A rectangular porous cavity saturated with (alumina-tantalum)/motor oil is considered. The top and bottom walls are kept at constant temperatures $T_0$ and $T_0+\text{Δ}T$, respectively. The Cartesian coordinate system used is composed of the unit vectors $e_x$, $e_y$ and $e_z$, whose y-axis follows the horizontal and vertical z-axis is collinear with gravity. The porous medium is subjected to a uniform magnetic field $B_0$. For this problem, we have neglected the effect of Joule heat and viscous dissipation on heat transfer, and have assumed that the induced magnetic field is very weak compared with $B_0$.

Referring to the work of Dèdèwanou in [30] [31] , of Cimpean in [32] and of Manjunatha in [33] , taking into account the above-mentioned criteria and assuming local thermal non-equilibrium between the liquid and solid phases of the skeleton in the porous medium, the coupled equations governing the convective flow of the hybrid nanoliquid are modeled as follows:

$\nabla \cdot q=0,$(1)

$\begin{array}{l}\frac{{\rho }_{hnf}}{\epsilon }\left[\frac{\partial q}{\partial t}+\frac{1}{\epsilon }\left(q\cdot \nabla \right)q\right]\\ =-\nabla P-\rho g+\frac{{\mu }_{hnf}}{\epsilon }{\nabla }^{2}q-\frac{{\mu }_{hnf}}{K}q+\frac{2{\rho }_{hnf}}{\epsilon }\left(q\wedge \Omega \right)+{\sigma }_{hnf}J\wedge B_0,\end{array}$(2)

${\left(\rho Cp\right)}_{hnf}\left[\epsilon \frac{\partial T}{\partial t}+\left(q\cdot \nabla \right)T\right]=\epsilon k_{hnf}{\nabla }^{2}T+Q\left(T-T_0\right),$(3)

$\left(1-\epsilon \right){\left(\rho Cp\right)}_s\frac{\partial T_s}{\partial t}=\left(1-\epsilon \right)k_s{\nabla }^2{T}_s+Q\left(T_s-T\right).$(4)

The nano-liquid considered complies with Faraday’s law, Ampere’s law, Maxwell’s law and the Gauss equations defined as:

$\nabla \wedge B_0={\mu }_e\left(J+{\epsilon }_{e}E\right),$(5)

$\nabla \wedge E=-\frac{\partial B_0}{\partial t},$(6)

$\nabla \cdot B_0=0,$(7)

$\nabla \cdot E=0,$(8)

$\nabla \cdot J=0,$(9)

with

$J={\sigma }_{hnf}\left(E+v\wedge B_0\right)$(10)

and ${\mu }_e$, ${\epsilon }_e$ are the magnetic permeability and absolute permeability of the fluid respectively.

In the above equations, $\nabla =\frac{\partial }{\partial x}{e}_x+\frac{\partial }{\partial y}{e}_y+\frac{\partial }{\partial z}{e}_z$ is the gradian vector and ${\nabla }^2=\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}+\frac{{\partial }^2}{\partial z^2}$ is the Laplacian operator. The indices $hnf$, $f$ and $s$ represent the properties of the nanoliquid, base liquid and solid respectively.

Given the extremely small size of the nanofragments, the interaction between the heat transfer fluid and the nanoparticles does not generate extracorporeal torque or other forces. It simply modifies the spatio-temporal properties, which are obtained from experimental data. Adopting the Boussinesq approximation, the equation of state in terms of density at temperature $T$ of the nano-liquid is written as follows:

$\rho ={\rho }_{hnf}\left[1-{\beta }_{hnf}\left(T-T_0\right)\right].$(11)

The physico-chemical parameters of the hybrid nano-liquids are

Density:

${\rho }_{hnf}={\phi }_1{\rho }_1+{\phi }_2{\rho }_2+\left(1-{\phi }_1-{\phi }_2\right){\rho }_f,$(12)

Thermal expansion coefficient:

${\left(\rho \beta \right)}_{hnf}={\phi }_1{\left(\rho \beta \right)}_1+{\phi }_2{\left(\rho \beta \right)}_2+\left(1-{\phi }_1-{\phi }_2\right){\left(\rho \beta \right)}_f.$(13)

Heat capacitance:

