Micropolar meets Newtonian. The Rayleigh–Bénard problem

Micropolar meets Newtonian. The Rayleigh–Bénard problem

Accepted Manuscript Micropolar meets Newtonian. The Rayleigh–Bénard problem Piotr Kalita, José A. Langa, Grzegorz Łukaszewicz PII: DOI: Reference: S...

NAN Sizes 0 Downloads 58 Views

Accepted Manuscript Micropolar meets Newtonian. The Rayleigh–Bénard problem Piotr Kalita, José A. Langa, Grzegorz Łukaszewicz

PII: DOI: Reference:

S0167-2789(17)30329-9 https://doi.org/10.1016/j.physd.2018.12.004 PHYSD 32070

To appear in:

Physica D

Received date : 14 June 2017 Revised date : 25 November 2018 Accepted date : 11 December 2018 Please cite this article as: P. Kalita, J.A. Langa and G. Łukaszewicz, Micropolar meets Newtonian. The Rayleigh–Bénard problem, Physica D (2018), https://doi.org/10.1016/j.physd.2018.12.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Micropolar meets Newtonian. The Rayleigh–Bénard problem. Piotr Kalitaa , José A. Langab , Grzegorz Łukaszewicza c a

Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland b Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain c Faculty of Mathematics, Informatics, and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland

Abstract We consider the Rayleigh–Bénard problem for the two-dimensional Boussinesq system for the micropolar fluid. Our main goal is to compare the value of the critical Rayleigh number, and estimates of the Nusselt number and the fractal dimension of the global attractor with those values for the same problem for the classical Navier–Stokes system. Our estimates reveal the stabilizing effects of micropolarity in comparison with the homogeneous Navier–Stokes fluid. In particular, the critical Rayleigh number for the micropolar model is larger then that for the Navier–Stokes one, and large micropolar viscosity may even prevent the averaged heat transport in the upward vertical direction. The estimate of the fractal dimension of the global attractor for the considered problem is better then that for the same problem for the Navier–Stokes system. Besides, we obtain also other results which relate the heat convection in the micropolar fluids to the Newtonian ones, among them, relations between the Nusselt number and the energy dissipation rate and upper semicontinuous convergence of global attractors to the Rayleigh–Bénard attractor for the considered model. Keywords: Rayleigh–Bénard problem, Boussinesq system, convective turbulence, micropolar fluid, attractor 2010 MSC: 76F35, 76E15, 37L30, 35Q30, 35Q79 1. Introduction and main results In this paper we consider the Rayleigh–Bénard problem for the two-dimensional Boussinesq approximation system for the micropolar fluid. The micropolar model is both a simple and significant generalization of the Navier–Stokes model of classical hydrodynamics. It has much more applications than the classical model due to the fact that the latter cannot describe (by definition) fluids with microstructure. In general, individual particles of such complex fluids (e.g., polymeric suspensions, blood, liquid crystals) may be of different shape, may shrink and expand, or change their shape, and moreover, they may rotate, independently of the rotation and movement of the fluid. To describe accurately the behavior of fluids with microstructure one needs a theory that takes into account geometry, deformation, Email addresses: [email protected] (Piotr Kalita), [email protected] (José A. Langa), [email protected] (Grzegorz Łukaszewicza ) a Corresponding author Preprint submitted to Physica D: Nonlinear Phenomena

November 26, 2018

and intrinsic motion of individual material particles. To account for these local structural aspects, many classical concepts such as the symmetry of the stress tensor or absence of couple stresses have required a reexamination, and while many principles of classical continuum mechanics still remain valid for this new class of fluids, they had to be augmented with additional balance laws and constitutive relations. In the framework of continuum mechanics several such theories have appeared, e.g., theories of simple microfluids, simple deformable directed fluids, micropolar fluids, dipolar fluids, to name some of them. Some of these theories are very general, while others are concerned with special types of material structure and/or deformation; the potential applicability of the various theories is diverse. It is clear that each particular theory has its advantages as well as disadvantages when considered from a particular point of view. Some theories may seem more sound, logical, justifiable, and useful than others. No one of them is universal. One of the bestestablished theories of fluids with microstructure is just the theory of micropolar fluids of A.C. Eringen [1]. Physically, micropolar fluids may represent fluids consisting of rigid, randomly oriented (or spherical) particles suspended in a viscous medium, where the deformation of the particles is ignored. This constitutes a substantial generalization of the Navier–Stokes model and opens a new field of potential applications including a large number of complex fluids. Let us point out some general features that make the model a favorite in both theoretical studies and applications. The attractiveness and power of the model of micropolar fluids come from the fact that it is both a significant and a simple generalization of the classical Navier–Stokes model. Only one new vector field, the angular velocity field of rotation of particles, denoted here by γ, is introduced. Correspondingly, only one (vector) equation is added—it represents the conservation of the angular momentum. While four new viscosities are introduced, if one of them, namely the microrotation viscosity, becomes zero, the conservation law of the linear momentum becomes independent of the presence of the microstructure. Thus, the size of the microrotation viscosity coefficient allows us to measure, in a certain sense, the deviation of flows of micropolar fluids from that of the Navier–Stokes model. Thanks to the simplicity of the model of micropolar fluids, in many classical applications (e.g., flows through the channel or between parallel plates) and under usual geometrical and dynamical assumptions made in such cases (e.g., symmetry, linearization of the equations), equations of micropolar fluids reduce to ones that can be explicitly (i.e., analytically) solved. Thus the solutions obtained, depending on several parameters coming from the viscosity coefficients, can be easily compared with solutions of the corresponding problems for the Navier–Stokes equations. In addition, as several experiments show, the former solutions better represent behavior of numerous real fluids (e.g., blood) than corresponding solutions of the classical model, especially when the characteristic dimensions of the flow (e.g., the diameter of the channel) become small. It well agrees with our expectations that the influence of the internal structure of the fluid is the greater, the smaller the characteristic dimension of the flow.

2

In non-dimensional variables the system of equations considered in this paper reads [1, 2, 3] 1 1 1 N (ut + (u · ∇)u) − ∆u + ∇p = 2 rot γ + e2 RaT Pr 1−N Pr 1−N div u = 0 M N N (γt + u · ∇γ) − L∆γ + 4 γ=2 rot u Pr 1−N 1−N Tt + u · ∇T − ∆T = 0 where Pr and Ra are the classical Prandtl and Rayleigh numbers, and L, M , and N are micropolar parameters. The most important of them is N = νr /(ν + νr ) which relates the micropolar model to the Navier–Stokes one. The effects of micropolarity depend crucially on the numerical value of parameter N , linking the micropolar viscosity νr with the usual kinematic viscosity ν. For N = 0 there are no effects of micropolarity on the dynamics of the flow (it holds also if the microrotation field is given by γ = rot u/2), and the above system (the first, second, and fourth equation) reduces to the usual Boussinesq system for classical hydrodynamics. Precise mathematical analysis, including the existence (and, in two-dimensional case, uniqueness) of weak solutions, regularizing effect of the associated semiflow and existence and finite dimensionality of global attractors for two-dimensional problems have been established in [2, 3, 4, 5]. Our motivation to undertake this research was manifold. First, the importance of the problem of heat convection in both theoretical studies of turbulence and practical studies in numerous applications in engineering, geophysics, climate modeling, biological sciences and many others (see [6, 7, 8, 9] and references therein). Second, the question of the validity of the micropolar model in this context, that could be confirmed or disconfirmed by purely mathematical considerations resulting in providing the estimates of the main characteristic nondimensional numbers of the theory. This is related to the problem of the understanding not only, say, the influence of the microrotation field on the flow dynamics but also of testifying the usefulness of the used mathematical tools (eg. the fractal dimension estimates) in fluid dynamics. Our intuition tell us that less turbulent flows should result in better fractal dimension estimates (of the global attractor) then more turbulent flows. It is long known by physicists (in most part by using some formal and not always precise arguments as well as from experiments) that the micropolar fluid flows are usually more stable then Navier–Stokes flows [10, 11, 12, 13, 14, 15, 16, 17]. Indeed, in the micropolar situation there is an extra dissipation, when compared to the Newtonian fluid, which corresponds to the friction between the rotating particles. It is a natural question to try to confirm this stabilizing effect for the considered important Rayleigh–Bénard problem by using the precise mathematical argument, and this has not yet been established. Thus, our main goal is to compare the estimates of the Nusselt number, critical Rayleigh number, and the fractal dimension of the global attractor for the micropolar Rayleigh–Bénard problem with that for the same problem for the classical Navier–Stokes system. Our results confirm the physical intuitions. Our estimates reveal the stabilizing effects of micropolarity in comparison with the homogeneous Navier–Stokes fluid. In particular, the critical Rayleigh number for the micropolar model is larger then that for the Navier–Stokes one, and large micropolar viscosity may even prevent the averaged heat transport in the upward vertical direction (the Nusselt number equals one). The estimate of the fractal dimension of the global attractor for the micropolar Rayleigh–Bénard problem is better then that for the same problem for the Navier–Stokes system. 3

Moreover, we obtain relations between the Nusselt number and the energy dissipation rate for the micropolar Rayleigh–Bénard problem. All estimates reduce to the known ones for the classical Rayleigh–Bénard problem when there are no micropolar effects. The plan of the paper is as follows. In Section 2 we formulate the micropolar Rayleigh–Bénard problem and state it in the nondimensional variables. The problem of existence and uniqueness of global in time solutions as well as existence of the global attractor for the problem have been already established, see [4], and we recall these results in this section. In Section 3 we provide the nonlinear stability analysis, estimate the critical Rayleigh number, and compare it with that for the classical Rayleigh–Bénard problem. In Section 4 we estimate the Nusselt number in terms of the Rayleigh number and micropolar parameters L and N , and analyze its dependence on these parameters. Section 5 is devoted to establishing relations between the Nusselt number and the energy dissipation rate parameter. The last one plays the role in the estimate of the attractor dimension. In Section 6 we estimate from above the (fractal) dimension of the global attractor for our problem using the method of Lyapunov exponents together with the estimate of the Nusselt number (instead of the maximum principle for temperature), which allows us to obtain better attractor dimension estimates. In particular, for the classical Boussinesq model we improve the attractor dimension estimate obtained in [18]. In Section 7 we prove the convergence of trajectories of the micropolar Rayleigh–Bénard problem to that of the classical Rayleigh–Bénard problem and, in consequence, we prove the upper-semicontinuous convergence of the corresponding attractors, when N → 0. In the last section we summarize our main results and outline some questions which may be a starting point for the continuation of this work. 2. Problem formulation, scalings and preliminaries. Problem formulation. Let Ω = (0, Lx ) × (0, h) ⊂ R2 . The boundary of Ω is divided into three parts ∂Ω = ΓB ∪ ΓT ∪ ΓL , where ΓB = (0, Lx ) × {0} is the bottom, ΓT = (0, Lx ) × {h} is the top, and ΓL = {0, Lx } × (0, h) is the lateral boundary. We use the notation   ∂γ ∂u2 ∂u1 ∂γ rot γ = ,− and rot u = − , ∂x2 ∂x1 ∂x1 ∂x2 for a scalar function γ defined on Ω, and a vector function u defined on Ω with values in R2 , respectively. We consider the following Rayleigh–Bénard problem for micropolar fluid, where (x, t) ∈ Ω × (0, ∞). ut + (u · ∇)u − (ν + νr )∆u + %−1 0 ∇p = 2νr rot γ + e2 αgT, div u = 0, j(γt + u · ∇γ) − α∆γ + 4νr γ = 2νr rot u, Tt + u · ∇T − χ∆T = 0.

(2.1) (2.2) (2.3) (2.4)

The unknowns in the above equations are the velocity field u : Ω × (0, ∞) → R2 , the pressure p : Ω × (0, ∞) → R, the temperature T : Ω × (0, ∞) → R, and the microrotation field γ : Ω × (0, ∞) → R. We assume the boundary conditions u = 0 and γ = 0 on ΓB ∪ ΓT , T = 0 on ΓT , and T = TB on ΓB . On ΓL we impose the periodic conditions on the functions u, T, γ and their normal derivatives as well as the pressure p such that all boundary integrals on ΓL in 4

the weak formulation cancel. All physical constants ν, νr , %0 , α, g, j, α, χ, TB are assumed to be positive numbers, save for νr , where we allow for the case νr = 0, in which the micropolar model reduces to the Boussinesq one, e2 = (0, 1). Finally we impose the initial conditions u(x, 0) = u0 (x),

T (x, 0) = T0 (x),

γ(x, 0) = γ0 (x)

for

x ∈ Ω.

The problem domain, partition of its boundary, as well as the boundary conditions are depicted in Fig. 1

Figure 1: Domain of the problem and partition of its boundary.

