Benard-Marangoni instability in a Maxwell-Cattaneo fluid

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29 October 1984

B E N A R D - M A R A N G O N I INSTABILITY IN A M A X W E L L - C A T T A N E O FLUID ¢ G. LEBON 1 and A. CLOOT

Institute of Physics, BS, Sart-Tilman, Libge University, 4000 Libge, Belgium Received 18 April 1984 Revised manuscript received 6 June 1984 The effects resulting from the substitution of the classical Fourier law of heat conduction by the Maxwell-Cattaneo law in B6nard's and Marangoni's problems are examined.

1. Introduction. In a recent note [1], Straughan and Franchi treated B6nard's convection when the Maxwell-Cattaneo heat flux law [2,3] is employed instead o f the usual Fourier law o f heat conduction. It was shown that overstability can only occur by heating the fluid from below but that stationary convection was possible even by heating the fluid from above, in contradiction with physics. The analysis by Straughan and Franchi rests on an extension on the Maxwell-Cattaneo relation in the form r(~ - I t . q ) : ~ / -

KVT,

(1)

where q is the heat flux vector, T the temperature, I¢ the skew-symmetric velocity gradient tensor, r a constant relaxation time, K the heat conductivity; an upper dot denotes the material time derivative. The quantity ~ - W" q represents the Jaumann time-derivative and is an objective quantity, in the sense o f Truesdell and Noll [4[. However, it has repeatedly been pointed out [ 5 - 7 ] that this expression is not a fortunate choice because it leads to results contradicted by the kinetic theory o f gases. To be specific, the constitutive equation for the heat flux derived from the Boltzmann equation contains a term like

q+W'q, One manuscript page was omitted in the printing of this paper (Phys. Lett. 104A (1984) 153). We apologize to the authors and readers for this error (the editors). 1 Also at: Department of Thermodynamics, University of Louvain, Louvain, Belgium. 0.375-9601/84/$ 03.00 ©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

wherein the sign in front o f 1¢" q is the opposite of that appearing in the Jaumann derivative. As shown by recent investigations [ 8 - 1 0 ] , agreement with the kinetic theory is achieved by selecting, instead o f Jaumann's, the following derivative ;i + W ' q

- 2a'q

,

(2)

which has been proven to be an objective quantity; in (2), f l represents the angular velocity o f the moving frame wherein the time-rate is measured. Since in the present analysis, we refer the fluid layer to an inertial frame, f l is zero and (2) reduces to + W.q.

(3)

The first objective of this note is to derive the consequences arising from the substitution o f the Jaumann derivative by expression (3) in Straughan and Franchi's analysis [ 1 ]. It will be shown in section 2 that when buoyancy is the single factor of instability (B6nard's instability), no stationary convection can develop in a fluid layer heated from above but oscillatory convection is possible. In section 3, the same expression (3) is used in Marangoni's problem where convection is driven by temperature-dependent surface tensions. All the results reported here concern only infinitesimally small perturbations.

2. The B~nard problem. Consider a horizontal quiescent fluid layer of thickness d and infinite lateral extent, submitted to a temperature drop AT between the lower and the upper faces; when A T reaches a critical value, convection is initiated. The equations 361

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of motion are non-dimensionalized by putting

z=z*d,

t =t*d2/X,

T =T-AT,

(4)

and are given by (omitting the stars):

p - I atV2w = R 7 2 T + V4w ,

(5a)

arT=w-

(5b)

V'q,

2Catq = --C(~zU - Vw) - q - V T ,

(5c)

with 2

v=(ax, y, az), v =ax2x

u is the velocity vector with cartesian components (u, o, w), P, R and C are the dimensionless Prandtl, Rayleigh and Cattaneo numbers respectively defined by

P=v/x,

R=o~ATd3/uX,

C = r x / 2 d 2.

(6)

Assume normal mode solutions of the form (w, T) = (W, 0) exp[ot + i(axX +aye)] ,

(7)

with a(ax, av) the horizontal wave-number, and introduce them in the set (5a)-(5c). After elimination of q, one obtains

(o/P)(D 2 - a2)W = - R a 2 0 + (D 2 - a2)2W,

(8a)

oO = W + (2Co + 1)-1 [(D 2 - a2)O - C(D 2 - a2)W] , (8b) where D stands for d/dz. We firstly examine the occurrence o f stationary convection for which the imaginary part of o vanishes. At marginal stability (o = 0), eqs. (8a) and (8b) can be combined into (D 2 - a2)3 W + Ra2 [ W - C(D 2 - aE)W] = 0 .

(9)

For two free surfaces, the associated boundary conditions are W=D2W=0=0,

atz=0andz

=1.

