Effects of Neumann and Robin boundaries on the thermal instability

Hajar Lagziri; Hanae El Fakiri; Abdelmajid El Bouardi

doi:10.1556/1848.2022.00577

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International Review of Applied Sciences and Engineering

Volume/Issue:: Volume 14: Issue 3

Effects of Neumann and Robin boundaries on the thermal instability

Authors:

Hajar Lagziri

Hajar Lagziri Department of Physics, Faculty of Science, University of Abdelmalek Essaadi, Tetouan, Morocco

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lagziri-hajar7@hotmail.fr https://orcid.org/0009-0006-4532-6735

Hanae El Fakiri

Hanae El Fakiri Department of Physics, Faculty of Science, University of Abdelmalek Essaadi, Tetouan, Morocco

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, and

Abdelmajid El Bouardi

Abdelmajid El Bouardi Department of Physics, Faculty of Science, University of Abdelmalek Essaadi, Tetouan, Morocco

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Pages:: 366–373

Online Publication Date:: 21 Mar 2023

Publication Date:: 10 Oct 2023

Article Category:: Research Article

DOI:: https://doi.org/10.1556/1848.2022.00577

Keywords:: porous medium; biot number; heat flux; local thermal non-equilibrium; linear stability

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Abstract

The thermo convective instability of the Darcy-Benard problem (DB) using Robin (third-kind) thermal conditions is investigated here. We consider a viscous Newtonian fluid saturating a porous layer in which the layer is sandwiched between two impermeable boundaries. The upper and the lower walls are modelled in the form of the Neumann (second-kind) and the Robin (third-kind) thermal conditions, respectively. The difference in the temperature distribution between both phases allows the lack of a local thermal equilibrium model to be present. As a consequence, the third kind of thermal condition brings about one extra dimensionless parameter of the Biot number to the usual one of the inter-heat transfer coefficient and the thermal conductivity ratio. The normal modes method adopted in a linear stability analysis gives rise to perturbed governing equations. The eigenvalue problem is handled numerically as a result of the perturbed governing equations leading to the marginal stability condition.

Abstract

1 Introduction

The Darcy-Benard problem (DB) [1], which embodies the thermo convective instability in porous media, becomes the cornerstone of many industrial and environmental applications. For instance, the onset of the convection cell accelerates the dissolution and the mixing of CO₂ in the aqueous phase, which, in turn, increases its storage lifespan in deep geological layers [2]. The emergence of this phenomenon does not only favor the process of CO₂ sequestration but also the extraction of the thermal energy from the geothermal reservoirs where the underground heating condition occurs. The main process behind the thermal instability in these fields is the prevailing effect of buoyant forces in comparison with resistive mechanisms that can be generated by viscous dissipation, thermal conductivity or other physical properties [3]. Whatever the medium in which instability arises, the motion of the fluid particles caused by buoyancy effects is always driven either by the concentration of solute particles or by the vertical temperature gradient [4, 5]. However, the materials manufactured by porous media are more effective to hasten the heat transfer process than the case of Newtonian clear fluid. The Rayleigh number R_c appears in the former configuration at the critical value of $R_{c} \approx 39$ while in the latter one at $R_{c} \approx 1707$ [6, 7].

The prevailing effects of the local thermal non-equilibrium (LTNE) render the thermal boundary conditions to be mismatched with the standard forms defined in the equilibrium one. As the temperature profile for the solid structure diverges from the fluid phase, several thermal boundaries are required to investigate the Darcy-Benard problem [8, 9]. On the other hand, whatever the type of these boundaries in a LTNE the marginal stability curves may tend to behave the same way as the standard one if the two dimensionless numbers defined by the internal heat transfer coefficient $H$ and the thermal conductivity ratio γ resulted in the discrepancy between thermal conductivities of phases and the loss or gain of heat from one phase to another have specific values or limits. For instance, there are two possible conditions at which the transition from LTNE to LTE is acceptable: an infinite limit of $H$ with a nonzero value of γ or finite and non-vanishing values of $H$ with large γ. The former case assures a rapid and higher amount of heat transferred between phases while the latter corresponds to a case in which the saturated fluid holds higher thermal conductance than the solid skeleton. A small $H$ or γ can reverse the approach of the LTE towards a non-equilibrium one [10–13].

