Nonlinear Stability for Thermal Convection in a Brinkman Porous Material with Viscous Dissipation

We investigate nonlinear stability in a model for thermal convection in a saturated porous material using Brinkman theory, taking into account viscous dissipation effects. There are (at least) two models for viscous dissipation available, and we include a derivation of one of these by assuming that the flow in the porous medium may be described by a theory for a mixture of an elastic solid and a linearly viscous fluid. A fully nonlinear stability result is provided when either of the viscous dissipation functions is taken into account, and it is shown that from the nonlinear energy stability viewpoint both models are, in a sense, equivalent.

In the context of flow in a Brinkman porous medium, there is some controversy over the form the relevant nonlinear viscous dissipation terms should appear, see, e.g., Al-Hadrami et al. (2003), Barletta (2008), Barletta et al. (2011b), Breugem and Rees (2006)

, Hooman
Communicated by Edith van der Wal.

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and Gurgenci (2007), Magyari and Rees (2006) and Nield (2000aNield ( , b, 2004Nield ( , 2007. It is not the purpose of this article to enter this controversy. We are interested in analyzing the fully nonlinear stability for the Brinkman theory, and so we allow both forms of nonlinear viscous dissipation. In fact, we show that, in a sense, from the viewpoint of nonlinear energy stability theory both forms of the viscous dissipation lead to exactly the same result and so in this sense are equivalent. In the next section, we include a derivation of the Brinkman equations for thermal convection in a porous medium which is based on the theory for a mixture of an elastic solid and a viscous fluid. The viscous dissipation terms arise naturally and are essentially the same form as those of Al-Hadrami et al. (2003). We point out that this form of viscous dissipation is also derived in the analysis of Breugem and Rees (2006), who perform a rigorous averaging procedure. The comments of Breugem and Rees (2006) on whether the form of the viscous dissipation is correct are very enlightening and are lucidly described on pages 1 and 2 of their article.
Our main goal is to derive a nonlinear energy stability analysis for thermal convection in a Brinkman porous material, and we employ both forms of viscous dissipation function. A recent very interesting development of Barletta and Mulone (2020) establishes an analogous nonlinear energy stability analysis for thermal convection in a Darcy porous material when the relevant viscous dissipation term is present. A detailed analysis of thermal convection in a Brinkman porous material neglecting viscous dissipation effects is contained in Rees (2002), and the validity of the Brinkman equations is discussed in Nield (2000b). We stress that even in the absence of viscous dissipation effects, the Darcy and Brinkman theories are very different. For example, Straughan (2016) shows that one may lead to stationary convection, whereas the other yields oscillatory convection for the same problem of resonant convection. Thus, the Brinkman and Darcy equations represent different physical phenomena.

Derivation of the Brinkman Equations from Mixture Theory
There are various approaches to presenting a theory for a mixture of an elastic solid and a viscous fluid. We here describe that of Eringen (1994Eringen ( , 2004. Actually, Eringen (1994Eringen ( , 2004 develops his theory for a mixture of a fluid, a solid, and a gas. We restrict attention to only a fluid and a solid. Throughout we employ standard indicial notation in conjunction with the Einstein summation convention. There are momentum equations for the fluid and for the solid, and letting s denote solid while f denotes fluid, these are (see also Straughan 2015a, pp. 20-22), where , x i , t ij , b i and p i denote the respective, density, particle position, Cauchy stress tensor, body force, and interaction force for the fluid (f) and solid (s). The notation ′′ denotes the second derivative following the motion of the particular particle. In addition, the equation of energy balance is h denote the internal energy, heat flux, fluid velocity, solid velocity, and externally supplied heat. The mixture density = f + s . Eringen (1994Eringen ( , 2004 develops his theory in terms of the deformation gradient x i ∕ X s K , the temperature, the fluid density, the invariants of d , the temperature gradient, and the difference of fluid and solid velocity v i is an objective quantity and so the velocity difference should enter the theory; from a mixture viewpoint, this term gives rise to the Darcy friction term. The full equations for a deformed elastic body are given in Eringen (1994Eringen ( , 2004, see also Straughan (2015a, pp. 20-22). In this work, we are interested in flow through an undeformed skeleton which does not move; hence, v s i ≡ 0 . Thus, we present a restricted constitutive theory to reflect this. The Cauchy stress t s ij ≡ 0 and so we require where p f is the fluid pressure, ̂ , k and b are here taken as constants, k being the thermal conductivity. The relevant fluid momentum equation now becomes where ̇v i is the material derivative, 0 is a reference density, and for an incompressible fluid the equation of continuity is The energy equation now has form Equations (4) and (6) are reduced assuming a Boussinesq approximation, cf. Breugem and Rees (2006), Nield and Barletta (2010), Straughan (2015a, pp. 16-21). We assume the acceleration may be neglected in (4), (cf. Barletta et al. 2011b), and now additionally omit the sub-or superscriptf. Adopting a Boussinesq approximation, Eqs. (4)-(6) may be reduced to, setting h = 0, where g is gravity, = (0, 0, 1) , Δ is the Laplacian, c p = ∕ T is the specific heat at constant pressure, and the density is constant apart from the buoyancy term where we have written f = 0 (1 − [T − T 0 ]) with 0 , T 0 being reference density and temperature values, and is the thermal expansion coefficient of the fluid. The term Ṫ is the material derivative (7) are the equations for thermal convection in a Brinkman theory incorporating viscous dissipation. Note that the b v i v i term is already present in the Darcy theory, see, e.g., Barletta et al. (2009a), Barletta and Mulone (2020), Nield and Barletta (2010). Furthermore, observe that as ̂→ 0 we recover the equations of Darcy theory when viscous dissipation is present, whereas when b → 0 the temperature equation assumes the correct form for nonisothermal flow in a clear fluid taking into account viscous dissipation effects.

