Mixed convection in a liquid-saturated densely packed porous medium using local thermal non-equilibrium model

Abstract

An analytic study of Darcy–Bénard convection in a Newtonian liquid-saturated porous medium in the presence of pressure gradient and heat source using local thermal non-equilibrium model (LTNE) is carried out. The presumption of LTNE hastens the convection onset and augments the amount of heat transport. The effect of increasing the porosity-modified ratio of thermal conductivity and heat source fosters the convection onset and inflates the amount of heat transport, while an opposite trend is noticed in case of the remaining parameters. The effect of LTNE ceases and results of Darcy–Bénard convection using LTE model are obtained as a limiting case for large values of ratio of thermal conductivities and interphase heat transfer coefficient. Asymptotic analysis is carried out to determine the point at which the effect of LTNE ceases.

Appendices

Appendix A

This section involves the asymptotic analysis in case of $H\rightarrow 0$, $H\rightarrow \infty$ and $\gamma \rightarrow \infty$ for the existence of heat source in the fluid phase by linear stability analysis (see Sect. (3.1)). (32) is expanded in powers of H, $\dfrac{1}{H}$ and $\dfrac{1}{\gamma }$ for the particular cases of $H\rightarrow 0$, $H\rightarrow \infty$ and $\gamma \rightarrow \infty$, respectively.

Linear stability analysis

1.1 Existence of heat source in fluid phase:

1.1.1 Asymptotic analysis $(H\rightarrow 0)$

In this case, $R_{D}$ takes the form

$$\begin{aligned} R_{D}=\frac{\delta ^2 \left( -Q_{f}+\delta ^2\right) }{a^2}\left( 1+\frac{ H}{\left( -Q_{f}+\delta ^2\right) }-\frac{\gamma \delta ^2 H^2}{\left( -Q_{f}+\delta ^2\right) \left( \delta ^4+a^2 \alpha ^2 \Pi ^2\right) }....\right) \end{aligned}$$

(57)

The critical wave number $a_{c}$ can be derived by minimizing $R_{D}$ with respect to a and equating $\dfrac{\partial R_{D}}{\partial a}=0$, keeping other parameters as constant, we obtain:

$$\begin{aligned} \begin{aligned}&\left( a^4-\pi ^4+\pi ^2 Q_f\right) \left( \delta ^4+a^2 \alpha ^2 \Pi ^2\right) ^2-\pi ^2 H\left( \delta ^4+a^2 \alpha ^2 \Pi ^2\right) ^2\\&\quad +\delta ^2 \gamma \left( \delta ^6+2 a^2 \pi ^2 \alpha ^2 \Pi ^2\right) H^2+..............=0. \end{aligned} \end{aligned}$$

(58)

Asymptotic expansion of a for small values of H, is given by:

$$\begin{aligned} a=a_{0}+a_{1}H+a_{2}H^{2}. \end{aligned}$$

(59)

where

$$\begin{aligned} a_{0}^{2}=\pi ^2-\pi \sqrt{Q_f} \end{aligned}$$

(60)

The first two corrections to the wave number are given by:

$$\begin{aligned} a_1=\frac{\pi ^2 \left( \delta _{0}^{2}+a_{0}^{2} \alpha ^2 \Pi ^2\right) }{4 a_0\left( \delta _{0}^{2}\left( 3 a_{0}^{4}+a_{0}^{2} \pi ^2-2\pi ^4+2 \pi ^2 Q_f\right) +\left( 2 a_{0}^{4}-\pi ^4+\pi ^2 Q_f\right) \alpha ^2 \Pi ^2\right) },\;\ a_2=\frac{\Delta _{11}}{\Delta _{12}}, \end{aligned}$$

(61)

where

$$\begin{aligned} \Delta _{11}&=\left( 2 a_{1} \delta _{0}^{4}\left( \delta _{0}^{2} \left( -33 a_{0}^{4} a_{1}+a_{0} \left( 4+9 a_0 a_1\right) \pi ^{2}+2 a_1 \pi ^4\right) -2 a_1 \pi ^2 \left( 7 a_{0}^{2}+\pi ^2\right) Q_{f}\right) \right. \\&\quad \left. -\delta _{0}^{8} \gamma -2 \alpha ^{2} \left( a_1\left( 45 a_{0}^{8} a_{1}+2 a_{0}^{5} \left( -3+28 a_0 a_1\right) \pi ^2-8 a_{0}^3 \pi ^4-2 a_0 \left( 1+6 a_0 a_1\right) \pi ^6\right. \right. \right. \\&\quad \left. \left. \left. -a_1 \pi ^8+a_1 \pi ^2 \left( 15 a_{0}^4+12 a_{0}^2 \pi ^2+\pi ^4\right) Q_f\right) +a_{0}^2 \pi ^2 \delta _{0}^2\gamma \right) \Pi ^2+2 a_{0}^2 a_1 \left( -14 a_{0}^4 a_1\right. \right. \\&\quad \left. \left. +2 a_0 \pi ^2+3 a_1\pi ^2 \left( \pi ^2-Q_f\right) \right) \alpha ^4 \Pi ^4\right) ,\\ \Delta _{12}&=\left( 4 a_0 \left( \delta _{0}^{4}+a_{0}^{2} \alpha ^2 \Pi ^2\right) \left( \delta _{0}^{2}\left( 3 a_{0}^{4}+a_{0}^{2} \pi ^2-2 \pi ^4+2 \pi ^2 Q_f\right) +\left( 2 a_{0}^{4}-\pi ^4+\pi ^2 Q_f\right) \alpha ^2 \Pi ^2\right) \right) . \end{aligned}$$

1.1.2 Asymptotic analysis $(H\rightarrow \infty )$

In this case, $R_{D}$ takes the form

$$\begin{aligned} R_D=\frac{\delta ^2 \left( \delta ^2(1+\gamma )-\text {Qf} \gamma \right) }{a^2 \gamma }-\frac{\delta ^2 \left( \delta ^4-a^2 \alpha ^2 \Pi ^2\right) }{a^2 \gamma ^2 H}+\frac{\delta ^4 \left( \delta ^4-3 a^2 \alpha ^2 \Pi ^2\right) }{a^2 \gamma ^3 H^2}.... \end{aligned}$$

(62)

Asymptotic expansion of a for $H\rightarrow \infty$ is given by:

$$\begin{aligned} a=a_0+a_1H^{-1}+a_2H^{-2} \end{aligned}$$

(63)

where

$$\begin{aligned} a_0{}^2 =\pi \sqrt{\frac{\left( \pi ^2 (1+\gamma )-Q_f\gamma \right) }{(1+\gamma )}} \end{aligned}$$

(64)

The first two nonzero corrections to $a_{c}$ are:

$$\begin{aligned} a_1=\frac{\Delta _{13}}{\Delta _{14}},\;\;\ a_2=\frac{\Delta _{15}}{\gamma \Delta _{14}}, \end{aligned}$$

