Ricerche mat. (2007) 56:229–239 DOI 10.1007/s11587-007-0016-3

A note on convection with nonlinear heat flux Brian Straughan

Received: 11 April 2007 / Accepted: 30 July 2007 / Published online: 22 November 2007 © Università degli Studi di Napoli 2007

Abstract The problem of thermal convection is investigated when the heat flux is a nonlinear function of the temperature gradient. A complete analysis of the linear instability problem is given. The nonlinear stability problem is studied in a case which is believed to be physically relevant and the stability threshold is compared directly to that found by linear instability theory. Keywords

Convection · Nonlinear heat flux · Stability · Instability

Mathematics Subject Classification (2000)

76E06 · 76E30

1 Introduction The problem of thermal convection (Bénard convection) where a cellular pattern is observed when a layer of fluid is heated from below is a classical problem of hydrodynamic instability theory. If the temperature difference across the layer is sufficiently large or the layer is deep enough then a well developed pattern forms and the mathematical theory for this is well documented, see, e.g. Flavin and Rionero [13], Straughan [34,35] and the references therein. The mathematical theory is usually based on a dimensionless number, Ra, called the Rayleigh number, defined by Ra =

αg ∆T d 3 νκ

Communicated by the Editor-in-Chief. B. Straughan (B) Department of Mathematical Sciences, Durham University, Durham DH1 3LE, UK e-mail: [email protected]

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where ∆T is the temperature difference across the layer, d is the depth of the layer, and α, g, ν, κ are (constant) values of thermal expansion coefficient, gravity, kinematic viscosity and thermal diffusivity. Although the classical theory adopts constant values for the parameters thermal expansion coefficient, kinematic viscosity and thermal diffusivity, in real fluids these functions often vary widely with temperature or even temperature gradient. Nonlinear stability and linear instability studies of flow with variable thermal expansion coefficient and kinematic viscosity, in particular, have occupied much attention, see, e.g. Ames and Payne [1,2], Budu [7], Capone [8], Capone and Gentile [9,10], Capone and Rionero [11], Diaz and Straughan [12], Flavin and Rionero [14,15], Payne and Song [22], Qin et al. [23], Straughan [32–34], Vaidya and Wulandana [37], Wall and Wilson [38], Webber [39], and the references therein. Furthermore, analyses of structural stability, i.e. the effect of changes of the thermal expansion coefficient, kinematic viscosity and thermal diffusivity on the solution to the governing partial differential equations have also attracted much attention, cf. Ames and Payne [1–3], Flavin and Rionero [13], Franchi and Straughan [16], and the references therein. For many real fluids the linear constitutive assumptions which lead to Navier– Stokes–Fourier theory are inadequate. For example, viscoelastic fluids certainly are not adequately described. Hutter [17] describes models for ice which have constitutive laws in which stress is not a linear function of temperature gradient. The paper by Zohdi and Wriggers [40] also questions the use of a linear Fourier-type heat flux–temperature gradient law for many classes of materials. The papers by Massoudi [20,21] present a careful and authoritative account and derivation of a suitable nonlinear heat flux law. Massoudi [20,21] clearly shows where a law more general than a Fourier one may well be necessary to describe the behaviour of many modern materials. In particular, on page 1604 he deals with the heat flux in a fluid and his equation (23) is one of the type we shall consider. Rodrigues and Urbano [31] develop an existence theory for a general nonlinear model of fluid behaviour and the heat flux law they adopt is one of form (1) q = −κ0 |∇T | p−2 ∇T, where κ0 is a positive constant, p ≥ 2 is a constant, and q is the heat flux vector. Of course, when p = 2 this reduces to Fourier’s law. A heat flux law like (1) may not be unrealistic to describe manufactured nanofluids which typically have metal oxide particles in suspension in a carrier fluid. Such fluids typically display very increased heat transfer properties and increased thermal conductivities, see, e.g. Maïga et al. [19], Vadasz et al. [36], Kim et al. [18]. In this paper we commence a study of thermal convection appropriate to a fluid governed by equation (1). One could argue that we should also adopt a nonlinear stress–velocity gradient law and investigate the two effects simultaneously. This is a viewpoint with which we do not disagree. However, as a first step we adopt a linear viscous fluid because we can then see the effects caused by the nonlinear heat flux theory more clearly. It will be seen that equation (1) leads to substantial deviations from the results of the classical Navier–Stokes–Fourier theory and this, we believe, justifies our investigation of (1) in isolation without the additional complication of a nonlinear stress–velocity gradient constitutive equation. Also, convective nonlinearities in

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231

systems can lead to dramatic behaviour such as solution blow-up in finite time, see, e.g. Ames et al. [4], Ames and Straughan [5,6], Payne and Song [22], Straughan [32]; it is thus preferable to analyse the effect of the nonlinearity in (1) upon convection directly. In Sect. 2 we develop a linear instability theory for the problem of a layer of fluid heated from below with the heat flux given by equation (1). Then, in Sect. 3 a nonlinear energy analysis is presented for the case when p = 4. Conclusions are reported in Sect. 4.