${\left(\rho Cp\right)}_{hnf}={\phi }_1{\left(\rho Cp\right)}_1+{\phi }_2{\left(\rho Cp\right)}_2+\left(1-{\phi }_1-{\phi }_2\right){\left(\rho Cp\right)}_f.$(14)

Thermal conductivity:

$\begin{array}{c}\frac{k_{hnf}}{k_f}=\left\{\frac{{\phi }_1{k}_1+{\phi }_2{k}_2}{{\phi }_1+{\phi }_2}+2k_f+2\left({\phi }_1{k}_1+{\phi }_2{k}_2\right)-2\left({\phi }_1+{\phi }_2\right)k_f\right\}\\ \times {\left\{\frac{{\phi }_1{k}_1+{\phi }_2{k}_2}{{\phi }_1+{\phi }_2}+2k_f-\left({\phi }_1{k}_1+{\phi }_2{k}_2\right)+\left({\phi }_1+{\phi }_2\right)k_f\right\}}^{-1}.\end{array}$(15)

Dynamic viscosity:

${\mu }_{hnf}={\mu }_f{\left(1-{\phi }_1-{\phi }_2\right)}^{-2.5}.$(16)

Electrical conductivity:

${\sigma }_{hnf}=\left\{1+\frac{3\left[\frac{{\phi }_1{\sigma }_{s1}+{\phi }_2{\sigma }_{s2}}{{\sigma }_{hf}}-\left({\phi }_1+{\phi }_2\right)\right]}{\left[\frac{{\phi }_1{\sigma }_{s1}+{\phi }_2{\sigma }_{s2}}{{\sigma }_{hf}\left({\phi }_1+{\phi }_2\right)}+2\right]-\left[\frac{{\phi }_1{\sigma }_{s1}+{\phi }_2{\sigma }_{s2}}{{\sigma }_{hf}}-\left({\phi }_1+{\phi }_2\right)\right]}\right\}{\sigma }_{hf}.$(17)

Subscripts 1 and 2 refer to aluminate and tantalum nanoparticles respectively. The porous medium thermal conductivity saturated with hybrid nanoliquid is [34] :

$k_{mhnf}=\epsilon k_{hnf}+\left(1-\epsilon \right)k_s.$(18)

Note that above 4% nanofragments volume fraction, the nanofluid becomes a non-Newtonian fluid, which would lead to a significant increase in viscosity and cancel out any benefit obtained in terms of improved thermal conductivity.

The porous medium saturated with nanoliquids occupies the domain $\Omega =\left\{\left(x,y,z\right)\in {ℝ}^3\right\}\times \left\{y\in \left(0,l\right)\right\}\times \left\{z\in \left(0,h\right)\right\}$. The nanoliquid flow is such that the convective rolls have axes parallel to the shortest dimension, i.e., the velocity component along the x-axis is zero. Thus, the equations governing nanofluid flow are reduced to 2-D. The stress-free and slip-free boundary conditions with the perturbation technique for the nanofluid flow problem are described as follows:

$\begin{array}{ll}T=T_s=T_0+\text{Δ}T\hfill & \text{at}z=0\hfill \\ T=T_s=T_0\hfill & \text{at}z=h\hfill \\ q=0\hfill & \text{at}z=0,h\hfill \end{array}\}$(19)

The Darcy-Benard convective transport model is adopted, assuming low porosity. So, the non-linear terms $\left(q\cdot \nabla \right)q$ and ${\nabla }^{2}q$ are negligible as the porosity of the medium is considered low. By non-dimensionalizing the transport and heat transfer equations using the following quantities

$Y=\frac{y}{h},Z=\frac{z}{h},\tau =\frac{{\alpha }_{w}t}{h^2},V=\frac{hv}{{\alpha }_w},W=\frac{hw}{{\alpha }_w},$(20)

$\tilde{p}=\frac{Kp}{{\mu }_f{\alpha }_w},\Theta =\frac{T}{\Delta T},{\Theta }_s=\frac{T_s}{\Delta T},$(21)

Pressure is eliminated in Equation (2). Thus, the vorticity transport equation is obtained after subtracting the resulting non-dimensional equations. The coupled partial equations are as follows:

$\frac{\partial V}{\partial Y}+\frac{\partial W}{\partial Z}=0$(22)

