4.1. Derivation of the equation

We now examine the stability of the basic solution (3.5) by substituting

\begin{equation*} p^{\,j} = 1-\frac{1-\alpha}{\alpha}z-\frac{z^2}{2} + \Pi^{\,j}, \quad T^{\,j} = 1 - z+ \Theta^{\,j}, \quad \rho^{\,j}=\frac{1-\alpha}{\alpha}+z+R^{\,j}, \end{equation*}

into (3.2), where $\Pi ^{\,j}, \Theta ^{\,j}, R^{\,j} (\boldsymbol {r}, t) \rightarrow 0$ are small perturbations of the $p$, $T$ and $\rho$. The fluid velocity is also assumed to be small, $\boldsymbol {u}^{\,j}\rightarrow 0$. After neglecting second-order small quantities and eliminating $q^{\,fm}$ between (3.2), we obtain the linearized equations

(4.1)\begin{gather} R^{\,j}=-\Theta^{\,j},\quad \boldsymbol{u}^{\,j} = - \kappa^{\,j} (\nabla \Pi^{\,j} +R^{\,j} \boldsymbol{e}_z ), \end{gather}

(4.2)\begin{gather} \left. \begin{array}{c@{}} \displaystyle \kappa^{\,j}\left(\nabla^2 \Pi^{\,j} - \nabla_z \Theta^{\,j}\right) = \dfrac{b^{\,j} \kappa^{m}}{ B_1} (\Pi^{\,f} - \Pi^m),\\ \displaystyle S^{\,j} \partial_t \Theta^{\,j} + \kappa^{\,j}(\nabla_z \Pi^{\,j} - \Theta^{\,j}) =\dfrac{\kappa^{\,j}}{{\textit{Ra}}^{\,j}} \nabla^2 \Theta^{\,j} - \dfrac{b^{\,j}\Lambda^{m}}{\overline{{\textit{Ra}}}B_2} (\Theta^{\,f} -\Theta^m), \end{array}\right\} \end{gather}

where (4.2) form a closed system of four linear partial differential equations for $\Pi ^{\,j}$ and $\Theta ^{\,j}$, and the fluid densities and velocities can be expressed in terms of $\Pi ^{\,j}$ and $\Theta ^{\,j}$ using (4.1).

Separating the variables, we present the solution to (4.2) in the form

(4.3)\begin{equation} \Pi^{\,j} = \Pi_0^{\,j}(z) \zeta , \quad \Theta^{\,j} = \Theta_0^{\,j}(z) \zeta , \quad \zeta = \exp(\mathrm{i}(k_x x +k_y y) + \Omega t), \end{equation}

where $k_x$ and $k_y$ are the wavenumbers and $\Omega$ is the growth rate. We assume that $k_x$ and $k_y$ are real numbers. This implies that the perturbations are not brought from infinity ($x, y \rightarrow \infty$), so the convection is not generated by an external force. Substituting (4.3) into (3.6) and (4.2), we obtain

(4.4)\begin{equation} \left. \begin{array}{c@{}} \displaystyle \kappa^{\,j}\left(\dfrac{\mathrm{d}^2\Pi_0^{\,j}}{\mathrm{d}z^2} - k_{xy}^2 \Pi_0^{\,j} - \dfrac{\mathrm{d}\Theta_0^{\,j}}{\mathrm{d}z}\right) - \dfrac{b^{\,j}\kappa^{m}}{B_1} (\Pi_0^{\,f} - \Pi_0^m) = 0, \\ \displaystyle \Omega S^{\,j} \Theta_0^{\,j} + \kappa^{\,j}\left(\dfrac{\mathrm{d}\Pi_0^{\,j}}{\mathrm{d}z} - \Theta_0^{\,j}\right) - \dfrac{\kappa^{\,j}}{{\textit{Ra}}^{\,j}} \left(\dfrac{\mathrm{d}^2\Theta_0^{\,j}}{\mathrm{d}z^2} - k_{xy}^2 \Theta_0^{\,j}\right) \\ \displaystyle \qquad\qquad\qquad\qquad\qquad\qquad\quad + \dfrac{b^{\,j}\Lambda^{m}}{\overline{{\textit{Ra}}}B_2} (\Theta_0^{\,f} - \Theta_0^m ) = 0, \end{array}\right\} \end{equation}

with the boundary conditions

(4.5)\begin{equation} \Theta_0^{\,j} = 0, \quad \frac{\mathrm{d}\Pi_0^{\,j}}{\mathrm{d}z} = 0\quad \mbox{on}\ z=0\quad \mbox{and}\quad z=1, \end{equation}

where $k_{xy}^2 = k_x^2 + k_y^2$ denotes the overall horizontal wavenumber. Without loss of generality, we assume that $k_{xy}\geq 0$ because this wavenumber enters (4.4) only as $k_{xy}^2$.

A non-trivial solution to the eigenvalue problem given by (4.4) and (4.5) exists for particular values of the spectral parameter $\Omega$. For boundary conditions considered in this paper (4.5), the non-trivial solution to the conjugate problem can be written in the following simple form (Davis Reference Davis1969):

(4.6)\begin{equation} \Theta_0^{\,j} = \delta \Theta^{\,j} \sin(k_z z),\quad \Pi_0^{\,j} =\delta \Pi^{\,j} \cos(k_z z), \quad k_z = {\rm \pi} n, \quad n=1,2,3,\ldots, \end{equation}

where $k_z$ is a real number that takes discrete values, and the constant amplitudes $\delta \Pi ^{\,j}$ and $\delta \Theta ^{\,j}$ are still unknown. To determine $\delta \Pi ^{\,j}$ and $\delta \Theta ^{\,j}$, we substitute (4.6) into (4.4). This yields

(4.7)\begin{equation} \left. \begin{array}{c@{}} \displaystyle \left(k^2 \kappa^{\,j} + \dfrac{\kappa^{m}}{B_1}\right) \delta \Pi^{\,j} + k_z \kappa^{\,j} \delta \Theta^{\,j} - \dfrac{\kappa^{m}}{B_1} \delta \Pi^i = 0, \\ \displaystyle - k_z \kappa^{\,j} \delta \Pi^{\,j} + \left(\Omega S^{\,j} - \kappa^{\,j} +k^2\dfrac{\kappa^{\,j}}{{\textit{Ra}}^{\,j}} +\dfrac{\Lambda^{m}}{\overline{{\textit{Ra}}}B_2} \right) \delta \Theta^{\,j} - \dfrac{\Lambda^{m}}{\overline{{\textit{Ra}}}B_2}\delta \Theta^i = 0, \end{array} \right\} \end{equation}

where we introduce the notation $k^2 = k_{xy}^2+k_z^2$ and $i=f,m$, $i\neq j$.