The majority of the analysis carried out in the Darcy-Benard problem is focused on the case where the local thermal equilibrium condition (LTE) takes place. In other words, the convection cell in a horizontal saturated porous layer is handled only when a Newtonian saturating fluid has a temperature field equal to a solid matrix. All these instability analyses are studied either with impermeable or permeable bounding planes beside different positions of the porous layer [14–16]. However, the feature of the LTE is not always true as we can encounter in some cases a disparity in the thermal conductivity or/and a weakness in the heat transfer coefficient, which can yield a relaxation of the local thermal equilibrium regime. From a practical standpoint, the local thermal non-equilibrium (LTNE) model has a vital effect on the cooling process of countless engineering devices [10, 17]. On the other hand, the approach of the LTNE can highlight difficulties in the modelling of thermal boundaries, especially in the case of isoflux [18–20]. The objective behind this work is to analyze the convective instability in the case where two impermeable walls are supposed to be modelled in the form of Newton's cooling law equation for Robin (third-kind) thermal conditions and uniform heat flux (Model A) for Neumann (second-kind) thermal conditions.

2 Mathematical model

The setup assumed here is the archetype of the Darcy-Benard configuration (DB) with two rigid walls [1]. In addition, two different temperature fields are allowed, which enables violation of local thermal equilibrium. The upper wall is heated by a heater to yield uniform heat flux (Model A) while the lower one is determined as Newton's cooling law equation. For definiteness, our investigation has been performed with porosity φ = 1/2. A sketch of the system geometry is drawn in Fig. 1. Besides, we presume that Darcy's law and the Oberbeck-Boussinesq approximation are applicable. The governing equations in the dimensionless forms are written as [19, 12]:

\nabla \times v = - R a \nabla \times (T_{f} e_{z}),

(1a)

χ \frac{\partial T_{s}}{\partial t} = \nabla^{2} T_{s} + H γ (T_{f} - T_{s}),

(1b)

\frac{\partial T_{f}}{\partial t} + v \cdot \nabla T_{f} = \nabla^{2} T_{f} - H (T_{f} - T_{s}) .

(1c)

Fig. 1.

The sketch of the Darcy-Benard configuration (DB)

Citation: International Review of Applied Sciences and Engineering 14, 3; 10.1556/1848.2022.00577

While the dimensionless boundary conditions used in Eqs (3) are summed up in Fig. 1 as:

z = 0 : \frac{\partial T_{f}}{\partial z} = B_{f} (T_{f} - 1), \frac{\partial T_{s}}{\partial z} = γ B_{f} (T_{s} - 1), w = 0,

(2a)

z = 1 : γ \frac{\partial T_{f}}{\partial z} + \frac{\partial T_{s}}{\partial z} + 1 + γ = 0, T_{s} = T_{f}, w = 0 .

(2b)

The notation of the vector v indicates the velocity field, T indicates the temperature [K] and the subscripts “s” and “f” indicate a saturating fluid phase and a solid matrix, respectively. In contrast, the dimensionless inter-heat transfer coefficient, thermal conductivity ratio, Biot number, and modified Rayleigh number in a porous medium are briefly the parameters noted as $H$ , γ, $B_{f}$ and $R a$ .

Eqs (1) and (2) are a virtue of scaling quantities alongside the curl operator applied to Darcy's law. The proposed scaling quantities are [19, 12]:

x^{*} \to x l, t^{*} \to t \frac{l^{2}}{α_{f}}, v^{*} \to (u, v, w) \frac{φ α_{f}}{l}, T_{s, f}^{*} \to T_{w} + Δ T T_{s, f}, λ_{f} = α_{f} {(ρ C)}_{f}, λ_{s} = α_{s} {(ρ C)}_{s},

∆ T = \frac{q_{w} l}{λ_{m}}, λ_{m} = (1 - φ) λ_{s} + φ λ_{f}, a_{m} = \frac{λ_{m}}{{(ρ C)}_{f}}, γ = \frac{φ λ_{f}}{(1 - φ) λ_{s}}, χ = \frac{α_{f}}{α_{s}},

H = \frac{{h l}^{2}}{φ λ_{f}}, R α = \frac{(1 + γ) β g ∆ T K l}{γ α_{m} v_{f}}, B_{f} = \frac{h_{w, f} l}{λ_{f}}

