Nonlinear thermal instability in a horizontal porous layer with an internal heat source and mass flow

Linear and nonlinear stability analyses of Hadley–Prats flow in a horizontal fluid-saturated porous medium with a heat source are performed. The results indicate that, in the linear case, an increase in the horizontal thermal Rayleigh number is stabilizing for both positive and negative values of mass flow. In the nonlinear case, a destabilizing effect is identified at higher mass flow rates. An increase in the heat source has a destabilizing effect. Qualitative changes appear in Rz as the mass flow moves from negative to positive for different internal heat sources.


Introduction
Thermal convection driven by an internal heat source with a horizontal mass flow has many practical applications such as underground energy transport, cooling of nuclear reactors, food processing, oil recovery, underground storage of waste products and thermal convection in clouds [1][2][3][4].
In the last few decades, convection involving internal heat sources has attracted particular research attention. Early experimental investigations include Schwiderski and Schwabh [5] and Tritton and Zarraua [6], with Roberts [7] and Thirlby [8] providing theoretical analyses. Parthiban and Patil [9] investigated thermal convection due to non-uniform heating boundaries with inclined thermal gradients in the presence of an internal heat source, followed by an extension to anisotropic porous layers by Parthiban and Patil [10]. The case of an inclined layer with internal heat source was analyzed by Barletta et al. [11], where both boundaries were isothermal and kept at the same temperatures. Rionero and Straughan [12] investigated the linear and nonlinear effects in the presence of variable gravity effect and heat generation. Extensive reviews of the theory and applications can be found in the article by Alex and Patil [13]. Hill [14] investigated a porous layer with concentration-based internal heat generation, with linear and nonlinear stability analyses of thermosolutal convection. Chamka et al. [15] analyzed the effect of an internal heat source or sink for hydromagnetic simultaneous heat and mass transfer by utilizing similarity solutions. Thermosolutal convection in a saturated anisotropic porous medium with internal heat generation is reported by Bhadauria [16]. Borujerdi et al. [17] examined the steady-state heat conduction with a uniform heat source where the solid and fluid phases are at different temperatures. Borujerdi et al. [18] study the influence of the Darcy number on the critical Rayleigh number in the onset of convection with uniform internal heating. A collection of comprehensive theories and experiments of thermal convection in porous media (with their practical applications) has been surveyed by Nield and Bejan [19]. Capone and Rionero [20] have studied the nonlinear stability of a convective motion in a horizontal porous layer which is driven by a temperature gradient. Several problems on nonlinear stability analyses using the energy method are discussed by Kaloni and his contributors [21][22][23][24][25]. In the Lyapunov sense, when the disturbance of the basic flow is unstable, linearized theory provides sufficient conditions, whereas nonlinear theory provides sufficient conditions for the disturbance to be asymptotically stable.
The aim of this article is to study the influences of both a heat source and a mass flow. The corresponding eigenvalue problems are solved numerically utilizing the shooting and Runge-Kutta method.

Mathematical analysis
An infinite shallow horizontal fluid-saturated porous medium with thickness d is considered. The z -axis is vertically upwards, and there is a net flow along the direction of the x -axis with magnitude M . The vertical temperature difference across the boundaries is θ . Further imposed is the horizontal temperature gradient vector (β θ x , β θ y ). The porous layer flow is governed by the Darcy law, where the linear Boussinesq approximation is assumed. Utilizing the equation of the conservation of energy, the governing equations in dimensional form are with the following boundary conditions: Here, the Darcy velocity is defined as q = (u , v , w ), P is the pressure, θ is temperature, and Q is an internal heat source. The subscripts f and m refer to the fluid and the porous medium, respectively. φ and K are the porosity and permeability of the porous layer. c, ρ, μ, k m and γ θ denote the specific heat, density, viscosity, thermal diffusivity, and thermal expansion coefficient in the porous medium, respectively. The following dimensionless variables are introduced to non-dimensionalize the governing equations: where Here, R z denotes the vertical thermal Rayleigh number. The horizontal thermal Rayleigh numbers are defined as follows: The previous scaling for dimensional variables and the horizontal thermal Rayleigh numbers were introduced by Weber [27] and used extensively by Nield [28]. Under these dimensionless variables, the governing Eqs. (1)- with the conditions of the plates being From Eqs. (8)- (10), we observe that all of the thermal Rayleigh numbers are involved in boundary conditions (11). The condition on temperature at both bounding planes give a linear variation of temperature. This spatial linear variation of temperature along the horizontal planes bounding a fluid layer is a physically more realistic situation than the strictly uniform heating (Capone and Rionero [20]). However, in the present problem uniform heating can be recovered by setting the horizontal thermal gradients to zero.