The first scaling. To reduce the number of parameters in the models and make the variables independent on the measurement units we introduce the scaling of unknowns, corresponding to the one for Newtonian fluids of [9] and [19]. In the scaling we use the following dimensionless numbers. • Rayleigh number Ra =

αgTB h3 . νχ

• Prandtl number Pr = χν . • Grashof number Gr =

Ra Pr

=

αgTB h3 . ν2

νr • Microrotation ratio N = ν+ν . Clearly, as ν > 0, νr ≥ 0, we have 0 ≤ N < 1. For the case r N = 0 the equations decouple, and we get the classical Raleigh–Bénard problem and the additional equation for the microrotation. Sometimes, for the ease of notation, in place of N N we use the constant K = 1−N = ννr , see [20]. Note that always K ≥ 0. We will use the two constants K and N interchangeably.

• Micropolar damping L = • Micropolar inertia M =

α . h2 ν

j . h2

The scaled variables and unknowns are given as x = hx0 ,

t=

h2 0 t, χ

u=

χ 0 u, h

p= 5

%0 χ2 0 p, h2

T = TB T 0 ,

γ=

χ 0 γ. h2

(2.5)

As x = hx0 , the set Ω is scaled to Ω0 = (0, Lx /h) × (0, 1). Since from now on we will work only with the scaled equations and variables, for the notation convenience we will always drop the primes and use the same symbols for scaled variables as for the nonscaled ones. Note that the periodic conditions on ΓL for all variables are always transformed to the corresponding periodic conditions on the scaled lateral boundary. Similarly, the homogeneous Dirichlet conditions are also always transformed to the corresponding homogeneous condition. The only boundary condition which scales nontrivially is the condition T = TB on ΓB . Note that the scaled ΓB is given by (0, Lx /h) × {0}. We shall denote A = Lx /h. System (2.1)–(2.4) in dimensionless variables takes the following form. For (x, t) ∈ Ω × (0, ∞), 1 1 (ut + (u · ∇)u) − (1 + K)∆u + ∇p = 2Krot γ + e2 RaT, Pr Pr div u = 0, M (γt + u · ∇γ) − L∆γ + 4Kγ = 2Krot u, Pr Tt + u · ∇T − ∆T = 0.

(2.6) (2.7) (2.8) (2.9)

The boundary condition for the temperature on ΓB is now given as T (x1 , 0, t) = 1

for

x1 ∈ (0, A).

(2.10)

We will compare the critical Rayleigh number, as well as the upper estimates of the Nusselt number and the fractal dimension of the global attractor for the above system, with the corresponding ones for the Newtonian fluid, given, in dimensionless form, by 1 1 (ut + (u · ∇)u) − ∆u + ∇p = e2 RaT, Pr Pr div u = 0, Tt + u · ∇T − ∆T = 0.

(2.11) (2.12) (2.13)

Background temperature. To make the boundary condition (2.10) homogeneous we will introduce the background temperature, which is a function τ : [0, 1] → R of the variable x2 , such that τ (1) = 0 and τ (0) = 1. We will replace the temperature with a new unknown θ such that T (x1 , x2 , t) = θ(x1 , x2 , t) + τ (x2 ), and solve the problem for θ. Such translation will make the boundary conditions for θ homogeneous both on the top and bottom boundary, i.e. θ(x1 , 1, t) = θ(x1 , 0, t) = 0 for

x1 ∈ (0, A).

After introducing the background temperature, system (2.6)–(2.9) takes the form 1 1 (ut + (u · ∇)u) − (1 + K)∆u + ∇¯ p = 2Krot γ + e2 Raθ, Pr Pr div u = 0, M (γt + u · ∇γ) − L∆γ + 4Kγ = 2Krot u, Pr θt + u · ∇θ + u2 τ 0 − ∆θ − τ 00 = 0, 6

(2.14) (2.15) (2.16) (2.17)

Rx where the pressure p has been replaced by p¯ = p + PrRa 0 2 τ (r) dr. In the sequel we will denote the translated pressure by the same letter p. Various choices of τ are possible, the simplest choice is τ (x2 ) = 1 − x2 , and with such choice system (2.14)–(2.17) takes the form 1 1 (ut + (u · ∇)u) − (1 + K)∆u + ∇p = 2Krot γ + e2 Raθ, Pr Pr div u = 0, M (γt + u · ∇γ) − L∆γ + 4Kγ = 2Krot u, Pr θt + u · ∇θ − ∆θ = u2 .

(2.18) (2.19) (2.20) (2.21)

The second scaling. In Sections 3 and 6 we will work with another rescaled version of system (2.18)–(2.21). We apply to the variables of the system (2.18)–(2.21) the following rescaling of time t, velocity u and microrotation γ r √ 1 RaPr t = t¯√ . (2.22) , u = u¯ RaPr, and γ = γ¯ M RaPr Using this rescaling (where we drop the bars), system (2.18)–(2.21) takes the form r r Pr 1 1 Pr ut + (u · ∇)u − (1 + K) ∆u + ∇p = 2K √ rot γ + e2 θ, Ra PrRa M Ra div u = 0, r r r L Pr 1 Pr 1 Pr γt + u · ∇γ − ∆γ + 4K γ = 2K √ rot u, M Ra M Ra M Ra 1 θt + u · ∇θ − √ ∆θ = u2 . RaPr

(2.23) (2.24) (2.25) (2.26)

Changed scaling is convenient for the nonlinear stability analysis of Section 3 as in this scaling the square root of the critical Rayleigh number is the first eigenvalue of the generalized eigenvalue problem obtained from the linearization of system (2.23)–(2.26). This is possible due to the fact that the√ constants in front of e2 θ in (2.23) and in front of u2 in (2.26) are both equal to one, and divisor Ra appears in all remaining linear terms save for gradient of pressure. The scaling will be also useful in the estimate of the global attractor fractal dimension in Section 6. Note that the composition of scalings (2.5) and (2.22) is consistent with the scaling of [18]. Function spaces. We have to introduce some notations. By (·, ·) we will denote the scalar product in L2 (Ω) or, depending on the context, in L2 (Ω)2 . Let Ve be the space of divergence-free functions which are restrictions to Ω of functions from C ∞ (R×[0, 1])2 which are equal to zero on R×{0, 1}, and, together with all their derivatives, A-periodic with respect to the first variable. Similarly, by f we define the space of all functions which are restrictions to Ω of functions from C ∞ (R × W [0, 1]) which are A-periodic with all derivatives with respect to the first variable and satisfying the homogeneous Dirichlet boundary conditions on bottom and top boundaries R × {0, 1}. Define the spaces V = {closure of Ve in H 1 (Ω)2 } and H = {closure of Ve in L2 (Ω)2 }, 7

f in H 1 (Ω) } W = {closure of W

and

f in L2 (Ω) }. E = {closure of W

Moreover, let P : L2 (Ω)2 → H be the Leray–Helmholz projection. The duality pairings between V and its dual V 0 as well as between W and its dual W 0 will be denoted by h·, ·i.

First Stokes and Laplacian eigenvalue. By λ1 we will always denote the first eigenvalue of ˜ 1 the first eigenvalue of the Stokes operator with the considered the scalar operator −∆ and by λ ˜ 1 = π 2 . The first eigenvalue of the Laplace boundary conditions. We will verify that λ1 = λ operator on W has the following characterization given by the Rayleigh quotient, see [21, Theorem 2, Section 6.5.1] k∇uk2L2 . λ1 = min u∈W,u6=0 kuk2 2 L We prove the following lemma. Lemma 2.1. For every u ∈ W there holds

π 2 kuk2L2 ≤ k∇uk2L2 ,

(2.27)

Moreover, taking u0 (x1 , x2 ) = sin(πx2 ), we obtain In consequence λ1 = π 2 .

u0 ∈ W

and

π 2 ku0 k2L2 = k∇u0 k2L2 .

(2.28)

f . Using the very well known characterization of the first Dirichlet Proof. Let us take u ∈ W eigenvalue of −uxx on interval [0, 1], cf., e.g., [22, Example 3.2.1] we obtain that for every x1 ∈ [0, A] here holds Z 1 Z 1 ∂u(x1 , x2 ) 2 2 2 |u(x1 , x2 )| dx2 ≤ π ∂x2 dx2 . 0 0 Integrating over (0, A) with respect to x1 we obtain

∂u 2 2 2

≤ k∇uk2 2 .

π kukL2 ≤ L ∂x2 L2 f in W , while (2.28) follows by the direct verification The assertion (2.27) follows from density of W 0 that u ∈ W and equality in (2.27) holds for u0 . The Rayleigh quotient characterization for the first Stokes eigenvalue has the form 2 e1 = min k∇ukL2 . λ u∈V,u6=0 kuk2 2 L

We prove the next result

Lemma 2.2. For every u ∈ V there holds

π 2 kuk2L2 ≤ k∇uk2L2 ,

(2.29)

Moreover, taking u0 (x1 , x2 ) = (u01 (x1 , x2 ), u02 (x1 , x2 )) = (sin(πx2 ), 0), we obtain e1 = π 2 . In consequence λ

u0 ∈ V

and

π 2 ku0 k2L2 = k∇u0 k2L2 .

8

(2.30)

Proof. Verification of (2.30) is, as in Lemma 2.1, straightforward. To get (2.29) take u = (u1 , u2 ) ∈ Ve . Again from [22, Example 3.2.1], for every x1 ∈ [0, A] we obtain Z 1 Z 1 ∂u1 (x1 , x2 ) 2 2 2 dx2 , |u1 (x1 , x2 )| dx2 ≤ π ∂x 2 0 0 Z 1 Z 1 ∂u2 (x1 , x2 ) 2 2 2 dx2 . π |u2 (x1 , x2 )| dx2 ≤ ∂x2 0 0 After adding the two inequalities and integrating with respect to x1 over interval (0, A) we obtain



∂u1 2

∂u2 2 2 2 2

π kukL2 ≤

∂x2 2 + ∂x2 2 ≤ k∇ukL2 . L L The assertion follows by density of Ve in V .

Note that all eigenfunctions and eigenvalues of the Stokes problem on parallelepipeds with the boundary data similar to ours are calculated in [23]. The method of [23] also works for a simpler two-dimensional case of parallelograms. Definition and existence of the weak solution. Existence of attractors. The weak form of (2.18)–(2.21) is obtained in a standard way. We assume that u0 ∈ H,

γ0 ∈ E,

and θ0 ∈ E,

and we look for u ∈ L2loc (0, ∞; V ), γ ∈ L2loc (0, ∞; W ), and θ ∈ L2loc (0, ∞; W ) with ut ∈ L2loc (0, ∞; V 0 ), γt ∈ L2loc (0, ∞; W 0 ), and θt ∈ L2loc (0, ∞; W 0 ) such that u(0) = u0 , γ(0) = γ0 , θ(0) = θ0 , and for all test functions v ∈ V , ξ ∈ W , η ∈ W and a.e. t > 0 there holds 1 (hut (t), vi + ((u(t) · ∇)u(t), v)) + (1 + K)(∇u(t), ∇v) = 2K(rot γ(t), v) + Ra(θ(t), v2 ), Pr (2.31) M (hγt (t), ξi + (u(t) · ∇γ(t), ξ)) + L(∇γ(t), ∇ξ) + 4K(γ(t), ξ) = 2K(rot u(t), ξ), (2.32) Pr hθt (t), ηi + (u(t) · ∇θ(t), η) + (∇θ(t), ∇η) = (u2 (t), η). (2.33) Existence, uniqueness, and regularity of weak solutions for the above problem has been established by Tarasi´nska [4, Theorem 3.1], also see [5], in the following theorem Theorem 2.3. The problem of heat convection in micropolar fluid has a unique weak solution. This solution has the regularity u ∈ L2loc (η, ∞; H 2 (Ω)2 ),

γ ∈ L2loc (η, ∞; H 2 (Ω)),

and

θ ∈ L2loc (η, ∞; H 2 (Ω)).

Moreover the mapping S(t)(u0 , γ0 , θ0 ) = (u(t), γ(t), θ(t)) is continuous as a map from H ×E ×E into itself and the family of mappings {S(t)}t≥0 is a semigroup.