(65)

where

$$\begin{aligned} \Delta _{13}& = {} a_{0}^6+3 a_{0}^{4} \pi ^2+3 a_{0}^{2} \pi ^4+\pi ^6-a_{0}^{4} \alpha ^2 \Pi ^2-a_{0}^{2} \pi ^2 \alpha ^2 \Pi ^2\\ \Delta _{14} & = 2 a_0 \gamma \left( 2 a_{0}^{2}+2 \pi ^2+2 a_{0}^{2} \gamma +2 \pi ^2 \gamma -Q_f \gamma \right) \\ \Delta _{15}& = a_0 \alpha ^2 \left( 3 a_{0} \left( a_{0}^{2}+\pi ^2\right) {}^2-2 \text {a1} \left( 2 a_{0}^{2}+\pi ^2\right) \gamma \right) \Pi ^2-a_{0}^{8}-4 a_{0}^{6} \pi ^2-6 a_{0}^{4} \pi ^4-4 a_{0}^{2} \pi ^6-\pi ^8\\{} & {} +6 a_{0}^{5} a_{1} \gamma +12 a_{0}^{3} a_1 \pi ^2 \gamma +6 a_0 a_1 \pi ^4 \gamma -6 a_{0}^{2} a_{1}^{2} \gamma ^2-2 a_{1}^{2} \pi ^2 \gamma ^2-6 a_{0}^{2} a_{1}^{2} \gamma ^3-2 a_{1}^{2} \pi ^2 \gamma ^3+a_{1}^{2} Q_{f} \gamma ^3. \end{aligned}$$

1.1.3 Asymptotic analysis $(\gamma \rightarrow \infty )$

In this case, $R_{D}$ takes the form

$$\begin{aligned}{} & {} R_D=\frac{\delta ^2\left( \delta ^2-Q_f\right) }{a^2}\left( 1+\frac{\delta ^2}{\left( \delta ^2-Q_f\right) \gamma }-\frac{\left( \delta ^4-a^2 \alpha ^2 \Pi ^2\right) }{\left( \delta ^2-Q_f\right) H \gamma ^2}.........\right) \end{aligned}$$

(66)

$$\begin{aligned}{} & {} a=a_0+a_1\gamma ^{-1}+a_2\gamma ^{-2} \end{aligned}$$

(67)

where

$$\begin{aligned} a_{0}^{2}=\pi ^2-\pi \sqrt{Q_f} \end{aligned}$$

(68)

The first two nonzero corrections to $a_{c}$ are:

$$\begin{aligned} a_1=\frac{\delta _{0}^{4}}{2a_{0}Q_{f}-4 a_{0} \delta _{0}^{2}},\;\;\ a_2=\frac{\Delta _{16}}{\Delta _{17}}, \end{aligned}$$

(69)

where

$$\begin{aligned} \begin{aligned} \Delta _{16}&=a_{0}^{6}-4 a_{0}^{3} a_1 H-4 a_0 a_1 H \pi ^2+\pi ^6+a_{1}^{2} H \left( -2 \pi ^2+Q_f\right) +a_{0}^{4} \left( 3 \pi ^2-\alpha ^2 \Pi ^2\right) \\&\quad -a_{0}^{2} \left( 6 a_{1}^{2} H-3 \pi ^4+\pi ^2 \alpha ^2 \Pi ^2\right) ,\\ \Delta _{17}&=4 a_0H \delta _{0}^{2}-2 a_0H Q_f. \end{aligned} \end{aligned}$$

Appendix B

This section involves the asymptotic analysis in case of $H\rightarrow 0$, $H\rightarrow \infty$ and $\gamma \rightarrow \infty$ for the existence of heat source in the fluid phase by linear stability analysis (see Sect. (3.2)). (33) is expanded in powers of H, $\dfrac{1}{H}$ and $\dfrac{1}{\gamma }$ for the particular cases of $H\rightarrow 0$, $H\rightarrow \infty$ and $\gamma \rightarrow \infty$, respectively.

1.1 Existence of heat source in the solid phase

1.1.1 Asymptotic analysis $(H\rightarrow 0)$

In this case, $R_{D}$ takes the form

$$\begin{aligned} R_D=\frac{\delta ^4}{a^2}\left( 1+\frac{ H}{\delta ^2}-\frac{\left( -Q_s \gamma +\gamma \delta ^2\right) H^2}{\delta ^2\left( Q_s{}^2-2 Q_s \delta ^2+\delta ^4+a^2 \alpha ^2 \Pi ^2\right) }......\right) \end{aligned}$$

(70)

minimizing $R_{D}$ given by Eq. (70) with respect to a, and by setting $\dfrac{\partial R_{D}}{\partial a}=0$, we get an expression as follows:

$$\begin{aligned} \begin{aligned}&\left( a^4-\pi ^4\right) \left( \left( \delta ^2-Q_s\right) {}^2+a^2 \alpha ^2 \Pi ^2\right) {}^2-\pi ^2 H\left( \left( \delta ^2-Q_s\right) {}^2+a^2 \alpha ^2 \Pi ^2\right) {}^2\\&\quad +\gamma \left( \left( \delta ^2-Q_s\right) {}^2\left( \delta ^4-\pi ^2 Q_s\right) +a^2 \left( 2 \pi ^2\delta ^2 -\left( a^2+2 \pi ^2\right) Q_s\right) \alpha ^2 \Pi ^2\right) H^2=0 \end{aligned} \end{aligned}$$

(71)

Asymptotic expansion of a for small values of H, is given by:

$$\begin{aligned} a=a_{0}+a_{1}H+a_{2}H^{2}. \end{aligned}$$

(72)

where

$$\begin{aligned} a_{0}^{2}=\pi ^2,\;\;\ a_1=\frac{\pi ^2 \left( \left( \delta _{0}^{2}-Q_s\right) {}^2+a_{0}^{2} \alpha ^2 \Pi ^2\right) }{\Delta _{21}},\;\;\ a_2=\frac{\Delta _{22}}{\left( \left( \delta _{0}^{2}-Q_s\right) ^2+a_{0}^{2} \alpha ^2 \Pi ^2\right) \Delta _{21}}, \end{aligned}$$

(73)

where

$$\begin{aligned} \Delta _{21}&=4 a_0 \left( 3 a_{0}^{6}+a_{0}^{2} \left( -\pi ^4-2 \pi ^2 Q_s+Q_s^2\right) +2 a_{0}^{4} \left( 2 \pi ^2-2 Q_s+\alpha ^2 \Pi ^2\right) -\pi ^4 \left( 2 \pi ^2-2 Q_s+\alpha ^2 \Pi ^2\right) \right) , \end{aligned}$$