Equations (4.7) form a homogeneous system of linear equations with respect to $\delta \Pi ^{\,j}$ and $\delta \Theta ^{\,j}$. There is a non-trivial solution if the matrix of coefficients for (4.7) is singular. By calculating and equating to zero the determinant of the matrix and carrying out identity transformations, we obtain the following dispersion relation:

(4.8)\begin{equation} D^{\,f}D^m+\left(D^{\,f}+D^m\right)Z^c = 0, \end{equation}

where we denote

(4.9a,b)\begin{equation} D^{\,j} = \Omega S^{\,j} + \kappa^{\,j} \left( \frac{k^2}{{\textit{Ra}}^{\,j}} - \frac{k_{xy}^2}{k^2} \right), \quad Z^c = \frac{\Lambda^{m}}{\overline{{\textit{Ra}}}B_2} - \frac{\kappa^{\,f} \kappa^m}{k^2 \kappa^{\,f} B_1 + 1} \frac{k_z^2}{k^2}. \end{equation}

The equation $D^{\,j} = 0$ is the dispersion relation in the HRL problem for the regular single porosity medium $\Phi ^{\,j}$, assuming zero mass and heat exchange with the adjacent medium.

Assuming that $D^{\,f}+D^m\neq 0$, (4.8) can be written as

(4.10)\begin{equation} \frac{D^{\,f} D^m}{D^{\,f} +D^m} =-Z^c. \end{equation}

The left-hand side of (4.10) is proportional to the harmonic mean of $D^{\,j}$, reflecting that the mass and heat transfer through DPM occur in parallel through $\Phi ^{\,f}$ and $\Phi ^m$. This is analogous to parallel conductors (Nield & Simmons Reference Nield and Simmons2007).

Taking into account (4.8) and (4.9), we derive from (4.1) and (4.7) the following relationships for the amplitudes:

(4.11)\begin{gather} D^{\,f}\delta\Theta^{\,f}+D^m\delta\Theta^m=0, \end{gather}

(4.12)\begin{gather} \delta\Pi^{\,j}=\frac{1}{\kappa^{\,j}k_z}\left(-\frac{\kappa^{\,j}}{D^{\,j}}\frac{k_z^2}{k^2}-\frac{1}{Z^c}\left(\frac{\Lambda^{m}}{\overline{{\textit{Ra}}} B_2}-Z^c\right)\right)D^{\,j}\delta\Theta^{\,j}, \end{gather}

(4.13)\begin{gather} \left.\begin{array}{c@{}} \displaystyle \delta u^{\,j}_z=\left(\dfrac{\kappa^{\,j}}{D^{\,j}}\dfrac{k_{xy}^2}{k^2}-\dfrac{1}{Z^c}\left(\dfrac{\Lambda^{m}}{\overline{{\textit{Ra}}} B_2}-Z^c\right)\right)D^{\,j}\delta\Theta^{\,j}, \\ \displaystyle \delta q^{\,fm}=\dfrac{\kappa^{m}}{B_1}\left(\delta \Pi^{\,f}-\delta \Pi^m\right)=\dfrac{1}{Z^c}\left(\dfrac{\Lambda^{m}}{\overline{{\textit{Ra}}} B_2}-Z^c\right)D^{\,f}\delta\Theta^{\,f}, \end{array}\right\} \end{gather}

where $\delta u^{\,j}_z$ and $\delta q^{\,fm}$ are the amplitudes of $u_z$ and $q^{\,fm}$ in the solution given by (4.3) and (4.6)

(4.14)\begin{equation} u^{\,j}_z=\delta u^{\,j}_z\sin(k_z z)\zeta,\quad q^{\,fm}=\delta q^{\,fm}\cos(k_z z)\zeta. \end{equation}

4.2. Geometric interpretation

In general, the growth rate $\Omega$ is a complex number. According to (4.3), if $\textrm {Re}\,\Omega >0$, then the perturbation grows, and if $\textrm {Re}\,\Omega <0$, then it fades. From (4.9), it follows that (4.8) is a quadratic equation with respect to $\Omega$

(4.15)\begin{equation} a_0 \Omega^2 + a_1 \Omega + a_2 = 0, \end{equation}

which coefficients are given by

(4.16a–c)\begin{gather} a_0 = S^{\,f}S^m, \quad a_1 = S^{\,f} Z^m + S^m Z^{\,f} + \overline{S} Z^c, \quad a_2 = Z^{\,f} Z^m + Z^{\,f} Z^c +Z^m Z^c, \end{gather}

(4.17)\begin{gather} Z^{\,j} = \kappa^{\,j} \left( \frac{k^2}{{\textit{Ra}}^{\,j}} - \frac{k_{xy}^2}{k^2} \right)=D^{\,j}-\Omega S^{\,j}, \quad j=f,m. \end{gather}

We apply the Routh–Hurwitz stability criterion to the characteristic polynomial (4.15) to determine the parameters corresponding to $\textrm {Re}\,\Omega \leq 0$. The inequality $a_0>0$ holds because $S^{\,j} > 0$. Therefore, the thermal stratification is stable if and only if all leading principal minors of the Hurwitz matrix are positive. This leads to the inequalities $a_1\geq 0$ and $a_2\geq 0$, or using the notations of (4.16b,c), to

(4.18a,b)\begin{equation} S^{\,f} Z^m + S^m Z^{\,f} + \overline{S} Z^c \geq 0,\quad Z^{\,f} Z^m +Z^{\,f} Z^c +Z^m Z^c \geq 0. \end{equation}

In the space $\lbrace Z^{\,f}, Z^m, Z^c \rbrace$, the equations $a_1 = 0$ and $a_2 = 0$ define a plane and a circular conical surface, respectively (figure 2). The origin is the only point of intersection of these surfaces. Indeed, if we express $Z^c$ from the equation $a_1 = 0$ and substitute it into the equation $a_2=0$, then we obtain

\[ (Z^{\,f})^2 S^m + (Z^m)^2 S^{\,f} = 0. \]