9

Hausdorff semidistance between two sets A and B in a Banach space X is denoted by distX (A, B) = sup inf ka − bkX . a∈A b∈B

A global attractor for a semigroup of mappings {S(t)}t≥0 leading from X to itself is a set A ⊂ X which is compact in X, invariant, i.e. S(t)A = A for every t ≥ 0, and attracting, i.e. limt→∞ distX (S(t)B, A) = 0 for every bounded B ⊂ X. Tarasi´nska has proved, see [4, Section 4 and 5] the following result Theorem 2.4. The solution semigroup {S(t)}t≥0 for a heat convection problem in a micropolar fluid has a global attractor A which is compact in H × E × E and bounded in V × W × W and has finite fractal (Hausdorff and upper box counting) dimension. Remark 2.5. In Section 7 we will consider the upper semicontinuous dependence of the global attractor on the parameter K (assuming that all other parameters are fixed). Hence we denote the attractors by AK for K ≥ 0. Note that the projection of A0 on the variables (u, θ) is the extensively studied Rayleigh–Bénard attractor [6, 7, 8, 9, 24]. Moreover, using the standard bootstrapping argument it is possible to prove that the weak solutions for the considered problem become smooth instantaneously, and the attractor contains in fact the smooth functions, see [25] for the argument in the case of the more general model of thermomicropolar fluids. Maximum principle for temperature. We conclude this section with the auxiliary result which will be used several times in the rest of the article. Multiplying (2.9) by (T − 1)+ and T − we get, respectively 1d k(T − 1)+ (t)k2L2 + k∇(T − 1)+ (t)k2L2 = 0, 2 dt 1d − kT (t)k2L2 + k∇T − (t)k2L2 = 0, 2 dt By the Poincaré inequality it follows d k(T − 1)+ (t)k2L2 + 2λ1 k(T − 1)+ (t)k2L2 ≤ 0, dt d − kT (t)k2L2 + 2λ1 kT − (t)k2L2 ≤ 0. dt The Gronwall lemma implies that for all t k(T − 1)+ (t)kL2 ≤ k(T − 1)+ (0)kL2 e−λ1 t . kT − (t)kL2 ≤ kT − (0)kL2 e−λ1 t .

We can decompose T (t) = T1 (t) + T2 (t), where T1 (t) = (T − 1)+ (t) − T − (t), whereas T2 (t) has values in [0, 1] for a.e. x ∈ Ω and T1 decays exponentially to zero according to the formula kT1 (t)kL2 ≤ (k(T − 1)+ (0)kL2 + kT − (0)kL2 )e−λ1 t . We deduce the following lemma

10

Lemma 2.6. If (u, γ, T ) is a solution to the system (2.6)–(2.9) with the considered boundary and initial data, then √ kT (t)kL2 ≤ A + 2kT0 kL2 e−λ1 t ,

where A = Lx /h·Ly /h. If the temperature T is translated by the background temperature τ (x2 ) = 1 − x2 according to the formula T (x1 , x2 , t) = θ(x1 , x2 , t) + τ (x2 ), then √ √ √ kθ(t)kL2 ≤ 2 A + 2kT0 kL2 e−λ1 t ≤ 2 A + 2(kθ0 kL2 + A)e−λ1 t . (2.34) In particular kT (t)kL2 and kθ(t)kL2 are bounded uniformly in time by the quantity which depends only on the initial data and A. 3. Nonlinear stability analysis and critical Rayleigh number In this section we will calculate the critical Rayleigh number using the simple nonlinear stability analysis as in [9]. Multiplying (2.23) by u, (2.25) by γ, and (2.25) by θ, and adding the three resultant equations to each other we get the following energy equation r

2  γ(t) Pr 1d 2 2 2

rot u(t) − 2 √ ku(t)kL2 + kγ(t)kL2 + kθ(t)kL2 + K 2 dt Ra M L2 r r Pr L Pr 1 k∇u(t)k2L2 + k∇γ(t)k2L2 + √ + k∇θ(t)k2L2 = 2(θ(t), u2 (t)). Ra M Ra RaPr (3.1) If we can prove that r r r

2

γ Pr Pr L Pr 2

rot u − 2 √ + K k∇ukL2 + k∇γk2L2

Ra Ra M Ra M L2 1 k∇θk2L2 − 2(θ, u2 ) ≥ C(kuk2L2 + kγk2L2 + kθk2L2 ), +√ RaPr for any (u, γ, θ) ∈ V × W × W with a constant C > 0, then any trajectory converges to 0 in H × E × E and the attractor becomes trivial. We will find the characterization of the critical Rayleigh number RaC (N, L), i.e. the highest possible value such that if Ra < RaC (N, L), then the above estimate holds with a constant C > 0 and the global attractor A is guaranteed to consist only of zero. We will also prove that the critical Rayleigh number given by the same nonlinear stability analysis is higher for the micropolar fluid (i.e. if N > 0 or, equivalently, K > 0) then for the Newtonian fluid (when N = 0 or, equivalently, K = 0). We always assume√that L > 0 as we need this for the global attractor existence. Multiplying the above inequality by RaPr, we get

2

γ

+ Prk∇uk2 2 + L Prk∇γk2 2 √ KPr rot u − 2 L L

M M L2 √ + k∇θk2L2 − 2 RaPr(θ, u2 ) ≥ C 0 (kuk2L2 + kγk2L2 + kθk2L2 ). The constant C 0 is positive provided 0<

inf

φ∈V ×W ×W kφk=1

  √ a(φ) − Ra b(φ) , 11

(3.2)

where we denote φ = (u, γ, θ), kφk2 = kuk2L2 + kγk2L2 + kθk2L2 ,

2

γ

+ Prk∇uk2 2 + L Prk∇γk2 2 + k∇θk2 2 , √ a(φ) = KPr rot u − 2 L L L

M M L2 √ b(φ) = 2 Pr(θ, u2 ).

We have the following Lemma 3.1. Let RaC be such that p RaC = Then inf

φ∈V ×W ×W kφk=1

inf

φ∈V ×W ×W kφk=1





a(φ) − a(φ) −

√ √

a(φ) . b(φ)

(3.3)

 Ra b(φ) > 0 ⇔ Ra < RaC .

(3.4)

inf

φ∈V ×W ×W b(φ)>0

 Ra b(φ) < 0 ⇔ Ra > RaC .

(3.5)

p Proof. For φ ∈ V × W × W and b(φ) > 0 we substitute ψ = φ/ b(φ) and deduce that p a(ψ). RaC = inf ψ∈V ×W ×W b(ψ)=1

We use the direct method of calculus of variations and we construct a minimizing sequence, which must be bounded in V × W × W and hence relatively compact in H × E √× E. It follows, that there exists a minimizer ψ ∈ V ×W ×W such that b(ψ ) = 1 and a(ψ ) = RaC . Since ψ0 6= (0, 0, 0) 0 0 √ 0 we deduce that RaC > 0. First assume that 0 < Ra < RaC . We must show that the infimum in (3.4) is positive. The direct method implies that the infimum is realized by a minimizer φ0 ∈ V ×W ×W with kφ0 k = 1. Without loss of generality we may assume that b(φ0 ) ≥ 0, indeed, if b(φ0 ) < 0, we can decrease the value of the functional by taking −θ in place of θ. If b(φ0 ) = 0, then the minimum must be positive as φ0 6= 0 and hence a(φ0 ) > 0. If b(φ0 ) > 0 we estimate ! √ √   √ a(φ0 ) p Ra Ra a(φ0 ) − Ra b(φ0 ) = √ b(φ0 ) − RaC + 1 − √ a(φ0 ) b(φ0 ) RaC RaC ! √ Ra ≥ 1− √ a(φ0 ) > 0. RaC If RaC = Ra, the same argument implies that   √ inf a(φ) − Rab(φ) ≥ 0, φ∈V ×W ×W kφk=1

12

and substituting φ = ψ0 we deduce that inf

φ∈V ×W ×W kφk=1



a(φ) −



 Rab(φ) = 0.

If Ra > RaC we substitute in place of φ in the expression under inf in (3.4) the function ψ0 /kψ0 k, whence we get    √ √ √  1  1 p a(ψ0 ) − Ra b(ψ0 ) = inf a(φ) − Rab(φ) ≤ RaC − Ra < 0. φ∈V ×W ×W kψ0 k2 kψ0 k2 kφk=1

and the proof is complete. We elaborate on the characterization of RaC . We obtain

2

γ √ KPr rot u − 2

+ Prk∇uk2L2 + p M L2 √ RaC = inf (u,γ,θ)∈V ×W ×W 2 Pr(θ, u2 )

L Prk∇γk2L2 M

+ k∇θk2L2

.

(u2 ,θ)>0

We can rescale variables u = √1Pr u0 , θ = √1Pr θ0 , and γ = critical Rayleigh number has the characterization p RaC (K, L) =

inf

(u,γ,θ)∈V ×W ×W (u2 ,θ)>0

√ √M γ 0 . Pr

Using this new variables, the

K krot u − 2γk2L2 + k∇uk2L2 + Lk∇γk2L2 + k∇θk2L2 , 2(θ, u2 )

so it depends only on K, L. Performing the same analysis for the Newtonian fluid, we get q k∇uk2L2 + k∇θk2L2 EW T . RaN = inf C (u,θ)∈V ×W 2(θ, u2 ) (u2 ,θ)>0

We have the following Lemma 3.2. If K > 0 and L > 0, then EW T RaN < RaC (K, L). C

Proof. Both infima are realized by some minimizers and in both cases u 6= 0 and θ 6= 0 for a minimizer (it follows by the fact that we can equivalently take infima with the constraint (θ, u2 ) = 1 and the minimizing sequences are bounded). We take the minimizer from Newtonian case and substitute it to micropolar formula. Then both terms krot u − 2γkL2 and k∇γkL2 cannot be equal to zero at the same time, and the assertion follows. We can make the calculations more explicit in the following   KLπ 2 4 Theorem 3.3. If Ra < π 1 + Lπ2 +4K then every solution (u(t), γ(t), θ(t)) converges to (0, 0, 0)   2 exponentially in H × E × E norm. In particular π 4 1 + LπKLπ ≤ RaC (K, L). 2 +4K 13

Proof. We explicitly estimate, for φ = (u, γ, θ) ∈ V × W × W ,

2 √

γ L 2 2

a(φ) − Ra b(φ) = E(φ) = KPr

rot u − 2 √M 2 + Prk∇ukL2 + M Prk∇γkL2 L √ 2 + k∇θkL2 − 2 RaPr(θ, u2 ) 4 ≥ (1 + K) Prk∇uk2L2 − √ KPrkγkL2 k∇ukL2 M √ 4 L RaPr 2 2 2 + PrKkγkL2 + Prλ1 kγkL2 + k∇θkL2 − 2 q k∇θkL2 k∇ukL2 , M M e λ1 λ1

e1 = π 2 , cf. Lemmas 2.1 and 2.2. It follows that where λ1 = λ   Ra K Prk∇uk2L2 − E(φ) ≥ 1 + K − e  δλ1 λ1 Pr + (4K + Lλ1 − 4K) kγk2L2 + (1 − δ)k∇θk2L2 , M

where , δ > 0 are arbitrary. All three constants in front of k∇uk2L2 , kγk2L2 , and k∇θk2L2 must be Lλ1 +4K 1 positive. We choose δ = 1+δ and  = 4K(1+δ) with δ > 0. Then the constants in front of kγk2L2 , and k∇θk2L2 are positive. We need 1 + K − (1 + δ) It is sufficient if  e Ra < λ1 λ1 1 + K −

Ra 4K 2 − (1 + δ) > 0. e1 Lλ1 + 4K λ1 λ

4K 2 Lλ1 + 4K

and the assertion holds.



 e = λ1 λ1 1 +

KLλ1 Lλ1 + 4K



,

Note that we have also proved the following Theorem 3.4 which follows as a corollary of the above result. EW T Theorem 3.4. If RaN is the critical Rayleigh number for the Newtonian fluid, then for K > 0 C and L > 0 EW T RaN < RaC (K, L). C

Moreover if either Ra ≤ π 4 , or Ra > π 4 , K >

Ra−π 4 , π4

and L > L0 (K) =

4 π2

1

π4 1 −K Ra−π 4

, then

(u(t), γ(t), θ(t)) converge to zero exponentially in H ×E×E. In particular limK→∞ RaC (K, L) = L→∞ ∞.