(74)

$$\begin{aligned} \Delta _{22}&=\left( -66 a_{0}^{10} a_{1}^{2}+8 a_{0}^{7} a_1 \pi ^2+12 a_{0}^{5} a_1 \pi ^2 \left( 2 \pi ^2-2 Q_s+\alpha ^2 \Pi ^2\right) +4 a_0a_1 \left( \pi ^3-\pi Q_s\right) ^2\right. \nonumber \\&\quad \left. \left( 2 \pi ^2-2 Q_s+\alpha ^2 \Pi ^2\right) +a_0={0}^{4} \left( \left( \pi ^2-Q_s\right) \left( 60 a_{1}^{2} \left( 2 \pi ^2-Q_s\right) Q_s+\left( -6 \pi ^2+Q_s\right) \gamma \right) \right. \right. \nonumber \\&\quad \left. \left. +\left( 2 \pi ^2-Q_s\right) \alpha ^2 \left( 30 a_{1}^{2} Q_s-\gamma \right) \Pi ^2\right) +4 a_{0}^{3} a_1 \pi ^2 \left( 6 \left( \pi ^2-Q_s\right) ^2+4 \left( \pi ^2-Q_s\right) \alpha ^2 \Pi ^2+\alpha ^4 \Pi ^4\right) \right. \nonumber \\&\quad \left. +2 a_{0}^{2} \left( \left( \pi ^2-Q_s\right) ^2 \left( 3 a_{1}^{2} \left( 5 \pi ^4+2 \pi ^2 Q_s-Q_s^2\right) -2 \pi ^2 \gamma \right) +\pi ^2 \left( \pi ^2-Q_s\right) \alpha ^2 \left( 12 a_{1}^{2} \pi ^2-\gamma \right) \Pi ^2\right. \right. \nonumber \\&\quad \left. \left. +3 a_{1}^{2} \pi ^4 \alpha ^4 \Pi ^4\right) -a_{0}^{8} \left( \gamma +90 a_{1}^{2} \left( 2 \pi ^2-2 Q_s+\alpha ^2 \Pi ^2\right) \right) +\left( \pi ^3-\pi Q_s\right) ^2 \left( \left( -\pi ^2+Q_s\right) \gamma \right. \right. \nonumber \\&\quad +\left. \left. 2 a_{1}^{2} \pi ^2 \left( 2 \pi ^2-2 Q_s+\alpha ^2 \Pi ^2\right) \right) -2 a_{0}^{6} \left( \left( 2 \pi ^2-Q_s\right) \gamma +14 a_{1}^{2} \left( 5 \pi ^4-12 \pi ^2 Q_s+6 Q_s^2\right. \right. \right. \nonumber \\&\quad +\left. \left. \left. 4 \left( \pi ^2-Q_s\right) \alpha ^2 \Pi ^2+\alpha ^4 \Pi ^4\right) \right) \right) \end{aligned}$$

(75)

1.1.2 Asymptotic analysis $(H\rightarrow \infty )$

In this case, $R_{D}$ takes the form

$$\begin{aligned} \begin{aligned} R_D&=\frac{\delta ^2}{a^2}\left( \frac{\left( \delta ^2(1-\gamma )-Q_s\right) }{\gamma }-\frac{\left( \delta ^4-2 Q_s\delta ^2+\text {Qs}^2-a^2 \alpha ^2 \Pi ^2\right) }{ \gamma ^2 H}\right. \\&\quad \left. +\frac{\left( \delta ^2-Q_s\right) \left( \left( \delta ^2-Q_s\right) {}^2-3 a^2\alpha ^2 \Pi ^2\right) }{\gamma ^3 H^2}.....\right) \end{aligned} \end{aligned}$$

(76)

Minimizing $R_{D}$ given by Eq. (76) with respect to a, and equating $\dfrac{\partial R_{D}}{\partial a}=0$, we get an expression as follows:

$$\begin{aligned} \begin{aligned}&2\gamma ^{-1} \left( a^4 (1+\gamma )-\pi ^2 \left( -Q_s+\pi ^2 (1+\gamma )\right) \right) \gamma ^{-1}+2 \gamma ^{-2}H^{-1}\left( -2 a^6+\left( \pi ^3-\pi Q_s\right) ^2\right. \\&\quad +\left. a^4 \left( -3 \pi ^2+2 Q_s+\alpha ^2 \Pi ^2\right) \right) +2 \gamma ^{-3}H^{-2}\left( a^2+\pi ^2-Q_s\right) ^2 \left( 3 a^4+2 a^2 \pi ^2-\pi ^4+\pi ^2 Q_s\right) \\&\quad +6 a^4 \left( -2 \left( a^2+\pi ^2\right) +Q_s\right) \alpha ^2 \Pi ^2=0. \end{aligned} \end{aligned}$$

(77)

Similarly, a for large values of H takes the form:

$$\begin{aligned} a=a_0+a_1H^{-1}+a_2H^{-2}, \end{aligned}$$

(78)

where

$$\begin{aligned} a_{0}^{2} = \frac{\pi \sqrt{\left( -Q_s+\pi ^2 (1+\gamma )\right) }}{\sqrt{(1+\gamma )}}, \end{aligned}$$

(79)

and

$$\begin{aligned} a_1=\frac{\delta _{0}^{2} \left( \left( \delta _{0}^{2}-Q_s\right) ^2-a_{0}^{2} \alpha ^2 \Pi ^2\right) }{2 a_0 \gamma \left( -Q_s+2 \delta _{0}^{2}(1+\gamma )\right) },\;\;\ a_2=\frac{\Delta _{24}}{\Delta _{23}}, \end{aligned}$$

(80)

where

$$\begin{aligned} \Delta _{23}&=2 a_0 \gamma \left( -Q_s+2 \delta _{0}^{2}(1+\gamma )\right) \\ \Delta _{24}&=\left( -a_{0}^{8}-\pi ^2 \left( \pi ^2-Q_s\right) ^3+6 a_{0}^{5} a_1 \gamma +a_{1}^{2} \left( -2 \pi ^2+Q_s\right) \gamma ^2-2 a_1{}^2 \pi ^2 \gamma ^3+4 a_0{}^3 a_1 \gamma \right. \\ &\qquad \left. \times \left( 3 \pi ^2-2 Q_s-\alpha ^2 \Pi ^2\right) -3 a_0{}^4 \left( 2 \pi ^2-Q_s\right) \left( \pi ^2-Q_s-\alpha ^2 \Pi ^2\right) +a_0{}^6 \left( -4 \pi ^2+3 \left( Q_s+\alpha ^2 \Pi ^2\right) \right) \right. \\ &\qquad \left. +2 a_0 a_1 \gamma \left( 3 \pi ^4+Q_s{}^2-\pi ^2 \left( 4 Q_s+\alpha ^2 \Pi ^2\right) \right) +a_0{}^2 \left( -6 a_1{}^2 \gamma ^2 (1+\gamma )-\left( \pi ^2-Q_s\right) \right. \right. \\ &\qquad \left.\times \left. \left( 4 \pi ^4+Q_s{}^2-\pi ^2 \left( 5 Q_s+3 \alpha ^2 \Pi ^2\right) \right) \right) \right) \end{aligned}$$

1.1.3 Asymptotic analysis $(\gamma \rightarrow \infty )$

In this case, $R_{D}$ takes the form

$$\begin{aligned} R_D=\frac{\delta ^4}{a^2}\left( 1+\frac{\left( \delta ^2-Q_s\right) }{\delta ^2\gamma }-\frac{\left( \delta ^4-2 \delta ^2 Q_s+Q_s^2-a^2 \alpha ^2 \Pi ^2\right) }{\delta ^2H \gamma ^2}.......\right) . \end{aligned}$$

(81)

Minimizing $R_{D}$ given by Eq. (81) with respect to a and by equating $\dfrac{\partial R_{D}}{\partial a}=0$, we get

$$\begin{aligned}{} & {} 2\left( a^4-\pi ^4\right) +2 \gamma ^{-1}\left( a^4-\pi ^4+\pi ^2 Q_s\right) +2 H^{-1} \gamma ^{-2}\left( -2 a^6+\left( \pi ^3-\pi Q_s\right) ^2\right. \nonumber \\{} & {} \quad \left. +a^4 \left( -3 \pi ^2+2 Q_s+\alpha ^2 \Pi ^2\right) \right) =0 \end{aligned}$$