(10)

Satisfaction o f (9) and (10) requires that R-

(n27r2 + a2)3 a 2 [1 + C(nEn 2 + a2)]

( n = 1 , 2 , 3 .... ) .

(11)

The lowest marginal stability curve R (a) corresponds to n = 1 ; R assumes its minimum critical value at the critical wave number a2 _ - 1 + [1 + Cn2(1 + Crt2)] 1]2

c

362

(12)

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For C = 0, (11)and (12) reduce to their classical values while by changing the sign in front o f C, one recovers the results reported in ref. [1]. It has been proved in thermodynamics [9-11 ] that the relaxation time, and consequently C, are positive quantities. It follows then from (11) that R cannot become negative, which precludes any onset of convection in a layer heated from above. As the critical Rayleigh number is smaller than its classical counterpart, we see that MaxwellCattaneo fluid is less stable than the Fourier one. It must however be realized that the corrections are minute as the values of C range from 10 -6 to 10 -8. Although exchange of stability has been demonstrated for a classical Fourier fluid, it is no longer guaranteed in a Maxwell-Cattaneo fluid. Hence, it is necessary to examine the behaviour of the system under oscillatory convection. This is performed by putting o = iol, where o I is real, in the balance equations (7) and (8). The procedure is classical and leads to the next result for o 2, R and a c2 : 02 _ (1 + e) + C(1 - P)(n2rr 2 + a 2 ) ,

(13)

2C2(1 - 2P) R =

(n2rr2 + a 2 ) [ l +P+2Cp2(nZrr2 +a2)] 2a2C2p(1 - 2P)

a2 = 7r(1 + e + 2rr2C2p2) 1]2 (2C)II2p

,

(14)

(15)

It follows from (13) and (14) that oscillatory modes appear, whatever the values o f C, for P < 112 while the corresponding R-values are negative. It may thus be concluded that by heating from above, instability can only be produced in the form o f oscillatory modes, at the condition that P < 1/2 holds. This is not an unrealistic value: mercury for instance is characterized by P = 0.025. Nevertheless, from a practical point o f view, it is highly improbable to observe this instability at it occurs for very large values of the critical Rayleigh and wave numbers, due to the presence of the C-terms at the denominators of (14) and (15): for C = 10 -6, calculations show that for a mercury layer, R c = - 2 . 1 6 X 1013 and a c = 302.

3. The Marangoni problem. Surface tension gradients are nowadays recognized as an essential agency of instability in thermal convection. In very thin layers and in microgravity situations, they play the dominant

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role with respect to buoyancy. The surface tension driven instability is known under the name o f Marangoni and is characterized by a hexagonal tessellation o f the fluid surface. Since standard descriptions o f Marangoni's effect (e.g. refs. [ 1 2 - 1 4 ] are exclusively based on Fourier's conduction law, we find it interesting to examine the influence o f a Maxwell-Cattaneo relation. The layer is now confined between a rigid bottom and a free top place, whereupon surface tensions are acting. Gravity effects are neglected (R = 0). The boundary conditions (10) are replaced by

29 October 1984

The critical Marangoni number at which stationary convection occurs depends on the parameters a, h and C and is given by Ma =

8a(a cosh a + h sinh a)(a - sinh a cosh a)

(a3 cosh a - s i n h 3 a ) - 4Ca2sinha(sinh2a - a2) " (24)

Eq. (17b) represents the boundary condition at a level surface submitted to a temperature-dependent surface tension, it involves the Marangoni number defined by

In the limiting case C = 0, one recovers Pearson's result [ 12] established for a classical Fourier fluid. In (27), the quantities (a - sinha cosha)and (a 3 cosh a - sinh3a) are negative while (a cosh + h sinh a) and (sinh2a - a 2) are positive. It follows that a marginal stability, Ma is necessarily positive. By selecting the Jaumann derivative as in ref. [1 ], one should have obtained a plus sign in front o f the quantity C in (27), allowing for negative Ma values and the occurrence o f steady convection by heating from above; to our knowledge this has never been observed experimentally. Like in B6nard's problem, oscillatory convective modes cannot be excluded a priori. But now the relevant equations cannot be solved analytically and are treated by using a Galerkin method. Let us select for W and 0 the simple trial functions

Ma-

W=z2(1-z)

W = DI4' = 0 = 0 ,

(16)

at z = 0 (lower rigid face), W=0,

D2W = - a z M a 0 ,

(17a,b)

at z = 1 (upper free face), DO + h O = O

(h>0).

(18)

(a~/aT)ATd plrx

where ~ is the surface tension and p the density of the fluid. Eq. (l 8) is Newton's law of cooling with h, the Biot heat transfer coefficient. Looking at stationary convection the linearized governing equations are given at marginal stability by (D 2 - a2) 2W = 0 ,

(19)

(D 2 - a2)(0 + Cw) = - W .

