable b,, 6,. This inequality is integrated to yield F(t) I exp(b, t)[F(O) + b,d]. F(t) is now bounded below by replacing Q by the intersection of a large sphere radius R with Q such a domain to be noted by R,, and F(0) is bounded using (26). We deduce that

Il~ll~~lI~~ll~~~,~~+~~~~~ where II * jl denotes the L2-norm on fiR weighted with g. The constants ~1,q and R are now related, as in c.f. .[7,8] to arrive at IIn II,: + IIVu II1:I_(k, $3

0<6<1.

(28)

It remains to establish the pointwise estimate for /U1.To this end we appeal to Lemma 2 of Galdi and Rionero [9]. Since the arguments leading to (28) may be repeated with the origin any point in Q, we are able to cover R and obtain

lu(x, t)] I f(q”. The theorem follows. 4. THERMAL

CONVECTION

To describe thermal convection in a fluid of third grade we first note that the momentum, continuity and constitutive equations continue to be (5), (6) and (1). We shall again assume p, tli, r2, /J constant although in some applications it may be convenient to allow them to depend on the temperature, as would be consistent with thermodynamics, see Cl]. In addition to the above equations we must add the energy equation pDa/Dt =T*L-divq+pr,

(29)

381

Fluid of third grade

where E, q and r are the internal energy, heat flux vector and heat supply, respectively. To manipulate equation (29) it is sufficient to consider no heat supply and so we first set r = 0 and then note that the Helmholtz free energy $ = E - VTis given by equation (4.15g) of Fosdick and Rajagopal [l] as ti = $(T)+~p-‘a,IA12.

(30)

We next see from [l] that the entropy tl satisfies

all/far),

q = - *r( = -

and so using this together with (30) and the relation E = J/ + qT the energy equation (29) may be reduced to pTDq/Dc = P - divq, (31) where P = $[~IAI’+(a, +a,)trA3 +BIA14]. (32) It is usual in thermal convection studies to assume ~~IAI’ is small in comparison to the derivative of the entropy and the heat flux terms in (31), c.f. Straughan [3, p. 5033 and using that argument, trA3 and I Ai4 will be even smaller and hence we are justified in neglecting P in (31); this is part of our Boussinesq approximation. Now, define c = - T$,,(> O), and we shall assume c is constant (specific heat). Then (31) becomes pcDT/Dt = kAT,

(33)

where we have adopted Fourier’s law q = -kVT,

(34)

for a constant thermal conductivity k. (In general, see Cl], q will depend also on A and Al, although for the application under consideration a linear constitutive law would appear reasonable.) We shall also adopt the Boussinesq approximation in the momentum equation and hence our final system of equations becomes: Dui/Dt = -gk,[l

-a(T-T,)]-p;‘Tji,j,

(35)

Dimi = 0,

(36)

DT/Dt =

KAT.

(37)

In deriving these equations we have assumed the density is modified only due to buoyancy effects and so have written the body force as phi = -~o~ki[l

TR)l,

-a(T-

where p,, is the constant density of the fluid at a reference temperature

T,, g is gravity,

k = (0,0, l), a is the coefficient of thermal expansion and further, in (37), K

=

k/p (= k/p,c,

the Boussinesq approximation). Equations (35)-(37), with T given by (1) are our basic system for thermal convection in a fluid of third grade. To investigate instability due to heating from below we suppose the fluid is contained in the layer z E (0, d). The stationary solution to (35)-(37) subject to specified temperatures on the boundaries T=

To,z=o;

T=

T,,z=d;

To > T,, To, T, constants, and the no-slip condition

v=o

on z = 0, d,

is v, = 0, T, = -b+

To,p, = pos(C1 -a(T,-

where the temperature gradient [ = (To - T,)/d.

TR}Iz+$Cz2),

(38)

F. FRANCHI and B. STRAUGHAE

382

To study the stabiiity/instability of (38) we write c’= u + v,, T = T, + 8, p = pS+ p and then from (35)-(37) derive equations for the perturbations u, 6, p which are not necessarily small. We non-dimensionaiize the equations by setting x = x’d,

u = U’U,

8 = T *8’, P’

U = v/d,

v = !JlPo, p = p’P”,

T# = (Pr~/ga)“%J, = ~p~v/d,

r, = pod’,%,

R = ~~~d4g/~v)‘12, Pr = V/K,

G = pod2b,,

t’ = tv/d2,

B = Pvlpod4,

where PI-, R2 are the Prandtl and Rayleigh numbers, r,, r2 are absorption numbers and B is a non-dimensional form of p. The non-dimensional equations for u and 6 are (omitting all primes) Dui/Dt = -p,i + kiR8 + AUi+ r;-‘(DAij/Dt + rz-‘(Ai,A,j),j

f Ai,L,j

+ L,iA,j),j

+ B[(trA2)&jl.jt

(39)

Ui,i = 0,

(40)

PrDB/Dt = Rw + A&

(41)

where now Lij = pi. j, Ai, =: Ui,j + ~1, i. We note now that the problem of linear instability reduces to that of the second grade fluid. Further, the proof of exchange of stabilities in [ 1l] still holds and so the critical Rayleigh number of linear theory Ri is just that given by classical theory, e.g. Ri 2 1708, for two fixed boundaries, c.f. Joseph [Ill. In fact, the linearized time independent part of (39H41) is symmetric. However, the strong result of Galdi and Straughan [lo] which shows the non-linear/linear stability boundaries are the same for a symmetric system does not hold here; the reason is that the condition (u, Nu) 2 0 of [lo] is not verified, a priori. We need, therefore, to investigate non-linear energy stability directly. In this context we now make the usual assumption that u, S, p are periodic functions in x, y. Define next E(t) =