We complete the section with the brief linear stability analysis. We will prove that if Ra > RaC (K, L), then the first eigenvalue of the linearized operator of the problem is negative and hence at RaC (K, L) there appears the onset of linear instability. Indeed, let us consider the eigenvalue 14

problem consisting in finding β ∈ R and functions u, γ, θ ∈ V × W × W such that for v ∈ V , ξ ∈ W , η ∈ W there holds r r 1 Pr Pr (∇u, ∇v) − 2K √ (rot γ, v) − (θ, v2 ) = β(u, v), (1 + K) Ra M Ra r r r L Pr 1 1 Pr Pr (∇γ, ∇ξ) − 4K (γ, ξ) − 2K √ (rot u, ξ) = β(γ, ξ), M Ra M Ra M Ra 1 √ (∇θ, ∇η) − (u2 , η) = β(θ, η). RaPr

every

(3.6) (3.7) (3.8)

The first eigenvalue β1 is given by the following infimum of the Rayleigh quotient, cf. [21, Section 6.5.1, Theorem 2], q q q Pr L Pr 1 Pr 2 2 2 √ k∇ukL2 + M Ra k∇γkL2 + RaPr k∇θkL2 + K Ra k∇u − 2 √γM k2 − 2(θ, u2 ) Ra β1 = inf . (u,γ,θ)∈V ×W ×W kuk2L2 + kγk2L2 + kθk2L2 Rearranging and denoting φ = (u, γ, θ) we obtain √   √ 1 1 a(φ) − Ra b(φ) √ β1 = √ inf = inf a(φ) − Ra b(φ) . kφk2 Ra Pr (u,γ,θ)∈V ×W ×W Ra Pr φ∈V ×W ×W kφk=1

Now, using assertion (3.5) of Lemma 3.1 we deduce the following result. Lemma 3.5. If Ra > RaC (K, L) then the equilibrium (u, γ, θ) = (0, 0, 0) becomes linearly unstable and hence at Ra = RaC (K, L) the steady state (u, γ, θ) = (0, 0, 0) undergoes the supercritical bifurcation from being nonlinearly stable at Ra < RaC (K, L) to being linearly unstable at Ra > RaC (K, L). Remark 3.6. We leave, as an open problem, the question of the nature of the equilibrium (u, γ, θ) = (0, 0, 0) at Ra = RaC (K, L). 4. Estimate on Nusselt number This section is devoted to the derivation of the upper bound of the Nusselt number, and its dependence on Ra and the micropolar parameters K and L. While there exist the numerical simulations which demonstrate that the Nusselt number decreases with increasing K, see, for instance [11, 12], the rigorous mathematical evidence of this fact was still missing. Throughout this section we use the notation h. . .it to denote time average Z 1 t hF (t)it = F (s) ds t 0 and the notation

1 hF (t)i = lim sup t→∞ t 15

Z

0

t

F (s) ds.

The Nusselt number in the dimensionless quantity which measures the averaged in time rate of the total heat transport to the conductive heat transport in the vertical direction. It measures how much heat is transported through motion of fluid and is defined as     convective heat transport convective heat transport + conductive heat transport = 1+ . Nu = conductive heat transport conductive heat transport For the moment, let us come back to the dimensional variables. Conductive heat transport is defined as Z Z ∂T conductive heat transport = − χe2 · ∇T dx = −χ dx. Ω Ω ∂x2 After a straightforward calculation we obtain Z Z Lx Z h ∂T ∂T −χ dx = −χ dx1 dx2 = TB Lx χ. 0 0 ∂x2 Ω ∂x2 The convective heat transfer is given by convective heat transport =

Z

u2 T dx.



We come back to the dimensionless variables according to the scaling (2.5). If we pass to those variables (denoted by primes) in the last quantity, we get Z Z Z χ 0 0 2 0 u2 T dx = u2 TB T h dx = TB χh T 0 u02 dx0 , Ω0 h Ω Ω0 whence

1 Nu = 1 + A

Z

Ω0

T 0 u02

0

dx



.

(4.1)

Remark 4.1. In Section 6 we will use the Nusselt number in the scaling (2.22) of the system (2.23)–(2.26). This scaled Nusselt number is given by Z  1 0 0 Nu = 1 + T u¯2 dx . A Ω0 √ Clearly, as u02 = u¯2 RaPr, the relation between Nu and Nu is as follows √ √ (Nu − 1)A = h(T 0 , u02 )i = RaPrh(T 0 , u¯2 )i = RaPr(Nu − 1)A. In the rest of this section we will always work with the dimensionless ”primed” variables (2.5) and we shall always drop the primes. For simplicity of notation we denote J(t) = (T (t), u2 (t)), so that Nu = 1 + A1 hJ(t)i. We shall estimate the Nusselt number from above using the background temperature method of Constantin and Doering (cf. [6, 7, 9]). We will work with the system (2.14)–(2.17) where we will choose the appropriate background temperature τ . Multiplying (2.14) by u(t) and integrating by parts we get 1 d ku(t)k2L2 + (1 + K)k∇u(t)k2L2 = 2K(rot γ(t), u(t)) + Ra(θ(t), u2 (t)). 2Pr dt 16

(4.2)

After multiplication of (2.16) by γ(t) and integration by parts we get M d kγ(t)k2L2 + Lk∇γ(t)k2L2 + 4Kkγ(t)k2L2 = 2K(rot u(t), γ(t)). 2Pr dt

(4.3)

Finally, if we multiply (2.17) by θ(t) and integrate by parts, we will get 1d kθ(t)k2L2 + (u2 (t), τ 0 θ(t)) + (∇(θ(t) + τ ), ∇θ(t)) = 0. 2 dt

(4.4)

The last equation can be equivalently rewritten as 1d kθ(t)k2L2 + (u2 (t), τ 0 T (t)) + k∇T (t)k2L2 − (θx2 (t) + τ 0 , τ 0 ) = 0, 2 dt where we have used the fact that (u2 , f ) = 0 for u ∈ V and sufficiently smooth f depending only on variable x2 . If, for the moment, we choose the simplest background temperature τ (x2 ) = 1−x2 , then the last equation has the form 1d kθ(t)k2L2 − (u2 (t), T (t)) + k∇T (t)k2L2 − A = 0. 2 dt Integrating the above equation from zero to t and dividing by t, we obtain

1 1 kθ(t)k2L2 + k∇T (t)k2L2 t = A + h(u2 (t), T (t))it + kθ0 k2L2 . t t

(4.5)

From the assertion (2.34) of Lemma 2.6 it follows that

1 kθ(t)k2L2 = 0. t→∞ t lim

Hence, passing with t to ∞ in (4.5), we get

k∇T (t)k2L2 = A + h(u2 (t), T (t))i .

(4.6)

We come back to the case where we can have an arbitrary background τ . From (4.4) we obtain Z 1 1d 1 1 1 0 2 2 kθ(t)kL2 + (u2 (t), τ θ(t)) + k∇T (t)kL2 − A |τ 0 (x2 )|2 dx2 + k∇θ(t)k2L2 = 0. (4.7) 2 dt 2 2 2 0 It is easy to prove the following relations, valid for u ∈ V and γ ∈ W , (rot u, γ) = (rot γ, u)

(4.8)

krot ukL2 = k∇ukL2 .

(4.9)

and Dividing (4.2) by Ra and using the fact that (T (t), u2 (t)) = (θ(t), u2 (t)), as well as (4.8) and (4.9), we get 1 d K 1 2K ku(t)k2L2 + krot u(t)k2L2 + k∇u(t)k2L2 = (rot u(t), γ(t)) + (θ(t), u2 (t)). 2PrRa dt Ra Ra Ra (4.10) 17

Dividing (4.3) by Ra gives M d L 4K 2K kγ(t)k2L2 + k∇γ(t)k2L2 + kγ(t)k2L2 = (rot u(t), γ(t)). 2PrRa dt Ra Ra Ra Adding the last two equations yields   K 1d 1 M 2 2 ku(t)kL2 + kγ(t)kL2 + krot u(t) − 2γ(t)k2L2 2 dt PrRa PrRa Ra 1 L + k∇u(t)k2L2 + k∇γ(t)k2L2 = (θ(t), u2 (t)). Ra Ra

(4.11)

(4.12)

Adding the above equation to (4.7), we get   1d 1 1 M 1 2 2 2 ku(t)kL2 + kγ(t)kL2 + kθ(t)kL2 + k∇T (t)k2L2 + k∇u(t)k2L2 2 dt PrRa PrRa 2 Ra 1 K L + (u2 (t), (τ 0 − 1)θ(t)) + k∇θ(t)k2L2 + krot u(t) − 2γ(t)k2L2 + k∇γ(t)k2L2 2 Ra Ra Z 1 1 |τ 0 (x2 )|2 dx2 . = A 2 0 Define H{u, γ, θ} =

1 1 L K k∇uk2L2 + (u2 , (τ 0 − 1)θ) + k∇θk2L2 + k∇γk2L2 + krot u − 2γk2L2 . Ra 2 Ra Ra

If only τ is chosen such that H{u, γ, θ} ≥ 0 for u ∈ V and γ, θ ∈ W , then   Z 1 d M 1 1 1 2 2 2 2 ku(t)kL2 + kγ(t)kL2 + kθ(t)kL2 + k∇T (t)kL2 ≤ A |τ 0 (x2 )|2 dx2 . dt PrRa PrRa 2 2 0 It follows that

Using (4.6) gives

k∇T (t)k2L2 ≤ A A + hJ(t)i ≤ A

or Nu ≤

Z

0

1

Z

0

Z

0

1

1

|τ 0 (x2 )|2 dx2 .

|τ 0 (x2 )|2 dx2 ,

|τ 0 (x2 )|2 dx2 ,

If we choose, following the lines of Section V of [7] (cf.  −1  1 + (1 − δ )x2 , τ (x2 ) = x2 ,   (1 − δ −1 )(x2 − 1),

where δ ≤ 1/2, it follows that

Nu ≤

2 − 3. δ

18

also [9], Section 10.3), 0 ≤ x2 ≤ δ, δ ≤ x2 ≤ 1 − δ, 1 − δ ≤ x2 ≤ 1, (4.13)

It remains to choose δ appropriately, such that H{u, γ, θ} ≥ 0. We have krot u − 2γk2L2 = k∇uk2L2 − 4(γ, rot u) + 4kγk2L2 ≥ k∇uk2 − 4kγkL2 k∇ukL2 + 4kγk2L2   2 2 2 ≥ k∇ukL2 (1 − 2ε) + kγkL2 4 − , ε with arbitrary  > 0. Using the Poincaré inequality λ1 kγk2L2 ≤ k∇γkL22 , which follows from Lemma 2.1, we deduce that the quantity H{u, γ, θ} is be estimated from below by 1 1 (1 + K(1 − 2)) k∇uk2L2 + (u2 , (τ 0 − 1)θ) + k∇θk2L2 Ra 2   1 1 + λ1 L + 2K 2 − kγk2L2 . Ra ε

H{u, γ, θ} ≥

The above estimate is valid for any ε > 0. We choose ε such that the term in front of kγk2L2 is equal to zero. It follows that we must choose ε=

2K . 4K + Lλ1

Then there holds H{u, γ, θ} ≥

C 1 k∇uk2L2 + (u2 , (τ 0 − 1)θ) + k∇θk2L2 Ra 2

with

LKλ1 . (4.14) 4K + Lλ1 The key observation is that with an appropriate choice of K and L, the second component in the above sum, and, in consequence the constant which multiplies k∇uk2L2 can be made arbitrarily large. Exactly as in (5.17) in [7] we have   δ c 1 2 2 0 |(u2 , (τ − 1)θ)| ≤ k∇ukL2 + k∇θkL2 with any c > 0. 4 4 c C = 1 + K(1 − 2) = 1 +

It follows that

C δc − Ra 16



k∇uk2L2

H{u, γ, θ} ≥



C δ2 − Ra 32

H{u, γ, θ} ≥ and, after choosing c = δ/2,



+





k∇uk2L2 .

The parenthesis in the last expression equals zero if we take √ √ 4 2 C δ = δ0 = √ . Ra Now we study two possibilities. 19

1 δ − 2 4c



k∇θk2L2 ,

(4.15)

The first possibility is that δ0 ≥ 1/2, i.e.



LKλ1 Ra ≤ 128 1 + 4K + Lλ1



(4.16)

.

Then we can take δ = 1/2 in (4.15) and obtain H{u, γ, θ} ≥ 0, whence (4.13) yields Nu ≤ 1, which means that if (4.16) holds, then we must have Nu = 1. In the second case the opposite inequality to (4.16) holds, i.e.   LKλ1 Ra > 128 1 + , (4.17) 4K + Lλ1 which means that Newtonian Rayleigh number is large enough, compared to K and L. Then we take δ = δ0 in (4.13), which gives √ √ 2 Ra √ Nu ≤ − 3, (4.18) 4 C where C is given by (4.14). Note that if K = 0 or L = 0, then C = 1, and we recover the estimate for the Newtonian fluids √ 2√ Ra − 3, (4.19) Nu ≤ 4 similar to the one obtained in [6, (10.3.33)] (also cf. [7, (5.21)]). In the general case (4.18) leads to the estimate √ √ s √ √ r 2 Ra 2 Ra KLλ1 N Lλ1 Nu ≤ 1− −3= 1− − 3. 4 4K + L(K + 1)λ1 4 4N + Lλ1 When L increases, the constant on the right-hand side is smaller and the estimate become better. It is also not hard to verify, that for a given L, the right-hand side of the above estimate is the decreasing function of N . Hence, when N grows from zero to one (or, equivalently, K grows from zero to infinity), the estimate becomes better. So both L and N have stabilizing effects. Note that if either N = 0 or L = 0 then the above bound reduces to the Newtonian one given by (4.19), and if both constants are nonzero, then the constant in the bound is always smaller that the one in the Newtonian case. Let us suppose that 128KLλ1 . (4.20) Ra − 128 ≤ 4K + Lλ1 Then, of course (4.16) holds and we have Nu = 1. If Ra ≤ 128 then Nu = 1 no matter what is the choice of L and K. If, in turn Ra > 128, then (4.20) is equivalent to 4 1 128 + ≤ Lλ1 K Ra − 128

This allows us to formulate the following theorem

Theorem 4.2. If Ra ≤ 128 then Nu = 1. Suppose that Ra > 128 is given. Then If K >

Ra 4 − 1 and L ≥ L0 (K, Ra) = 2 128 π 20

1 128 Ra−128



1 K

then Nu = 1.