(82)

Similarly, a for large values of $\gamma$ takes the form:

$$\begin{aligned} a=a_0+a_1\gamma ^{-1}+a_2\gamma ^{-2} \end{aligned}$$

(83)

where

$$\begin{aligned} a_{0}^{2}=\pi ^2,\;\;\ a_1=\frac{Q_s-\delta _{0}^{2}}{4 a_0},\;\;\ a_2=\frac{\Delta _{25}}{4 a_0 H \delta _{0}^{2}}, \end{aligned}$$

(84)

where

$$\begin{aligned} \Delta _{25} & = a_{0}^{6}-4 a_{0}^{3} a_1 H+\pi ^2 \left( -2 a_{1}^{2} H+\left( \pi ^2-Q_s\right) ^2\right) +2 a_0a_1 H \left( -2 \pi ^2+Q_s\right) +a_{0}^{4} \left( 3 \pi ^2-2 Q_s-\alpha ^2 \Pi ^2\right) \nonumber \\{} & {} +a_{0}^{2} \left( -6 a_1{}^2 H+3 \pi ^4+Q_s^2-\pi ^2 \left( 4 Q_s+\alpha ^2 \Pi ^2\right) \right) . \end{aligned}$$

(85)

Appendix C

This section involves the asymptotic analysis in case of $H\rightarrow 0$, $H\rightarrow \infty$ and $\gamma \rightarrow \infty$ for the existence of heat source in the fluid phase by nonlinear stability analysis (see Sect. (4.1.1)). (47) is expanded in powers of H, $\dfrac{1}{H}$ and $\dfrac{1}{\gamma }$ for the particular cases of $H\rightarrow 0$, $H\rightarrow \infty$ and $\gamma \rightarrow \infty$, respectively.

1.1 Nonlinear stability analysis

1.1.1 Existence of heat source in fluid phase

1.1.2 Asymptotic analysis $(H\rightarrow 0)$

In this case, $Nu_{w}$ takes the form:

$$\begin{aligned} Nu_{w}=G_{1}+G_{2}H+G_{3}H^{2}+..., \end{aligned}$$

(86)

where $\delta _{0}^{2}=a_{0}^{2}+\pi ^{2}$,

$$\begin{aligned} G_1&=1-\frac{2 \left( \delta _{0}^{2} \left( \delta _{0}^{2}-Q_f\right) -a_{0}^{2} R_D\right) \epsilon }{a_{0}^2 R_D}, \end{aligned}$$

(87)

$$\begin{aligned} G_2&=\frac{1}{2 a_{0}^{3} \pi ^2 R_D}\left( a_{0}^{5} \gamma (-1+\epsilon )-8 a_{0}^{4} a_1 \pi ^2 \epsilon +8 a_1 \pi ^4 \left( \pi ^2-Q_f\right) \epsilon +a_0 \pi ^2 \left( -Q_f \gamma (-1+\epsilon )+\pi ^2\right. \right. \nonumber \\&\quad \left. \left. (-\gamma +(-4+\gamma ) \epsilon )\right) +a_{0}^{3} \left( -\left( Q_f+R_D\right) \gamma (-1+\epsilon )+2 \pi ^2 (-\gamma +(-2+\gamma ) \epsilon )\right) \right) , \end{aligned}$$

(88)

$$\begin{aligned} G_3&=\frac{1}{8 a_{0}^{4} \pi ^4 R_D}\left( -a_{0}^{6} \gamma ^2 (-1+\epsilon )+48 a_{1}^{2}\pi ^6 \left( -\pi ^2+Q_f\right) \epsilon +a_{0}^{4} \left( -2 \pi ^2 (-2+\gamma ) \gamma (-1+\epsilon )\right. \right. \nonumber \\&\quad \left. \left. +\left( Q_f+R_D\right) \gamma ^2 (-1+\epsilon )-16 a_{1}^{2} \pi ^4\epsilon \right) -8 a_{0}^{5} \pi ^2 \left( 4 a_2\pi ^2\epsilon +a_1(\gamma -\gamma \epsilon )\right) +a_{0}^{2} \pi ^2 \gamma \left( Q_f\gamma (-1+\epsilon )\right. \right. \nonumber \\&\quad \left. \left. +\pi ^2 (-4+\gamma +20 \epsilon -\gamma \epsilon )\right) +8 a_0 \pi ^4 \left( 4 a_2 \pi ^2 \left( \pi ^2-Q_f\right) \epsilon +a_1\left( \text {Qf} \gamma (-1+\epsilon )+\pi ^2 (\gamma +4 \epsilon -\gamma \epsilon )\right) \right) \right) . \end{aligned}$$

(89)

1.1.3 Asymptotic analysis $(H\rightarrow \infty )$

In this case, $Nu_{w}$ takes the form:

$$\begin{aligned} Nu_{w}=G_{4}+G_{5}\frac{1}{H}+G_{6}\frac{1}{H^{2}}+..., \end{aligned}$$

(90)

where $\delta _{0}^{2}=a_{0}^{2}+\pi ^{2}$,

$$\begin{aligned} G_4&=\frac{\left( 3 a_{0}^{2} R_D-2 \delta _{0}^{2} \left( \delta _{0}^{2}-Q_f\right) \right) \gamma -2 \delta _{0}^{4}}{a_{0}^{2} R_D \gamma },\end{aligned}$$

(91)

$$\begin{aligned} G_5&=\frac{1}{a_{0}^{3} R_D \gamma ^2}\left( 2 \left( a_{0}^{7}-2 a_{0}^{4} a_1\gamma (1+\gamma )+2 a_1 \pi ^2 \gamma \left( -Q_f \gamma +\pi ^2 (1+\gamma )\right) +a_{0}^{5} \pi ^2 (7+4 \gamma \right. \right. \nonumber \\&\quad \left. \left. -4 (1+\gamma ) \epsilon )+a_{0}^{3} \pi ^2 \left( 4 \left( Q_f+R_D\right) \gamma (-1+\epsilon )+\pi ^2 (11+8 \gamma -8 (1+\gamma ) \epsilon )\right) +a_0\pi ^4 \right. \right. \nonumber \\&\quad \left. \left. \left( 4 Q_f \gamma (-1+\epsilon )+\pi ^2 (5+4 \gamma -4 (1+\gamma ) \epsilon )\right) \right) \right) , \end{aligned}$$