This equation is satisfied only if both $Z^{\,f} = 0$ and $Z^m = 0$, and according to (4.16b,c), $Z^c = 0$. Thus, the plane $a_1=0$ splits the conical surface $a_2 = 0$ into the upper and lower parts, which contain the intervals $Z^c>0$ and $Z^c<0$ of the axis $Z^c$, respectively. The inequality $a_1\geq 0$ constrains the upper region above the plane, and the inequality $a_2\geq 0$ constrains the region inside the conical surface (figure 2). Therefore, the parameters of stable layers, in which convection does not occur, coincide with the interior of the upper conical surface given by (4.18). If $\overline {{\textit {Ra}}} \rightarrow 0$, then all solutions to (4.8) satisfy inequalities (4.18) because if the dimensional temperature gradient $\alpha _0$ approaches 0, then the thermal stratification is stable. This means that the locus of the solution for each $n =\mbox {const}$ is a curve lying inside the conical surface at $\overline {{\textit {Ra}}}\rightarrow 0$ (figure 2, curve 1). Each curve can be parameterized by either $k_{xy}$ or $\Omega$. With rising $\overline {{\textit {Ra}}}$, i.e. with increasing temperature gradient $\alpha _0$, one of these curves (further analysis of solutions to (4.8) shows that it is with $n=1$) approaches the boundary of (4.18b), and it first intersects the boundary at the critical Rayleigh number, $\overline {{\textit {Ra}}}_c$. If $\overline {{\textit {Ra}}} > \overline {{\textit {Ra}}}_c$, then a segment of such curve corresponding to growing perturbations lies outside the stability region (4.18). At $\overline {{\textit {Ra}}} =\overline {{\textit {Ra}}}_c$, the locus touches exactly the conical surface $a_2 = 0$ and not the plane surface $a_1 = 0$. Therefore, the inequality (4.18b) is more relevant than (4.18a) because it is this inequality that is violated at $\overline {{\textit {Ra}}} =\overline {{\textit {Ra}}}_c$. The critical number $\overline {{\textit {Ra}}}_c$ is the minimum $\overline {{\textit {Ra}}}$ for which the equation $a_2=0$ has a solution.

Figure 2.

Sketch of the locus of solutions to the dispersion equation in the $\lbrace Z^{\,f}, Z^m, Z^c \rbrace$ space (curve 1) and key surfaces defining the onset of convection. The coordinate axes belong to the conical surface $a_2=0$.

According to (4.15) and (4.16), the monotonic (not oscillatory) instability of the basic state (3.6) develops at near-critical parameters, $\overline {{\textit {Ra}}}\rightarrow \overline {{\textit {Ra}}}_c+$. This means that if $\textrm {Re}\,\Omega \rightarrow 0$ (or $a_2 \rightarrow 0$), then $\textrm {Im}\,\Omega \equiv 0$. Indeed, if $a_2 \rightarrow 0$, then the discriminant of (4.15) is not negative, and therefore, both solutions to (4.15) are real numbers.

6.1. Asymptotic $B\rightarrow 0$

Let us consider the layer stability in the asymptotic case $B\rightarrow 0$ corresponding to local equilibrium between $\Phi ^{\,j}$. If $B\rightarrow 0$, then according to (4.9b), $Z^c\rightarrow \infty$. Consequently, the dispersion equation (4.8) can be satisfied only if $D^{\,f}+D^m\rightarrow 0$ because $Z^{\,f}$ and $Z^m$ are independent of $B$. Substituting (3.4) and (4.9a) into the equation $D^{\,f}+D^m=0$, we obtain the dispersion relation at $B=0$

(6.2)\begin{equation} \Omega\overline{S}+\frac{k^2}{\overline{{\textit{Ra}}}}-\frac{k_{xy}^2}{k^2}=0. \end{equation}

This equation is the dispersion relation for EPM (Horton & Rogers Reference Horton and Rogers1945). Therefore, the conduction state (3.6) at $B=0$ is stable at $\overline {{\textit {Ra}}}\leq 4{\rm \pi} ^2$ and unstable otherwise at $\overline {{\textit {Ra}}}>4{\rm \pi} ^2$. This yields

(6.3)\begin{equation} \overline{{\textit{Ra}}}_c\rightarrow4{\rm \pi}^2,\quad k_c\rightarrow{\rm \pi}\quad\mbox{as}\ B\rightarrow 0, \end{equation}

where $k_c$ is the critical overall horizontal wavenumber $k_{xy}$. The relations (6.3) are the consequence of DPM equivalence to EPM in the case of local equilibrium at $B\rightarrow 0$.

6.2. Asymptotic $B\rightarrow \infty$

We now consider the system behaviour in the asymptotic case $B\rightarrow \infty$ corresponding to infinite time required for reaching local equilibrium. Thus, the heat and mass exchange between $\Phi ^{\,j}$ are absent, and the media can be considered isolated from each other. The local equilibrium is never reached in the case of $B\rightarrow \infty$. As noted in § 3, a realistic porous medium cannot be characterized with large values of $B$, but from the mathematical point of view, the asymptotic $B\rightarrow \infty$ allows for simple conclusions to be reached that, as we will show, do extrapolate into the region of finite $B\sim 1$.

If $B\rightarrow \infty$, then according to (4.9b), $Z^c\rightarrow 0$ and (4.8) reduces to

\[ D^{\,f}D^m=0 \]

or according to (4.17), to

\[ Z^{\,f}Z^m=0 \]

in the case of neutral stability ($\Omega =0$). Thus, in the asymptotic case $B\rightarrow \infty$, the convection develops at the minimum $\overline {{\textit {Ra}}}$ for which either $Z^{\,f}=0$ or $Z^m=0$ is satisfied. If the condition $Z^{\,f}=0$ (or $Z^m=0$) is reached first with increasing $\overline {{\textit {Ra}}}$, then the state (3.6) in $\Phi ^{\,f}$ (or $\Phi ^m$) becomes unstable and convection appears in $\Phi ^{\,f}$ (or $\Phi ^m$). At the same time, the stratification in the adjacent medium $\Phi ^m$ (or $\Phi ^{\,f}$) remains stable. According to (5.3), $Z^{\,j}=0$ is satisfied if ${\textit {Ra}}^{\,j}={\textit {Ra}}^{\,j}_c=4{\rm \pi} ^2$ and $k_{xy}=k_z={\rm \pi}$. Consequently, the following asymptotic relation holds:

(6.4)\begin{equation} \overline{{\textit{Ra}}}_c\rightarrow \min\left({\textit{Ra}}^{\,f}_c\frac{\Lambda^{\,f}}{\kappa^{\,f}},{\textit{Ra}}^m_c\frac{\Lambda^m}{\kappa^m}\right),\quad k_c\rightarrow{\rm \pi}\quad\mbox{as}\ B\rightarrow\infty. \end{equation}

Substituting (6.1) into (6.4) yields

(6.5)\begin{equation} \overline{{\textit{Ra}}}_c\rightarrow4{\rm \pi}^2 \min\left(\frac{\Lambda}{\kappa},\frac{1-\Lambda}{1-\kappa}\right)\leq 4{\rm \pi}^2\quad\mbox{as}\ B\rightarrow\infty. \end{equation}

The minimum of the fractions in (6.5) is not larger than unity for $0\leq \kappa \leq 1$ and $0\leq \Lambda \leq 1$, and it reaches a maximum at $\kappa =\Lambda$, i.e. if $\overline {K}$ and $\overline {\lambda }$ are distributed in equal proportions between $\Phi ^{\,j}$. Then, according to (6.5), $\overline {{\textit {Ra}}}_c$ reaches the maximum value of $4{\rm \pi} ^2$ in the case of $\kappa =\Lambda$ as compared to other values of $\kappa$ and $\Lambda$. Therefore, we can conclude for $B\rightarrow \infty$ that $\overline {{\textit {Ra}}}_c$ for DPM does not exceed that for EPM. The onset of convection in DPM occurs for smaller (or equal at $\kappa =\Lambda$) vertical temperature gradients $\alpha _0$ than that in EPM.

