Keywords

5.1 Introduction

In the industrial sector, materials with higher porosity such as metal foams are often used to enhance heat exchange between two bodies or structures. Furthermore, their large surface area and lightweight make them good candidates for recycling energy efficiently. In this case, the usual Darcy’s law becomes no longer suitable for describing fluid motion and, it is necessary to adopt the Brinkman-Darcy model instead (Nield 2017; Dubey and Murthy 2019; Bouachir et al. 20121; Caprone and Rionero 2016).

Moreover, many man-made materials like porous media have anisotropic mechanical and thermal properties. Because of that, thermal instability can be managed by the permeability and thermal conductivity of the medium (Storesletten and Rees 1997; Tyvand and Storesletten 2015; Govender and Vadasz 2007; Mahjoob and Vafai 2008). Rotation also plays a vital role in thermoconvective instability, for example rotating machinery and centrifugal filtration processes. A reference to a rotating frame must be established to investigate the situations in which the solid matrix rotates (Vadasz 2016, 2019; Govender and Vadasz 2007). Otherwise, thermal instability or convection in the subjects of rotating solid matrix, free and forced convection layer with cavities, a non-Darcian model with open boundaries or saturating Oldroyd-B fluid are ones of the several papers that deal with the convection phenomenon (Malashetty et al. 2006; Shivakumara et al. 2006; Rees 2002; Baytas 2004; Saeid 2004). Most papers in the last decade are predicated on the idea that the Newtonian fluid phase and the solid medium are everywhere under the same temperature, which is known as the thermal equilibrium regime (LTE). However, this assumption becomes inadequate for many real-world applications, especially when high-speed flows or significant temperature gradients between both phases are present (Fathi-Kelestani et al. 2020; Gandomkar and Gray 2018; Lagziri and Bezzazi 2019; Barletta 2019). In such cases, it is necessary to consider a two-field energy equation model to represent each phase separately, and this emerges as a virtue of no thermal equilibrium behaviour (LTNE). In addition, it is anticipated that the model of LTNE will have an important role in future technology consisting of porous media such as computer chips, tube refrigerators, heat exchangers and others (Mahjoob and Vafai 2008; Pulvirenti et al. 2020).

The chapter studies the emergence of thermal instability cells in a non-Darcian flow using mixed thermal boundary conditions and a thermal non-equilibrium regime. The influence of these two features on instability behaviour is investigated in detail.

5.2 Mathematical Modeling

A homogeneous fluid-saturated porous layer sandwiched between two rigid impermeable walls is shown in Fig. 5.1. An external heat source is imposed at the lower boundary with two different external heat transfer coefficients \(h_f\) and \(h_s\). In other words, Robin’s thermal boundary conditions are considered. On the other hand, uniform perfect conducting temperature \(T_1\) is applied at the upper layer. The local thermal nonequilibrium model and linear Oberbeck Boussinesq approximation are both pertinent here. The Brinkman-Darcy equation describes the saturating non-Darcian flow in a solid matrix. Following these descriptions the Mathematical equations of the problem are

$$\begin{aligned}&{{\nabla }^*}\cdot \vec {{u}^*}=0, \end{aligned}$$
(5.1a)
$$\begin{aligned}&\rho \left[ \dfrac{1}{\phi }\dfrac{\partial {\vec {u^*}}}{\partial {t^*}}+\dfrac{1}{\phi ^2}(\vec {u^*}\cdot \nabla ^*)\vec {u^*}\right] +\dfrac{\mu }{K}~\vec {{u}^*}=-{{\nabla }^*} P^*+{\mu }'{{\nabla }^{* 2}}\vec {{u}^{*}} +{\rho _f}{\beta }{g}({T}^{*}_{f}-T_{0}) \vec {ez}, \end{aligned}$$
(5.1b)
$$\begin{aligned}&(1-{\phi }) (\rho C)_s\dfrac{\partial {{T}^{*}_{s}}}{\partial {t^{*}}}=(1-{\phi }){\alpha _s}~{{\nabla }^{*}}^2 {T}^{*}_{s}+{h}({T}^{*}_{f}-{T}^{*}_{s}), \end{aligned}$$
(5.1c)
$$\begin{aligned}&{\phi }(\rho C)_f \dfrac{\partial {{T}^{*}_{f}}}{\partial {t^{*}}}+(\rho C)_f (\vec {{u}^{*}}\cdot {{\nabla }^{*})}{T}^{*}_{f}=\phi {\alpha _f}~{{\nabla }^{*}}^2 {T}^{*}_{f}-{h}({T}^{*}_{f}-{T}^{*}_{s}). \end{aligned}$$
(5.1d)