Otherwise we have the bound √ √ s √ √ r 2 Ra 2 Ra KLλ1 N Lλ1 Nu ≤ 1− −3= 1− − 3. 4 4K + L(K + 1)λ1 4 4N + Lλ1 In particular, for the Newtonian fluid, if Ra > 128, we have the bound √ √ 2 Ra Nu ≤ − 3. 4 Obtained bound for Nu as a function of K and L for Ra = 10000 is depicted in Fig.2. Rayleigh number, below which we have Nu = 1, as a function of K and L is presented in Fig.3. Note that the formula (4.16) is very similar to the formula of Theorem 3.3 which provides the Rayleigh number below which the trajectories always converge to zero, which clearly also implies that Nu = 1. The difference between two formulas is that in (4.16) we have the constant 128 and in Theorem 3.3 we have the constant π 4 ≈97.41. This arises  the question  what is the asymptotic behavior of  LKπ 2 LKπ 2 4 trajectories if π 1 + 4K+Lπ2 ≤ Ra ≤ 128 1 + 4K+Lπ2 , when we know that Nu = 1, but we cannot deduce from Theorem 3.3 that the flow stabilizes to a pure conduction state.

Figure 2: Upper bound on Nusselt number as a function of N and L which follows from Theorem 4.2 for Ra = 10000. The corresponding bound for the Newtonian fluid (N = 0) is equal approximately to 32.36

5. Relations between Nusselt number and energy dissipation rate In this section we will derive the estimates which relate the Nusselt number with the energy dissipation rates. We start from the derivation of these relations in the ”primed” variables (2.5) of the system (2.6)–(2.9) (or the translated systems (2.14)–(2.17) or (2.18)–(2.21)). In the special case of N = 0 we will recover the known relation for Newtonian fluids. Then we will derive the estimates from above by the Nusselt number on the momentum, angular momentum, and thermal energy dissipation rates in the ”barred” variables (2.22) of the system (2.23)–(2.26). These relations will be useful in the estimates of the attractor fractal dimension in Section 6. 21

Figure 3: Rayleigh number below which the Nusselt number is guaranteed to be equal to one, as a function of N and L, according to Theorem 4.2.

Relations for the first scaling. Integrating the energy equation (4.12) in time from zero to t and dividing by t we obtain   M kγ0 k2L2 1

1 ku0 k2L2 1 2 2 + k∇u(t)kL2 + Lk∇γ(t)kL2 t ≤ h(θ(t), u2 (t))it + . Ra 2 PrRa t PrRa t

Note that the quantity (θ(t), u2 (t)) does not depend on the choice of the background τ . Indeed, in fact, we have (θ(t), u2 (t)) = (T (t), u2 ), as (u2 , f ) = 0 for u ∈ V and sufficiently smooth f depending only on the variable x2 . Passing with t to infinity we obtain

It follows that

1

k∇uk2L2 + Lk∇γk2L2 ≤ h(T, u2 )i = A(Nu − 1). Ra

k∇uk2L2 ≤ RaA(Nu − 1).

(5.1)

To derive the above inequality we have used (4.12), which we have derived by adding (4.11) and (4.10). If, in turn, we subtract (4.11) from (4.10), using the fact that krot ukL2 = k∇ukL2 , we obtain 1 d 1+K ku(t)k2L2 + k∇u(t)k2L2 2PrRa dt Ra M d L 4K = kγ(t)k2L2 + k∇γ(t)k2L2 + kγ(t)k2L2 + (θ(t), u2 (t)). 2PrRa dt Ra Ra It follows that M d 1 d 1+K kγ(t)k2L2 + (θ(t), u2 (t)) ≤ ku(t)k2L2 + k∇u(t)k2L2 . 2PrRa dt 2PrRa dt Ra Integrating this inequality from zero to t and dividing by t we obtain h(θ(t), u2 (t))it ≤

1+K

M kγ0 k2L2 1 ku(t)k2L2 k∇u(t)k2L2 t + + . Ra 2PrRa t 2PrRa t 22

(5.2)

From (4.12) we deduce   1 1d 1 M L 2 2 ku(t)kL2 + kγ(t)kL2 + k∇u(t)k2L2 + k∇γ(t)k2L2 ≤ (θ(t), u2 (t)), 2 dt PrRa PrRa Ra Ra whence, by the Poincaré inequality   e1 λ 1d 1 M Lλ1 Ra 2 2 ku(t)kL2 + kγ(t)kL2 + ku(t)k2L2 + kγ(t)k2L2 ≤ kθ(t)k2L2 . e 2 dt PrRa PrRa 2Ra Ra 2λ1

Using the assertion (2.34) of Lemma 2.6 it follows that   e1 λ 1d 1 M Lλ1 2 2 ku(t)kL2 + kγ(t)kL2 + ku(t)k2L2 + kγ(t)k2L2 ≤ C, 2 dt PrRa PrRa 2Ra Ra

where C depends on the problem data and the initial temperature, and by the Gronwall lemma we obtain ku(t)kL2 ≤ C,

where C depends on the initial data and the constants present in the problem. Hence, we can pass with t to +∞ in (5.2) to obtain (keeping in mind that we can replace (θ(t), u2 (t)) with (T (t), u2 (t))), 1+K

A(Nu − 1) = h(T, u2 )i ≤ k∇uk2L2 . (5.3) Ra Summarizing, we get from (5.1) and (5.3) the following Theorem 5.1. For micropolar fluids there holds the following relation between the energy dissipation rate and the Nusselt number

Ra A(Nu − 1) ≤ k∇uk2L2 ≤ RaA(Nu − 1). 1+K

In particular if the fluid is Newtonian, then K = 0, and we recover the well known relation (cf, e.g., [9, Equation (10.3.9)])

k∇uk2L2 = RaA(Nu − 1).

Relations for the second scaling. In Section 6 we will need to estimate from above by the Nusselt number the rates of dissipation of momentum, angular momentum, and temperature. Since we will work with ”barred” variables (2.22), our starting point will be system (2.23)–(2.26). Multiplying (2.23) by u, (2.25) by γ, adding both equations, integrating the resultant equation in time (keeping in mind that (T (t), u2 (t)) = (θ(t), u2 (t))), and dividing by t we obtain r   ku0 k2L2 kγ0 k2L2 Pr L 2 2 k∇u(t)kL2 + k∇γ(t)kL2 ≤ h(T (t), u2 (t))it + + . Ra M 2t 2t t

Passing with time to infinity and using Remark 4.1 we get the following estimates for the energy dissipation rates r  r  L Ra Ra 1 k∇u(t)k2L2 + k∇γ(t)k2L2 ≤ h(T, u2 )i = A(Nu − 1) = A(Nu − 1). (5.4) M Pr Pr Pr 23

We derive the similar estimate for the dissipation of thermal energy. From (2.26) we obtain 1d 1 kθ(t)k2L2 + √ k∇θ(t)k2L2 = (u2 (t), θ(t)). 2 dt RaPr Hence



√ √ k∇θ(t)k2L2 ≤ RaPr h(T, u2 )i = RaPrA(Nu − 1) = A(Nu − 1).

(5.5)

(5.6)

6. Fractal dimension of attractors

The aim of this section is to calculate the upper bound of the fractal dimension as the function of the dimensionless constants of the model and refer it to the corresponding estimate of [18] for the Newtonian fluids. As a byproduct we improve the bound of [18] even in Newtonian case, as we use the previously derived bound on the Nusselt number instead of the maximum principle for temperature, see [18, Remark 6.4]. The technique to estimate the global attractor fractal (upper box-counting) dimension from above that we use here is based on the estimation of the Lyapunov exponents for the semigroup and has already been used for micropolar fluids in [4] to show that the said dimension of the attractor is in fact finite. It is known that the first zero of the sum of global Lyapunov exponents bounds from above the Hausdorff dimension, see, for instance, [26]. The fact that same bound holds for the fractal dimension is proved in [27] under the additional condition that the sum of the first n global Lyapunov exponents is majorized by a concave function. The upper bound of a global attractor fractal dimension is given by the first positive zero of this function. The proof of this fact for the general case, where it is required that the (quasi)differential is continuous on the attractor, is given in [28]. Conditions needed to apply both results of [27] and [28] are satisfied in the case treated in this paper. System (2.23)–(2.26) has the following weak form, for the test functions z ∈ V and ζ, ξ ∈ W , r r 2K Pr Pr (∇u, ∇z) = √ (rot γ, z) + (θ, z2 ), (6.1) hut , zi + ((u · ∇)u, z) + (1 + K) Ra M Ra r r r L Pr 4K Pr 2K Pr hγt , ζi + (u · ∇γ, ζ) + (∇γ, ∇ζ) + (γ, ζ) = √ (rot u, ζ), (6.2) M Ra M Ra M Ra 1 hθt , ξi + (u · ∇θ, ξ) + √ (∇θ, ∇ξ) = (u2 , ξ), (6.3) RaPr and the equations are assumed to hold for a.e. t > 0. The equations define a semigroup {S(t)}t≥0 of the mappings S(t) : H × E × E → H × E × E, which has a global attractor A, cf. Theorem 2.4. The algorithm to estimate the attractor fractal dimension is described, for example in [26, Section V]. In particular we use [26, Theorem 3.1, page 369], which needs that the semigroup is uniformly differentiable, i.e., that for any t > 0 and u ∈ A there exists the linear operator Lt (u) ∈ L(H × E × E) such that lim →0

sup u,v∈A 0
kS(t)u − S(t)v − Lt (u)(u − v)kH×E×E = 0, ku − vkH×E×E

24

and sup kLt (u)kL(H×E×E) < ∞. u∈A

Since uniform differentiability of the semigroup for the given problem has been verified in [4, Section 5.2], we only calculate the sum of first n Lyapunov exponents and derive from them the fractal dimension upper bound, keeping carefully track of the dependence of various estimates on the physical constants. To this end, we assume that (u(t), γ(t), θ(t)) for t ∈ R is the complete trajectory of the system (6.1)–(6.3) which lies inside its global attractor A and we linearize the system around this trajectory. The linearized system, the unknowns of which will be denoted by (w(t), ψ(t), η(t)), governs the behavior in time of the small perturbation of the data taken at t = 0 in the linear approximation. The linearized system takes the form r Pr (∇w, ∇z) hwt , zi + ((w · ∇)u, z) + ((u · ∇)w, z) + (1 + K) Ra r 2K Pr =√ (rot ψ, z) + (η, z2 ), (6.4) M Ra r r L Pr 4K Pr hψt , ζi + (w · ∇γ, ζ) + (u · ∇ψ, ζ) + (∇ψ, ∇ζ) + (ψ, ζ) M Ra M Ra r Pr 2K (rot w, ζ), (6.5) =√ M Ra 1 hηt , ξi + (w · ∇θ, ξ) + (u · ∇η, ξ) + √ (∇η, ∇ξ) = (w2 , ξ), (6.6) RaPr where (z, ζ, ξ) ∈ W × W × W is the test function and the equations hold for a.e. t > 0. The above system of three equations can be rewritten as (w(t), ψ(t), η(t))t + C(u(t), γ(t), θ(t))(w(t), ψ(t), η(t)) = 0, where C(u, γ, θ) is a linear operator. We assume that the triples of functions {(wj , ψj , ηj )}nj=1 belong to V × W × W and are orthonormal in H × E × E, i.e., in L2 induced scalar product, and we need to estimate from below the quantity n X (C(u(t), γ(t), θ(t))(wj , ψj , ηj ), (wj , ψj , ηj )). j=1

We calculate, taking into account that u is divergence free and some of the terms disappear (C(u(t), γ(t), θ(t))(wj , ψj , ηj ), (wj , ψj , ηj )) = ((wj · ∇)u(t), wj ) + (wj · ∇γ(t), ψj ) + (wj · ∇θ(t), ηj ) r r Pr L Pr 1 2 k∇wj kL2 + k∇ψj k2L2 + √ k∇ηj k2 − 2(wj2 , ηj ) + Ra M Ra RaPr r   Pr 4 4 2 2 +K k∇wj kL2 − √ (rot wj , ψj ) + kψj kL2 . Ra M M 25

