We consider a horizontal layer of porous medium confined between the planes

$z=0$ and

$z=H$ where the

*z*-axis is vertically upwards. Each boundary wall is assumed to be permeable to the throughflow and perfectly thermally conducting. Radiation heat transfer between the sides of walls is negligible when compared with other modes of heat transfer. The size of nanoparticles is small as compared to the pore size of the matrix. The nanoparticles are spherical, and the nanofluid is incompressible and laminar. It is assumed that nanoparticles are suspended in the nanofluid using either a surfactant or surface charge technology, preventing the agglomeration and deposition of these on the porous medium. The porosity of the medium is denoted by

*ϵ* and the permeability by

*K*. The temperatures at the lower and upper wall are

${T}_{1}$ and

${T}_{0}$ with

${T}_{1}>{T}_{0}$. The nanoparticle volume fractions are

${\varphi}_{0}$ at the lower wall and

${\varphi}_{1}$ at the upper wall, and it is assumed that the difference

${\varphi}_{1}-{\varphi}_{0}$ is small in comparison with

${\varphi}_{0}$. A uniform magnetic field of strength

*B* is imposed normal to the plate. Using the modified Brinkman model and the Oberbeck-Boussinesq approximation, the conservation equations for mass, momentum, energy and nanoparticles are as follows:

$\mathrm{\nabla}\cdot \mathbf{v}=0,$

(1)

$\frac{\rho}{\u03f5}\frac{d\mathbf{v}}{dt}=-\mathrm{\nabla}\mathbf{p}+[\varphi {\rho}_{p}+(1-\varphi )\left\{\rho (1-\beta (T-{T}_{0}))\right\}]\mathbf{g}+\tilde{\mu}{\mathrm{\nabla}}^{2}\mathbf{v}-\frac{\mu}{K}\mathbf{v}-{\sigma}_{m}{B}^{2}\mathbf{v},$

(2)

${(\rho c)}_{m}\frac{\partial T}{\partial t}+{(\rho c)}_{f}\mathbf{v}\cdot \mathrm{\nabla}T={k}_{m}{\mathrm{\nabla}}^{2}T+\u03f5{(\rho c)}_{p}({D}_{B}\mathrm{\nabla}\varphi \cdot \mathrm{\nabla}T+\frac{{D}_{T}}{{T}_{1}}\mathrm{\nabla}T\cdot \mathrm{\nabla}T),$

(3)

$\frac{\partial \varphi}{\partial t}+\frac{1}{\u03f5}\mathbf{v}\cdot \mathrm{\nabla}\varphi ={D}_{B}{\mathrm{\nabla}}^{2}\varphi +\frac{{D}_{T}}{{T}_{1}}{\mathrm{\nabla}}^{2}T,$

(4)

where $\mathbf{v}=(u,v,w)$ is the velocity vector, *ρ* is the density of the fluid, *t* is the time, **p** is the hydraulic pressure, *ϕ* is the volume fraction of nanoparticles, ${\rho}_{p}$ is the density of nanoparticles, *β* is the coefficient of thermal expansion, $\tilde{\mu}$ is effective viscosity, *μ* is viscosity and ${\sigma}_{m}$ is electric conductivity. In the energy equation, ${(\rho c)}_{m}$ is the heat capacity of the fluid in the porous medium, ${(\rho c)}_{p}$ is the heat capacity of nanoparticles and ${k}_{m}$ is thermal conductivity. In the equation of continuity for nanoparticles, ${D}_{B}$ is the Brownian diffusion coefficient, given by the Einstein-Stokes equation and ${D}_{T}$ is the thermophoretic diffusion coefficient of nanoparticles.