2 Linear instability theory Consider a layer of viscous, heat conducting fluid contained between the planes z = 0 and z = d, with gravity in the negative z-direction. The bottom plane is maintained at a higher temperature T1◦ C than the top plane (z = d) which is maintained at temperature T2◦ C (T2 < T1 ). With a linear stress law, a Boussinesq approximation in the buoyancy term, and the heat flux given by (1), the partial differential equations governing the behaviour of the fluid in the layer z ∈ (0, d) are ∂vi 1 ∂p ∂vi = − + ν∆vi + gαT ki , + vj ∂t ∂x j ρ0 ∂ xi ∂vi = 0, ∂ xi
 ∂T ∂T + vi = κ0 ∇ · |∇T | p−2 ∇T , ∂t ∂ xi

(2) (3) (4)

where vi , T and p are the velocity, temperature and pressure fields. The fluid is viscous and so no-slip conditions are applied at a fixed surface and stress-free ones at a free surface. With either of these boundary conditions the equations possess the stationary solution v¯i = 0, T¯ = −βz + T1 ,

(5)

where β is the temperature gradient, β=

T1 − T2 ∆T = , d d

and the corresponding pressure p¯ is found from (2). We here study the stability and instability of solution (5). To do this we introduce perturbations (u i , θ, π ) to the velocity, temperature and pressure fields of form vi = v¯i + u i , T = T¯ + θ,

p = p¯ + π.

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With k = (0, 0, 1) the perturbation quantities are found to satisfy the equations ∂u i ∂u i 1 ∂π = − + ν∆u i + gαθ ki , +uj ∂t ∂x j ρ0 ∂ xi ∂u i = 0, ∂ xi    p−2 ∂θ ∂θ   + ui = βw + κ0 ∇ ·  − βk + ∇θ  (−βk + ∇θ ) , ∂t ∂ xi

(6) (7) (8)

where we have set w = u 3 . These equations are conveniently non-dimensionalised with the variables d2 , u i = U u i∗ , ν ρ0 νU ν P= , π = Pπ ∗ , U= , d d  p−2   ∆T   κ = κ0 |∇T | p−2 = κ0  , d  x = x∗ d,

t = t∗

T = ∆T,

Pr =

ν κ

Ra =

θ = T θ ∗,

gα∆T d 3 , νκ

where T , U and P are temperature, velocity and pressure scales and Pr is the Prandtl number. Observe that we are using the temperature difference across the layer as a temperature scale, although it is convenient to choose another scale in the nonlinear analysis of the next section. With the above non-dimensionalisation the equations (6)–(8) become (omitting all stars) ∂u i ∂π Ra ∂u i +uj θ ki , = − + ∆u i + ∂t ∂x j ∂ xi Pr ∂u i = 0, ∂ xi      p−2 ∂θ ∂θ   = Pr w + ∇ ·  − k + ∇θ  + ui Pr (−k + ∇θ ) . ∂t ∂ xi

(9) (10) (11)

In our analysis of linearised instability we let p = 2m, m ∈ N. The next stage is to linearise (9)–(11) and then seek a time dependence like eσ t . This results in the system σ ui = − ∂u i = 0, ∂ xi

∂π Ra θ ki , + ∆u i + ∂ xi Pr

σ Pr θ = Pr w + (2m − 2)

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(12) (13)

∂ 2θ + ∆θ. ∂z 2

(14)

Convection with nonlinear heat flux

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For either fixed or stress free boundary conditions on u i , since θ is zero on z = 0, 1 and a plane tiling periodicity is assumed in the horizontal directions, it is straightforward to show that σ ∈ R and then the strong form of the principle of exchange of stabilities holds. Henceforth, we therefore, take σ = 0. The pressure perturbation is eliminated from (12) by taking curlcurl of that equation and retaining the third component of the result. This leads to the system Ra ∗ ∆ θ = 0, Pr ∂ 2θ Pr w + (2m − 1) 2 + ∆∗ θ = 0, ∂z