(3)

The superscript of the star notation refers to dimensional variables. α is the thermal diffusivity [m² s⁻¹], T_w is the temperature of the lower wall [K], ρ is the density [kg m⁻³], $h$ is the inter-phase volumetric heat transfer coefficient [W (m³ K⁻¹)], e_z unit vector in the z-direction, C is the heat capacity per unit of mass [J (kg K⁻¹)], $q_{w}$ is the heat flux at the upper wall [W m⁻²], ν_f is the kinematic viscosity [m² s⁻¹], g is the modulus of the gravitational acceleration [m s⁻²], β is thermal expansion coefficient [K⁻¹], t is the time [s] and 1 is the thickness of the layer [m]. The subscript “w” means the wall while the symbol “m” denotes the effective. λ is the thermal conductivity [W (m K⁻¹)], $h_{w, f}$ is the superficial heat transfer coefficient of the fluid phase [W (m² K⁻¹)], $K$ is the permeability of the medium [m²].

3 Basic state

The basic profile assumed for Eqs (1) and (2) is a steady state with zero velocity that fluid exists in a solid skeleton. Consequently, the basic profile is:

v_{b} = 0

(4)

T_{s, b} = \frac{- (2 B_{f} (1 - z) γ \sqrt{Λ} + {(1 + γ)}^{\frac{3}{2}} \sqrt{\frac{Λ}{1 + γ}}) \cosh \sqrt{Λ} - B_{f} (1 + γ (γ + B_{f} (z - 1) (1 + γ)) \sinh (z - 1) \sqrt{Λ}}{B_{f} (2 γ \sqrt{Λ} \cosh \sqrt{Λ} + B_{f} γ (1 + γ) \sinh \sqrt{Λ}}

(5)

T_{f, b} = \frac{- (2 B_{f} (1 - z) γ \sqrt{Λ} + {(1 + γ)}^{\frac{3}{2}} \sqrt{\frac{Λ}{1 + γ}}) \cosh \sqrt{Λ} + B_{f} (1 + γ (γ + B_{f} (z - 1) (1 + γ)) \sinh (z - 1) \sqrt{Λ}}{B_{f} (2 γ \sqrt{Λ} \cosh \sqrt{Λ} + B_{f} γ (1 + γ) \sinh \sqrt{Λ}}

(6)

Where: $Λ = H (1 + γ)$

We have used the subscript “b” as a symbol of the basic flow.

The first derivative of

T_{f, b}

that will be adopted in the governing equations is

T_{f, b}^{'} = - 1 + \frac{(- 1 + γ) \sqrt{Λ} \cosh [(- 1 + y) \sqrt{Λ}]}{2 γ \sqrt{Λ} \cosh [\sqrt{Λ}] + B f γ (1 + γ) \sinh [\sqrt{Λ}]}

(7)

3.1 Basic state for local thermal equilibrium cases

The cases where the temperature of solid structure matches the fluid phase can emerge through two different limits:

a)The limiting case of $H \to \infty$ with $γ \approx O (1)$ that refers to the basic temperature profile of both phases as

T_{s b} = T_{f b} = - \frac{1 + γ + 2 γ B_{f} (1 - z)}{2 γ B_{f}} .

(8)

b)The case of $γ \to \infty$ with $H \approx O (1)$ whose basic temperature profile is,

T_{s b} = T_{f b} = 1 - z - \frac{1}{B_{f}} .

(9)

3.2 Basic state for local thermal non-equilibrium cases

There are two limits in which we can have two autonomous temperature profiles:

H \to 0

with

γ \approx O (1)

γ \to 0

with

H \approx O (1)

.The basic temperature profile for both phases in the former limit is

T_{f b} = - \frac{1 + γ + {B_{f}}^{2} (- 1 + z) γ (1 + γ) + B_{f} (z + (- 1 + z) γ + γ^{2})}{B_{f} γ (2 + B_{f} + B_{f} γ)},

(10a)

T_{s b} = - \frac{1 + B_{f} + γ + B_{f} (1 + B_{f}) (- 1 + z) γ + B_{f} (B_{f} (- 1 + z) + z) γ^{2}}{B_{f} γ (2 + B_{f} + B_{f} γ)} .