We note that in equation (

2),

$\frac{d}{dt}=\frac{\partial}{\partial t}+\frac{1}{\u03f5}(\mathbf{v}\cdot \mathrm{\nabla}),\phantom{\rule{2em}{0ex}}\mathrm{\nabla}=(\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}),\phantom{\rule{2em}{0ex}}{\mathrm{\nabla}}^{2}=(\frac{{\partial}^{2}}{\partial {x}^{2}},\frac{{\partial}^{2}}{\partial {y}^{2}},\frac{{\partial}^{2}}{\partial {z}^{2}})$

is the convective derivative. We introduce non-dimensional variables as

$\begin{array}{r}({x}^{\prime},{y}^{\prime},{z}^{\prime})=\left(\frac{x,y,z}{H}\right),\phantom{\rule{2em}{0ex}}({u}^{\prime},{v}^{\prime},{w}^{\prime})=\left(\frac{u,v,w}{\kappa}\right)H,\phantom{\rule{2em}{0ex}}{t}^{\prime}=\frac{t\kappa}{\sigma {H}^{2}},\\ {p}^{\prime}=\frac{pK}{\mu \kappa},\phantom{\rule{2em}{0ex}}{T}^{\prime}=\frac{T-{T}_{1}}{{T}_{0}-{T}_{1}},\phantom{\rule{2em}{0ex}}{\varphi}^{\prime}=\frac{\varphi -{\varphi}_{0}}{{\varphi}_{1}-{\varphi}_{0}},\end{array}$

(5)

where

$\kappa ={k}_{m}/{(\rho c)}_{p}$ is thermal diffusivity of the fluid. It is assumed that the wall is heated by convection from a hot fluid with temperature

${T}_{w}$ and heat transfer coefficient

${h}_{f}$. Under these conditions, the thermal field is written as

$-{k}_{m}\frac{\partial T}{\partial z}{|}_{z=0}={h}_{f}({T}_{0}-{T}_{w}),\phantom{\rule{2em}{0ex}}T{|}_{z=H}={T}_{1},$

(6)

while the boundary conditions for velocity and nanoparticle concentration are

$w(0)=V,\phantom{\rule{2em}{0ex}}\varphi (0)={\varphi}_{0},\phantom{\rule{2em}{0ex}}w(H)=V\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\varphi (H)={\varphi}_{1}.$

(7)

After substituting equations (

5) into (1)-(4), the resulting nondimensional equations are written as follows (ignoring primes):

$\mathrm{\nabla}\cdot \mathbf{v}=0,$

(8)

$\frac{1}{\mathit{Va}}\frac{d\mathbf{v}}{dt}=-\mathrm{\nabla}\mathbf{p}-\mathit{Rm}{\stackrel{\u02c6}{e}}_{z}+\mathit{Ra}T{\stackrel{\u02c6}{e}}_{z}-\mathit{Rn}\varphi {\stackrel{\u02c6}{e}}_{z}+\tilde{\mathit{Da}}{\mathrm{\nabla}}^{2}\mathbf{v}-\mathbf{v}-M\mathbf{v},$

(9)

$\frac{\partial T}{\partial t}+\mathbf{v}\cdot \mathrm{\nabla}T={\mathrm{\nabla}}^{2}T+\frac{{N}_{B}}{\mathit{Le}}\mathrm{\nabla}\varphi \cdot \mathrm{\nabla}T+\frac{{N}_{A}{N}_{B}}{\mathit{Le}}\mathrm{\nabla}T\cdot \mathrm{\nabla}T,$

(10)

$\frac{1}{\sigma}\frac{\partial \varphi}{\partial t}+\frac{1}{\u03f5}\mathbf{v}\cdot \mathrm{\nabla}\varphi =\frac{1}{\mathit{Le}}{\mathrm{\nabla}}^{2}\varphi +\frac{{N}_{A}}{\mathit{Le}}{\mathrm{\nabla}}^{2}T.$

(11)

The boundary conditions are written as

$\begin{array}{r}w=Q,\phantom{\rule{2em}{0ex}}\frac{\partial T}{\partial z}=-\mathit{Bi}(1-T),\phantom{\rule{2em}{0ex}}\varphi =0\phantom{\rule{1em}{0ex}}\text{at}z=0,\\ w=Q,\phantom{\rule{2em}{0ex}}T=0,\phantom{\rule{2em}{0ex}}\varphi =1\phantom{\rule{1em}{0ex}}\text{at}z=0.\end{array}$