The temperature and velocity conditions suggested at the boundary are

$$\begin{aligned}&\qquad \qquad \qquad \qquad \quad {{z}^{*}}=d:\quad {w}^{*}=0, {u}^{*}=0, \quad T^*_s=T^*_f=T_1. \end{aligned}$$
(5.2a)
$$\begin{aligned}&{{z}^{\star }}=0:\quad {w}^{*}=0, {u}^{*}=0,\quad k_{s}\dfrac{\partial {{T}^{*}_{s}}}{\partial {z^{*}}}= h_{s}({T}^{*}_{s}-T_{0}),\quad k_{f}\dfrac{\partial {{T}^{*}_{f}}}{\partial {z^{*}}}= h_{f}({T}^{*}_{f}-T_{0}) \end{aligned}$$
(5.2b)
Fig. 5.1
figure 1

Sketch of Brinkman model with mixed thermal boundary conditions

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The superscript of the star notation refers to dimensional variables. The vector \(\vec {u}\) means the velocity field. Otherwise, we have also other physical properties such as the thermal diffusivity \(\alpha \) with \([m^2/s]\), the effective viscosity \(\mu '\) with \([N.s/m^2]\), the dynamic viscosity of the fluid \(\mu \) with \([N.s/m^2]\), the coefficient of thermal expansion \(\beta \) the pressure P, the temperature at the lower wall \(T_0\), the reference density \(\rho \) with \([Kg/m^3]\), the inter-phase volumetric heat transfer coefficient h with \([W/(m^3 K)]\), the heat capacity per unit of mass C with [J/(KgK)], the porosity \(\phi \), the time t with [s], the thickness of the layer d with [m], the thermal conductivity k with [W/(mK)], the superficial heat transfer coefficient between both phases \(h_{s,f}\) with \([W/(m^2 K)]\) and the permeability K with \([m^2]\).

The dimensionless expression of the governing equations is,

$$\begin{aligned} \nabla \times (D^{-1}\vec {u}- \nabla ^2 \vec {u})=R D^{-1}[{\nabla }\times {T_f}] \vec {ez}, \nonumber \\ {\lambda }\dfrac{\partial {T_s}}{\partial {t}}=\nabla ^2 {T_s}+{\gamma }H({{T_f}}-{{T_s}}), \nonumber \\ \dfrac{\partial {{T_f}}}{\partial {t}}+\vec {u} \cdot \nabla T_f=\nabla ^2{T_f}+H({{T_s}}-{{T_f}}), \end{aligned}$$
(5.3)

We get the resulting boundary conditions as

$$\begin{aligned}&z=1: \quad {w}=0,\quad {u}=0, \quad T_s=T_f= 0, \nonumber \\z=0: \quad {w}=0&,\quad {u}=0,\quad \dfrac{\partial {T_s}}{\partial {z}}=B_{s}(T_{s}-1),\quad \dfrac{\partial {T_f}}{\partial {z}}=B_{f} (T_{f}-1). \end{aligned}$$
(5.4)

The notations “s” and “f” signify the saturating Newtonian fluid phase and solid structure. The dimensionless forms of R and H mean the modified thermal Rayleigh number and the inter-heat transfer coefficient while \(B_s\) and \(B_f\) describe the Biot number of the solid matrix and fluid phase. Besides, we have \(\gamma \) and D whose physical meanings are the thermal conductivity ratio and the Darcy number respectively.