If $\kappa >\Lambda$, then the first fraction within the brackets in (6.5) is smaller than the second one, i.e. convection occurs in $\Phi ^{\,f}$ at $\overline {{\textit {Ra}}}\rightarrow \overline {{\textit {Ra}}}_c+$ (${\textit {Ra}}^{\,f}_c=4{\rm \pi} ^2>{\textit {Ra}}^m_c$). If $\kappa <\Lambda$, then the second fraction in (6.5) is larger, i.e. convection occurs in $\Phi ^m$ (${\textit {Ra}}^m_c=4{\rm \pi} ^2>{\textit {Ra}}^{\,f}_c$). Thus, we can expect that for any finite and sufficiently large $B$, where $\overline {{\textit {Ra}}}\rightarrow \overline {{\textit {Ra}}}_c+$, and $\kappa >\Lambda$ (or $\kappa <\Lambda$), the convection pattern in DPM is determined by the flow in the medium $\Phi ^{\,f}$ (or $\Phi ^m$) because it is the stratification in this medium that is unstable at $B\rightarrow \infty$.

6.3. Case $\kappa =\Lambda$

If $\overline {K}$ and $\overline {\lambda }$ are distributed between $\Phi ^{\,f}$ and $\Phi ^m$ in equal proportions, i.e. $\kappa =\Lambda$, then we derive from (4.9b), (4.17), and (6.1)

(6.6)\begin{align} Z^{\,f}=\kappa\left(Z^{\,f}+Z^m\right),\quad Z^m=(1-\kappa)\left(Z^{\,f}+Z^m\right),\quad Z^c=\frac{1-\kappa}{\overline{{\textit{Ra}}}B}-\frac{\kappa(1-\kappa)}{\kappa B k^2+1}\frac{k_z^2}{k^2}. \end{align}

Substituting (6.6) and $\Omega =0$ into the dispersion equation (4.8) yields

(6.7)\begin{equation} \left(Z^{\,f}+Z^m\right)\left(\kappa\left(Z^{\,f}+Z^m\right)+\frac{1}{\overline{{\textit{Ra}}}B}-\frac{\kappa}{\kappa B k^2+1}\frac{k_z^2}{k^2}\right)=0. \end{equation}

Equation (6.7) is satisfied if either

(6.8 a,b)\begin{equation} Z^{\,f}+Z^m=0\quad\mbox{or}\quad\kappa\left(Z^{\,f}+Z^m\right)+\frac{1}{\overline{{\textit{Ra}}}B}-\frac{\kappa}{\kappa B k^2+1}\frac{k_z^2}{k^2}=0. \end{equation}

Which of the two conditions in (6.8) is satisfied first with increasing $\overline {{\textit {Ra}}}$ determines $\overline {{\textit {Ra}}}_c$ at $\kappa =\Lambda$. Equation (6.8a) is the dispersion relation (6.2) for EPM at $\Omega =0$. It has solutions if $\overline {{\textit {Ra}}}\geq 4{\rm \pi} ^2$.

Substituting (6.6) into (6.8b) and collecting the terms, we obtain

(6.9)\begin{equation} \frac{\overline{{\textit{Ra}}}}{k_z^2}=\frac{A^2}{A-1},\quad\mbox{where}\ A=\frac{\kappa B k^2+1}{\kappa B k_z^2}>1. \end{equation}

According to (6.9), $A$ is an increasing function of $k^2$, whereas $\overline {{\textit {Ra}}}$ reaches a minimum of $4k_z^2=4{\rm \pi} ^2$ at $A=2$ (figure 3a), i.e. at

(6.10)\begin{equation} k_{xy}^2=k_z^2-\frac{1}{\kappa B}={\rm \pi}^2-\frac{1}{\kappa B}. \end{equation}

Equation (6.10) has a solution at $\kappa B\geq {\rm \pi} ^{-2}$. Consequently, if $\kappa B\geq {\rm \pi} ^{-2}$, then (6.8) are satisfied simultaneously at $\overline {{\textit {Ra}}}=4{\rm \pi} ^2$, and if $\kappa B<{\rm \pi} ^{-2}$, then (6.8a) is satisfied before (6.8b) with increasing $\overline {{\textit {Ra}}}$. This yields

(6.11)\begin{equation} \overline{{\textit{Ra}}}_c=4{\rm \pi}^2,\quad k_{c}={\rm \pi}\quad\mbox{or}\quad k_{c}=\sqrt{{\rm \pi}^2-\frac{1}{\kappa B}}\quad\mbox{at}\ \kappa=\Lambda. \end{equation}

Figure 3(b) shows the critical wavenumbers (6.11) as solutions to (6.7). Straight line 1 and curve 2 are the solutions with $Z^c>0$ and $Z^c<0$ to (6.8a) and (6.8b), respectively. The critical Rayleigh number is $4{\rm \pi} ^2$ for any such solution. As we have shown, only (6.8a) has a solution at $\kappa B<{\rm \pi} ^{-2}$. If $B\rightarrow 0$ or $B\rightarrow \infty$, then in accordance with (6.3) and (6.4), $k_c$ tends to ${\rm \pi}$ in both solutions.

Figure 3.

(a) Value of $\overline {{\textit {Ra}}}$ against $A$ as defined by (6.9). (b) The critical wavenumbers at $\kappa =\Lambda$.

In the case of $k_c={\rm \pi}$ and $\overline {{\textit {Ra}}}={\textit {Ra}}^{\,f}={\textit {Ra}}^m=4{\rm \pi} ^2$, both $Z^{\,f}=0$ and $Z^m=0$, i.e. the conduction state in both media becomes unstable. Substituting $Z^{\,f}=0$, $Z^m=0$ and $\Omega =0$ into (4.11)–(4.13), we obtain

\[ \delta\Theta^{\,f}=\delta\Theta^m,\quad \delta\Pi^{\,f}=\delta\Pi^m, \quad \delta q^{\,fm}=0. \]

Thus, the temperature and pressure amplitudes in the media are identical at $k_c={\rm \pi}$, $\kappa =\Lambda$, any $B$ and $\overline {{\textit {Ra}}}=\overline {{\textit {Ra}}}_c$, meaning that the heat and mass transfer between $\Phi ^{\,j}$ is absent. Consequently, the flow pattern is identical to that in EPM.