Thermal Convection Equations
Suppose the saturated porous material is contained between the horizontal planes z = 0 and z = d with the upper and lower temperatures retained at the constant values, Define the temperature gradient, , by Then, the steady (conduction) solution in whose stability we are interested is given by where the steady pressure, p(z) , is a quadratic function derived from (7) 1 .
To analyze thermal convection, we develop a stability analysis for the solution to (7). Thus, let (u i , , ) be a perturbation to the steady solution Then, the nonlinear perturbation equations are given by Note that ̂ is the Brinkman coefficient, whereas b represents the Darcy coefficient. Equations (9) may be non-dimensionalized in a standard way, see Straughan (2008), p. 163, to derive a non-dimensional version in terms of the Rayleigh number to obtain where Eqs. (10) hold on the domain {(x, y) ∈ ℝ 2 } × {z ∈ (0, 1)} × {t > 0} , and , A, B are the non-dimensional forms of ̂ and the viscous dissipation coefficients.
In this work, we assume the surfaces z = 0, d are free from stress, and then, the boundary conditions are given by u, v, w) , and additionally the solution satisfies a plane tiling periodicity with a wave number a, Straughan (2004), p. 51, Straughan (2008), p. 152. The periodic cell so defined will be denoted by V. To exclude rigid motions, we require

