# Thermal instability in a non-Darcy porous medium saturated with a nanofluid and with a convective boundary condition

- Sachin Shaw
^{1}and - Precious Sibanda
^{1}Email author

**2013**:186

https://doi.org/10.1186/1687-2770-2013-186

© Shaw and Sibanda; licensee Springer 2013

**Received: **10 April 2013

**Accepted: **5 August 2013

**Published: **20 August 2013

## Abstract

In this paper, we investigate the effect of vertical throughflow on the onset of convection in a horizontal layer of a non-Darcy porous medium saturated with a nanofluid. A normal mode analysis is used to find solutions for the fluid layer confined between parallel plates with free-rigid boundaries. The criterion for the onset of stationary and oscillatory convection is derived. The analysis incorporates the effects of Brownian motion, thermophoresis and a convective boundary condition. The effects of the concentration Rayleigh number, Lewis number, Darcy number and modified diffusivity ratio on the stability of the system are investigated.

### Keywords

throughflow convective boundary condition oscillatory convection linear stability analysis## 1 Introduction

In the last several years, an innovative technique for improving the heat transfer characteristics by adding ultra fine metallic particles in common fluids such as water and oil has been investigated. The term nanofluid refers to these kinds of fluids which have applications in automotive industries, energy saving devices, nuclear rectors, *etc.*, Choi [1]. Nanoparticles also have medical applications including cancer therapy and nano-drug delivery (Shaw and Murthy [2]). Buongiorno [3] noted that the absolute velocity of nanoparticles can be viewed as the sum of the base fluid velocity and a relative (slip) velocity. He concluded that in the absence of turbulence, Brownian diffusion and thermophoresis would be important. A lot of work has been done on nanofluids; see, for instance, Nadeem *et al.* [4], Malik *et al.* [5], Nadeem *et al.* [6] and the review by Das *et al.* [7].

The effect of a magnetic field on flow and heat transfer problems is important in industrial applications, such as in the buoyant upward gas-liquid flow in packed bed electrodes (Takahashi and Alkire [8]), sodium oxide-silicon dioxide glass melt flows (Guloyan [9]), reactive polymer flows in heterogeneous porous media [10], electrochemical generation of elemental bromine in porous electrode systems (Qi and Savinell [11]).

The convective boundary condition is more general and realistic in engineering and industrial processes such as transportation cooling processes, material drying, *etc.* It therefore seems appropriate to use the convective boundary condition to study other boundary layer flow situations. Aziz [12] studied the Blasius flow over a flat plate with a convection thermal boundary condition. Ishak [13] investigated the effects of suction and injection on a flat surface with convective boundary condition. The study of the boundary layer flows over flat surfaces under convective surface boundary condition has attracted the attention of many researchers, such as Makinde and Aziz [14], Yao *et al.* [15].

Nanofluids have great potential as coolants due to their enhanced thermal conductivities. The enhancement of effective thermal conductivity was confirmed by experiments conducted by many researchers (Masuda *et al.* [16]). Instability of nanofluids in natural convection was studied by Tzou [17]. Tzou [18] studied thermal instability of nanofluids in natural convection. Thermal instability in nanofluids in a porous medium has been a topic of interest due to potential applications of such flows in food and chemical processes, petroleum industry, bio-mechanics and geophysical problems. Buongiorno’s model was applied to the problem of the onset of instability in a porous medium layer saturated with a nanofluid by Nield and Kuznetsov [19, 20]. Kuznetsov and Nield [21] used the Brinkman model to study thermal instability in a horizontal porous layer saturated with a nanofluid. Other related studies of thermal instability in a porous medium saturated with a nanofluid include those by Kuznetsov and Nield [22] and Nield and Kuznetsov [23–26].

In this study, we extend the work by Nield and Kuznetsov [26] to a non-Darcy porous medium saturated with a nanofluid. We have considered a convective boundary condition in place of an isothermal condition. The effect of the Biot number, magnetic parameter, Brinkman-Darcy parameter on thermal instability has been studied.

## 2 Mathematical formulation

*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:

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.

*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

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

### 2.1 Basic solution

*z*direction only and has the form

*Le*is very large (of order 10

^{2}to 10

^{3}, see Buongiorno [3]), equations (9)-(11) now reduce to

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

### 2.2 Perturbation solution

*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

## 3 Normal modes and stability analysis

*l*and

*m*are wave numbers in the

*x*and

*y*directions and

*n*is the growth rate of the disturbances. Substituting into the differential equations, we obtain

*N*algebraic equations in 3

*N*unknowns. The vanishing of the determinant of coefficients produces an eigenvalue equation for the system. Regarding

*Ra*as the eigenvalue, we find

*Ra*in terms of the other parameters. It is interesting to note that the convective boundary condition is applicable for rigid-free and rigid-rigid boundary conditions. In this study we mainly focus on rigid-free boundary and discuss the case of stationary and oscillatory convection. The vanishing of the shear-stresses tangent to the surface and continuity equation gives the boundary conditions

*W*, Θ and Φ are of the form

*z*from $z=0$ to $z=1$, the system of equations may be written as

For the non-trivial solution, the determinant of the augmented matrix is equal to zero, *s* is a dimensionless growth factor. We put $s=i\omega $, where *ω* is real and is the dimensionless frequency.