(92)

$$\begin{aligned} G_6&=\frac{1}{a_{0}^{4} R_D \gamma ^3}2 \left( 4 a_{0}^{7} a_1 \gamma -a_{0}^{10}-3 a_1{}^2 \pi ^2 \gamma ^2 \left( -Q_f \gamma +\pi ^2 (1+\gamma )\right) +4 a_{0}^{8} \pi ^2 (-2+\epsilon )\right. \nonumber \\ & \quad \ \left. -2 a_{0}^{6} \pi ^4 (17+8 \gamma -2 (7+4 \gamma ) \epsilon )-a_{0}^{4} \left( a_{1}^{2} \gamma ^2 (1+\gamma )+16 \pi ^4 \left( Q_f+R_D\right) \gamma (-1+\epsilon )\right. \right. \nonumber \\&\quad \ \left. \left. +4 \pi ^6 (12+8 \gamma -11 \epsilon -8 \gamma \epsilon )\right) -2 a_{0}^{5} \gamma \left( a_2 \gamma (1+\gamma )+a_1 \pi ^2 (-7-4 \gamma +4 (1+\gamma ) \epsilon )\right) \right. \nonumber \\&\quad \ \left. -a_{0}^{2} \pi ^6 \left( 16 Q_f \gamma (-1+\epsilon )+\pi ^2 (21+16 \gamma -4 (5+4 \gamma ) \epsilon )\right) +2 a_0\pi ^2 \gamma \left( a_2 \gamma \left( -Q_f \gamma \right. \right. \right. \nonumber \\&\quad \left. \left. \left. +\pi ^2 (1+\gamma )\right) +a_1 \pi ^2 \left( -4 Q_f \gamma (-1+\epsilon )+\pi ^2 (-5-4 \gamma +4 (1+\gamma ) \epsilon )\right) \right) \right) . \end{aligned}$$

(93)

1.1.4 Asymptotic analysis $(\gamma \rightarrow \infty )$

In this case, $Nu_{w}$ takes the form:

$$\begin{aligned} Nu_{w}=G_{7}+G_{8}\frac{1}{\gamma }+G_{9}\frac{1}{\gamma ^{2}}+..., \end{aligned}$$

(94)

where $\delta _{0}^{2}=a_{0}^{2}+\pi ^{2}$,

$$\begin{aligned} G_7&=3-\frac{2 \delta _{0}^{2}\left( \delta _{0}^{2}-Q_f\right) }{a_{0}^{2} R_D} \end{aligned}$$

(95)

$$\begin{aligned} G_8&=\frac{1}{a_{0}^{3} H R_D}\left( 2 \left( 2 a_1 H \pi ^2 \left( \pi ^2-Q_f\right) -2 a_0{}^4 a_1 H-a_{0}^{5} \left( H+4 \pi ^2 (-1+\epsilon )\right) -a_0 \pi ^4 \left( H+ \right. \right. \right. \nonumber \\&\quad \left. \left. \left. 4 \left( \pi ^2-Q_f\right) (-1+\epsilon )\right) -2 a_{0}^{3} \pi ^2 \left( H+2 \left( 2 \pi ^2-Q_f-R_D\right) (-1+\epsilon )\right) \right) \right) \end{aligned}$$

(96)

$$\begin{aligned} G_9&=\frac{1}{a_{0}^{4} H^2 R_D}2 \left( a_{0}^{8} H+3 a_{1}^{2} H^2 \pi ^2 \left( -\pi ^2+Q_f\right) -2 a_{0}^{5} H \left( a_2H+a_1\left( H+4 \pi ^2 (-1+\epsilon )\right) \right) \right. \nonumber \\&\quad \ \left. +2 a_0 H \pi ^2 \left( a_2H \left( \pi ^2-Q_f\right) +a_1 \pi ^2 \left( H+4 \left( \pi ^2-Q_f\right) (-1+\epsilon )\right) \right) +a_{0}^{4} \left( -a_{1}^{2} H^2+\pi ^4 \right. \right. \nonumber \\ &\quad \ \left. \left. \left( H (11-8 \phi )+16 \left( 2 \pi ^2-Q_f-R_D\right) (-1+\epsilon )\right) \right) +a_{0}^{6} \pi ^2 \left( H (7-4 \epsilon )+16 \pi ^2 (-1+\epsilon )\right) \right. \nonumber \\&\quad \ \left. +a_0{}^2 \pi ^6 \left( H (5-4 \phi )+16 \left( \pi ^2-Q_f\right) (-1+\epsilon )\right) \right. \end{aligned}$$

(97)

Appendix D

This section involves the asymptotic analysis in case of $H\rightarrow 0$, $H\rightarrow \infty$ and $\gamma \rightarrow \infty$ for the existence of heat source in the fluid phase by nonlinear stability analysis (see Sect. (4.1.2)). (56) is expanded in powers of H, $\dfrac{1}{H}$ and $\dfrac{1}{\gamma }$ for the particular cases of $H\rightarrow 0$, $H\rightarrow \infty$ and $\gamma \rightarrow \infty$, respectively.

1.1 Existence of heat source in solid phase

1.1.1 Asymptotic analysis $(H\rightarrow 0)$

In this case, $Nu_{w}$ takes the form:

$$\begin{aligned} Nu_{w}=M_{1}+M_{2}H+M_{3}H^{2}+..., \end{aligned}$$

(98)

where $\delta _{0}^{2}=a_{0}^{2}+\pi ^{2}$,

$$\begin{aligned} M_1&=1-\frac{2 \left( \delta _c^4-a_0^2 R_D\right) \epsilon }{a_0^2 R_D}, \end{aligned}$$

(99)

$$\begin{aligned} M_2&=\frac{1}{2 a_0^3 \pi ^2 R_D}\left( a_0 \left( 1+Q_s\right) \left( \delta _c^4-a_0^2 R_D\right) \gamma -\left( 4 a_0^3 \pi ^2+8 a_0^4 a_1 \pi ^2+4 a_0 \pi ^4-8 a_1 \pi ^6+a_0(1+\text {Qs})\right. \right. \nonumber \\&\quad \left. \left. \left( \delta _c^4-a_0^2 R_D\right) \gamma \right) \epsilon \right) , \end{aligned}$$

(100)

$$\begin{aligned} M_3&=\frac{1}{8 a_0^4 \pi ^4 R_D}\left( a_0^6\left( 1+Q_s\right) \gamma ^2 (-1+\epsilon )-48 a_1^2\pi ^8 \epsilon +a_0^2 \pi ^4 \left( 1+Q_s\right) \gamma (4+\gamma (-1+\epsilon )+12 \epsilon )\right. \nonumber \\&\quad \left. -8 a_0^5 \pi ^2 \left( a_1\left( 1+Q_s\right) \gamma (-1+\epsilon )+4 a_2 \pi ^2 \epsilon \right) +a_0^4 \left( 2 \pi ^2 \left( 1+Q_s\right) (-2+\gamma ) \gamma (-1+\epsilon )\right. \right. \nonumber \\&\quad \left. \left. -\left( 1+Q_s\right) R_D \gamma ^2 (-1+\epsilon )-16 a_1^2 \pi ^4 \epsilon \right) +8 a_0 \pi ^6 \left( 4 a_2 \pi ^2\epsilon +a_1\left( \left( 1+Q_s\right) \gamma (-1+\epsilon )+4 \epsilon \right) \right) \right) \end{aligned}$$

(101)

1.1.2 Asymptotic analysis $(H\rightarrow \infty )$

In this case, $Nu_{w}$ takes the form:

$$\begin{aligned} Nu_{w}=M_{4}+M_{5}\frac{1}{H}+M_{6}\frac{1}{H^{2}}+..., \end{aligned}$$