∆2 w +

(15) (16)

where ∆∗ = ∂ 2 /∂ x 2 + ∂ 2 /∂ y 2 . Let now a be the wavenumber of the perturbation and then suppose the (x, y) parts of (w, θ ) satisfy the relation ∆∗ g + a 2 g = 0, which is consistent with a plane tiling cellular shape, e.g. hexagons. Then, from (15), (16) we find Pr (D 2 − a 2 )2 W − Ra a 2 Θ = 0, Pr W + (2m − 1)D 2 Θ − a 2 Θ = 0,

(17)

where D = d/dz and W (z) and Θ(z) are the z-dependent parts of w and θ. At this stage we must solve these equations subject to boundary conditions of free or fixed type. As this is a first study we restrict attention to two free surfaces to obtain the effect of the nonlinear heat flux law. Thus, we solve (17) subject to the boundary conditions W = D 2 W = Θ = 0, z = 0, 1. It is not difficult to show that the solution which yields the smallest Rayleigh number has a z-dependence like sin π z. Then Ra is found to satisfy  (π 2 + a 2 )2 (2m − 1)π 2 + a 2 Ra = . a2

(18)

The critical wavenumber is found by minimizing Ra in a 2 to find 2 = ac2 = acrit

π 2  2 4m + 12m − 7 − 2m + 1 . 4

The critical Rayleigh number is then found by substitution of ac2 in (18). For m = 1 we recover ac2 = π 2 /2 and Ra = 27π 4 /4 which are the classical values. It is seen that ac2 is an increasing function of m with asymptotic value ac2 = π 2 for very large m. The critical Rayleigh number increases indefinitely. The values of Ra and ac2 for

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234 Table 1 Critical values of Ra and a 2 against m.

B. Straughan

m

Ra

a2

1

657.511

4.934802

2

1487.804

6.771937

3

2286.792

7.555818

4

3076.962

8.011530

5

3863.225

8.313456

6

4647.385

8.529392

7

5430.276

8.691943

8

6212.338

8.818925

9

6993.829

8.920958

10

7774.909

9.004788

11

8555.682

9.074917

12

9336.221

9.134467

13

10116.58

9.185673

14

10896.79

9.230183

15

11676.88

9.269233

16

12456.87

9.303773

17

13236.79

9.334544

18

14016.63

9.362133

19

14796.42

9.387009

20

15576.16

9.409556

Fig. 1 Critical values of the Rayleigh number Ra against nonlinearity parameter m

m = 1, . . . , 20 are given in Table 1 and Figs. 1 and 2 plot Ra against m and ac2 against m, respectively. It might at first sight appear that the curve in Fig. 1 is linear in m. However, upon closer inspection this is seen not to be so, although the curve approaches a linear one for large m, with slope 8π 4 .

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Fig. 2 Critical values of the square of the wavenumber a 2 against nonlinearity parameter m

3 Nonlinear stability To make a direct comparison of nonlinear energy stability theory with linear instability theory we concentrate on the case p = 4, or m = 2. The severity of the nonlinearity makes the analysis challenging in the full nonlinear situation and hence we restrict attention to one case to obtain an idea of what to expect. To obtain a direct comparison between the critical Rayleigh and wave numbers of linear instability theory with those of nonlinear energy stability theory it is convenient to adopt a different non-dimensionalisation to that of Sect. 2, one which renders the linearised system immediately symmetric. The governing equations are again (6)–(8) and the scalings are the same as given there except we choose a different temperature and Rayleigh number scale, namely

(T ) p =

βνd p−2 U 2 , κ0 gα

R=

gαd 2 νU



βνd p−2 U 2 κ0 gα

1/ p ,

where Ra = R 2 is the now the Rayleigh number. Then, the non-dimensionalised system of partial differential equations governing (u i , θ, π ) becomes ∂u i ∂u i ∂π +uj = − + ∆u i + Rθ ki , ∂t ∂x j ∂ xi ∂u i = 0, ∂ xi      p−2 ∂θ ∂θ   = Rw + ∇ ·  − λk + ∇θ  + ui Pr (−λk + ∇θ ) , ∂t ∂ xi

(19) (20) (21)

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where the parameter λ is given by λ=