$\begin{array}{l}\left[\left(\frac{{\rho }_{hnf}}{{\rho }_f}\right)\left(\frac{{\mu }_f}{{\mu }_{hnf}}\right)\frac{1}{Va}\frac{\partial }{\partial t}+1\right]{\nabla }^{2}W\\ =\left(\frac{{\mu }_f}{{\mu }_{hnf}}\right)\left(\frac{{\left(\rho \beta \right)}_{hnf}}{{\left(\rho \beta \right)}_f}\right)R{a}_f\frac{{\partial }^2\text{Θ}}{\partial Y^2}+\left(\frac{{\rho }_{hnf}}{{\rho }_f}\right)\left(\frac{{\mu }_f}{{\mu }_{hnf}}\right)\sqrt{T{a}_D}\frac{\partial \zeta }{\partial z}\\ -\left(\frac{{\sigma }_{hnf}}{{\sigma }_f}\right)\left(\frac{{\mu }_f}{{\mu }_{hnf}}\right)H{a}^2{\nabla }^{2}W,\end{array}$(23)

$\begin{array}{l}\frac{{\left(\rho Cp\right)}_{hnf}}{{\left(\rho Cp\right)}_f}\left[V_N\frac{\partial \Theta }{\partial \tau }+\frac{\gamma +1}{\gamma }\left(V\frac{\partial \Theta }{\partial Y}+W\frac{\partial \Theta }{\partial Z}\right)\right]\\ =\frac{k_{hnf}}{k_f}\left(\frac{{\partial }^2\Theta }{\partial Y^2}+\frac{{\partial }^2\Theta }{\partial Z^2}\right)-Q_0\left(\Theta -{\Theta }_s\right),\end{array}$(24)

$V_N{B}_N\frac{\partial {\Theta }_s}{\partial \tau }=\left(\frac{{\partial }^2{\Theta }_s}{\partial Y^2}+\frac{{\partial }^2{\Theta }_s}{\partial Z^2}\right)-\gamma Q_0\left({\Theta }_s-\Theta \right).$(25)

The vorticity $\zeta$ verifies the equation:

$\begin{array}{l}\left[\left(\frac{{\rho }_{hnf}}{{\rho }_f}\right)\left(\frac{{\mu }_f}{{\mu }_{hnf}}\right)\frac{1}{Va}\frac{\partial }{\partial t}+\left(1+\left(\frac{{\sigma }_{hnf}}{{\sigma }_f}\right)\left(\frac{{\mu }_f}{{\mu }_{hnf}}\right)H{a}_f^2\right)\right]\zeta \\ =\left(\frac{{\rho }_{hnf}}{{\rho }_f}\right)\left(\frac{{\mu }_f}{{\mu }_{hnf}}\right)\sqrt{T{a}_D}\frac{\partial w}{\partial z}.\end{array}$(26)

From Equations (23) and (26), we obtain:

$\left[{\left(\frac{M_1}{Va}\frac{\partial }{\partial t}+M_0\right)}^2{\nabla }^2+M_1^{2}T{a}_D\frac{{\partial }^2}{\partial z^2}\right]W=M_{3}R{a}_{Df}\left(\frac{M_1}{Va}\frac{\partial }{\partial t}+M_0\right)\frac{{\partial }^2\text{Θ}}{\partial Y^2}.$(27)

In Equations (22)-(25), $Va$ is the Vadasz number, $R{a}_{Df}$ is the thermal Darcy-Rayleigh number, $Q_0$ is the coefficient of scaled inter-phase heat transport, $\gamma$ is the ratio of porosity-weighted conductivity, $V_N$ is the Vafai number and $B_N$ is the Barletta number expressed as:

$\begin{array}{l}Va=\frac{\epsilon {\mu }_f{h}^2}{{\rho }_f{\alpha }_{w}K},R{a}_f=\frac{{\rho }_{f}g{\beta }_{f}hK\left(\text{Δ}K\right)}{{\alpha }_f{\mu }_f}\left(\frac{k_f}{k_w}\right),Q_0=\frac{Q{h}^2}{\epsilon k_f},\\ \gamma =\frac{\epsilon k_f}{\left(1-\epsilon \right)k_s},V_N=\frac{{\alpha }_w}{{\alpha }_f},B_N=\frac{{\alpha }_f}{{\alpha }_s},{\alpha }_w=\frac{k_w}{{\left(\rho Cp\right)}_w},\\ {\alpha }_s=\frac{k_s}{{\left(\rho Cp\right)}_s},T{a}_D={\left(\frac{2\text{Ω}{\rho }_{f}K}{k{\mu }_f}\right)}^2,M_0=\left(1+M_{2}H{a}_f^2\right),\\ M_1=\left(\frac{{\rho }_{hnf}}{{\rho }_f}\right)\left(\frac{{\mu }_f}{{\mu }_{hnf}}\right),M_2=\left(\frac{{\sigma }_{hnf}}{{\sigma }_f}\right)\left(\frac{{\mu }_f}{{\mu }_{hnf}}\right),{\alpha }_f=\frac{k_f}{{\left(\rho Cp\right)}_f},\\ M_3=\left(\frac{{\mu }_f}{{\mu }_{hnf}}\right)\left(\frac{{\left(\rho \beta \right)}_{hnf}}{{\left(\rho \beta \right)}_f}\right),M_4=\left(\frac{{\left(\rho Cp\right)}_f}{{\left(\rho Cp\right)}_{hnf}}\right)\left(\frac{k_{hnf}}{k_f}\right).\end{array}$(28)

The boundary conditions to which the equations governing nanoliquids in non-dimensional form are subject are defined by:

$\begin{array}{ll}W=0,\Theta =0,{\Theta }_s=1\hfill & \text{at}Z=0,\hfill \\ W=0,\Theta =0,{\Theta }_s=0\hfill & \text{at}Z=1.\hfill \end{array}\}$(29)

In the basic state, heat is transported only by conduction, and physical quantities vary only along the Y axis, as follows:

$V_b=0,W_b=0,\Theta ={\Theta }_b\left(Z\right)\text{and}\Theta ={\Theta }_{sb}\left(Z\right)$(30)

Using the conduction solutions defined in Equation (30), Equations (22)-(25) generate other equations that are solved with the boundary conditions of Equation (29); we obtain

${\Theta }_b\left(Z\right)=1-Z\text{and}{\Theta }_{sb}\left(Z\right)=1-Z.$(31)

Equation (31) shows that the solid and nanoliquid temperatures have the same expression and do not depend on $Q_0$ and $\gamma$, confirming that the basic state is a conduction state. After having carried out small disturbances of the basic flow, which are:

$\begin{array}{l}V=V^{\prime }\left(\tau ,Y,Z\right),W=W^{\prime }\left(\tau ,Y,Z\right),\\ \Theta ={\Theta }_b\left(Z\right)+{\Theta }^{\prime }\left(\tau ,Y,Z\right)\text{and}{\Theta }_s={\Theta }_b\left(Z\right)+{\Theta }^{\prime }\left(\tau ,Y,Z\right).\end{array}\}$(32)

From Equation (32), the equations of motion and heat-flux in the perturbed state take the following form, after omitting prime numbers and neglecting a perturbed quantity for simplicity:

$\left[{\left(\frac{M_1}{Va}\frac{\partial }{\partial t}+M_0\right)}^2{\nabla }^2+M_1^{2}T{a}_D\frac{{\partial }^2}{\partial z^2}\right]W=M_{3}R{a}_D\left(\frac{M_1}{Va}\frac{\partial }{\partial t}+M_0\right)\frac{{\partial }^2\text{Θ}}{\partial Y^2}$(33)

$V_N\frac{\partial \text{Θ}}{\partial \tau }+\frac{\gamma +1}{\gamma }\left[\left(V\frac{\partial }{\partial Y}+W\frac{\partial }{\partial Z}\right)\text{Θ}-W\right]=M_4\left(\frac{{\partial }^2\text{Θ}}{\partial Y^2}+\frac{{\partial }^2\text{Θ}}{\partial Z^2}\right)-Q_0\left(\text{Θ}-{\text{Θ}}_s\right),$(34)

$V_N{B}_N\frac{\partial {\text{Θ}}_s}{\partial \tau }=\left(\frac{{\partial }^2{\text{Θ}}_s}{\partial Y^2}+\frac{{\partial }^2{\text{Θ}}_s}{\partial Z^2}\right)+\gamma Q_0\left(\text{Θ}-{\text{Θ}}_s\right).$(35)

In the absence of nano-fragments ( $\phi =0$), the nanoliquid parameters $M_1$, $M_2$, $M_3$ and $M_4$ are all equal to 1. The above condition reduces to governing equations in the perturbed state for simple viscous fluid flow.