6.4. Case $k_{xy}=0$ (the plane flow regime)

The dispersion equation (4.8) admits non-trivial solutions with zero horizontal wavenumbers, which do not exist in the case of homogeneous medium. To determine the Rayleigh numbers for which such solutions are possible, we substitute $k_{xy}=0$ and $\Omega =0$ into (4.8), designate $\overline {{\textit {Ra}}}$ as $\overline {{\textit {Ra}}}_p$, and thereby obtain

(6.12)\begin{equation} \overline{{\textit{Ra}}}_p=\left(\kappa B k_z^2+1\right)\left(\Lambda B k_z^2+1\right)\frac{1-\Lambda}{\kappa(1-\kappa)B}. \end{equation}

If $\overline {{\textit {Ra}}}>\overline {{\textit {Ra}}}_p$, then the problem admits the growing plane perturbations with $k_{xy}=0$. In what follows, we refer to such a flow pattern as the convection regime $P$. For $k_z={\rm \pi}$ ($n=1$), it is schematically shown in figure 4 (in central panel $P$). The flow pattern consists of only the cross-medium convection cell (figure 1). For instance, the fluid can flow upward through $\Phi ^{\,f}$ ($u_z^{\,f}>0$), then into $\Phi ^m$ ($q^{\,fm}>0$) near the upper boundary of the layer ($z=1$), then downward through $\Phi ^m$ ($u_z^m<0$) and, finally, it flows back into $\Phi ^m$ ($q^{\,fm}<0$) near the lower boundary ($z=0$), thereby closing the cross-medium circulation. At the same time, according to (4.11), the inequality $\Theta ^{\,f}\geq \Theta ^m$ holds at any position vector $\boldsymbol {r}$. The circulation can occur in the opposite direction, and then, $\Theta ^{\,f}\leq \Theta ^m$.

Figure 4.

Possible convection regimes at $\overline {{\textit {Ra}}}\rightarrow \overline {{\textit {Ra}}}_c+$. The isotherms $\Theta ^{\,j}=1/2$ and the direction of flow in convection cells are schematically shown in the upper panel for every regime. The pressure distributions, $\Pi ^{\,j}(x)$, at the lower boundary $z=0$ are shown in the lower panels.

At near-critical conditions, the plane flow regime appears if and only if $\overline {{\textit {Ra}}}_c=\overline {{\textit {Ra}}}_p$, where $\overline {{\textit {Ra}}}_p$ is given by (6.12) with $k_z={\rm \pi}$. In general, with increasing $\overline {{\textit {Ra}}}$, the thermal stratification can first become unstable with respect to other perturbations with $k_{xy}>0$. Consequently, the inequality $\overline {{\textit {Ra}}}_c\leq \overline {{\textit {Ra}}}_p$ holds, and the opposite inequality $\overline {{\textit {Ra}}}>\overline {{\textit {Ra}}}_p$ is a sufficient condition for instability.

7.1. Flow regime classification

The numerical investigation of solutions to the dispersion equation (4.8) for $\Omega =0$ and all possible $\kappa$, $\Lambda$ and $B$ shows that 9 flow regimes can arise at near-critical conditions. First, we discuss in general the convection patterns of these regimes, leaving for §§ 7.2 and 7.3 the description of parameters $\kappa$, $\Lambda$ and $B$ at which they occur.

The 9 regimes are further referred to as $F$, $F_z$, $C_{fx}$, $C_f$, $C_m$, $C_{mx}$, $M_z$, $M$ and $P$ (figures4 and 5). The capital $F$ or $M$ refers to the fractures ($\Phi ^{\,f}$) or matrix ($\Phi ^m$) medium that is unstable at $\overline {{\textit {Ra}}}\rightarrow \overline {{\textit {Ra}}}_c$, respectively. The capital $C$ refers to the counter-rotating pattern that appears at $Z^c<0$ (see § 5). The capital $P$ corresponds to the plane regime considered in § 6.4. The regimes whose designation includes either the capital $F$ (or $M$) or the subscript $f$ (or $m$) are controlled mostly by the convective heat transfer through $\Phi ^{\,f}$ (or $\Phi ^m$). The regimes whose designation differs only by the presence of the subscript $x$ (or $z$) are the same except of the sign for $u^{\,j}_x$ (or $u^{\,j}_z$) in one of the media. For further discussion it is convenient to organize the regimes into 4 groups, namely, the co-rotating regimes $F$ and $M$; the counter-rotating regimes $C_f$ and $C_m$; the intermediate regimes $F_z$, $C_{fx}$, $C_{mx}$ and $M_z$ that appear in transition from co-rotating to counter-rotating patterns; and the regime $P$ (figure 5).

Figure 5.

Possible transitions of the flow regimes. The dashed lines group the regimes in pairs of identical convection patterns in $\Phi ^{\,j}$ but with different amplitudes $\delta \Pi ^{\,j}$ and $\delta \Theta ^{\,j}$ for $j=f,m$. The labels $u_x^{\,j}$, $u_z^{\,j}$, $q^{\,fm}$ and $\delta \Theta ^{\,j}$ designate transitions in which only the corresponding parameter changes sign.

The flow patterns and corresponding distributions of $\Theta ^{\,j}$ and $\Pi ^{\,j}$ are schematically shown in figure 4. The diagram in figure 5 shows which flow pattern transitions are possible with continuous changes of the quantities $\kappa$, $\Lambda$ and $B$. For example, the regime $F$ can change only to $M$, $F_z$ or $P$. In other words, the region of parameters in the $\{\kappa ,\Lambda ,B\}$ space corresponding to the regime $F$ can have boundaries only with the regions corresponding to $M$, $F_z$ and $P$.

The regimes $F$ and $F_z$ correspond to the inequality $Z^{\,f}<0$ at $\overline {{\textit {Ra}}}=\overline {{\textit {Ra}}}_c$, i.e. if the locus of solutions to (4.8) touches the conical surface $a_2=0$ at $Z^{\,f}<0$ (figure 2). Numerical analysis of the solutions to (4.8) indicates that these regimes can arise only if $\kappa \geq \Lambda$, i.e. if $\Phi ^{\,f}$ determines the flow pattern (${\textit {Ra}}^{\,f}\geq 4{\rm \pi} ^2\geq {\textit {Ra}}^m$). This is in agreement with the theoretical analysis of the asymptotic case $B\rightarrow \infty$ discussed in § 6.2. In the regimes $F$ and $F_z$, the convection in DPM is triggered by the unstable conduction state in $\Phi ^{\,f}$ and the flow in $\Phi ^m$ appears only due to heat and mass transfer between the media. Therefore, and according to (4.11), in which $D^{\,j}=Z^{\,j}$ at $\Omega =0$, the absolute value of the temperature amplitude in $\Phi ^{\,f}$ is larger than that in $\Phi ^m$ ($|\delta \Theta ^{\,f}|>|\delta \Theta ^m|$).