Steady-state solution
The flow governing Eqs. (8)-(10), subject to (11), have a basic state solution of the form with Here D = d dz , and we have a net flow M in the horizontal direction such that The solution in the form of flow velocity and temperature in the medium is then given by

Perturbation equations
We consider the perturbations in the form q = q s + q, θ = θ s + θ and P = P s + P. By substituting these perturbations in the dimensionless governing Eqs. (8)-(10), we get ∂θ ∂t + q s · ∇θ + q· ∇θ s + q· ∇θ = ∇ 2 θ where The conditions at the plates are Note that (19) shows that there are no normal velocity and temperature perturbations at the plates.

Linear stability analysis
To perform a linear stability analysis, we neglect the nonlinear terms from Eq. (18). The linearized perturbations equations are then ∂θ ∂t + q s · ∇θ + q· ∇θ s = ∇ 2 θ where The conditions at the plates are Adopting a normal mode solution to Eqs. (20)-(22) of the form and further eliminating P yields Equations (25) and (26), subject to w = θ = 0 at both the plates z = 1 2 and z = − 1 2 , constitute an eigenvalue problem for the vertical thermal Rayleigh number R z with a, R x , R y , k and l as parameters. In the above, α = √ k 2 + l 2 is the overall wave number. Numerical results are presented in Sect. 7.

Nonlinear stability analysis
In this section, our nonlinear analysis via energy functional is presented as follows. We multiply Eqs. (17) and (18) by q and θ , respectively, and integrate over Ω, where Ω denotes a typical periodicity cell. This yields the following identities: Here ||· || and · denote the norm and inner product on L 2 (Ω). We adopt the energy functional (cf. [29]) with coupling parameter ξ > 0. The system of Eqs. (27) and (28) where We define where H is the space of all admissible solutions to Eqs. (16)- (18). If 0 < n < 1, it follows that The classical Poincare inequality ||q − q Ω || L p (Ω) ≤ C||∇q|| L p (Ω) , where Ω is a open connected locally compact Hausdorff space and use of q Ω = 1 |Ω| Ω q (y) dy yields Equation (35) then guarantees that E (t) → 0 as t → ∞ for 0 < n < 1. Applying the arithmetic-geometric mean inequality on Eq. (27) yields From Eqs. (36) and (29), it follows that the decay of ||q|| is implied by the decay of E (t), and hence, the system is stable. From the above, we have identified that the nonlinear stability requires the critical argument at n = 1. The corresponding Euler-Lagrange system with the maximum problem Eq. (33) is Here δ is a Lagrange multiplier introduced because q is divergence free. We consider R z as the eigenvalue and estimate the maximum variation of R z with optimal choice of ξ . Equations (37) and (38) yield Equation (39) is important and also noted that if R x = R y = 0 and Q = 0, we get Equation (40) is the same as the expressions reported by Guo and Kaloni [21]. We solve the system of Eqs. (37) and (38) in the presence of the critical value n = 1. To evaluate this system numerically, we apply curlcurl of Eq. (37) and further use the third component of the resulting equation with R x , R y · (k, l) = 0, (Nield [26] and Kaloni and Qiao [22]), i.e., the horizontal thermal Rayleigh number vector is orthogonal to the wave number vector. We substitute (42) in Eqs. (37), (38) and (41) and eliminate u, v and δ to obtain The system of Eqs. (43) and (44) is evaluated with the boundary conditions w = θ = 0 at z = ± 1 2 . The critical vertical thermal Rayleigh number is obtained as R z = max ξ min α 2 R z .