(12)

The dimensionless parameters in equations (

8)-(

12) are the Prandtl number

*Pr*, the Darcy number

*Da*, the Vadasz number

*Va*, the density Rayleigh number

*Rm*, the Rayleigh-Darcy number

*Ra*, the concentration Rayleigh number

*Rn*, the Brinkman-Darcy number

$\tilde{\mathit{Da}}$, the magnetic parameter

*M*, the Lewis number

*Le*, the modified diffusivity ratio

${N}_{A}$, the modified particle-density increment

${N}_{B}$, the Peclet number

*Q* and the Biot number

*Bi*. These parameters are defined, respectively, by

$\begin{array}{c}\mathit{Pr}=\mu /\rho \kappa ,\phantom{\rule{2em}{0ex}}\mathit{Da}=K/{H}^{2},\phantom{\rule{2em}{0ex}}\mathit{Va}=\frac{\u03f5\mathit{Pr}}{\mathit{Da}},\hfill \\ \mathit{Rm}=\frac{{\rho}_{p}{\varphi}_{0}+\rho (1-{\varphi}_{0})gH}{\mu \kappa},\phantom{\rule{2em}{0ex}}\mathit{Ra}=\frac{\rho g\alpha H({T}_{0}-{T}_{1})}{\mu \kappa},\phantom{\rule{2em}{0ex}}\mathit{Rn}=\frac{({\rho}_{p}-\rho )({\varphi}_{1}-{\varphi}_{0})gH}{\mu \kappa},\hfill \\ \tilde{\mathit{Da}}=\frac{\tilde{\mu}K}{\mu {H}^{2}},\phantom{\rule{2em}{0ex}}M=\frac{{\sigma}_{m}{B}_{0}^{2}\kappa}{\mu},\phantom{\rule{2em}{0ex}}\mathit{Le}=\frac{\kappa}{{D}_{B}},\phantom{\rule{2em}{0ex}}{N}_{A}=\frac{{D}_{T}({T}_{0}-{T}_{1})}{{D}_{B}{T}_{1}({\varphi}_{1}-{\varphi}_{0})},\hfill \\ {N}_{B}=\frac{{(\rho c)}_{p}({\varphi}_{1}-{\varphi}_{0})}{{(\rho c)}_{f}},\phantom{\rule{2em}{0ex}}Q=\frac{HV}{{\alpha}_{m}},\phantom{\rule{2em}{0ex}}\mathit{Bi}=\frac{{h}_{f}H}{{k}_{m}}.\hfill \end{array}$

We note here that the parameter *Rm* is a measure of the basic static pressure gradient.

### 2.1 Basic solution

A time-independent quiescent solution is obtained in the

*z* direction only and has the form

$\mathbf{v}={v}_{b}\equiv (0,0,Q),\phantom{\rule{2em}{0ex}}T={T}_{b}(z),\phantom{\rule{2em}{0ex}}\varphi ={\varphi}_{b}(z).$

(13)

Assumptions that

*Le* is very large (of order 10

^{2} to 10

^{3}, see Buongiorno [

3]), equations (

9)-(

11) now reduce to

$\tilde{\mathit{Da}}\frac{{d}^{2}{v}_{b}}{d{z}^{2}}-\frac{d{P}_{b}}{dz}-\mathit{Rm}+\mathit{Ra}{T}_{b}-\mathit{Rn}{\varphi}_{b}-{v}_{b}-M{v}_{b}=0,$

(14)

$\frac{{d}^{2}{T}_{b}}{d{z}^{2}}+\frac{{N}_{B}}{\mathit{Le}}\frac{d{\varphi}_{b}}{dz}\frac{d{T}_{b}}{dz}+\frac{{N}_{A}{N}_{B}}{\mathit{Le}}{\left(\frac{d{T}_{b}}{dz}\right)}^{2}=0,$