(10b)

While in the latter one it is

T_{f b} = \frac{e^{- \sqrt{H} z} (e^{2 \sqrt{H}} - e^{2 \sqrt{H} z} + e^{\sqrt{H} z} (1 + B_{f} (- 1 + z)) + e^{\sqrt{H} (2 + z)} (- 1 + B_{f} - B_{f} z)) (- 1 + \coth [\sqrt{H}])}{2 B_{f}},

(11a)

T_{s b} = 1 - \frac{1}{B_{f}} - z .

(11b)

The basic temperature profile for all cases is expressed following the condition of φ = 1/2. Therefore, the Biot number of the solid phase can be replaced by the fluid one through the relation of $B_{s} = γ B_{f}$ . For instance, some authors [20] have adopted this relation as a condition in the numerical method to look for the critical value of $B_{s} \neq B_{f}$ without mentioning anything related to the basic temperature profile for φ = 1/2. This in turn exhibits a discrepancy between our basic temperature profiles and the one cited in [20], knowing that they have employed the same thermal boundary conditions as we did.

4 Stability analysis

The basic solution is investigated by using the perturbation process which defines the main flow of each variable as a sum between the basic state and the distributed flow. More precisely,

T_{f, s} = T_{f s, b} + ε {\tilde{T}}_{f s}, v = v_{b} + ε \tilde{v}

(12)

According to linearization method, all the second orders of ε are cancelled in Eqs (1) and (2). Thus, we can obtain:

\nabla \times \tilde{v} = R a \nabla \times ({\tilde{T}}_{f} e_{z}),

(13a)

χ \frac{\partial {\tilde{T}}_{s}}{\partial t} = \nabla^{2} {\tilde{T}}_{s} + H γ ({\tilde{T}}_{f} - {\tilde{T}}_{s}),

(13b)

\frac{\partial {\tilde{T}}_{f}}{\partial t} + \tilde{w} T_{f, b}^{'} = \nabla^{2} {\tilde{T}}_{f} + H ({\tilde{T}}_{s} - {\tilde{T}}_{f}),

(13c)

z = 0 : \frac{\partial {\tilde{T}}_{f}}{\partial z} = B_{f} ({\tilde{T}}_{f} - 1), \frac{\partial {\tilde{T}}_{s}}{\partial z} = γ B_{f} ({\tilde{T}}_{s} - 1), \tilde{w} = 0

(13d)

z = 1 : γ \frac{\partial {\tilde{T}}_{f}}{\partial z} + \frac{\partial {\tilde{T}}_{s}}{\partial z} + 1 + γ = 0, {\tilde{T}}_{s} = {\tilde{T}}_{f}, \tilde{w} = 0 .

(13e)

The governing equations are invariable with respect to rotation around the z-axis. We consider velocity perturbations in the two-dimensional plane (x, z), in the form of perturbed stream function

\tilde{ψ} (x, z)

\tilde{u} = \frac{\partial \tilde{ψ}}{\partial z}, \tilde{w} = - \frac{\partial \tilde{ψ}}{\partial x}

(14)

The normal modes method allows us to tackle the perturbed equations by expressing the small secondary flow as

{\tilde{ψ} (x, z, t), {\tilde{T}}_{f s} (x, z, t)} = {i Ψ (z), θ_{f s} (z)} e^{i (k x - ω t)}

(15)

Hence the symbols of

Ψ (z)

and

θ (z)

are used to describe the perturbation amplitude functions. The wave number is defined by the symbol

k

while the growth rate and the angular frequency are noted with

ω_{i} = I m {ω}

and

ω_{r} = R e {ω}

respectively. The complex parameter ω is defined as the sum of the imaginary and real parts of ω.

The linear stability analysis is focused only on those modes whose growth rate is neither growing nor decaying with time. This feature implies the condition of

ω_{i} = 0

. Therefore, substituting Eqs (14) and (15) into Eqs (13) with

ω_{i} = 0

yields that,

Ψ^{″} - k^{2} Ψ + k R a θ_{f} = 0

(16a)

θ_{s}^{″} - k^{2} θ_{s} + γ H (θ_{f} - θ_{s}) + i ω_{r} χ θ_{s} = 0

(16b)

θ_{f}^{″} - k^{2} θ_{f} + H (θ_{s} - θ_{f}) + k T_{f}^{'} Ψ + i ω_{r} θ_{f} = 0

(16c)

z = 0 : θ_{f}^{'} = B_{f} θ_{f}, θ_{s}^{'} = γ B_{f} θ_{s}, Ψ = 0

(16d)

z = 1 : γ θ_{f}^{'} + θ_{s}^{'} = 0, θ_{s} = θ_{f}, Ψ = 0

(16e)

The two primes in Eqs (16) mean a second derivative with respect to z.