Obviously, the orthonormality of (wj , ψj , ηj ) implies that 2(wj2 , ηj ) ≤ kwj k2L2 + kηj k2L2 ≤ 1. We deal with the last term. For any δ > 0 we obtain 4 4 4 √ (rot wj , ψj ) ≤ √ k∇wj kL2 kψj kL2 ≤ δk∇wj k2L2 + kψj k2L2 . δM M M We choose the value δ such that   1 1 Lλ1 + 4K 1 − = 0, 2 δ where λ1 = π 2 is the first eigenfunction of the scalar operator −∆ with the boundary conditions given in the definition of the problem, see Lemma 2.1. This holds, provided that δ=

8K . λ1 L + 8K

We get the bound (C(u(t), γ(t), θ(t))(wj , ψj , ηj ), (wj , ψj , ηj )) ≥ ((wj · ∇)u(t), wj ) + (wj · ∇γ(t), ψj ) + (wj · ∇θ(t), ηj ) r r   Pr 1 L Pr 1 Kλ1 L 2 + k∇wj kL2 1 + + k∇ψj k2L2 + √ k∇ηj k2 − 1. Ra λ1 L + 8K 2 M Ra RaPr For the ease of further calculations we introduce a constant D =1+

KLλ1 , Lλ1 + 8K

which allows us to write the above estimate in the simpler form (C(u(t), γ(t), θ(t))(wj , ψj , ηj ), (wj , ψj , ηj )) ≥ ((wj · ∇)u(t), wj ) + (wj · ∇γ(t), ψj ) + (wj · ∇θ(t), ηj ) r r Pr 1 L Pr 1 2 +D k∇wj kL2 + k∇ψj k2L2 + √ k∇ηj k2 − 1. Ra 2 M Ra RaPr The Lieb–Thirring inequality in our case takes the form, see [26], Z



ρ2 (x) dx =

Z



Pn

n X (|wi |2 + |ψi |2 + |ηi |2 ) i=1

!2

dx ≤ C

n X i=1

(k∇wi k2L2 + k∇ψi k2L2 + k∇ηi k2L2 ),

where ρ(x) = i=1 (|wi |2 + |ψi |2 + |ηi |2 ). Here and below the positive constant C changes from line to line and depends only on the shape of the domain and not its size, so it can depend on the rescaled length A of the rectangle (see [29, 30] for versions of the Lieb–Thirring inequality where 26

this dependence is stated explicitly). Since the sequences (wi ), (ψi ), (ηi ) are suborthonormal in L2 , cf. [31, 32], we also have !2 !2 Z Z n n n n X X X X 2 2 2 |wi | |ψi | dx ≤ C k∇wi kL2 , dx ≤ C k∇ψi k2L2 , Ω

i=1

Z



n X i=1

|ηi |2

!2



i=1

dx ≤ C

n X i=1

i=1

i=1

k∇ηi k2L2 ,

We estimate the trilinear terms n X j=1

Z

n X

((wj · ∇)u(t), wj ) ≥ −

Z



n X

|∇u|

1 (wj · ∇γ, ψj ) ≥ − |∇γ| |wj ||ψj | dx ≥ − 2 Ω j=1 j=1

Z



n X j=1

|wj |2 dx,

|∇γ|

n X j=1

|wj |2 +

n X j=1

|ψj |2

Z Z n n n n X X X X 1 2 |∇θ| (wj · ∇θ, ηj ) ≥ − |∇θ| |wj ||ηj | dx ≥ − |wj | + |ηj |2 2 Ω Ω j=1 j=1 j=1 j=1

!

!

dx,

dx.

We can use the Lieb–Thirring inequalities to get the estimates v r r uX n n X X u n D Pr Ra 1 2 2 k∇wj kL2 ≥ − k∇uk2L2 , ((wj · ∇)u, wj ) ≥ −k∇ukL2 t k∇wi kL2 − C 6 Ra Pr D j=1 j=1 i=1 n X j=1

v  v uX uX n n u u (wj · ∇γ, ψj ) ≥ −Ck∇γkL2 t k∇wj k2L2 + t k∇ψj k2L2  r

≥−

j=1

Pr Ra

n  X i=1

j=1

L D k∇wi k2L2 + k∇ψi k2L2 6 4M



−C

r

Ra Pr



 M 1 2 2 k∇γkL2 + k∇γkL2 , D L

 v v uX uX n n n X u u (wj · ∇θ, ηj ) ≥ −Ck∇θkL2 t k∇wj k2L2 + t k∇ηj k2L2  j=1

r

≥−

j=1

Pr Ra

n  X i=1

j=1

D 1 k∇wi k2L2 + k∇ηi k2L2 6 2Pr

27



−C

r

Ra Pr



 1 2 2 k∇θkL2 + Prk∇θkL2 . D

Combining this estimates we obtain n X

(C(u(t), γ(t), θ(t))(wj , ψj , ηj ), (wj , ψj , ηj ))

j=1

r

 n  Pr X L 1 2 2 2 Dk∇wj kL2 + k∇ψj kL2 + k∇ηj k − n Ra j=1 2M Pr r       Ra 1 1 2M 1 2 2 2 −C k∇u(t)kL2 + + k∇γ(t)kL2 + + Pr k∇θ(t)kL2 Pr D D L D r n X 1 Pr DL ≥ (k∇wj k2L2 + k∇ψj k2L2 + k∇ηj k2L2 ) − n 2 Ra DLPr + 2DM + L j=1 r       Ra 1 1 1 2M 2 2 2 −C k∇γ(t)kL2 + k∇u(t)kL2 + + + Pr k∇θ(t)kL2 . Pr D D L D R Now, as Ω ρ(x) dx = n, we get, by the Lieb–Thirring inequality 1 ≥ 2

n X n ≤ CA (k∇wi k2L2 + k∇ψi k2L2 + k∇ηi k2L2 ), 2

i=1

and hence

n X (C(u(t), γ(t), θ(t))(wj , ψj , ηj ), (wj , ψj , ηj )) j=1

C ≥ A

r

DL Pr n2 − n Ra DLPr + 2DM + L r       Ra 1 1 2M 1 2 2 2 −C k∇u(t)kL2 + + + Pr k∇θ(t)kL2 . k∇γ(t)kL2 + Pr D D L D

Following [18, 26], to use the method of Lyapunov exponents, we should estimate from above the expression * n + X − (C(u(t), γ(t), θ(t))(wj , ψj , ηj ), (wj , ψj , ηj )) . j=1

This estimate, dependent only on n and the problem data, is also an estimate of the sum of the first n Lyapunov exponents. We get * n + X − (C(u(t), γ(t), θ(t))(wj , ψj , ηj ), (wj , ψj , ηj )) j=1

r DL C Pr ≤− n2 + n Ar Ra DLPr + 2DM + L       Ra 1 1 2M 1 2 2 2 +C hk∇γkL2 i + hk∇ukL2 i + + + Pr hk∇θkL2 i . Pr D D L D 28

Using the energy dissipation estimates (5.4) and (5.6), we get r       Ra 1 1 2M 1 2 2 2 C hk∇ukL2 i + + hk∇γkL2 i + + Pr hk∇θkL2 i Pr D D L D   √ M 2M 2 1 1 + + + + 1 (Nu − 1) ≤ CA RaPr DPr2 DLPr2 L2 Pr2 DPr   √ 1 M2 ≤ CA RaPr + + 1 (Nu − 1), DPr2 L2 Pr2 where in the last estimate we have used the fact that D > 1. It follows that * n + X − (C(u(t), γ(t), θ(t))(wj , ψj , ηj ), (wj , ψj , ηj )) j=1

C ≤− A

r

√ Pr DL n2 + n + CA RaPr Ra DLPr + 2DM + L



 1 M2 + + 1 (Nu − 1). DPr2 L2 Pr2

The right-hand side of the above inequality is a concave function of n. Hence, by [27, Corollary 2.2] the attractor fractal dimension is bounded from above by the positive zero of the above expression, i.e., r   Ra M 1 Pr + + dF (A) ≤ CA Pr L D r r r √ √ 1 M2 M 1 1 4 4 Ra 2 + CA RaPr + 2 + Pr Nu − 1 Pr + + . Pr D L Pr L D It follows that r r √ r   √ M Ra M 1 Ra 1 M2 1 2 dF (A) ≤ C(A) Pr + + + C(A) + 2 + Pr Nu − 1 Pr + + , Pr L D Pr D L L D

where we write C(A) to stress that the constant C depends on the length A of the rectangle, without expressing this dependence explicitly We deduce r √    r Ra 1 Ra √ M 1 M 1 M dF (A) ≤ C(A) Pr + + + C(A) Nu − 1 Pr + +√ Pr + + . Pr L D Pr L L D D   KLπ 2 KLπ 2 Since D1 = 1 − 8K+(K+1)Lπ 1 − 8K+(K+1)Lπ 2 ≤ 2 2 , introducing the notation H(K, L) = 1 − we obtain the bound

KLπ 2 , 4K + (K + 1)Lπ 2



 M dF (A) ≤ C(A) Gr Pr + + H(K, L) L !  r √ Gr M p M +√ Pr + + H(K, L) Pr + + H(K, L) Nu − 1 . L L Ra √

Using the previously obtained bound for Nu, we get 29

Theorem 6.1. The following bound holds for the fractal dimension of the attractor for the RayleighBénard convection in micropolar fluid,   √ M Gr Pr + + H(K, L) dF (A) ≤ C(A) L !  r p Gr M p M +√ + H(K, L) + H(K, L) 4 H(K, L) , Pr + Pr + 4 L L Ra where H(K, L) = 1 −

KLπ 2 , 4K + (K + 1)Lπ 2

is a function decreasing in both variables. In particular, H(0, L) = H(K, 0) = 1

and

lim

K→∞,L→∞

H(K, L) = 0.

The value of the obtained upper bound in comparison with the one for the Newtonian case for Ra = Gr = 100000, and M/L = A = Pr = 1, assuming that C = 1, is presented in Fig.4, where in place of the constant K we use, equivalent to it, constant N , which belongs to [0, 1) and not [0, ∞).

Figure 4: Upper bound on the global attractor fractal dimension which follows from Theorem 6.1 as a function of N and L for C = 1, A = 1, Ra = 100000, Pr = 1 and M/L = 1. The corresponding bound for the case of Newtonian fluid for the same data is equal to, approximately 16537.87. Newtonian bound is lower that the maximum ”micropolar” bound in the plot due to influence of micropolar inertia term M/L. For large N and L, however, the bound is visibly lower than the corresponding Newtonian one.

The obtained result is discussed in several remarks. Remark 6.2. Clearly, increasing L (micropolar damping) and N (microrotation viscosity) makes the upper bound smaller, and increasing M (micropolar inertia) makes the bound higher. 30

Remark 6.3. If the Rayleigh number is less than the critical Rayleigh number (which depends on K and L), then A = {0}. This case holds, in particular, if the assumptions of Theorem 3.4 hold, 4 i.e. Ra ≤ π 4 , or Ra > π 4 , K > Ra−π , and L > L0 (K) = π42 π4 1 1 . Clearly, in such case π4 Ra−π 4

−K

dF (A) = 0. We deduce that the estimate obtained in Theorem 6.1 is not optimal, as we do not recover the fact that if K and L are sufficiently large, then the dimension should be equal to zero. Remark 6.4. Suppose, we do not know the Rayleigh number but we know the Prandtl number. L For a given Pr we can choose K, L, M large enough such that the Prandtl number dominates over terms with H(K, L) and M/L. Then the bound simplifies to √ 3 5p dF (A) ≤ C(A)( GrPr + Gr 4 Pr 4 4 H(K, L)),

and the influence of the second term can be made as small as we want by choosing appropriate K, √ L to make the expression H(K, L) arbitrarily small. So, the estimate is dominated by the term GrPr. Remark 6.5. If K = 0 and M = 0, i.e. there are no micropolar effects, we obtain the bound   √ 3 Gr 2 Gr (1 + Pr) + √ dF (A) ≤ C(A) (1 + Pr) . 4 Ra

This is an improvement over the estimate of Foias, Manley, and Temam [18], who obtained the bound  √ Gr (1 + Pr) + Gr (1 + Pr)2 . dF (A) ≤ C(A) √ Indeed, we can assume that Ra > π 4 , because otherwise we know that dF (A) = 0. Then 4 Ra > π and we get the bound √  3 Gr (1 + Pr) + Gr (1 + Pr) 2 . dF (A) ≤ C(A)

If we additionally assume that the Prandtl number satisfies Pr ≥ 1, we obtain the better bound given as follows √  3 5 GrPr + Gr 4 Pr 4 , dF (A) ≤ C(A) whereas Foias, Manley, and Temam [18], in analogous situation obtain the bound √  dF (A) ≤ C(A) GrPr + GrPr2 .