### 3.1 Stationary convection

*i.e.*, $N=1$, which gives the non-oscillatory stability boundary as

We calculate the corresponding critical Rayleigh number ${({\mathit{Ra}}_{s})}_{\mathrm{crit}}$ using the above critical value of *α* for stationary convection.

### 3.2 Oscillatory convection

*ω*to be real, it is necessary that

*Ra*(independent of

*ω*) and

*ω*(independent of

*Ra*) are written as

*α*for oscillatory convection is calculated from

The critical Rayleigh number ${({\mathit{Ra}}_{\mathrm{osc}})}_{\mathrm{crit}}$ is obtained using the above critical value of *α* for oscillatory convection.

## 4 Results and discussion

The effects of the Biot number, magnetic parameter, Darcy number and porosity on the stationary Rayleigh number are shown in Figure 1, where it is evident that ${\mathit{Ra}}_{s}$ increases with the magnetic parameter but decreases with the Biot number. Hence the magnetic parameter exerts a stabilizing influence on the stationary convection regime, but Biot number destabilizes stationary convection. The Darcy number has a stabilizing effect on stationary convection, while the porosity parameter has a destabilizing influence on the stationary convection for fixed Biot numbers. This finding is in line with the results reported by Chand and Rana [27].

*λ*, the modified particle-density increment and the Lewis number on the stationary Rayleigh number. The stationary Rayleigh number increases with ${N}_{A}$, ${N}_{B}$ and the Lewis number, and this helps to stabilize the stationary convection regime. On the other hand,

*λ*destabilizes stationary convection. Thus, in the absence of the Darcy-Brinkman and magnetic parameters, the stability boundary depends on ${N}_{B}$ and ${N}_{A}$ as earlier suggested by Nield and Kuznetsov [26]. The throughflow (due to Peclet number

*Q*) also assists in increasing the critical stationary Rayleigh number.

*Bi*,

*Va*and

*Q*. Hence the magnetic parameter stabilizes the oscillatory convection, while the other parameters are destabilizing to the oscillatory regime.

*Rn*so that it is positive when the particle density increases upwards (the destabilizing situation). From Figure 6, we note that

*Ra*takes a negative value when

*Rn*is sufficiently large. In this case, the destabilizing effect of nanoparticle concentration is so large that the bottom of the fluid layer must be cooled relative to the top to produce a state of neutral stability as earlier found by Kuznetsov and Nield [21] in the absence of a magnetic field and higher Biot numbers. In the present problem, a state of neutral stability appeared when $\mathit{Rn}=0.4$.

## 5 Conclusion

In this study we used linear stability to investigate the onset of thermal instability in a non-Darcy porous medium saturated with a nanofluid, and with a convective boundary condition. We have determined the effects of various embedded parameters such as the Biot number, the magnetic parameter and the Darcy number on the critical Rayleigh number for the onset of both oscillatory and stationary thermal instabilities. We have shown that increasing the Darcy number and the magnetic parameter has the effect of increasing the critical Rayleigh number for the onset of thermal instabilities in the case of stationary convection, while increasing the Biot number and the porosity is destabilizing to the stationary regime. The modified diffusivity ratio, particle density increment and the Lewis number help to stabilize stationary convection. Oscillatory convection was found not to be as sensitive to the fluid and physical parameters as stationary convection.

## Declarations

### Acknowledgements

The authors wish to thank the University of KwaZulu-Natal for financial support.