(102)

where $\delta _{0}^{2}=a_{0}^{2}+\pi ^{2}$,

$$\begin{aligned} M_4&=\frac{1}{a_0^3 R_D \gamma }\left( 4 a_1 \pi ^2 Q_s \gamma \left( -1+Q_s(-1+\epsilon )\right) +2a_0^5 \left( 1+Q_s+\gamma \right) \left( -1+Q_s (-1+\epsilon )\right) +2 a_0 \pi ^4\right. \nonumber \\&\quad \left. \left( -\left( 1+Q_s\right) \left( 1+5 Q_s+\gamma \right) +Q_s \left( 5+5 Q_s+\gamma \right) \epsilon \right) +a_0^3 \left( R_D \gamma \left( 3+2 Q_s-2 Q_s\epsilon \right) +4 \pi ^2\right. \right. \nonumber \\&\quad \left. \left. \left( -\left( 1+Q_s\right) \left( 1+3 Q_s+\gamma \right) +Q_s \left( 3+3 Q_s+\gamma \right) \epsilon \right) \right) \right) ,\end{aligned}$$

(103)

$$\begin{aligned} M_5&=\frac{1}{a_0^4R_D \gamma ^2}2 \left( -a_0^8 \left( 1+Q_s\right) \left( -1+Q_s (-1+\epsilon )\right) +2 a_0^5 a_1 \gamma \left( 1+Q_s+\gamma \right) \left( -1+Q_s(-1+\epsilon )\right) \right. \nonumber \\&\quad \left. +3 a_1^2 \pi ^2 Q_s \gamma ^2 \left( 1+Q_s-Q_s \epsilon \right) -a_0^6 \pi ^2 \left( 1+Q_s\right) \left( -7-4 \gamma +7 Q_s (-1+\epsilon )+4 (1+\gamma ) \epsilon \right) -a_0^2 \pi ^6\right. \nonumber \\&\quad \left. \left( 1+Q_s\right) \left( -5-4 \gamma +21 Q_s (-1+\epsilon )+4 (1+\gamma ) \epsilon \right) -a_0^4 \pi ^2 \left( 1+Q_s\right) \left( -4 R_D \gamma (-1+\epsilon )+\pi ^2 \right. \right. \nonumber \\&\quad \left. \left. \left( -11-8 \gamma +27 Q_s (-1+\epsilon )+8 (1+\gamma ) \epsilon \right) \right) +2 a_0 \pi ^2 \gamma \left( a_2 Q_s \gamma \left( -1+Q_s (-1+\epsilon )\right) +a_1 \pi ^2\right. \right. \nonumber \\&\quad \left. \left. \left( 1+\gamma +Q_s \left( 6+5 Q_s+\gamma -\left( 5+5 Q_s+\gamma \right) \epsilon \right) \right) \right) \right) , \end{aligned}$$

(104)

$$\begin{aligned} M_6&=-\frac{1}{a_0^5 R_D \gamma ^3}2 \left( -a_0^{11} \left( 1+Q_s\right) \left( -1+Q_s (-1+\epsilon )\right) +4a_0^8 a_1 \left( 1+Q_s\right) \gamma \left( -1+Q_s(-1+\epsilon )\right) \right. \nonumber \\&\quad \left. -4a_0^9 \pi ^2 \left( 1+Q_s\right) \left( -2+2 Q_s (-1+\epsilon )+\epsilon \right) -2 a_0^7\pi ^4 \left( 1+Q_s\right) \left( -17+17 Q_s (-1+\epsilon )+8 \right. \right. \gamma \nonumber \\&\quad \left. \left. (-1+\epsilon )+14 \epsilon \right) -a_0^3 \pi ^8 \left( 1+Q_s\right) \left( -21+85 Q_s(-1+\epsilon )+16 \gamma (-1+\epsilon )+20 \epsilon \right) +4 a_1^3 \pi ^2 Q_s \gamma ^3 \right. \nonumber \\&\quad \left. \left( 1+Q_s-Q_s \epsilon \right) +a_0^5 \left( -a_1^2 \gamma ^2 \left( 1+Q_s+\gamma \right) \left( -1+Q_s(-1+\epsilon )\right) +16 \pi ^4 \left( 1+Q_s\right) R_D \gamma \right. \right. \nonumber \\&\quad \left. \left. (-1+\epsilon )-4 \pi ^6 \left( 1+Q_s\right) \left( -12+28 Q_s(-1+\epsilon )+8 \gamma (-1+\epsilon )+11 \epsilon \right) \right) +2 a_0^6 \gamma \left( -a_2 \gamma \right. \right. \nonumber \\&\quad \left. \left. \left( 1+Q_s+\gamma \right) \left( -1+Q_s(-1+\epsilon )\right) +a_1\pi ^2 \left( 1+Q_s\right) \left( -7-4 \gamma +7 Q_s (-1+\epsilon )+4 (1+\gamma ) \epsilon \right) \right) \right. \nonumber \\&\quad \left. -3 a_0 a_1 \pi ^2 \gamma ^2 \left( 2 a_2Q_s \gamma \left( 1+Q_s-Q_s \epsilon \right) +a_1 \pi ^2 \left( -\left( 1+Q_s\right) \left( 1+5 Q_s+\gamma \right) +Q_s\left( 5+5 Q_s+\right. \right. \right. \right. \nonumber \\&\quad \left. \left. \left. \left. \gamma \right) \epsilon \right) \right) -2 a_0^2 \pi ^4 \gamma \left( a_1 \pi ^2 \left( 1+Q_s\right) \left( -5-4 \gamma +21 Q_s (-1+\epsilon )+4 (1+\gamma ) \epsilon \right) +a_2 \gamma \left( 1+\gamma +Q_s \right. \right. \right. \nonumber \\&\quad \left. \left. \left. \left( 6+5 Q_s+\gamma -\left( 5+5 Q_s+\gamma \right) \epsilon \right) \right) \right) \right) . \end{aligned}$$

(105)

1.1.3 Asymptotic analysis $(\gamma \rightarrow \infty )$

In this case, $Nu_{w}$ takes the form:

$$\begin{aligned} Nu_{w}=M_{7}+M_{8}\frac{1}{\gamma }+M_{9}\frac{1}{\gamma ^{2}}+..., \end{aligned}$$

(106)

where $\delta _{0}^{2}=a_{0}^{2}+\pi ^{2}$,

$$\begin{aligned} \begin{aligned} M_7=1+\frac{2 \left( \delta _0^2 \left( \delta _0^2-H Q_s\right) -a_0^2 R_D\right) \left( -1+Q_s (-1+\epsilon )\right) }{a_0^2 R_D}, \end{aligned} \end{aligned}$$

(107)

$$\begin{aligned} \begin{aligned} M_8&=\frac{1}{a_0^3 R_DH}\left( 2 \left( 2 a_1H \left( a_0^4-\pi ^4+H \pi ^2 Q_s\right) \left( -1+Q_s (-1+\epsilon )\right) +a_0 \left( 1+Q_s\right) \left( -\pi ^4 \left( 4 \pi ^2\right. \right. \right. \right. \\ &\qquad \left. \left. \left. \left. (-1+\epsilon )+H \left( 1+5 Q_s-5 Q_s\epsilon \right) \right) -2 a_0^2 \pi ^2 \left( 2 \left( 2 \pi ^2-R_D\right) (-1+\epsilon )+H \left( 1+3 Q_s-3 Q_s\epsilon \right) \right) \right. \right. \right. \\ & \qquad \left. \left. \left. -a_0^4 \left( 4 \pi ^2 (-1+\epsilon )+H \left( 1+Q_s-Q_s\epsilon \right) \right) \right) \right) \right) , \end{aligned} \end{aligned}$$