5 The principle of exchange of instabilities

As the principal of exchange of instabilities cannot be proven analytically owing to the variable coefficients that arise in the expression of $T_{f, b}^{'} (z)$ in Eq. (7), we will be obligated to verify its validity through the numerical solution presented in section (6) by using the eigenvalue problem written in Eqs (16). To satisfy this principle we must obtain numerically the value of $ω_{r} = R e {ω} = 0$ . In other words, the fluid layer has to be without oscillatory instability behaviour.

According to the outcomes of Table 1 the condition of

ω_{r} = R e {ω} = 0

is fulfilled as all the values are nearer to zero, thus, the principle exchange of instabilities is held. Consequently, we can neglect

ω_{r}

from the eigenvalue problem Eqs (16) and write that

Ψ^{″} - k^{2} Ψ + k R a θ_{f} = 0

(17a)

θ_{s}^{″} - k^{2} θ_{s} + γ H (θ_{f} - θ_{s}) = 0

(17b)

θ_{f}^{″} - k^{2} θ_{f} + H (θ_{s} - θ_{f}) + k T_{f}^{'} Ψ = 0

(17c)

z = 0 : θ_{f}^{'} = B_{f} θ_{f}, θ_{s}^{'} = {γ B}_{f} θ_{s}, Ψ = 0

(17d)

z = 1 : γ θ_{f}^{'} + θ_{s}^{'} = 0, θ_{s} = θ_{f}, Ψ = 0

(17e)

Table 1.

The real values of $ω_{r}$ calculated by the numerical method of the Runge-Kutta solver and shooting method

$H = 10^{- 2}$ and $B_{f} = 10$				$H = 10^{2}$ and $B_{f} = 10$
$γ = 10^{- 1}$		$γ = 10$		$γ = 10^{- 1}$		$γ = 10$
$k$	$ω_{r}$	$k$	$ω_{r}$	$k$	$ω_{r}$	$k$	$ω_{r}$
1	$2.260839 \times 10^{- 21}$	1	$1.049347 \times 10^{- 18}$	1	$1.657385 \times 10^{- 18}$	1	$9.989764 \times 10^{- 6}$
2	$9.074241 \times 10^{- 22}$	2	$4.309620 \times 10^{- 18}$	2	$8.040071 \times 10^{- 16}$	2	$9.948204 \times 10^{- 6}$
3	$3.932491 \times 10^{- 21}$	3	$9.037235 \times 10^{- 18}$	3	$2.885237 \times 10^{- 15}$	3	$9.582209 \times 10^{- 6}$
4	$7.540579 \times 10^{- 21}$	4	$1.569522 \times 10^{- 17}$	4	$3.826143 \times 10^{- 14}$	4	$9.666727 \times 10^{- 6}$
5	$1.298220 \times 10^{- 20}$	5	$2.674431 \times 10^{- 17}$	5	$7.646567 \times 10^{- 15}$	5	$9.269420 \times 10^{- 6}$
6	$2.027311 \times 10^{- 20}$	6	$4.297010 \times 10^{- 17}$	6	$4.574078 \times 10^{- 15}$	6	$9.209937 \times 10^{- 6}$

6 Numerical solutions

The numerical solutions proposed for the eigenvalue problem Eqs (17) are based on the two combined procedures carried out between the Runge-Kutta solver and the finding-root algorithm of the shooting method. To apply both of them, we need to switch Eqs (17) into an initial value problem by providing other boundary conditions to Eq. (17d),

Ψ (0) = 1, θ_{f} = s_{1}, θ_{s} = s_{2}

(18)

We take into consideration the normalisation condition of $Ψ (0) = 1$ . The two unknown's parameters s₁ and s₂ are defined as values for $θ_{s} (0)$ and $θ_{f} (0)$ , in the meantime they are assigned as one of those input parameters fixed in the shooting method. This latter is also implemented via the function Find Root of the Mathematica10 software. In general, the process employed here allows us to develop accurate values for s₁, s₂ and $R a$ . The resulting function of $R a (k)$ is employed to draw the marginal stability curves for every input parameter (H, γ, B_f).