The dimensionless parameters that emerge in (5.3) and (5.4) are:

$$\begin{aligned} {\lambda }=\dfrac{\alpha _f}{\alpha _s},\quad k_m=(1-{\phi })k_s+{\phi }k_f,\quad {\gamma }=\dfrac{{\phi }{k_f}}{(1-{\phi }){k_s}}, \nonumber \\ H=\dfrac{{h}{{d}^2}}{{\phi }{k_f}},\quad B_{s,f}=\dfrac{h_{s,f} d}{k_{s,f}},\quad {\alpha _f}=\dfrac{k_f}{({\rho }C)_f}, \quad {\alpha _m}=\dfrac{k_m}{({\rho }C)_f}, \nonumber \\ \quad D=\dfrac{\mu ' K}{\mu d^2},\quad R=\dfrac{{\rho _f}{\beta }{\Delta {T}}{g}{K}{d}}{{\nu }{k_f} \phi }, \quad \Delta {T}=\dfrac{\phi (T_1-T_0) d}{k_m}. \end{aligned}$$
(5.5)

where the rescaling variables applicable in the set of governing equations are,

$$\begin{aligned} {\nabla }^{*}\rightarrow {\nabla } \dfrac{1}{d}, \quad {t}^{*} \rightarrow \dfrac{{d}^2}{\alpha _f}, \quad \vec {u^{*}} \rightarrow \vec {u} \frac{{\phi }{\alpha _f}}{d}, \quad {T}^{*}_{s,f} \rightarrow T_{0}+ {T_{s,f}}{\Delta {T}}. \end{aligned}$$
(5.6)

5.3 Basic Profile

We consider a motionless flow whose basic state is

$$\begin{aligned} \vec {u}_b=0,\quad T_{f,b}=T_{fb}(z), \quad T_{s,b}=T_{sb}(z). \end{aligned}$$
(5.7)

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

5.3.1 Linear Stability Analysis

Let us disturb the basic flow by writing that

$$\begin{aligned}&\vec {u}=\vec {u}_b+\epsilon \vec {U} , \nonumber \\&T_{fs}=T_{sf,b} (z)+\epsilon {\theta _{s,f}}. \end{aligned}$$
(5.8)

We are concerned only with the first-order terms of disturbances, therefore, the linearized form of the equations is

$$\begin{aligned}&\qquad \quad \nabla \times (D^{-1}\vec {U}- \nabla ^2 \vec {U})=R D^{-1}[{\nabla }\times {\theta _f}] \vec {ez}=0, \end{aligned}$$
(5.9a)
$$\begin{aligned}&\qquad \quad \quad \qquad \qquad {\theta _s}''-a^2 {\theta _s}+{\gamma }H({\theta _f}-{\theta _s})=0, \end{aligned}$$
(5.9b)
$$\begin{aligned}&\qquad \qquad \qquad \quad {\theta _f}''-a^2 {\theta _f}+H({\theta _s}-{\theta _f})-a T'_{f,b} {\vec {U}}=0, \end{aligned}$$
(5.9c)
$$\begin{aligned}&\qquad \quad \quad \qquad z=1: \quad {W}=0,\quad {U}=0, \quad {\theta _s}={\theta _f}=0. \end{aligned}$$
(5.9d)
$$\begin{aligned}&z=0: \quad {W}=0,\quad {U}=0, \quad {\theta _s}'- {\theta _s} B_{s}=0,\quad {\theta _f}'-{\theta _f} B_{f}=0, \end{aligned}$$
(5.9e)

Now we apply the normal modes method by defining the functions as

$$\begin{aligned}&\vec {U}(x,z,t)=\Re \{i \hat{\vec {U}}(z) e^{i(a x - \omega t)}\} , \nonumber \\&\theta _{f}(x,z,t)=\Re \{\theta (z) e^{i(a x - \omega t)}\}, \nonumber \\&\theta _{s}(x,z,t)=\Re \{\varphi (z) e^{i(a x - \omega t)}\} \end{aligned}$$
(5.10)

Hence the symbols of \(\hat{\vec {U}}\), \(\theta \) and \(\varphi \) are used to describe the perturbed functions with respect to z. The wave number is defined by the symbol a while the growth rate and the angular frequency are noted with \({\omega _r}\) and \({\omega _i}\) respectively. The complex parameter \(\omega \) is defined as the sum of the imaginary and real parts.