(108)

$$\begin{aligned} \begin{aligned} M_9&=\frac{1}{a_0^4R_D}2 ((a_0^4 a_1^2+2 a_0^5 a_2+3 a_1^4 \pi ^4-2 a_0 a_2 \pi ^4+(-3 a_1^4+2 a_0 a_2) H \pi ^2 Q_s) (-1+Q_s \\&\qquad (-1+\epsilon ))-1/H^2a_0^4(1+Q_s) (a_0^6 H (-1+Q_s (-1+\epsilon ))+a_0^4 \pi ^2 (-16 \pi ^2 (-1+\epsilon )+H\\ & \qquad (-7+7 Q_s (-1+\epsilon )+4\epsilon ))+\pi ^6 (-16 \pi ^2 (-1+\epsilon )+H (-5+21 Q_s (-1+\epsilon )+4 \epsilon ))+a_0^2\pi ^4\\& \qquad (-16 (2 \pi ^2-R_D) (-1+\epsilon )+H (-11+27 Q_s (-1+\epsilon )+8 \epsilon )))+2H^{-1} a_0 a_1 (1+Q_s)\\&(\pi ^4 (4 \pi ^2 (-1+\epsilon )+H (1+5 Q_s-5 Q_s\epsilon ))-a_0^4 (4 \pi ^2 (-1+\epsilon )+H (1+Q_s-Q_s\epsilon )))). \end{aligned} \end{aligned}$$

(109)

Appendix E

This section is concerned with obtaining the amplitudes of convection for the existence of heat source in the fluid phase by weakly nonlinear stability analysis by solving the algebraic Eqs. (40)–(44) (see Sect. (4.1.1)).

1.1 Weakly nonlinear stability analysis

1.1.1 Existence of heat source in the fluid phase

The amplitudes of convection are:

$$\begin{aligned} B= & {} \frac{a_cF_1A}{r \left( \delta _{c}^{4} \left( \delta _{c}^{2}+H-Q_f\right) +H \delta _{c}^{2}F_2\gamma +H^2 \left( \delta _{c}^{2}-Q_f\right) \gamma ^2+a_{c}^{2} F_2\alpha ^2 \Pi ^2\right) }, \end{aligned}$$

(110)

$$\begin{aligned} B_1= & {} -\frac{\pi \left( 4 \pi ^2+H \gamma \right) a_{c}^{2} F_1A^2 }{4 r \left( 16 \pi ^4-4 \pi ^2 Q_f+H F_6\right) \left( \delta _{c}^{4} F_3+H \delta _{c}^{2}F_4\gamma +H^2 F_5\gamma ^2+a_{c}^{2} F_3 \alpha ^2 \Pi ^2\right) }, \end{aligned}$$

(111)

$$\begin{aligned} C= & {} \frac{a_c H \gamma F_1A }{r F_2\left( \delta _{c}^{4} F_3+H \delta _{c}^{2}F_4\gamma +H^2 F_5\gamma ^2+a_{c}^{2} F_3\alpha ^2 \Pi ^2\right) }, \end{aligned}$$

(112)

$$\begin{aligned} C_1= & {} -\frac{H \gamma \pi a_{c}^{2} F_1A^2 }{4 r \left( 16 \pi ^4-4 \pi ^2 Q_f+H F_6\right) \left( \delta _{c}^{4} F_3+H \delta _{c}^{2}F_4\gamma +H^2 F_5\gamma ^2+a_{c}^{2} F_3 \alpha ^2 \Pi ^2\right) }, \end{aligned}$$

(113)

$$\begin{aligned}{} & {} \left. \begin{array}{l} F_1=\left( \left( \delta _{c}^{2}+H \gamma \right) ^2+a_{c}^{2} \alpha ^2 \Pi ^2\right) ,\\ F_2= \left( \delta _{c}^{2}+H \gamma \right) ,\\ F_3=\left( \delta _{c}^{2}+H-Q_f\right) ,\\ F_4=\left( 2 \delta _{c}^{2}+H-2 Q_f\right) ,\\ F_5=\left( \delta _{c}^{2}-Q_f\right) ,\\ F_6=\left( -Q_f \gamma +4 \pi ^2 (1+\gamma )\right) \end{array} \right\} , \end{aligned}$$

(114)

$$\begin{aligned} A^2= & {} \frac{4 r \left( 16 \pi ^4+H F_6-4 \pi ^2 Q_f\right) \left( H \gamma \left( H \gamma F_5+F_4 \delta _c^2\right) +F_3 \left( \alpha ^2 \Pi ^2 a_c^2+\delta _c^4\right) \right) }{\pi ^2 \left( 4 \pi ^2+H \gamma \right) a_c^2 F_1}\nonumber \\{} & {} \quad \left( 1-\frac{F_1 \left( F_2 F_3-H^2 \gamma \right) }{r F_2 \left( H \gamma \left( H \gamma F_5+F_4 \delta _c^2\right) +F_3 \left( \alpha ^2 \Pi ^2 a_c^2+\delta _c^4\right) \right) }\right) , \end{aligned}$$

(115)

Appendix F

This section is concerned with obtaining the amplitudes of convection for the existence of heat source in the solid phase by weakly nonlinear stability analysis by solving the algebraic Eqs. (49)–(53) (see Sect. (4.1.2)).

1.1 Existence of heat source in the solid phase

The amplitudes of convection are:

$$\begin{aligned} B= & {} \frac{A E_1}{a_c r \left( H^2 \gamma ^2 \delta _c^2+E_3+E_4\right) }, \end{aligned}$$

(116)

$$\begin{aligned} B_1= & {} \frac{A^2 \pi \left( 4 \pi ^2+H \gamma \right) E_1}{4 r \left( H^2 \gamma ^2 \delta _c^2+E_3+E_4 E_2\right) }, \end{aligned}$$

(117)

$$\begin{aligned} C= & {} \frac{A H \gamma E_1 \left( 1+Q_s\right) }{a_c r \left( H \gamma +\delta _c^2\right) \left( H^2 \gamma ^2 \delta _c^2+E_3+E_4\right) }, \end{aligned}$$

(118)

$$\begin{aligned} C_1= & {} \frac{A^2 H \pi \gamma E_1 \left( 1+Q_s\right) }{4 r \left( H^2 \gamma ^2 \delta _c^2+E_3+E_4E_2\right) }, \end{aligned}$$

(119)

$$\begin{aligned}{} & {} \left. \begin{array}{l} E_1=\left( a_c^4 \alpha ^2 \Pi ^2+a_c^2 \left( H \gamma +\delta _c^2-Q_s\right) {}^2\right) ,\\ E_2=\left( -4 \pi ^2 \left( H+4 \pi ^2+H \gamma \right) +H^2 \gamma Q_s\right) ,\\ E_3=a_c^2 \alpha ^2 \left( H+\delta _c^2\right) \Pi ^2+H \gamma \left( H+2 \delta _c^2\right) \left( \delta _c^2-Q_s\right) ,\\ E_4=\left( H+\delta _c^2\right) \left( \delta _c^2-Q_s\right) {}^2\\ \end{array} \right\} .\end{aligned}$$