Similarly, the regimes $M$ and $M_z$ correspond to the inequality $Z^m<0$ at $\overline {{\textit {Ra}}}=\overline {{\textit {Ra}}}_c$, i.e. if the locus touches the conical surface $a_2=0$ at $Z^m<0$. As in the asymptotic case $B\rightarrow \infty$, these regimes can arise only if $\kappa \leq \Lambda$, i.e. if $\Phi ^m$ determines the flow pattern (${\textit {Ra}}^m\geq 4{\rm \pi} ^2\geq {\textit {Ra}}^{\,f}$). The convection in these regimes is triggered by the unstable state in $\Phi ^m$, and the flow in $\Phi ^{\,f}$ appears only due to heat and mass transfer between the media. Therefore, and according to (4.11), the absolute value of the temperature amplitude in $\Phi ^m$ is larger than that in $\Phi ^{\,f}$ ($|\delta \Theta ^m|>|\delta \Theta ^{\,f}|$).

According to (4.11), in which $D^{\,j}=Z^{\,j}$ at $\Omega =0$, the temperature amplitudes $\delta \Theta ^{\,f}$ and $\delta \Theta ^m$ have identical signs for $F$, $F_z$, $M$ and $M_z$. Thus, higher (or lower) temperatures in $\Phi ^{\,f}$ correspond to higher (or lower) temperatures in $\Phi ^m$ in all 4 regimes.

The other 5 regimes – $C_{fx}$, $C_f$, $C_m$, $C_{mx}$ and $P$ – correspond to the inequality $Z^c<0$ at $\overline {{\textit {Ra}}}=\overline {{\textit {Ra}}}_c$, i.e. if the locus of solutions to (4.8) touches the surface $a_2=0$ at $Z^c<0$. These regimes can arise at both $\kappa >\Lambda$ and $\kappa <\Lambda$. Therefore, the convection is not determined by only $\Phi ^{\,f}$ or $\Phi ^m$, which can result in more complicated flow patterns. For example, according to (4.11), the temperature amplitudes in $\Phi ^{\,f}$ and $\Phi ^m$ have opposite signs in these regimes ($\delta \Theta ^{\,f}\delta \Theta ^m<0$), i.e. higher (or lower) temperatures in $\Phi ^{\,f}$ correspond to lower (or higher) temperatures in $\Phi ^m$ (figure 4). Furthermore, a noteworthy observation is that the conduction states in $\Phi ^{\,f}$ and $\Phi ^m$ considered apart (isolated from each other) are stable because $Z^{\,f}>0$ and $Z^m>0$. The convection appears only if $\Phi ^{\,f}$ and $\Phi ^m$ are merged to form DPM. We can interpret this as that the heat transfer between the media triggers the onset of convection in regimes $C_{fx}$, $C_f$, $C_m$, $C_{mx}$ and $P$.

The co-rotating regimes $F$ and $M$ are characterized with qualitatively similar flow fields, which are a simple generalization of the cellular convection cells in homogeneous porous medium, i.e. EPM (Nield & Bejan Reference Nield and Bejan2017). Horizontal and vertical components of the fluid velocities in $\Phi ^{\,f}$ and $\Phi ^m$ for any position vector $\boldsymbol {r}$ have equal signs corresponding to co-rotating cells in both media (figure 4). The regimes $F$ and $M$ differ only in the direction of fluid flow between $\Phi ^{\,j}$ and the amplitudes $\delta \Pi ^{\,j}$ and $\delta \Theta ^{\,j}$. In the regime $F$ (or $M$), the convection in $\Phi ^{\,f}$ (or $\Phi ^m$) dominates over that in $\Phi ^m$ (or $\Phi ^{\,f}$), leading to larger absolute values of $\delta \Theta ^{\,f}$ and $\delta \Pi ^{\,f}$ (or $\delta \Theta ^m$ and $\delta \Pi ^m$). At the same time, in the regime $F$, the medium $\Phi ^{\,f}$ withdraws fluid from $\Phi ^m$ at the bottom of ascending currents due to lower pressure $\Pi ^{\,f}$ ($q^{\,fm}<0$ at $z=0$) and returns it back to $\Phi ^m$ at the top of the currents ($q^{\,fm}>0$ at $z=1$).

Let us consider how the flow pattern changes with varying $\kappa$, $\Lambda$ and $B$ in the following sequence of transitions (corresponding to the counter-clockwise movement around $P$ in figures 4 and 5):

(7.1)\begin{equation} F\rightarrow F_z\rightarrow C_{fx}\rightarrow C_f. \end{equation}

The transition from the co-rotating regime $F$ to the intermediate regime $F_z$ is characterized by increasing intensity of the fluid withdrawal at the bottom of the ascending current. When entering the regime $F_z$, the withdrawal becomes so intense that it reverts the vertical direction of flow in $\Phi ^m$. Now, the fluid in $\Phi ^m$ converges to the bottom of ascending currents in $\Phi ^{\,f}$, which can be considered as sink zones for the medium $\Phi ^m$. Similarly, the top of ascending currents in $\Phi ^{\,f}$ are the source zones for $\Phi ^m$ from which the flow diverges through $\Phi ^m$. Thus, the transition from $F$ to $F_z$ breaks down convection cells in $\Phi ^m$ and leads to the appearance of new cross-media cells as in the $P$ regime. At every $x,y=\mbox {const}$ in the cross-media cells, the fluid ascends through $\Phi ^{\,f}$ (or $\Phi ^m$) and flows to the adjacent medium $\Phi ^m$ (or $\Phi ^{\,f}$) at the top of ascending currents, then it flows down in the descending currents through the medium $\Phi ^m$ (or $\Phi ^{\,f}$), and ultimately flows back to $\Phi ^{\,f}$ (or $\Phi ^m$) at $z=0$, closing the circulation (figure 1).

The intermediate regimes $F_z$ and $C_{fx}$ have identical convection patterns and different temperature distributions in $\Phi ^m$. Transition from $F_z$ to $C_{fx}$ is characterized by increasing influence of convective heat transfer through $\Phi ^m$ as compared to that between $\Phi ^{\,j}$. Consequently, the temperature amplitude $\delta \Theta ^m$ changes its sign when entering $C_{fx}$, and the inequality $\delta \Theta ^{\,f}\delta \Theta ^m<0$ holds in the regime $C_{fx}$. Now, the descending (or ascending) currents in $\Phi ^m$ move the isotherms $\Theta ^m=\mbox {const}$ downward (or upward).