In the meantime, the velocity components can be expressed in the stream functions as,

$$\begin{aligned} \hat{U}=\dfrac{\partial {\psi }}{\partial {z}}, \quad \hat{W}=-\dfrac{\partial {\psi }}{\partial {x}}. \end{aligned}$$
(5.11)

Otherwise, the definition of \(T'_{fb}\) is

$$\begin{aligned} T'_{fb}=\dfrac{-B_f B_s (\gamma +1) \sinh (\varOmega )+(-\varOmega ) (B_f \gamma +B_s) \cosh (\varOmega )+(B_s-B_f) \varOmega \cosh (\varOmega z)}{(B_f (B_s \gamma +B_s+1)+B_s \gamma ) \sinh (\varOmega )+\varOmega (B_f \gamma +B_s+\gamma +1) \cosh (\varOmega )}. \end{aligned}$$
(5.12)

With \(\varOmega =\sqrt{(1+\gamma )H}\).

By substituting (5.10) and (5.11) into (5.13) the set of equations become,

$$\begin{aligned}&-{\psi }''''+(2 a^2 +D^{-1}){\psi }''-a^2(a^2+D^{-1}){\psi }+a D^{-1}R {\theta } =0, \end{aligned}$$
(5.13a)
$$\begin{aligned}&\qquad \qquad \qquad \qquad \quad {\varphi }''-a^2 {\varphi }+{\gamma }H({\theta }-{\varphi })=0, \end{aligned}$$
(5.13b)
$$\begin{aligned}&\qquad \qquad \qquad \qquad {\theta }''-a^2 {\theta }+H({\varphi }-{\theta })-a T'_{f,b} {\psi }=0, \end{aligned}$$
(5.13c)
$$\begin{aligned}&\qquad \qquad \qquad \quad \quad z=1: \quad {W}=0,\quad {U}=0, \quad {\varphi }={\theta }=0. \end{aligned}$$
(5.13d)
$$\begin{aligned}&z=0: \quad {W}=0,\quad {U}=0, \quad {\varphi }'- {\varphi } B_{s}=0,\quad {\theta }'-{\theta } B_{f}=0, \end{aligned}$$
(5.13e)

As our motivation is to seek the marginal curves, the imaginary part of \(\omega \) has to be neglected. Meanwhile, the principle of exchange of instabilities is achieved numerically thus we can eliminate both parts of \(\omega \) and write (5.13).

5.4 Numerical Solutions

The numerical method adopted for dealing with (5.13) is the shooting method and Range-Kutta solver. In general, this latter required the definition of extra boundary conditions as a first step to manage (5.13) in the form of an initial value problem. Thus, we can add

$$\begin{aligned} \psi ''(0)=\zeta _1,\quad \psi '''(0)=\zeta _2,\quad \varphi (0)= \zeta _{3},\quad {\theta (0)}=1. \end{aligned}$$
(5.14)

The condition noted by \(\theta (0)=1\) is included as a virtue of the homogeneity in the governing equations. The parameters of \(\zeta _1, \zeta _2\) and \(\zeta _3\) are considered as unknowns with real values. The next step here is to define these unknown constants together with R for any given value of \(H, \gamma , a, D, B_s\) and \(B_f\) through the use of the shooting method and boundary conditions of the upper wall. The shooting method consists in employing the root-finding algorithm in Mathematica 10 to determine the value pair of \((R_c,a_c)\).