(120)

$$\begin{aligned} A^2= & {} \frac{4 \left( -16 \pi ^4+H^2 Q_s \gamma -4 H \pi ^2 (1+\gamma )\right) }{a^2 \pi ^2 \delta _c^2 \left( 4 \pi ^2+H \gamma \right) E_1}*\left( \delta _c^2 \left( \delta _c^2 \left( \delta _c^2+H\right) +H \left( \delta _c^2-H \text {Qs}\right) \gamma \right) E_1 \right. \nonumber \\{} & {} \quad \left. -\text {ra}^2\delta _c^2 \left( \alpha ^2 \Pi ^2 a_c^2 \left( H+\delta _c^2\right) +\left( H \gamma -Q_s+\delta _c^2\right) \left( -Q_s \left( H+\delta _c^2\right) +\delta _c^2 \left( H+H \gamma +\delta _c^2\right) \right) \right) \right) . \end{aligned}$$

(121)

Fig. 1

Schematic representation

Fig. 2

Variation of a $R_{Dc}$ and b $a_{c}$ with $\Pi$, in the presence of fluid heat source for various values of H and $\gamma$ when $Q_{f}=2$

Fig. 3

Variation of a $R_{Dc}$ and b $a_{c}$ with $\Pi$, in the presence of solid internal heat source for various values of H and $\gamma$ when $Q_{s}=2$

Fig. 4

Variation of $R_{Dc}$ with $\log _{10}H$, in the existence of fluid heat source for different values of a $\gamma$ when $\Pi =5$ and b) $\Pi$ when $\gamma =1$ with $\alpha =0.5$ and $Q_{f}=1$, (c) $Q_{f}$ when $\Pi =5, \gamma =1$ and $\alpha =0.5$

Fig. 5

Variation of $a_{c}$ with $\log _{10}H$, in the existence of fluid heat source for different values of a $\gamma$ when $\Pi =5$ and b $\Pi$ when $\gamma =1$ with $\alpha =0.5$ and $Q_{f}=1$, c $Q_{f}$ when $\Pi =5, \gamma =1$ and $\alpha =0.5$

Fig. 6

Variation of $R_{Dc}$ with $\log _{10}H$, in the existence of solid heat source for different values of a $\gamma$ when $\Pi =5$ and b $\Pi$ when $\gamma =1$ with $\alpha =0.5$ and $Q_{s}=1$, c $Q_{s}$ when $\Pi =5, \gamma =1$ and $\alpha =0.5$

Fig. 7

Variation of $a_{c}$ with $\log _{10}H$, in the existence of solid heat source for different values of a $\gamma$ when $\Pi =5$ and b $\Pi$ when $\gamma =1$ with $\alpha =0.5$ and $Q_{s}=1$, c $Q_{s}$ when $\Pi =5, \gamma =1$ and $\alpha =0.5$

Fig. 8

Variation of $R_{Dc}$ with a $Q_{f}$ and b $Q_{s}$ for various values of $\Pi$ when $H=10, \gamma =1$ and $\alpha =0.5$

Fig. 9

Variation of $Nu_{w}$ with $R_{D}$ in the existence of fluid heat source a H with $\gamma =0.1, Q_{f}=1, \alpha =0.5, \epsilon =0.4$ and $\Pi =2$ and b $\gamma$ with $H=10$, $Q_{f}=1, \alpha =0.5, \epsilon =0.4$ and $\Pi =2$, c $\Pi$ with $Q_{f}=1$, $H=100, \epsilon =0.4, \gamma =1$ and $\alpha =0.5$ and d $Q_{f}$ with $\Pi =2$, $H=10, \epsilon =0.4, \gamma =0.1$ and $\alpha =0.5$

Fig. 10

Variation of $Nu_{w}$ with $R_{D}$ in the existence of solid heat source a H with $\gamma =0.1$, $Q_{s}=1, \alpha =0.5, \epsilon =0.4$ and $\Pi =2$ and b $\gamma$ with $H=10$, $Q_{s}=1, \alpha =0.5, \epsilon =0.4$ and $\Pi =2$, c $\Pi$ with $Q_{s}=1$, $H=10, \epsilon =0.4, \gamma =10$ and $\alpha =0.5$, and d $Q_{s}$ with $\Pi =2$ when $H=10, \epsilon =0.4, \gamma =0.1$ and $\alpha =0.5$

Fig. 11

Variation of $Nu_{w}$ with $R_{D}$ for small values of H with a Fluid internal heat source when $\gamma =0.1$, $\alpha =0.5, \epsilon =0.4, Q_{f}=1$ and $\Pi =2$ and b Solid internal heat source when $\gamma =0.1$, $Q_{s}=1, \alpha =0.5, \epsilon =0.4$ and $\Pi =2$

Fig. 12

Variation of $Nu_{w}$ with $R_{D}$ for large values of H with a Fluid internal heat source when $\gamma =2$, $\alpha =0.5, \epsilon =0.4, Q_{f}=1$ and $\Pi =20$ and b Solid internal heat source when $\gamma =20$, $Q_{s}=1, \alpha =0.5, \epsilon =0.4$ and $\Pi =200$

Fig. 13

Variation of $Nu_{w}$ with $R_{D}$ for large values of $\gamma$ with a Fluid internal heat source when $H=20$, $\alpha =0.5, \epsilon =0.4, Q_{f}=1$ and $\Pi =2$ and b Solid internal heat source when $H=3$, $Q_{s}=1, \alpha =0.5, \epsilon =0.4$ and $\Pi =5$

Table 1 Exact and asymptotic values of critical Darcy–Rayleigh $(R_{Dc})$ for different values of $Q_{f}$ and small values of H for $\gamma =1, \alpha =0.5$ and $\Pi =2$.E-exact value, A-asymptotic value

Table 2 Exact and asymptotic values of $(R_{Dc})$ for various values of $Q_{s}$ and small values of H for $\gamma =1, \alpha =0.5$ and $\Pi =2$.(E-exact value, A-asymptotic value)

Table 3 Exact and asymptotic values of $(R_{Dc})$ for various values of $Q_{s}$ and large values of H for $\gamma =1, \alpha =0.5$ and $\Pi =2$

Table 4 Exact and asymptotic values of $(R_{Dc})$ for various values of $Q_{f}$ and large values of H for $\gamma =1, \alpha =0.5$ and $\Pi =2$

Table 5 Exact and asymptotic values of $(R_{Dc})$ for various values of $Q_{s}$ and large values of $\gamma$ for $H=10, \alpha =0.5$ and $\Pi =2$

Table 6 Exact and asymptotic values of $(R_{Dc})$ for various values of $Q_{f}$ and large values of $\gamma$ for $H=10, \alpha =0.5$ and $\Pi =2$

Table 7 Exact meaning of $H \rightarrow 0$ and $H \rightarrow \infty$, for large values of $\gamma$

Table 8 Exact meaning of $\gamma \rightarrow \infty$, for large values of H

Table 9 Comparison of Postelnicu and the present paper for $Q_{f}=Q_{s}=0$ and $\alpha =0.5$

Table 10 Comparison of Banu and Rees [1] and the present study for $\Pi =Q_{f}=Q_{s}=0$ and $\alpha =0.5$