Table 2 shows a comparison of the critical values noted for work [20] in the case where Robin and Neumann thermal boundary conditions are used with the presence of the free surface, and the critical values calculated for our problem with the same thermal conditions but with rigid impermeable walls. If we look at the modified Rayleigh number defined in this paper Eq. (3) is the same as the one employed in [12] and not the one used by [20]. The incongruence in the definition will make the comparison of the results a little bit unclear or hardly seen. To overcome this discrepancy, we calculate the critical values for both $R_{c}$ and $R a_{c}$ by considering the same conditions γ, $B_{f}$ and $H$ , mentioned in [20]. From Table 1 we can remark that the values of $R_{c}$ and $R a_{c}$ in our problem differ from one another. This is somehow contrasted with what arises in the Brinkman problem whose $R_{c}$ and $R a_{c}$ are nearly equal (see [11]). As a consequence, the parameter γ has a more vital impact on Darcy's configuration than Brinkman one. On the other hand, the results in [20] point out that as much as γ decreases, the $R_{c}$ goes closer to zero, in other words, the basic flow becomes alone unstable when $γ \to 0$ . Overall, this limiting case makes the effect of other parameters on the instability behaviour unseen, and this why we have taken by preference the definition of $R a_{c}$ rather than $R_{c}$ . The difference between the critical values in both studied cases does not depend only on the definition of $R_{c}$ but also on the hydrodynamic boundary conditions employed by each problem. As is well known the instability in a porous layer emerges at low critical values in the case of an open boundary than an impermeable wall. This in general upholds the finding results in Table 2.

Table 2.

Comparison of Celli et al. [20] results with the study problem

B_f = 1 and H = 0.1
The work of Celli et al. [20]			The studied problem
γ	$k_{c}$	$R_{c}$	γ	$k_{c}$	$R_{c}$	$R a_{c} = \frac{1 + γ}{γ} R_{c}$
1	1.40561	5.01644	1	$2.06170$	$12.22724$	$24.45449$
0.5	1.51745	3.68189	0.5	$2.23459$	$8.82849$	$26.48548$
0.25	1.62350	2.38060	0.25	$2.38631$	$5.62602$	$28.13013$

7 Results and discussion

The frames in Figs 2 –5 present the trends of Ra_c and k_c versus B_f with various values of γ while the marginal curves drawn for γ and H with fixed B_f are displayed in Fig. 6. The dashed line describes the critical value of Ra_c and k_c obtained for cases close to the LTE model. We recall that stability always exists in the region situated below the concave shape of the neutral curves, while instability emerges in the area above the concave curves. In Fig. 6 the right frame reveals that decreasing the values of γ tend the marginal stability curves to move continuously upward, therefore prescribing a more stable state. In addition, the results that emerge from Figs 2 –5 ensure a non-monotonic increase of the critical values with respect to γ. Furthermore, the effect of γ→0 does not seem to neglect the influence of H as Ra_c = ∞ with H→∞. This feature enables the basic state to behave as adiabatic conditions. Regardless of the effect of H, the limit of γ→∞ leads the values of Ra_c and k_c to be nearly as Ra_c = 27.10, k_c = 2.33 apart from the ones in Fig. 3 and in Fig. 5 whose critical values are different, this refers to the definition of Ra_c. Therefore, a higher value of γ can be regarded as one of the destabilizing factors in this case. Otherwise, the growth of H can bridge the gap between LTNE and LTE when γ has a finite value and B_f →∞. In other words, the parameters of H, B_f and γ can have stabilising or destabilising effect as they are based on the thermal conductivity of the solid and fluid phase besides the inter-phase volumetric heat transfer coefficient h.

Fig. 2.

The plots of Ra_c and k_c versus B_f for H = 1/10

Citation: International Review of Applied Sciences and Engineering 14, 3; 10.1556/1848.2022.00577

Fig. 3.

The trends of Ra_c and k_c versus B_f for H = 10

Citation: International Review of Applied Sciences and Engineering 14, 3; 10.1556/1848.2022.00577

Fig. 4.