(15)

$\frac{{d}^{2}{\varphi}_{b}}{d{z}^{2}}+{N}_{A}\frac{{d}^{2}{T}_{b}}{d{z}^{2}}=0.$

(16)

Solving equations (

15) and (

16) with boundary condition (12) gives

${T}_{b}=-\frac{\mathit{Bi}({e}^{Qz}-{e}^{Q})}{Q-\mathit{Bi}(1-{e}^{Q})},$

(17)

${\varphi}_{b}=\frac{{e}^{\lambda z}-1}{{e}^{\lambda}-1}(1+\frac{\delta ({e}^{Q}-1)}{Q(Q-\lambda )})+\frac{\delta}{Q(Q-\lambda )({e}^{\lambda}-1)}(1-{e}^{Qz}({e}^{\lambda}-1)),$

(18)

where

$\lambda =\frac{QLe}{\epsilon}$ and

$\delta =\frac{{N}_{A}\mathit{Bi}{Q}^{2}}{Q-\mathit{Bi}(1-{e}^{Q})}$. In the limit

$Q\to 0$, we obtain

${T}_{b}=\frac{\mathit{Bi}}{1+\mathit{Bi}}(1-z),\phantom{\rule{2em}{0ex}}{\varphi}_{b}=z.$

(19)

As

$\mathit{Bi}\to \mathrm{\infty}$, the thermal boundary condition at the lower plate reduces to

$T(0)=1$ (isothermal condition), and the base solution becomes

${T}_{b}=1-z\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\varphi}_{b}=z;$

(20)

the same results were obtained by Nield and Kuznetsov [26].

### 2.2 Perturbation solution

To study the stability of the system, we superimpose infinitesimal perturbations on the basic state solution,

$\mathbf{v}={\mathbf{v}}_{b}+{\mathbf{v}}^{\prime},\phantom{\rule{2em}{0ex}}p={p}_{b}+{p}^{\prime},\phantom{\rule{2em}{0ex}}T={T}_{b}+{T}^{\prime},\phantom{\rule{2em}{0ex}}\varphi ={\varphi}_{b}+{\varphi}^{\prime}.$

(21)

Substituting (21) in equations (

8)-(

11), and linearizing by neglecting products of primed quantities, we get the following equations:

$(\frac{\partial {w}^{\prime}}{\partial x}+\frac{\partial {w}^{\prime}}{\partial y}+\frac{\partial {w}^{\prime}}{\partial z})=0,$

(22)

$\frac{1}{\mathit{Va}}\frac{\partial {w}^{\prime}}{\partial t}=-\frac{\partial {p}^{\prime}}{\partial z}+\mathit{Ra}{T}^{\prime}{\stackrel{\u02c6}{e}}_{z}-\mathit{Rn}{\varphi}^{\prime}{\stackrel{\u02c6}{e}}_{z}+\tilde{\mathit{Da}}(\frac{{\partial}^{2}{w}^{\prime}}{\partial {x}^{2}}+\frac{{\partial}^{2}{w}^{\prime}}{\partial {y}^{2}}+\frac{{\partial}^{2}{w}^{\prime}}{\partial {z}^{2}})-{w}^{\prime}-M{w}^{\prime},$

(23)

$\begin{array}{r}\frac{\partial T}{\partial t}+Q\frac{\partial {T}^{\prime}}{\partial z}+\frac{d{T}_{b}}{dz}{w}^{\prime}=(\frac{{\partial}^{2}{T}^{\prime}}{\partial {x}^{2}}+\frac{{\partial}^{2}{T}^{\prime}}{\partial {y}^{2}}+\frac{{\partial}^{2}{T}^{\prime}}{\partial {z}^{2}})\\ \phantom{\frac{\partial T}{\partial t}+Q\frac{\partial {T}^{\prime}}{\partial z}+\frac{d{T}_{b}}{dz}{w}^{\prime}=}+\frac{{N}_{B}}{\mathit{Le}}(\frac{d{\varphi}_{b}}{dz}\frac{\partial {T}^{\prime}}{\partial z}+\frac{d{T}_{b}}{dz}\frac{\partial {\varphi}^{\prime}}{\partial z})+\frac{2{N}_{A}{N}_{B}}{\mathit{Le}}\frac{d{T}_{b}}{dz}\frac{\partial {T}^{\prime}}{\partial z},\end{array}$