(20)

(21)

0 = E ( e az + Fe - a z ) + Ma 0p ,

(22)

A, B, E and F are easily derivable functions of a and h while the particular solution 0p is expressed by

0=z3(l-~z3).

(25)

In the classical case (C = 0), the functions (25) yield, for the critical Marangoni and wave numbers, the values Ma c = 7 9 . 5 7 ,

a c---2,

which are very close to the exact ones [12,15] Ma c = 7 9 . 6 1 ,

ac =1.99.

Therefore, it seems reasonable to use the same set (25) for non-zero but small values o f C. We have obtained for a 2 and Ma the following expressions:

With the normalization 0(1) = 1, analytical solutions can directly be written down in the form W = Ma A sinh az + Ma Bz sinh a z ,

and

910(i~ao+ LC)(l~23o M + N K / P ) - ~33oMNC °2 =

113C 2 [(K/P)L - s2-iT6 2a M]

' (26)

Ma = 2 [(t'~o + L C ) ( M N - sls-~2$oa2K/p) + -~-63oCo2 (ssJ'~-6M + K N / P ) ] / 23

a 2 [(a"~'6 + L C ) 2 + (~'~o Ca I )2 ] ,

(27)

Op = (A/2a2)(sinh az - 2a 2 cosh az )

- (B/4a3)(1 - 2az + 2a2z 2) cosh az - C ( B / A ) ( 1 - 2az) cosh a z .

(23)

where K, L, M, N are polynomial functions o f a 2. The analysis simplifies considerably in the two limiting cases P = 0 and P = ~. 363

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For P = O, a 2 is unconditionally positive while Ma is a negative quantity with order of magnitude Ma "" - 1014 for C = 10 - 8 . Theoretically, surstability is possible by heating from above but practically, the large values of Ma prevent us for observing this mechanism of instability. Similar results are obtained for P = oo: overstability can only occur in the heated from above case, with very large values for Ma. It clearly appears that in MaxweU-Cattaneo fluid, oscillatory convection does not play an important practical role. It follows from the above considerations that it is crucial to select appropriately the objective time derivative in the Maxwell-Cattaneo fluid model. A wrong choice can lead to unrealistic results. In that respect, Jaumann's time derivative must be rejected for at least two reasons. Firstly, as shown above, it is inconsistent with common observations in B~nard's and Marangoni's convection. Moreover, as proved elsewhere [ 5 - 7 ] , it leads to results denied by the kinetic theory of gases. In contrast, the choice (2) of the objective time-derivative is reliable not only from the kinetic point of view [ 8 - 1 0 ] but also from hydrodynamic stability considerations. The authors wish to thank Dr. Straughan (Glasgow) for making a preprint of its work [1] available to us prior to publication.

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References [1] B. Straughan and F. Franchi, Proc. Roy. Soc. Edinburgh (1984), to be published. [2] C. Cattaneo, Atti. Sere. Mat. Fis. Univ. Modena 3 (1984) 83. [3] J.C. Maxwell, Philos. Trans. R. Soc. A157 (1867) 49. [4] C. Truesdell and W. Noll, The non-linear field theories, in: Encyclopedia of Physics, Vol. III/3 (Springer, Berlin, 1965). [5] L Mliller, Arch. Rat. Mech. Anal. 45 (1972) 241. [6] D. Edelen and J. MacLennan, Int. J. Eng. Sci. 11 (1973) 813. [7] G. Lebon, Bull. Acad. R. Sci. Belg. 64 (1978) 456. [8] I. Murdoch, Arch. Rat. Mech. Anal. 83 (1983) 185. [9] G. Lebon and M.S. Boukary, submitted for publication (1984). [ 101 i. Mliller,in: Recent developments in non-equilibrium thermodynamics, eds. J. Casas, D. Jou and G. Lebon, Lecture Notes in Physics, Vo1199 (Springer, Berlin, 1984). [ 11 ] G. Lebon and D. Jou, J. Chem. Phys., 77 (1982) 970. [12] J.R. Pearson, J. Fluid Mech. 4 (1958) 489. [ 13] J. Casas and G. Lebon, eds., Stability of thermodynamic systems, Lecture Notes in Physics, Vol. 164 (Springer, Berlin, 1982). [14] H. Oertl and J. Zierep, eds., Convective transport and instability phenomena (Braun, Karlsruhe, 1981). [15] A. Nield, J. Fluid Mech 19 (1964) 341. [16] E. Guyon and J. Pantaloni, C.R. Acad. Sci. 209B (1980) 301. [171 G. Lebon and A. Cloot, Acta Mechanica43 (1982) 141.