The trends of Ra_c and k_c versus B_f for H = 1/100

Citation: International Review of Applied Sciences and Engineering 14, 3; 10.1556/1848.2022.00577

Fig. 5.

The plots of Ra_c and k_c versus B_f for H = 100

Citation: International Review of Applied Sciences and Engineering 14, 3; 10.1556/1848.2022.00577

Fig. 6.

Marginal stability curves relative to H→0 (right frame) and γ→0 (left frame)

Citation: International Review of Applied Sciences and Engineering 14, 3; 10.1556/1848.2022.00577

8 Conclusion

The studied configuration of the modified Darcy-Benard problem is an approach to the design of metal foams saturated with Newtonian fluid. The solid matrix of the metal foams has thermal conductivity infinitely higher than its counterpart fluid phase, which means the presence of the LTNE model. This design is used as a heat exchanger to enhance the transfer of heat from the cooling Newtonian fluid to the solid body, as a result it can be a good alternative to the finned surfaces. According to the numerical results, the stability effects are much more dominant in γ→0 than in γ→∞ when no heat is transferred between the two phases. Besides, the curves $R a$ relative to H→0 exhibit more sensitivity via the value of γ when B_f →∞. Broadly speaking, the behaviour of γ, H and B_f can hasten the performance of heat exchange in metal foams between the porous layer and a saturating fluid.

Acknowledgement

This research was not funded by any grant.

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International Review of Applied Sciences and Engineering

Print ISSN:: 2062-0810

Online ISSN:: 2063-4269

Editorial Board

Mohammad Nazir AHMAD, Institute of Visual Informatics, Universiti Kebangsaan Malaysia, Malaysia

Murat BAKIROV, Center for Materials and Lifetime Management Ltd., Moscow, Russia

Nicolae BALC, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

Umberto BERARDI, Toronto Metropolitan University, Toronto, Canada

Ildikó BODNÁR, University of Debrecen, Debrecen, Hungary

Sándor BODZÁS, University of Debrecen, Debrecen, Hungary

Fatih Mehmet BOTSALI, Selçuk University, Konya, Turkey

Samuel BRUNNER, Empa Swiss Federal Laboratories for Materials Science and Technology, Dübendorf, Switzerland

István BUDAI, University of Debrecen, Debrecen, Hungary

Constantin BUNGAU, University of Oradea, Oradea, Romania

Shanshan CAI, Huazhong University of Science and Technology, Wuhan, China

Michele De CARLI, University of Padua, Padua, Italy

Robert CERNY, Czech Technical University in Prague, Prague, Czech Republic

Erdem CUCE, Recep Tayyip Erdogan University, Rize, Turkey

György CSOMÓS, University of Debrecen, Debrecen, Hungary

Tamás CSOKNYAI, Budapest University of Technology and Economics, Budapest, Hungary

Anna FORMICA, IASI National Research Council, Rome, Italy

Alexandru GACSADI, University of Oradea, Oradea, Romania

Eugen Ioan GERGELY, University of Oradea, Oradea, Romania

Janez GRUM, University of Ljubljana, Ljubljana, Slovenia

Géza HUSI, University of Debrecen, Debrecen, Hungary

Ghaleb A. HUSSEINI, American University of Sharjah, Sharjah, United Arab Emirates

Nikolay IVANOV, Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia

Antal JÁRAI, Eötvös Loránd University, Budapest, Hungary

Gudni JÓHANNESSON, The National Energy Authority of Iceland, Reykjavik, Iceland

László KAJTÁR, Budapest University of Technology and Economics, Budapest, Hungary

Ferenc KALMÁR, University of Debrecen, Debrecen, Hungary

Tünde KALMÁR, University of Debrecen, Debrecen, Hungary

Milos KALOUSEK, Brno University of Technology, Brno, Czech Republik

Jan KOCI, Czech Technical University in Prague, Prague, Czech Republic

Vaclav KOCI, Czech Technical University in Prague, Prague, Czech Republic

Imre KOCSIS, University of Debrecen, Debrecen, Hungary

Imre KOVÁCS, University of Debrecen, Debrecen, Hungary

Angela Daniela LA ROSA, Norwegian University of Science and Technology, Trondheim, Norway