(24)

$\begin{array}{r}\frac{1}{\sigma}\frac{\partial {\varphi}^{\prime}}{\partial t}+\frac{Q}{\u03f5}\frac{\partial {\varphi}^{\prime}}{\partial z}+\frac{1}{\u03f5}\frac{d{\varphi}_{b}}{dz}{w}^{\prime}=\frac{1}{\mathit{Le}}(\frac{{\partial}^{2}{\varphi}^{\prime}}{\partial {x}^{2}}+\frac{{\partial}^{2}{\varphi}^{\prime}}{\partial {y}^{2}}+\frac{{\partial}^{2}{\varphi}^{\prime}}{\partial {z}^{2}})\\ \phantom{\frac{1}{\sigma}\frac{\partial {\varphi}^{\prime}}{\partial t}+\frac{Q}{\u03f5}\frac{\partial {\varphi}^{\prime}}{\partial z}+\frac{1}{\u03f5}\frac{d{\varphi}_{b}}{dz}{w}^{\prime}=}+\frac{{N}_{A}}{\mathit{Le}}(\frac{{\partial}^{2}{T}^{\prime}}{\partial {x}^{2}}+\frac{{\partial}^{2}{T}^{\prime}}{\partial {y}^{2}}+\frac{{\partial}^{2}{T}^{\prime}}{\partial {z}^{2}}),\end{array}$

(25)

with the boundary conditions

$\begin{array}{r}{w}^{\prime}=0,\phantom{\rule{2em}{0ex}}\frac{\partial {T}^{\prime}}{\partial z}=Bi{T}^{\prime},\phantom{\rule{2em}{0ex}}{\varphi}^{\prime}=0\phantom{\rule{1em}{0ex}}\text{at}z=0,\\ {w}^{\prime}=0,\phantom{\rule{2em}{0ex}}{T}^{\prime}=0,\phantom{\rule{2em}{0ex}}{\varphi}^{\prime}=0\phantom{\rule{1em}{0ex}}\text{at}z=1.\end{array}$

(26)

The derivatives of

${T}_{b}$ and

${\varphi}_{b}$ are

$\frac{d{T}_{b}}{dz}=-\frac{BiQ{e}^{Qz}}{Q-\mathit{Bi}(1-{e}^{Q})},$

(27)

$\frac{d{\varphi}_{b}}{dz}=\frac{\lambda {e}^{\lambda z}}{{e}^{\lambda}-1}(1-\frac{\delta ({e}^{Q}-1)}{Q(Q-\lambda )})-\frac{\delta Q}{Q-\lambda}{e}^{Qz}.$

(28)

For regular fluids, the parameters

*Rn*,

${N}_{A}$ and

${N}_{B}$ are zero and the third term in equation (

25) is absent since

$d{\varphi}_{b}/dz=0$. For

$\tilde{\mathit{Da}}=0$, and in the absence of a magnetic field, the equations reduce to the familiar Horton-Roger-Lapwood problem with throughflow. Taking the curl of equation (

23) and simplifying, we obtain

$\frac{1}{\mathit{Va}}\frac{\partial}{\partial t}{\mathrm{\nabla}}^{2}{w}^{\prime}-\tilde{\mathit{Da}}{\mathrm{\nabla}}^{4}{w}^{\prime}+(1+M){\mathrm{\nabla}}^{2}{w}^{\prime}=\mathit{Ra}{\mathrm{\nabla}}_{H}^{2}{T}^{\prime}+\mathit{Rn}{\mathrm{\nabla}}_{H}^{2}{\varphi}^{\prime}.$

(29)