The improvement was possible due to the fact that we used the estimate on the Nusselt number instead of the maximum principle for the temperature. 7. Convergence of trajectories and upper-semicontinuous convergence of attractors In Section 2 we have reminded the result of Tarasi´nska [4] on the existence of global attractor for the heat convection problem in micropolar fluid. We fix parameters Ra, Pr, M, L and study the dependence of this global attractor on the parameter N (or, equivalently K = N/(1 − N )), so we denote this global attractor by AK . We will prove that the global attractors AK converge upper-semicontinuously to the global attractor of the limit problem when K = 0, i.e. we will prove the following 31

Theorem 7.1. Let Ra, Pr, M, L be fixed positive constants. We denote by AK the global attractor of the problem (2.31)–(2.33) for a given K ≥ 0. Then lim distH×E×E (AK , A0 ) = 0.

K→0+

The limit problem, for K = 0, in the weak form consists in finding u ∈ L2loc (0, ∞; V ), γ ∈ L2loc (0, ∞; W ), and θ ∈ L2loc (0, ∞; W ) with ut ∈ L2loc (0, ∞; V 0 ), γt ∈ L2loc (0, ∞; W 0 ), and θt ∈ L2loc (0, ∞; W 0 ), such that u(0) = u0 ∈ H, γ(0) = γ0 ∈ E, θ(0) = θ0 ∈ E and for all test functions v ∈ V , ξ ∈ W , η ∈ W and a.e. t > 0 there holds 1 (hut (t), vi + ((u(t) · ∇)u(t), v)) + (∇u(t), ∇v) = Ra(θ(t), v2 ), Pr M (hγt (t), ξi + (u(t) · ∇γ(t), ξ)) + L(∇γ(t), ∇ξ) = 0, Pr hθt (t), ηi + (u(t) · ∇θ(t), η) + (∇θ(t), ∇η) = (u2 (t), η).

(7.1) (7.2) (7.3)

Equations (7.1) and (7.3) constitute the classical Rayleigh–Bénard problem and can be solved only for u and θ, while (7.2) is the additional equation which can be solved for γ once we know u. It follows that the projection of the global attractor A0 on the variables u, θ is simply the Rayleigh– Bénard attractor. In general, the results on the upper-semicontinous convergence of the global attractors are standard and the techniques to prove them are well established, see for instance [33, 34, 26]. For micropolar fluids without heat convection the corresponding result was already obtained in [3]. We prove the upper-semicontinuous convergence result to make our comparison between the heat convection in micropolar and Newtonian fluid more complete. We will use the following theorem, [33, Theorem 2.5.2], Theorem 7.2. Let {SK (t)}t≥0 be a family of semigroups on the Banach space X parameterized by K ≥ 0. Assume that the following three assertions hold. (i) For every K ≥ 0 the semigroup {SK (t)}t≥0 has S a global attractor AK . (ii) There exists a bounded set B ⊂ X such that K∈[0,1] AK ⊂ B. (iii) For any  > 0 and t ≥ 0 there exists K0 (, t) < 1 such that for every K ∈ [0, K0 ] and for every x ∈ AK there holds kSK (t)x − S0 (t)xkX ≤ . Then lim+ distX (AK , A0 ) = 0. K→0

To prove Theorem 7.1 we should verify assertions (ii) and (iii) of the above theorem. We do this in the following two lemmas. Lemma 7.3. There exists a bounded set B ⊂ H × E × E such that [ AK ⊂ B. K∈[0,1]

In particular assertion (ii) of Theorem 7.2 holds.

32

√ ¯ ∈ AK . Lemma 2.6 implies that kθk ¯ L2 ≤ 2 A. As AK are invariant then Proof. Let (¯ u, γ¯ , θ) ¯ = (u(t), γ(t), θ(t)) with (u, γ, θ) for any t > 0 there exists (u0 , γ0 , θ0 ) ∈ AK such that (¯ u, γ¯ , θ) solving (2.31)–(2.33) with the initial data (u0 , γ0 , θ0 ). Testing (2.31) with u(t) and (2.32) with γ(t) and adding the two equations we get, after standard transformations M d 1 d ku(t)k2L2 + k∇u(t)k2L2 + kγ(t)k2L2 + Lk∇γ(t)k2L2 ≤ Rakθ(t)kL2 ku(t)kL2 . 2Pr dt 2Pr dt √ As kθ(t)kL2 ≤ 2 A for all t from the maximum principle (as the whole trajectory belongs to the global attractor), by the Poincaré inequality we obtain √ 1 1 M d 1 d ku(t)k2L2 + k∇u(t)k2L2 + λ1 ku(t)k2L2 + kγ(t)k2L2 +Lk∇γ(t)k2L2 ≤ Ra2 Aku(t)kL2 . 2Pr dt 2 2 2Pr dt We deduce M d 4ARa2 1 d ku(t)k2L2 + k∇u(t)k2L2 + kγ(t)k2L2 + 2Lk∇γ(t)k2L2 ≤ . Pr dt Pr dt λ1 The Poincaré and Gronwall inequalities imply that, for t large enough, ku(t)kL2 ≤ C,

kγ(t)kL2 ≤ C,

where C depends on Pr, Ra, A, M, L, but not on K. It also follows that Z t Z t 2 k∇u(s)kL2 ds ≤ C(1 + t), k∇u(s)k2L2 ds ≤ C(1 + t), 0

(7.4)

0

where, similarly C depends on Pr, Ra, A, M, L, but not on K. Testing (2.33) with θ(t) it is not hard to verify that Z t k∇θ(s)k2L2 ds ≤ C(1 + t), (7.5) 0

with C also depending only Pr, Ra, A, M, L, but not on K. The lemma is proved.

Lemma 7.4. Let K ∈ [0, 1] and let (u0 , γ0 , θ0 ) ∈ AK . Let (uK , γ K , θK ) be the trajectory (in AK ) of the problem (2.31)–(2.33) with the initial data (u0 , γ0 , θ0 ) and let (u0 , γ 0 , θ0 ) be the trajectory of (7.1)–(7.3) with the same data. Then there exists a constant C dependent on A, Pr, Ra, M, L, but not on K, such that for every t > 0 kuK (t) − u0 (t)k2L2 + kγ K (t) − γ 0 (t)k2L2 + kθK (t) − θ0 (t)k2L2 ≤ KC (1 + t) eC(1+t) . In particular the assertion (iii) of Theorem 7.2 holds. Proof. Let the initial data (u0 , γ0 , θ0 ) belong to AK and let (uK , γ K , θK ) be the trajectory (in AK ) of the problem (2.31)–(2.33) with this initial data and (u0 , γ 0 , θ0 ) be the trajectory of (7.1)–(7.3) with the same data. Subtracting the corresponding equations from each other and testing them with

33

the differences of both solutions we obtain 1 d K 1 ku (t) − u0 (t)k2L2 + ((uK (t) · ∇)uK (t) − (u0 (t) · ∇)u0 (t), uK (t) − u0 (t)) 2Pr dt Pr K + k∇u (t) − ∇u0 (t)k2L2 + K(∇uK (t), ∇uK (t) − ∇u0 (t))

0 = 2K(rot γ K (t), uK (t) − u0 (t)) + Ra(θK (t) − θ0 (t), uK 2 (t) − u2 (t)), M d K M kγ (t) − γ 0 (t)k2L2 + (uK (t) · ∇γ K (t) − u0 (t) · ∇γ 0 (t), γ K (t) − γ 0 (t)) 2Pr dt Pr K + Lk∇γ (t) − ∇γ 0 (t)k2L2 + 4K(γ K (t), γ K (t) − γ 0 (t)) = 2K(rot uK (t), γ K (t) − γ 0 (t)), 1d K kθ (t) − θ0 (t)k2L2 + (uK (t) · ∇θK (t) − u0 (t) · ∇θ0 (t), θK (t) − θ0 (t)) + k∇θK (t) − ∇θ0 (t)k2L2 2 dt K 0 0 = (uK 2 (t) − u2 (t), θ (t) − θ (t)).

Denoting v = uK − u0 , ξ = γ K − γ 0 , and η = θK − θ0 , we get 1 d 1 kv(t)k2L2 + ((v(t) · ∇)uK (t), v(t)) + k∇v(t)k2L2 + K(∇uK (t), ∇v(t)) 2Pr dt Pr = 2K(γ K (t), rot v(t)) + Ra(η(t), v2 (t)), M d M kξ(t)k2L2 + (v(t) · ∇ξ(t), γ K (t)) + Lk∇ξ(t)k2L2 + 4K(γ K (t), ξ(t)) = 2K(rot uK (t), ξ(t)), 2Pr dt Pr 1d kη(t)k2L2 + (v(t) · ∇η(t), θK (t)) + k∇η(t)k2L2 = (v2 (t), η(t)). 2 dt Using the Ladyzhenskaya and Schwartz inequalities we arrive at 1 d kv(t)k2L2 + k∇v(t)k2L2 ≤ 2Kkγ K (t)kL2 k∇v(t)kL2 + Rakη(t)kL2 kv(t)kL2 2Pr dt C + Kk∇uK (t)kL2 k∇v(t)kL2 + kv(t)kL2 k∇v(t)kL2 k∇uK (t)kL2 , Pr M d kξ(t)k2L2 + Lk∇ξ(t)k2L2 ≤ 2Kk∇uK (t)kL2 kξ(t)kL2 + 4Kkγ K (t)kL2 kξ(t)kL2 2Pr dt CM 1/2 1/2 1/2 1/2 kv(t)kL2 k∇v(t)kL2 k∇ξ(t)kL2 kγ K (t)kL2 k∇γ K (t)kL2 , + Pr 1d 2 kη(t)kL2 + k∇η(t)k2L2 2 dt 1/2 1/2 1/2 1/2 ≤ kv(t)kL2 kη(t)kL2 + Ckv(t)kL2 k∇v(t)kL2 k∇η(t)kL2 kθK (t)kL2 k∇θK (t)kL2 . We denote by C a generic constant independent of K. Using the fact that γ K (t) and θK (t) belong to the global attractor, so, by Lemma 7.3, after calculations which use the Cauchy and Poincaré inequalities, we get d kv(t)k2L2 + Prk∇v(t)k2L2 ≤ C(K 2 + kη(t)k2L2 + Kk∇uK (t)k2L2 + kv(t)k2L2 k∇uK (t)k2L2 ), dt d LPr kξ(t)k2L2 + k∇ξ(t)k2L2 ≤ C(Kk∇uK (t)k2L2 + K 2 + kv(t)kL2 k∇v(t)kL2 k∇γ K (t)kL2 ), dt M d 2 kη(t)kL2 + k∇η(t)k2L2 ≤ C(kv(t)k2L2 + kv(t)kL2 k∇v(t)kL2 k∇θK (t)kL2 ). dt 34

Adding all three above inequalities we obtain, after standard calculations, d (kv(t)k2L2 + kξ(t)k2L2 + kη(t)k2L2 ) ≤ CK(1 + k∇uK (t)k2L2 ) + Ckη(t)k2L2 dt + Ckv(t)k2L2 (1 + k∇uK (t)k2L2 + k∇γ K (t)k2L2 + k∇θK (t)k2L2 ). The Gronwall lemma implies that kv(t)k2L2 + kξ(t)k2L2 + kη(t)k2L2   Z t Rt K 2 K 2 K 2 K 2 k∇u (s)kL2 ds eC (t+ 0 k∇u (s)kL2 +k∇γ (s)kL2 +k∇θ (s)kL2 ds) . ≤ CK t + 0