The transition from $C_{fx}$ to $C_f$ is characterized by a further increasing influence of the reversed flow direction in $\Phi ^m$. At the transition, the horizontal velocity in $\Phi ^m$ changes its sign, and new cellular convection with opposite direction of circulation forms in $\Phi ^m$. Thus, the flow pattern in the regime $C_f$ consists of counter-rotating cells in the media $\Phi ^{\,f}$ and $\Phi ^m$. At the same time, the cross-medium convection cells also exist (figure 4).

The flow fields in regimes $C_f$ and $C_m$ are qualitatively similar, with different amplitudes of pressure and temperature. The temperature amplitudes $\delta \Theta ^{\,f}$ and $\delta \Theta ^m$ have opposite signs ($\delta \Theta ^{\,f}\delta \Theta ^m<0$) reflecting the counter-rotating behaviour of convection. Here, the dominant feature is the convective heat transfer through the media that moves the isotherms in $\Phi ^{\,f}$ and $\Phi ^m$ in opposite directions. The heat transfer between $\Phi ^{\,j}$ does not significantly affect $\delta \Theta ^{\,j}$ because it is secondary. By definition, the boundary between $C_f$ and $C_m$ corresponds to equal absolute values of $\delta \Theta ^{\,f}$ and $\delta \Theta ^m$. The inequalities $|\delta \Theta ^{\,f}|>|\delta \Theta ^m|$ and $|\delta \Theta ^{\,f}|<|\delta \Theta ^m|$ hold in regimes $C_f$ and $C_m$, respectively. Transition from $C_f$ to $C_m$ is characterized by increasing influence of convective heat transfer through $\Phi ^m$ as compared to that through $\Phi ^{\,f}$. From now on, if we consider the regimes $C_m$, $C_{mx}$, $M_z$ and $M$ (figure 4), the convection in $\Phi ^m$ determines the flow pattern.

The transition sequence

\[ M\rightarrow M_z\rightarrow C_{mx}\rightarrow C_m \]

is characterized by the processes in the media $\Phi ^m$ and $\Phi ^{\,f}$, which mirror the processes in the media $\Phi ^{\,f}$ and $\Phi ^m$ in the sequence (7.1) described in detail above. In short, when entering $M_z$ from $M$, the vertical velocity $u_z^{\,f}$ in $\Phi ^{\,f}$ changes sign. When entering $C_{mx}$, the temperature amplitude $\delta \Theta ^{\,f}$ changes sign. When entering $C_m$, the horizontal velocity $u^{\,f}_x$ in $\Phi ^{\,f}$ reverts direction and cellular convection in $\Phi ^{\,f}$ with an opposite direction of circulation forms (figure 5).

7.2. Solution map at $B=\mbox {const}$

Now, we constrain the regions of the $\{\kappa ,\Lambda ,B\}$ space corresponding to each convection regime. First, we consider the numerical solutions to the dispersion equation (4.8) for fixed $B$ and different $\kappa$ and $\Lambda$ (figure 6). The numerical solution indicates that only the transitions shown in figure 5 are possible. The straight line $\kappa =\Lambda$ for any $B$ is the boundary between a pair of flow regimes. In agreement with the asymptotic case $B\rightarrow \infty$ (see § 6.2), regimes $F$ and $F_z$ can arise only at $\kappa \geq \Lambda$, i.e. below the straight line $\kappa =\Lambda$, whereas $M$ and $M_z$ can arise only at $\kappa \leq \Lambda$, i.e. above the straight line $\kappa =\Lambda$. The regimes $C_{fx}$, $C_f$, $C_m$, $C_{mx}$ and $P$ can arise near the straight line $\kappa =\Lambda$ and in a wider region at $\Lambda >\kappa >1/2$. To make the boundaries of the flow regimes near the straight line $\kappa =\Lambda$ more distinct, we show the solution map in both the $\{\kappa ,\Lambda \}$ plane (figure 6a,c) and the $\{\xi _1,\xi _2\}$ plane (figure 6b,d), where

\[ \xi_1=\frac{1}{\sqrt{2}}\left(\kappa+\Lambda-1\right),\quad \xi_2=\frac{1}{\sqrt{2}}\left(\Lambda-\kappa\right). \]

Thus, the origin of the $\{\xi _1,\xi _2\}$ plane is the point $\kappa =\Lambda =1/2$ in the $\{\kappa ,\Lambda \}$ plane (figure 6a).

Figure 6.

Solutions to the dispersion equation for (a,b) $B=0.1$ and (c,d) $B=1$. The solution maps for other values of $B$ are also shown in the panels (b,d). The solid lines separate different flow regimes. The dashed lines in (a,c) are the contour lines of $\overline {{\textit {Ra}}}_c$ with step 5.

As discussed in § 6.1, if $B\rightarrow 0$ (i.e. if $\Phi ^{\,j}$ are under local equilibrium), then the solution map contains only two regions $\kappa >\Lambda$ and $\kappa <\Lambda$ corresponding to the regimes $F$ and $M$, respectively. When $B$ increases from 0 (i.e. the degree of local non-equilibrium increases), the regions $M_z$, $P$ and $C_{mx}$ appear at $\kappa \rightarrow 1$, $\Lambda \rightarrow 1$ and $\kappa <\Lambda$ (figure 6b, $B=0.03$). When $B$ increases to 0.1, the regions $M_z$, $C_{mx}$ and $P$ expand downward and leftward along the straight line $\kappa =\Lambda$ ($\xi _2=0$), and the area of the region $P$ reaches a maximum at $B\approx 0.1$ (figure 6b).

When $B$ increases from 0.1 to 1, the region $Z^c<0$, corresponding to the counter-rotating and plane regimes, expands across the straight line $\kappa =\Lambda$ into the region $\kappa >\Lambda$ and $\kappa <1/2$ (figure 6c,d). In addition, the region $C_m$ appears inside the envelope curve bounding the region $P$. Then, the boundary between $P$ and $C_m$ moves downward and leftward (along the straight line $\kappa =\Lambda$ corresponding to equal distributions of $\overline {K}$ and $\overline {\lambda }$ between $\Phi ^{\,j}$), which results in the disappearance of the region $P$ at $\kappa <\Lambda$. At $\kappa >\Lambda$, the regions $F_z$, $C_{fx}$ and $C_{f}$ appear, with the region $F_z$ growing downward from $C_{fx}$. A more detailed evolution of the solution map near the straight line $\kappa =\Lambda$ when $B$ increases from 0.025 to 2.0 is shown in figure 7.

Figure 7.

Boundaries of the convection regimes in the $\{\xi _1,\xi _2\}$ plane. See figure 6 for reference between the solution map and the convection regimes. The conditions $Z^{\,f}<0$, $Z^m<0$, $Z^c<0$ and $k_{xy}=0$ are satisfied in the corresponding regions bounded by the solid lines.