Results and discussion
The onset of thermal convection in a fluid-saturated porous layer in the presence of mass flow and an internal heat source effect is analyzed using both linear and nonlinear stability theory. Both cases are studied based on the classical normal mode technique. We treat the vertical thermal Rayleigh number as the eigenvalue R z . Here, the critical vertical thermal Rayleigh number R z is defined as the minimum of all R z values as the wave number α is varied. The vector of wave number is defined as α = (k, l, 0). To achieve the stationary convection boundary, we set σ = 0 (the removal of the oscillatory mode is discussed in "Appendix"), with R x , R y · (k, l) = 0. The longitudinal disturbances are characterized by k = 0. In the same way, transverse disturbances are characterized by l = 0. In Table 1, R z l and R z e indicate the linear and nonlinear critical thermal Rayleigh number (R z ). α l and α e indicate the critical wave number in linear and nonlinear cases.
From Table 1, it is observed that when Q = 0 and M = 0, in the linear case, the results are in very good agreement with earlier published results in the literature, Nield [26]. For an increase in the value of Q from 0 to 2, the critical value of R z is reduced as seen in Table 1 in both cases. Hence, the heat flow parameter causes destabilization in the medium. A fixed notation is used to represent the curves corresponding to the linear and nonlinear results. Dotted lines represent linear stability results, and solid lines represent nonlinear stability results in Figs. 1, 2 and 3.   Fig. 1 at R x = 10 and R y = 0. It is observed that as Q increases the critical values of R z decrease. However, as M increases from −2 to 2, at a higher value of M, the critical R z value is lower than at lower values of M in both the cases as seen in Fig. 1. For both positive and negative values of M, R z decreases with increasing values of Q. It indicates that increasing the heat source has a strongly destabilizing effect. This is due to the fact that the global temperature of the system is increasing with increasing heat source and causes the instability. As increasing the heat source, the threshold critical region between linear and nonlinear results is increasing as seen in Fig. 1.
The response of R z with varying R x is shown in Fig. 2 for negative and positive values of M. It is noted that, at M = −2 in the linear case, the critical value of R z is higher than all the remaining values of M. When R x is increased, the corresponding R z values also increase for all M values. This indicates that the flow rate is strongly stabilizing in the linear case as compared to the nonlinear case seen in Fig. 2. It is also interesting to observe that as R x increases the critical value of R z also increases up to a certain value of R x ; thereafter, the R z value decreases for M = 2 in the nonlinear case seen in Fig. 2. It means that the flow rate is strongly destabilizing at higher values of R x and M in the nonlinear case. Figure 3 shows the response of R z with mass flow rate M in the presence of different values of Q = 0, 1, 2 for R x = 10.
As the mass flow rate (M) increases, R z also increases up to certain values of M and thereafter decreases for all values of Q as demonstrated in Fig. 3. Q has a strongly stabilizing effect up to a certain value of M, then after strongly destabilizing the flow in both cases. It is seen that by increasing the magnitude of horizontal mass flow in both negative and positive directions the critical value of R z is decreased.

Conclusions
We have analyzed the instability of thermal convection in Hadley-Prats flow subject to an internal heat source using linear and nonlinear stability analysis. The results yield the following conclusions: -An increase in the internal heat source causes a strong destabilization in all cases, as it raises the global temperature of the system. -In the presence of horizontal mass flow, the flow is stabilizing at higher horizontal Rayleigh numbers in the linear case, whereas it is destabilizing in the nonlinear case at larger mass flows. -Qualitative changes appear in R z as the mass flow moves from negative to positive for different internal heat sources.