Using (7.4) and (7.5), we obtain kv(t)k2L2 + kξ(t)k2L2 + kη(t)k2L2 ≤ KC (1 + t) eC(1+t) , and the proof of the Lemma is complete. This ends the proof of Theorem 7.1. 8. Summary of the main results. In this section we give a brief overview of the main results of the article. In Section 3 we compared the critical Rayleigh number for micropolar and Newtonian fluids. Using the nonlinear stability analysis we proved that if the actual Rayleigh number is below the critical Rayleigh number, then every trajectory (u(t), γ(t), T (t)) converges in H × E × E to the equilibrium in which no convection occurs, i.e. to the point (0, 0, 1 − x2 ). We have also proved that if Ra ≤ π 4 or if Ra > π 4 and K and L are large enough, than the Rayleigh number is below the critical one, and the above global asymptotic stability of pure conduction equilibrium holds. This means that if micropolar effects are strong enough than the pure conduction state is globally asymptotically stable. We proved that the critical Rayleigh number for micropolar fluids (for K, L > 0) is always greater then the critical Rayleigh number for the Newtonian fluid and that at this critical Rayleigh number the trivial equilibrium changes from nonlinearly stable to linearly unstable. We mention here that similar relation was established by using linear stability methods in [35], cf. also [13], where it was found also that no subcritical instability region exists and the critical Rayleigh number as derived from the energy method is identical to that of the linear limit. We observe, however, that most of the papers devoted to linear analysis of the problem considered the dynamically free boundary conditions which are easier to analyze. In Section 4 we obtained an upper bound on the averaged heat flux in the vertical direction (measured by the Nusselt number), using the background flow method. In particular we obtained the following estimate √ √ r √ √ s 2 Ra KLλ1 2 Ra N Lλ1 1− −3= 1− − 3. Nu ≤ 4 4K + L(K + 1)λ1 4 4N + Lλ1 Since the bound decreases with increasing N (or, equivalently, K) and L, it follows that the maximum possible amount of average heat exchange by convection (at least in terms of the upper bound) 35

decreases when micropolar effects increase. Since we always have 1 ≤ Nu, we can deduce from the above upper bound the values of N (sufficiently close to one) and L (large enough), depending on the Rayleigh number, for which the average convection is equal to zero. Clearly, the fact that micropolar effects of sufficient strength can make convection disappear follows from the nonlinear stability analysis by energy method, but the explicit bounds we obtained from the Nusselt number and the background flow method are a little better than the ones that followed from the energy method. In Section 5 we derived the following relation between the energy dissipation rate and the Nusselt number Z 1 t Ra A(Nu − 1) ≤ lim sup k∇u(s)k2L2 ds ≤ RaA(Nu − 1). 1+K t t→∞ 0 In case K = 0 the bound reduces to the known one for the Newtonian case. Section 6 was devoted to the upper bounds on the fractal dimension of the global attractor . Using the method of Lyapunov exponents we obtained an estimate showing that an increasing of the micropolar inertia M increases the bound and an increasing of the constants K and L decreases it. We related this estimate to the one known for the Newtonian fluid and we deduced that the micropolar effects have stabilizing influence in terms of the upper bound of the fractal dimension of the global attractor. Finally, in Section 7 we completed our analysis by showing that the projections of global attractors for the micropolar fluid on variables (u, θ) converge upper semi-continuously to the global attractor for the Newtonian fluid when K → 0 (or, equivalently, N → 0) and all other parameters are fixed. In the end, we list a few questions that we hope to answer in the future work. Question 1. There is a gap between the range of K, L for which we proved that all trajectories converge to the conductive steady state and the range for which we obtained Nu = 1. Hence we may ask, if it is possible that Nu = 1 for a given trajectory which does not converge a steady state. The second question that arises is whether it is true that the attractor consists of only one point when Nu = 1 for every trajectory. Question 2. It is already well known, cf., e.g., [36] that for the Newtonian fluid under the infinite Prandtl number regime it is possible to obtain the physical bound of the Nusselt number in terms of √ √ 3 Ra rather that Ra scaling. For the micropolar case, using the method of [36], the same scaling has been established recently in [37], namely,   1/3    3 4K + L 4K + L 1/3 Nu ≤ √ Ra ln Ra 3 4K + L + KL 4K + L + KL 5 under the assumption that f = Ra Ra



4K + L 4K + L + KL



≥ 80.

If K or L equals zero, the estimate reduces to that obtained in [36] for the corresponding Newtonian case. Observe, however, that enlarging K and L we can make Nu arbitrarily small. As concerns Newtonian fluids and finite Prandtl numbers, several heuristic physical consider√ 3 ations predict the Ra scaling for some regimes of large Prandtl numbers. cf. e.g. [38]. Quite 36

recent precise results, cf. [19] and [39] confirm that it is really the case. These √ results used a perturbative argument as the background flow method can not give the bound with 3 Ra, cf. [40] for finite Prandtl number. We leave here as an open, and in our opinion very interesting, problem if the methods of [19] and [39] give the improved bounds also for the micropolar fluid case and, if yes, how this improved bounds depend on the micropolar constants K and L. √ We also leave open the possibility to use the vorticity equation to improve the bound Ra in two dimensional case for the micropolar model, cf. [41]. It is also interesting and open line of research to use numerical simulations to predict the optimal form of the bounds, cf. [42, 43, 44]. Question 3. In [13] it is stated that for values of K sufficiently close to one (no matter what L is) the pure conductive state attracts all trajectories. Can nonlinear stability analysis yield such result? Question 4. The attractor dimension estimates derived in Section 6 do not cover the situation, when Ra is large enough and all trajectories converge to pure conductive steady state. The dimension of the attractor in such case should be clearly zero. Is it possible to obtain from the Lyapunov exponents method the estimates that would cover this situation? Question 5. Data assimilation algorithms allow to recover, asymptotically, the unknown solution from the discrete measurements. Recently, Farhat, Jolly, Lunassin, and Titi [45, 46] showed that it is possible to construct the data assimilation algorithm for 2D Rayleigh–Bénard problem and even recover whole trajectory (velocity and temperature) by measuring only horizontal velocity. The natural question that appears is, which data in the micropolar model is sufficient to measure in the data assimilation algorithm to recover the whole solution. Acknowledgments We thank anonymous reviewers for their careful reading and valuable comments. This work was supported by National Science Center (NCN) of Poland under projects No. DEC-2017/25/B/ST1/00302 and UMO-2016/22/A/ST1/0007. References [1] A. Eringen, Theory of micropolar fluids, J. Math. Mech. 16 (1966) 1–16. [2] G. Łukaszewicz, Micropolar Fluids - Theory and Applications, Birkhäuser Basel, 1999. [3] G. Łukaszewicz, Long time behavior of 2D micropolar fluid flows, Mathemathical and Computer Modelling 34 (2001) 487–509. [4] A. Tarasi´nska, Global attractor for heat convection problem in a micropolar fluid, Math. Meth. Appl. Sci. 29 (2006) 1215–1236. [5] A. Tarasi´nska, Deterministic and statistical solutions of micropolar fluid equations, Phd thesis, University of Warsaw (2010). [6] P. Constantin, C. Doering, Heat transfer in convective turbulence, Nonlinearity 9 (1996) 1049–1060. [7] P. Constantin, C. Doering, Variational bounds on energy dissipation in incompressible flows. III. Convection, Physical Review E 53 (1996) 5957–5981. 37

[8] P. Constantin, C. Foias, R. Temam, On the dimension of the attractors in two-dimensional turbulence, Physica D: Nonlinear Phenomena 30 (1988) 284–296. [9] C. Doering, J. Gibbon, Applied Analysis of the Navier-Stokes equations, Cambridge Texts in Applied Mathematics, 1995. [10] R. Russo, M. Padula, A note on the bifurcation of micropolar fluid motion, Bollettino dell’Unione Matematica Italiana A 17 (1980) 85–90. [11] O. Aydin, I. Pop, Natural convection in a differentially heated enclosure filled with a micropolar fluid, International Journal of Thermal Sciences 46 (2007) 963–969. [12] G. Bourantas, V. Loukopoulos, Modeling the natural convective flow of micropolar nanofluids, International Journal of Heat and Mass Transfer 68 (2014) 35–41. [13] G. Ahmadi, Stability of a micropolar fluid layer heated from below, Int. J. Engng Sci 14 (1976) 81–89. [14] G. Lebon, C. Perez-Garcia, Convective instability of a micropolar fluid layer by the method of energy, Int. J. Engng Sci 19 (1981) 1321–1329. [15] C. Perez-Garcia, J. Rubí, C. Casas-Vázquez, On the stability of micropolar fluids, J. NonEquilib. Thermodyn. 6 (1981) 65–78. [16] A. Datta, V. Sastry, Thermal instability of a horizontal layer of micropolar fluid heated from below, Int. J. Engng Sci 14 (1976) 631–637. [17] B. Straughan, The Energy Method, Stability, and Nonlinear Convection, Springer, 2003. [18] C. Foias, O. Manley, R. Temam, Attractors for the Bénard problem: existence and physical bounds on their fractal dimensions, Nonlinear Analysis: Theory, Methods and Applications 11 (1987) 939–967. [19] X. Wang, Bound on vertical heat transport at large Prandtl number, Physica D 237 (2008) 854–858. [20] L. Payne, B. Straughan, Critical Rayleigh numbers for oscillatory and nonlinear convection in an isotropic thermomicropolar fluid, International Journal of Engineering Science 27 (1989) 827–836. [21] L. Evans, Partial Differential Equations, 2nd edn., American Mathematical Society, Providence, RI, 2010. [22] A. Georgescu, L. Palese, Stability criteria for fluid flows, World Scientific Publishing Co. Pte. Ltd., 2010. [23] B. Rummler, The eigenfunctions of the Stokes operator in special domains. ii, ZAMM Z. angew. Math. Mech. 77 (1997) 669–675.

38

[24] T. Ma, S. Wang, Dynamic bifurcation and stability in the Rayleigh–Bénard convection, Commun. Math. Sci. 2 (2004) 159–193. [25] P. Kalita, G. Łukaszewicz, J. Siemianowski, On Rayleigh–Bénard problem for thermomicropolar fluid, Topological Methods in Nonlinear Analysis (2018) DOI: 10.12775/TMNA.2018.012. [26] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer, 1997. [27] V. Chepyzhov, A. Ilyin, A note on the fractal dimension of attractors of dissipative dynamical systems, Nonlinear Anal. 44 (2001) 811–819. [28] V. Chepyzhov, A. Ilyin, On the fractal dimension of invariant sets; applications to Navier– Stokes equations, Discrete Contin. Dyn. Syst. 10 (2004) 117–135. [29] C. Doering, X. Wang, Attractor dimension estimates for two-dimensional shear flows, Physica D: Nonlinear Phenomena 123 (1998) 206–222. [30] M. Boukrouche, G. Łukaszewicz, An upper bound on the attractor dimension of a 2D turbulent shear flow in lubrication theory, Nonlinear Analysis: Theory, Methods & Applications 59 (2004) 1077–1089. [31] J.-M. Ghidaglia, M. Marion, R. Temam, Generalization of the Sobolev–Lieb–Thirring inequalities and applications to the dimension of attractors, Differential Integral Equations 1 (1988) 1–21. [32] A. Ilyin, Lieb–Thirring integral inequalities and their applications to attractors of the Navier– Stokes equations, Sbornik: Mathematics 196 (2005) 33–66. [33] J. Hale, Asymptotic behavior of dissipative systems, American Mathematical Society, 1899. [34] G. Raugel, Global attractors in partial differential equations, in: Handbook of dynamical systems, Vol. 2, North-Holland, 2002, pp. 995–982. [35] J. Dhiman, P. Sharma, G. Singh, Convective stability analysis of a micropolar fluid layer by variational method, Theoretical and Applied Mechanics Letters 1 (2011) 042004. [36] C. Doering, F. Otto, M. Reznikoff, Bounds on vertical heat transport for infinite-Prandtlnumber Rayleigh–Bénard convection, J. Fluid Mech. 560 (2006) 229–241. [37] M. Caggio, P. Kalita, G. Łukaszewicz, K. Mizerski, Vertical heat transport at infinite Prandtl number for the micropolar fluid, in preparation. [38] S. Grossman, D. Lohse, Thermal convection for large Prandtl numbers, Physical Review Letters 86 (2001) 3316–3319. [39] A. Choffrut, V. Nobili, F. Otto, Upper bounds on Nusselt number at finite Prandtl number, Journal of Differential Equations 260 (2016) 3860–3880. 39

[40] V. Nobili, F. Otto, Limitations of the background field method applied to Rayleigh–Bénard convection, Journal of Mathematical Physics 58 (2017) 093102. [41] J. Whitehead, C. Doering, Ultimate state of two-dimensional Rayleigh–Bénard convection between free-slip fixed-temperature boundaries, Phys. Rev. Lett. 106 (2011) 244501. [42] S. Plasting, R. Kerswell, Improved upper bound on the energy dissipation rate in plane Couette flow: the full solution to Busse’s problem and the Constantin–Doering–Hopf problem with one-dimensional background field, Journal of Fluid Mechanics 477 (2003) 363–379. [43] B. Rollin, Y. Dubief, C. Doering, Variations on Kolmogorov flow: turbulent energy dissipation and mean flow profiles, Journal of Fluid Mechanics 670 (2011) 204–213. [44] B. Wen, G. Chini, R. Kerswell, C. Doering, Time-stepping approach for solving upper-bound problems: Application to two-dimensional Rayleigh–Bénard convection, Phys. Rev. E 92 (2015) 043012. [45] A. Farhat, M. Jolly, E. Titi, Continuous data assimilation for the 2D Bénard convection through velocity measurements alone, Physica D 303 (2015) 59–66. [46] A. Farhat, E. Lunasin, E. Titi, Continuous data assimilation algorithm for a 2D Bénard convection through horizontal velocity measurements alone, Journal of Nonlinear Science 27 (2017) 1065–1087.

40

Highlights • Rayleigh-B´enard convection is compared between micropolar and Newtonian fluids. • Micropolar effects make the convective flow more stable. • Critical Rayleigh number for micropolar fluid is higher. • Nusselt number upper bound for micropolar fluid is lower. • Global attractor dimension upper bound for micropolar fluids is lower.

1