When $B$ increases from 1 to 100, all regions except $F$, $M_z$ and $C_{mx}$ shrink to the straight line $\kappa =\Lambda$. The solution map for $B>100$ is indistinguishable from the map at $B=100$ shown in figure 6(d). According to (4.11) and (4.12), the amplitudes $\delta \Theta ^{\,f}$ and $\delta \Pi ^{\,f}$ tend to zero in the region $\kappa <\Lambda$ at $\Omega =0$ and $B\rightarrow \infty$ ($Z^m=0$ or $Z^c=0$), and the amplitudes $\delta \Theta ^m$ and $\delta \Pi ^m$ tend to zero in the region $\kappa >\Lambda$ ($Z^{\,f}=0$ or $Z^c=0$). Thus, the convection only in $\Phi ^m$ (or $\Phi ^{\,f}$) develops if $\kappa <\Lambda$ (or $\kappa >\Lambda$) and $B\rightarrow \infty$. This observation agrees with the analytical investigation of the case $B\rightarrow \infty$ carried out in § 6.2.

As shown in figure 6(a), there is a sharp transition to the regime $P$ from any other regime. When entering the regime $P$, the wavenumber $k_{c}$ sharply decreases to 0 and the solution bifurcates to $P$. At the same time, $\overline {{\textit {Ra}}}_c$ changes continuously. Other transitions, which are not into $P$ or across the straight line $\kappa =\Lambda$ (see (6.11)), are continuous rather than sharp bifurcations.

In general, all except two boundaries between the convection regimes at $B=\mbox {const}$ are curved lines, which cannot be described with a finite readable relationship. The first exception is the straight line $\kappa =\Lambda$. Let us prove that the boundary between $M_z$ and $C_{mx}$, characterized with the condition $\delta \Theta ^{\,f}=0$ (figure 5), is the straight line $\kappa =\mbox {const}$, i.e. the second exception. The parameters of any point belonging to this line satisfy the following system of equations:

(7.2a)\begin{equation} Z^m=0,\quad Z^c=0,\end{equation}

(7.2b)\begin{equation} \frac{\mathrm{d}}{\mathrm{d}k^2}\left(Z^{\,f}Z^m+Z^{\,f}Z^c+Z^mZ^c\right)=0.\end{equation}

Equations (7.2a) indicate that the line is the boundary between the regimes $M_z$ ($Z^m<0$ and $Z^c>0$) and $C_{mx}$ ($Z^m>0$ and $Z^c<0$). The dispersion equation (4.8) is satisfied by (7.2a). Equation (7.2b) indicates that the locus of solutions to (4.8) touches the surface $a_2=0$ (figure 2). Here, $k^2$ is considered as the locus parameter. With account for (7.2a), (7.2b) simplifies to

(7.3)\begin{equation} \frac{\mathrm{d}}{\mathrm{d}k^2}\left(Z^m+Z^c\right)=0. \end{equation}

Differentiating (4.9b) and (4.17), we obtain

(7.4)\begin{equation} \frac{\mathrm{d} Z^m}{\mathrm{d}k^2}=\frac{1-\Lambda}{\overline{{\textit{Ra}}}}-(1-\kappa)\frac{k_z^2}{k^2},\quad \frac{\mathrm{d} Z^c}{\mathrm{d}k^2}=(1-\kappa)\kappa\frac{2k^2\kappa B+1}{(k^2\kappa B+1)^2}\frac{k_z^2}{k^4}. \end{equation}

Substituting (4.9b), (4.17) and (7.4) into (7.2a) and (7.3), we derive the system of three algebraic equations. After excluding $\overline {{\textit {Ra}}}$ and $k_{xy}^2$ from that system and carrying out identity transformations, we obtain

(7.5)\begin{equation} \kappa=\frac{Bk_z^2+1}{2Bk_z^2+1}. \end{equation}

For $B=\mbox {const}$, (7.5) gives a vertical straight line in the plane $\{\kappa ,\Lambda \}$ between the regimes $M_z$ and $C_{mx}$. According to (7.5), if $B=0$, then $\kappa =1$, and with increasing $B$, the boundary between $M_z$ and $C_{mx}$ moves leftward in the plane $\{\kappa ,\Lambda \}$ approaching the straight line $\kappa =1/2$ in the limit $B\rightarrow \infty$.

7.3. Solutions at $\kappa =\mbox {const}$ or $\Lambda =\mbox {const}$

The solutions to (4.8) for fixed $\kappa =0.7$ and $\Lambda =0.7$ are shown in figures 8 and 9, respectively. The style of the contour lines shows the type of solution, either $Z^{\,f}<0$ (regimes $F$ and $F_z$) or $Z^m<0$ (regimes $M$ and $M_z$), or $Z^c<0$ (regimes $C_{fx}$, $C_{f}$, $C_m$ and $C_{mx}$), or $k_{xy}=0$ (regime $P$). If $B\rightarrow 0$, then, in agreement with (6.5), $\overline {{\textit {Ra}}}_c\rightarrow 4{\rm \pi} ^2$ and $k_c\rightarrow {\rm \pi}$. For any $B$, the critical Rayleigh number is not higher than $4{\rm \pi} ^2$, and thus, $\overline {{\textit {Ra}}}_c$ for DPM does not exceed $\overline {{\textit {Ra}}}_c$ for EPM (figures 8a and 9a). This result demonstrates that DPM differs from BDPM, in which LTNE can increase the value of $\overline {{\textit {Ra}}}_c$ (Nield & Kuznetsov Reference Nield and Kuznetsov2006). The critical wavenumber $k_c$ can be either higher or lower than ${\rm \pi}$. When entering the regime $P$, $k_c$ abruptly decreases to 0 and the solution bifurcates into the plane flow pattern (figures 8b and 9b).

Figure 8.

Solutions to the dispersion equation at $\kappa =0.7$ against variable $B$. The dotted curve is given by (6.12) at $\Lambda =0.8$.

Figure 9.

Solutions to the dispersion equation at $\Lambda =0.7$ against variable $B$. The dotted curve is given by (6.12) at $\kappa =0.6$.

The labels $\Lambda =0.7+$ and $\kappa =0.7-$ mean that $\Lambda$ and $\kappa$ are slightly less than or higher than 0.7, respectively. The contour lines $\Lambda =0.7+$ at $\kappa B\geq {\rm \pi} ^{-2}$ and $\kappa =0.7-$ at $\Lambda B\geq {\rm \pi} ^{-2}$ correspond to the second solution for $k_c$ in (6.11). As shown in figure 3(b), the contour lines $\Lambda =0.7-$ and $\kappa =0.7+$ would be straight lines $k_c={\rm \pi}$ corresponding to the first solution for $k_c$ in (6.11).

The numerical solutions to (4.8) indicate that the conclusions of this section are valid for any constant value of $\kappa$ or $\Lambda$ other than 0.7.