Éva LOVRA, Univeqrsity of Debrecen, Debrecen, Hungary

Elena LUCCHI, Eurac Research, Institute for Renewable Energy, Bolzano, Italy

Tamás MANKOVITS, University of Debrecen, Debrecen, Hungary

Igor MEDVED, Slovak Technical University in Bratislava, Bratislava, Slovakia

Ligia MOGA, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

Marco MOLINARI, Royal Institute of Technology, Stockholm, Sweden

Henrieta MORAVCIKOVA, Slovak Academy of Sciences, Bratislava, Slovakia

Phalguni MUKHOPHADYAYA, University of Victoria, Victoria, Canada

Balázs NAGY, Budapest University of Technology and Economics, Budapest, Hungary

Husam S. NAJM, Rutgers University, New Brunswick, USA

Jozsef NYERS, Subotica Tech College of Applied Sciences, Subotica, Serbia

Bjarne W. OLESEN, Technical University of Denmark, Lyngby, Denmark

Stefan ONIGA, North University of Baia Mare, Baia Mare, Romania

Joaquim Norberto PIRES, Universidade de Coimbra, Coimbra, Portugal

László POKORÁDI, Óbuda University, Budapest, Hungary

Roman RABENSEIFER, Slovak University of Technology in Bratislava, Bratislava, Slovak Republik

Mohammad H. A. SALAH, Hashemite University, Zarqua, Jordan

Dietrich SCHMIDT, Fraunhofer Institute for Wind Energy and Energy System Technology IWES, Kassel, Germany

Lorand SZABÓ, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

Csaba SZÁSZ, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

Ioan SZÁVA, Transylvania University of Brasov, Brasov, Romania

Péter SZEMES, University of Debrecen, Debrecen, Hungary

Edit SZŰCS, University of Debrecen, Debrecen, Hungary

Radu TARCA, University of Oradea, Oradea, Romania

Zsolt TIBA, University of Debrecen, Debrecen, Hungary

László TÓTH, University of Debrecen, Debrecen, Hungary

László TÖRÖK, University of Debrecen, Debrecen, Hungary

Anton TRNIK, Constantine the Philosopher University in Nitra, Nitra, Slovakia

Ibrahim UZMAY, Erciyes University, Kayseri, Turkey

Andrea VALLATI, Sapienza University, Rome, Italy

Tibor VESSELÉNYI, University of Oradea, Oradea, Romania

Nalinaksh S. VYAS, Indian Institute of Technology, Kanpur, India

Deborah WHITE, The University of Adelaide, Adelaide, Australia

International Review of Applied Sciences and Engineering
Address of the institute: Faculty of Engineering, University of Debrecen
H-4028 Debrecen, Ótemető u. 2-4. Hungary
Email: irase@eng.unideb.hu

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2023
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Scimago H-index	11
Scimago Journal Rank	0.249
Scimago Quartile Score	Architecture (Q2) Engineering (miscellaneous) (Q3) Environmental Engineering (Q3) Information Systems (Q4) Management Science and Operations Research (Q4) Materials Science (miscellaneous) (Q3)
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Scopus Cite Score	2.3
Scopus CIte Score Rank	Architecture (Q1) General Engineering (Q2) Materials Science (miscellaneous) (Q3) Environmental Engineering (Q3) Management Science and Operations Research (Q3) Information Systems (Q3)
Scopus SNIP	0.751

International Review of Applied Sciences and Engineering
Publication Model	Gold Open Access Online only
Submission Fee	none
Article Processing Charge	1100 EUR/article
Regional discounts on country of the funding agency	World Bank Lower-middle-income economies: 50% World Bank Low-income economies: 100%
Further Discounts	Limited number of full waivers available. Editorial Board / Advisory Board members: 50% Corresponding authors, affiliated to an EISZ member institution subscribing to the journal package of Akadémiai Kiadó: 100%
Subscription Information	Gold Open Access

International Review of Applied Sciences and Engineering
Language	English
Size	A4
Year of Foundation	2010
Volumes per Year	1
Issues per Year	3
Founder	Debreceni Egyetem
Founder's Address	H-4032 Debrecen, Hungary Egyetem tér 1
Publisher	Akadémiai Kiadó
Publisher's Address	H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
Responsible Publisher	Chief Executive Officer, Akadémiai Kiadó
ISSN	2062-0810 (Print)
ISSN	2063-4269 (Online)