After decoupling the temperatures $\text{Θ}$ and ${\text{Θ}}_s$ from Equations (34) and (35), we obtain a new local thermal non-equilibrium equation with three phase shifts:

$\begin{array}{l}\left(1+C_{PV}\frac{\partial }{\partial \tau }\right)\frac{\partial \Theta }{\partial \tau }+\left[1-\frac{1}{\gamma Q_0}{\nabla }^2+C_{DN}\frac{\partial }{\partial \tau }\right]\left[\left(V\frac{\partial }{\partial Y}+W\frac{\partial }{\partial Z}\right)\Theta -W\right]\\ =M_4\left[1-\frac{1}{Q_0\left(1+\gamma \right)}{\nabla }^2+C_{BS}\frac{\partial }{\partial \tau }\right]{\nabla }^2\Theta ,\end{array}$(36)

The new parameters of the equation are expressed as follows:

$C_{PV}=\frac{V_N{B}_N}{\left(\gamma +B_N\right)Q_0},C_{DN}=\frac{V_N{B}_N}{\gamma Q_0}\text{and}{C}_{BS}=\frac{V_N\left(1+B_N\right)}{\left(1+\gamma \right)Q_0}.$(37)

In Equation (36), when $Q_0$ and $\gamma Q_0$ respectively tend towards 1, the parameters $C_{PV}$, $C_{DN}$, $C_{BS}$ and the theme $\frac{1}{\gamma Q_0}$ tend towards zero. Under this condition, Equation (36) reduces to that of the heat flux according to Fourier’s law (local thermal equilibrium) found in [30] . Moreover, in the case where the constants $\gamma$ and $Q_0$ have finite values, we obtain an equation that has the characteristics of the hyperbolic heat equation derived from Cattaneo’s flux law. The parameters $C_{PV}$, $C_{DN}$, $C_{BS}$ correspond to the relaxation/delay times caused by the phase shifts between the relevant quantities. To better understand the pathways leading to local thermal equilibrium from the local thermal non-equilibrium formulation, we introduce Rees numbers into the Darcy Bénard convection problem with LTNE defined as follows:

$\gamma Q_0=R_N\cos \left(\xi \right),Q_0=R_N\sin \left(\xi \right)$(38)

with the effective thermal resistance of the medium

$R_N=h^{2}Q\sqrt{\left(R_l^2+R_s^2\right)}$(39)

and the other Rees number $\xi =\text{tan}\left(\frac{R_l}{R_s}\right)$ that represents the phase-lag between the effective thermal resistance of the solid phases $R_s=\frac{1}{\left(1-\epsilon \right)k_s}$ and of the fluid $R_l=\frac{1}{\epsilon k_l}$.

We now want to divide Equation (36) into two equations with a first-order time derivative. To this end, we introduce new parameter changes by posing:

${\xi }_1=R_N\cos \left(\xi \right),{\xi }_2=R_N\cos \xi \left(1+\tan \xi \right),p_1=\frac{C_{DN}}{C_{PV}},p_2=\frac{C_{BS}}{C_{PV}}.$(40)

We thus obtain two new coupled equations governing the heat-flux which are:

$\frac{\partial \text{Θ}}{\partial \tau }-\chi +p_1\left[\left(V\frac{\partial }{\partial Y}+W\frac{\partial }{\partial Z}\right)\text{Θ}-W\right]=M_4{p}_2{\nabla }^2\text{Θ},$(41)

$\begin{array}{l}{C}_{PV}\frac{\partial \chi }{\partial \tau }+\chi +\left(1-p_1-\frac{1}{{\xi }_1}{\nabla }^2\right)\left[\left(V\frac{\partial }{\partial Y}+W\frac{\partial }{\partial Z}\right)\text{Θ}-W\right]\\ =M_4\left(1-p_2-\frac{1}{{\xi }_2}{\nabla }^2\right){\nabla }^2\text{Θ}.\end{array}$(42)

In Equation (42), the Cattaneo-Nield, Cattaneo-Vadasz and Cattaneo-Straughan numbers can be given early in new expressions.