Table 5.1 Critical values of \(R_c\) and \(a_c\) for fixed \(\gamma =1\) and variable values of H, \(B_s\), \(B_f\) and D
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5.5 Discussion and Results

Table 5.1 exhibits the critical values of the modified Rayleigh number and wave number for different values of \(B_s\), \(B_f\) and H in the cases where \(D=0.01\) and \(D=0.1\). Overall, we note that the Darcy number consists in having the viscous diffusion at the region nearer to the boundary layers. In other words, as much as the viscous effects decrease the fluid can flow and move more rapidly and easily without resistance, thereby the onset of convection can be yielded at a small critical Rayleigh number. Besides, we mention the values of \(H=10\) and \(H=0.1\) as the two approaches of the thermal equilibrium and non-equilibrium one. A small value of H leads the heat to be poorly exchanged between the two phases whereas for a higher one the ability to heat transfer becomes extremely large. Otherwise, the range assumed for \(B_s\) and \(B_f\) extends from \(10^{-2}\) to \(10^{2}\) to recover the both thermal conditions of uniform heat flux and perfect conducting temperature. The finding results of Table 5.1 show that the thermal stability increases in the cases where the combined effect of LTNE and fluid inertia is present. In addition, we can notice that even both Biot numbers can have the tendency to emerge stable behaviour.

Table 5.2 Comparison of \(R_c\) and \(a_c\) with others results for the case of \(D=0\) and \(\gamma =1\)
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Table 5.3 Comparison of \(R_c\) and \(a_c\) with others results for the case of \(D=1\) and \(\gamma =1\)
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The results computed by our numerical method in Tables 5.2, 5.3 for the case of \(D=0\) (Darcian flow) and \(D=1\) display a good congruence with those of Postelnicu (2008) and Shivakumara (2010). These two tables confirm that the fluid inertia can retard the fluid motion which decreases later the buoyancy effects in the medium.

The neutral curves evaluated numerically for various values of \(\gamma \) and fixed \(D=0.01\) are drawn in Fig. 5.2. We remind that stability takes place in the regions situated below the concave curves. In fact, all these curves follow the same standard shape of the well-known Benard problem. Therefore, if we look at the behaviour of these curves in the function of \(\gamma \) and both Biot numbers we can notice that stability effects increase with the reduction of these two parameters. The small value of \(\gamma \) manages the heat to be transported only through the solid structure, this in turn slows the onset of convection especially when both phases at the upper layer have no ability to enter or outer the heat with the external environment. Otherwise, Figs. 5.3, 5.4 display the variation of \(R_c\) and \(a_c\) with respect to H for \(D=0.01\). The broken lines in both figures describe the critical values at thermal equilibrium. The results extracted from both figures confirm that the stability effects prevail more in the case of uniform heat flux as \(R_c=11427.15\) and \(a_c=2.5\) for \(\gamma =10\).

Fig. 5.2
figure 2

Neutral curves for \(H=100\) and \(D=0.01\)

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Fig. 5.3
figure 3

Plots for \(R_c\) and \(a_c\) versus H with \(B_f=B_s=1000\) and \(D=0.01\)

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Fig. 5.4
figure 4

Plots for \(R_c\) and \(a_c\) versus H with \(B_f=B_s=0.001\) and \(D=0.01\)

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5.6 Conclusion

The combined effect of non Darcian model and LTNE regime in a porous layer with mixed thermal boundary conditions is investigated in this chapter. The root-finding algorithm of the shooting method and Runge-Kutta solver are considered to solve numerically the eigenvalue problem tackled by linear stability analysis. Briefly, the finding results may be summed up as

  • The thermal non-equilibrium (LTNE) regime with weak heat exchange at the upper layer by fluid phase creates more stability than the solid one.

  • The growth of Darcy’s number hastens the stability as a result of the fluid inertia. In other words, the buoyancy effects become less dominant in front of inertia effects.

  • The reduction in both Biot numbers brings about stabilizing effects in the medium either in LTE or LTNE model.