Nonlinear Stability
Let (⋅, ⋅) and ‖ ⋅ ‖ be the inner product and norm on the Hilbert space L 2 (V).
To prove nonlinear stability, we multiply (10) 1 by u i and integrate over V to find after using the boundary conditions (11) where the arithmetic-geometric mean inequality has been employed. Select now = 1∕R to obtain Next, multiply (10) 1 by −Δu i and integrate over V to now obtain with the aid of (11), Note that in deriving (13), the boundary term ∮ V Δw dA is encountered, where V is the boundary of V. This is where the boundary conditions (11) are necessary to ensure this is zero. Upon using the arithmetic-geometric mean inequality on (13), one may then derive The next step involves multiplying (10) 1 by u i , multiplying (10) 3 by , integrating each over V, and adding to obtain where and From Eq. (15) one proceeds to (13) ‖∇ ‖ 2 + ‖Δ ‖ 2 = R( , Δw).
where with H being the space of admissible solutions. The Euler-Lagrange equations are found from this, and one may show they are the same as the linearized version of (10). Then, the nonlinear critical Rayleigh number boundary is the same as the linear one, details may be found in, e.g., Straughan (2008, pp. 163-166).
To establish nonlinear stability, it remains to handle the nonlinear term N. To do this, we require the Poincaré inequality the Wirtinger inequality see Galdi and Straughan (1985), and the Sobolev inequalities and which may be established as in Galdi and Straughan (1985) and Payne and Straughan (2009), inequality (50), where c 1 and c 2 are constants. We also need the estimate, Galdi and Straughan (1985), inequality (5.27), We now bound the term N in (17). Firstly, by the Cauchy-Schwarz inequality Then, We integrate by parts the last term in this expression and use the conditions (11) to find To bound the last term, note where the Cauchy-Schwarz inequality has also been used. Furthermore where we have employed (23). Utilizing (25)-(28), we may see that Observe now from (12) and (14) where √ 2 = min{1, −1 } . Using the form for D in (16) together with (30) and inequality (20), we now derive where Suppose now R < R E so that a = 1 − R∕R E > 0 . Then, from (18) using (31) one derives Require now ‖ (0)‖ < a∕h . Then, by a continuity argument, see, e.g., Straughan (2004, pp. 15, 16), from (32) we see that ‖ (t)‖ ≤ ‖ (0)‖ and so (32) may be replaced by (32), we now show and so ‖ (t)‖ decays exponentially. Using (30), it follows ‖ ‖, ‖∇ (t)‖, and ‖Δ (t)‖ also decay exponentially and we obtain nonlinear stability. In addition, using (24) we obtain decay of sup V | | and so the perturbation velocity decays pointwise in time.
The conditions for nonlinear stability are The first, R < R E , corresponds to the linear instability threshold, cf. Straughan (2004, pp. 163-166). The second places a restriction on the magnitude of the temperature perturbation. Due to the severe nonlinearities of the viscous dissipation terms in (10) 3 , we expect such a restriction.
Remark 1 A nonlinear energy stability analysis for the analogous problem for the Darcy equations has recently been provided by Barletta and Mulone (2020). The equations for the Darcy problem follow from (10) by setting and B equal to zero, and there are no boundary conditions on u ,z , v ,z in (11). Their analysis hinges on use of the Sobolev inequality in ℝ 3 and a bound for ∇ in terms of ∇ . The bound in Barletta and Mulone (2020) involves employing the differential equation on the boundary in its derivation. We here include an alternative derivation. For the Darcy system, (10) 1 is replaced by where now we only have w = 0 on z = 0, 1 , but there is still periodicity in x and y. From (36), we take ∕ x j and then multiply by u i,j − u j,i to find, cf. Payne and Straughan (1996, pp. 230-232) and Straughan (2015b, pp. 117, 118), since (u i,j − u j,i ) ,ij = 0 . Then, we use the skewness of u i,j − u j,i to write where in the last line the arithmetic-geometric mean inequality is used. Now take = R to find Expanding these expressions, Note since w = 0 on z = 0, 1 . Then, This is the bound of Barletta and Mulone (2020). Note that we require more than this for the full system (10).
Remark 2 Breugem and Rees (2006) note that the viscous dissipation term in the Darcy formula can yield very high temperatures. The possibility of thermal runaway is discussed. Without the Brinkman terms, the Darcy version of Eq. (7) may be written in non-dimensional form as cf. Barletta and Mulone (2020). It is to be expected high temperatures will be achieved due to the very strong forcing term Bv i v i . Indeed, one should ask whether finite time blow-up could occur, since, for example, it is well known that such blow-up does occur for equations like with suitable boundary conditions and C > 0 constant. We present a heuristic one-dimensional model arising from (37) which suggests that while T might become very large, blow-up in finite time will not occur. Consider the onedimensional version of (37) without the pressure term where Δ is now the second derivative with respect to x. Eliminate w and derive Let this equation be defined on (0, 1) × {t > 0} with T(x, 0) given and T = 0 at x = 0, 1 . By using a weighted energy, Straughan (2004, pp. 16-19), shows that ‖T(t)‖ 2 remains bounded and T has a steep boundary layer near x = 1 . In fact, where = ARa , is the asymptotic behavior of T as t → ∞ . There is a trade-off between the convective term TT x and the viscous dissipation term BRa 2 T 2 . In fact, the convective term is preventing the viscous dissipation from inducing the solution to blow-up in a finite time.

Nonlinear Stability Analysis, Alternative Viscous Dissipation
There is an alternative viscous dissipation function which replaces the d ij d ij term by one of the form −u i Δu i , see Nield (2007), Barletta et al. (2011b). In this case, the perturbation equations (10) are replaced by (37) 1 3 where the domain and boundary conditions are as in Sect. 4. A nonlinear energy analysis proceeds exactly as in Sect. 4, and we arrive at Eq. (15) excepting now The B term is handled exactly as in Sect. 4. For the A term, we integrate by parts to find Observe that the form of the right-hand side of (41) is very similar to the right-hand side of (26). Indeed, the right-hand side of (41) is bounded in the same way as (27) and (28) to then arrive at (31). Thus, the analysis of energy stability for the alternative viscous dissipation function turns out to lead to exactly the same nonlinear stability result. In this sense, the two formulations are equivalent, although the actual solutions (u i , , ) may not be. Hooman and Gurgenci (2007) suggest employing a viscous dissipation function which is essentially a combination of the forms in Sects. 4 and 5. In our notation, this would give rise to a set of nonlinear perturbation equations which replace (10) of form, A nonlinear energy stability analysis may be worked out in a straightforward manner by following the techniques in Sects. 4 and 5. Again, the critical Rayleigh number found is the same as the one of linear instability theories.

Conclusions
We have analyzed the nonlinear stability for a solution to the Brinkman equations for thermal convection in a porous material incorporating viscous dissipation terms. The viscous dissipation consists of a term which is present in the analogous problem employing Darcy theory. Additionally, there is a term in the energy balance equation which is due entirely to the presence of the Brinkman contribution in the momentum equation. Barletta and Mulone (2020) recently established a rigorous nonlinear stability result for the Darcy problem. We here extend the method to derive an analogous rigorous nonlinear stability result for the Brinkman theory. In both cases, the nonlinear stability threshold is shown to be exactly the same as the linear instability one. For both situations, the size of the initial temperature field has to be suitably small to obtain rapid exponential decay in time. Given the (39)