$( <[u12> + Pr(e2)

++r,-‘(IA12)),

(42)

where r, > 0, and where (a ) denotes the integral over a period cell V. By differentiating E and using (39)-(41) to substitute for the resulting time derivatives we derive the energy equation, dE/dt = 2R(&v)

- o(6?)-f([A12)

-$(r,-’

f r2-1)(trA3)

-$B(IA14),

(43)

where D( .) denotes the Dirichlet integral. The form x = (/AIt> +(rlV1 f rzS’)(arA3)

+ B(lA14>,

(44)

which arises in (43) is important. For this form, from Lemma 3 of Fosdick and Rajagopal [l J we deduce that I(CQ+ d/pod21 I CWWp~d4P2, which is the same as (4.15~) of [l], Therefore, we may assert that x 2: 0, courtesy of thermodynami~al arguments. Because of this, however, to analyse non-linear stability from (43) we begin by examining two limiting cases. (i) aI +cc,=O There is a precedent for having such a condition since it is consistent with thermodynamics for a fluid of second grade, although thermodynamics does not necessarily require it to hold for a fluid of third grade. If, however, CL~ + a1 = 0 then we may obtain a non-linear stability result which is stronger than the classical or second grade case. To see this we observe that in this situation (43) reduces to dE/dt = ~R(~w)--D(~)--~(u)-~~(IA~~). (45)

383

Fluid of third grade

From this we obtain immediately that the critical Rayleigh number of non-linear energy stability theory, R& satisfies Ri = Ri. This strong result, which shows sub-critical instabilities are not possible, was first proved for a Newtonian fluid (grade one) by Joseph (see e.g. [l l]), and for a fluid of second grade by Straughan [3]. But, for the third grade fluid we may additionally assert that (46) E L’(0, cx)). In fact, we believe it should be possible to prove from (46) the “almost pointwise” result that Ilull,cxlEL1(O, co). Unfortunately,

a proof of this fact has so far eluded us.

(ii) I%,+ azl = ~(24~~) In this situation there are always choices of A such that the form x may equal zero, see Fosdick and Rajagopal Cl], Lemma 2, hence all we are able to deduce from (43) is that dE/dt I 2R(Bw) -O(e). This relation evidently does not allow us to deduce non-linear asymptotic stability. it remains to analyse the intermediate case: (iii) 0 c la, -t a2 I -z J(24@c() First let us observe that by use of Lemma 3 of Fosdick and Rajagopal Cl] and the arithmetic-geometric mean inequality,

x~U

-12, +a21/2’JPod2,/6)

+(IA14>UW~~d4-~la1

+a21/&M2J6).

for y > 0 to be chosen. Let us further choose y = 2&/6/la, O
(47)

+ a21p,,d2 and set

1-lr,fcr212/24~~<1.

With this choice, employing (47) in (43) we find that dE/dt I 2R (6~) - o(0) - ED(U). From this inequality we deduce that dE/dt<

-DR(R-I--RiI)

(48)

where D = D(B) + &D(U) and R;’

= m;x(I/D),

(49)

with I = 2(Bw}, and where N denotes the space of admissible solutions. Define now e112u= rp and then (49) becomes &2&

1

=

m;x CWWl(W)

+ D&9)1.

(50)

The maximum probIem on the right hand side of (SO)is just the one which appears in the second grade theory or classical Newtonian theory, see [3, 91. Hence from (50) we may assert that R; =

ER;.

This result, of course, demonstrates that the non-linear Rayleigh number we have found is lower than that for a second grade fluid. Acknowledgement-This paper was written while B.S. was a guest at the University of Bologna. We are grateful to the Italian Ministry of Education for the financial support in the form of the 60% Research Project.

384

F. FRANCHI and B. STRAUGHAN REFERENCES

1. R. L. Fosdick and K. R. Rajagopal, Thermodynamics and stability of fluids of third grade, Proc. R. Sot. Lend. A 339, 351-377 (1980). 2. J. E. Dunn and i. L. Fosdick, Thermodynamics, stability and boundedness of fluids of complexity 2 and fluids of second grade. Archsration. Mech. Analysis 56, 191-252 (1974). 3. B. Straughan, Energy stability in the Bbnard problem for a fluid of second grade. Z. angew. bfarh. Phys. 34, 502-509 (1983). 4. R. L. Fosdick and B. Straughan, Catastrophic instabilities and related results in a fluid of third grade. Int. J. Non-linear Mech. 16, 191-198 (1981). 5. S. Rionero and G. P. Galdi, On the uniqueness of viscous fluid motions. Archs ration. Mech. Analysis 62, 295-301 (1976). 6. G. P. Galdi and P. Maremonti, A uniqueness theorem for viscous fluid motions in exterior domains. Archs ration. Mech. Analysis 91, 375-384 (1986). 7. G. P. Galdi and B. Straughan, Stability of solutions to the Navier-Stokes equations backward in time. Archs ration. Mech. Analysis, 101, 107-114 (1988). 8. G. P. Galdi and S. Rionero, Continuous dependence theorems for the Navier-Stokes equations in unbounded domains by the weight function method. Q. J. Mech. appl. Marh. 32, 149-161 (1979). 9. G. P. Galdi and S. Rionero, Local estimates and stability of viscous flows in exterior domains. Archs ration. Mech. Analysis 81, 333-347 (1983). 10. G. P. Galdi and B. Straughan, Exchange of stabilities, symmetry and non-linear stability. Archs ration. Mech. Analysis 89, 21 l-228 (1985). I I. D. D. Joseph, Stability ofFfluid Motions If. Springer, Berlin (1976).