∆T . T

Restrict now attention to the case p = 4. Then multiply (19) by u i , (21) by θ and integrate over a period cell V to obtain, with ·, ( , ) denoting norm and inner product on L 2 (V ), d 1 u2 = R(θ, w) − ∇u2 , dt 2 d 1 Pr θ 2 = R(w, θ ) − λ2 ∇θ 2 − 2λ2 θz 2 dt 2

+3λ θz |∇θ |2 d V − |∇θ |4 d V , V

(22)

(23)

V

where θz = ∂θ/∂z. There are two problems with developing an energy theory from (22) and (23). One is to find the optimal way to add (22) to (23). The other arises directly from the nonlinear heat flux, and is to remove the cubic term on the right of (23). By the symmetry of the linearised problem we believe a coupling parameter of 1 is a good choice, i.e. add (22) and (23) directly. A general discussion of symmetry and the logic behind the optimal choice of coupling parameters may be found in Chap. 4 of Straughan [34]. There would appear to be some scope in handling the cubic term in (23). We here use the arithmetic-geometric mean inequality to see that

θz |∇θ | d V ≤

|∇θ |4 d V +

2

3λ V

V

9λ2 θz 2 . 4

This in (23) yields 3λ2 d 1 Pr θ 2 ≤ R(w, θ ) − θz 2 − λ2 ∇ ∗ θ 2 , dt 2 4

(24)

where ∇ ∗ = (∂/∂ x, ∂/∂ y). By adding (22) and (24) we are led to our energy inequality dE ≤ R I − D, dt where we have defined E, I and D by 1 1 u2 + Pr θ 2 , 2 2 I (t) = 2(θ, w), 3λ2 D(t) = ∇u2 + θz 2 + λ2 ∇ ∗ θ 2 . 4 E(t) =

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(25)

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By standard arguments in energy stability theory, see, e.g. Straughan [34], one can now show that the critical Rayleigh number of nonlinear energy stability [according to inequality (25)] is I 1 , (26) = max H D RE where H is the space of admissible solutions. If R < R E then unconditional (i.e. for all initial data) nonlinear stability is obtained, cf. [34]. To determine R E requires resolution of (26) and from this the Euler-Lagrange equations are found to be 3λ2 θzz + λ2 ∆∗ θ + R E w = 0, 4 ∆u i + R E θ ki = µ,i ,

(27)

where µ is a Lagrange multiplier. These should be contrasted with the corresponding equations of linear instability theory which in the current non-dimensional form are 3λ2 θzz + λ2 ∆∗ θ + Rw = 0, ∆u i + Rθ ki = π,i . Use of normal modes in (27) leads to the system (D 2 − a 2 )2 W − R E a 2 Θ = 0, 3λ2 2 D Θ − λ2 a 2 Θ + R E W = 0. 4 For two free surfaces one then finds R 2E =

λ2 ([3π 2 /4] + a 2 )(π 2 + a 2 )2 . a2

From this, for two stress free surfaces one finds the critical values a 2E

√  105 − 3 = π 2 = 0.45293π 2 , 16

and Ra E = R 2E(crit) = 5.60659λ2 π 4 . The corresponding values from linear instability theory are a L2 = 6.771937 = 0.68614π 2 , Ra L2 = 15.27377λ2 π 4 .

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4 Conclusions We have studied stability in a model for Bénard convection which employs the nonlinear heat flux law of Rodrigues and Urbano [31]. The findings are that the nonlinearity has a pronounced effect. As the parameter p in (1) is increased beyond 2 the critical wavenumber of linear instability theory increases reaching an asymptotic value of twice that of classical Navier–Stokes–Fourier theory. This means that for p > 2 shorter wavelength instabilities are responsible for convection and narrower convection cells should be observed with this theory. The effect of the nonlinearity is to strongly increase the critical Rayleigh number and thus make it harder for the system to convect. A nonlinear energy stability analysis has been developed for the case p = 4. Here it is seen that there is much work to be done on the nonlinear stability problem in general. The nonlinear threshold for stability is just over one third of that found for linear instability. This means that one cannot rule out sub-critical instabilities with this theory. A more sophisticated approach to energy theory may yield a sharper stability boundary. In this regard, one may consider developing the recent method of Rionero [24–30], with the problem of this article in mind. Acknowledgments I am indebted to Professor Jose Francisco Rodrigues for bringing his paper [31] with Professor Urbano to my attention: this paper stimulated the present work. I am also very grateful to Professor G. Mulone for help with the latex macros.

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