## Authors’ Affiliations

## References

- Choi SUS: Enhancing thermal conductivity of fluids with nanoparticles. 231. In
*Development and Applications of Non-Newtonian Flows*. Edited by: Siginer DA, Wang HP. ASME, New York; 1995:99-105. and FED vol. 66Google Scholar - Shaw S, Murthy PVSN, Sibanda P: Magnetic drug targeting in permeable microvessel.
*Microvasc. Res.*2013, 85: 77-85.View ArticleGoogle Scholar - Buongiorno J: Convective transport in nanofluids.
*J. Heat Transf.*2006, 128: 240-250. 10.1115/1.2150834View ArticleGoogle Scholar - Nadeem S, Rehman A, Ali M: The boundary layer flow and heat transfer of a nanofluid over a vertical slender cylinder.
*J. Nanoeng. Nanosyst.*2012, 226: 165-173.Google Scholar - Malik MY, Hussain A, Nadeem S: Flow of a non-Newtonian nanofluid between coaxial cylinders with variable viscosity.
*Z. Naturforsch. A*2012, 67a: 255-261.Google Scholar - Nadeem S, Mehmood R, Akbar NS: Non-orthogonal stagnation point flow of a nano non-Newtonian fluid towards a stretching surface with heat transfer.
*Int. J. Heat Mass Transf.*2013, 55: 3964-3970.Google Scholar - Das SK, Choi SUS, Yu W, Pradeep T:
*Nanofluids: Science and Technology*. Wiley, New York; 2007.View ArticleGoogle Scholar - Takahashi K, Alkire R: Mass transfer in gas-sparged porous electrodes.
*Chem. Eng. Commun.*1985, 38: 209-227. 10.1080/00986448508911307View ArticleGoogle Scholar - Guloyan YA: Chemical reactions between components in the production of glass-forming melt.
*Glass Ceram.*2003, 60: 233-235. 10.1023/A:1027395310680View ArticleGoogle Scholar - Liu H, Thompson KE: Numerical modelling of reactive polymer flow in porous media.
*Comput. Chem. Eng.*2002, 26: 1595-1610. 10.1016/S0098-1354(02)00130-8View ArticleGoogle Scholar - Qi J, Savinell RF: Analysis of flow-through porous electrode cell with homogeneous chemical reactions: application to bromide oxidation in brine solutions.
*J. Appl. Electrochem.*1993, 23: 873-886. 10.1007/BF00251022View ArticleGoogle Scholar - Aziz A: A similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition.
*Commun. Nonlinear Sci. Numer. Simul.*2009, 14: 1064-1068. 10.1016/j.cnsns.2008.05.003MathSciNetView ArticleGoogle Scholar - Ishak A: Similarity solutions for flow and heat transfer over a permeable surface with convective boundary condition.
*Appl. Math. Comput.*2010, 217: 837-842. 10.1016/j.amc.2010.06.026MathSciNetView ArticleMATHGoogle Scholar - Makinde OD, Aziz A: MHD mixed convection from a vertical plate embedded in a porous medium with a convective boundary condition.
*Int. J. Therm. Sci.*2010, 49: 1813-1820. 10.1016/j.ijthermalsci.2010.05.015View ArticleGoogle Scholar - Yao S, Fang T, Zhong Y: Heat transfer of a generalized stretching/shrinking wall problem with convective boundary conditions.
*Commun. Nonlinear Sci. Numer. Simul.*2011, 16: 752-760. 10.1016/j.cnsns.2010.05.028View ArticleMATHGoogle Scholar - Masuda H, Ebata A, Teramae K, Hishinuma N: Alteration of thermal conductivity and viscosity of liquid by dispersing ultra-fine particles.
*Netsu Bussei*1993, 7: 227-233. 10.2963/jjtp.7.227View ArticleGoogle Scholar - Tzou DY: Instability of nanofluids in natural convection.
*J. Heat Transf.*2008., 130: Article ID 072401Google Scholar - Tzou DY: Thermal instability of nanofluids in natural convection.
*Int. J. Heat Mass Transf.*2008, 51: 2967-2979. 10.1016/j.ijheatmasstransfer.2007.09.014View ArticleMATHGoogle Scholar - Nield DA, Kuznetsov AV: Thermal instability in a porous medium layer saturated by a nanofluid.
*Int. J. Heat Mass Transf.*2009, 52: 5796-5801. 10.1016/j.ijheatmasstransfer.2009.07.023View ArticleMATHGoogle Scholar - Kuznetsov AV, Nield DA: Effect of local thermal non-equilibrium on the onset of convection in a porous medium layer saturated by a nanofluid.
*Transp. Porous Media*2010, 83: 425-436. 10.1007/s11242-009-9452-8View ArticleGoogle Scholar - Kuznetsov AV, Nield DA: Thermal instability in a porous medium layer saturated by a nanofluid: Brinkman model.
*Transp. Porous Media*2010, 81: 409-422. 10.1007/s11242-009-9413-2MathSciNetView ArticleGoogle Scholar - Kuznetsov AV, Nield DA: The onset of double-diffusive nanofluid convection in a layer of a saturated porous medium.
*Transp. Porous Media*2010, 85: 941-951. 10.1007/s11242-010-9600-1MathSciNetView ArticleGoogle Scholar - Nield DA, Kuznetsov AV: The onset of convection in a horizontal nanofluid layer of finite depth.
*Eur. J. Mech. B, Fluids*2010, 29: 217-223. 10.1016/j.euromechflu.2010.02.003MathSciNetView ArticleMATHGoogle Scholar - Nield DA, Kuznetsov AV: The onset of convection in a layer of cellular porous material: effect of temperature-dependent conductivity arising from radiative transfer.
*J. Heat Transf.*2010., 132: Article ID 074503Google Scholar - Nield DA, Kuznetsov AV: The onset of double-diffusive convection in a nanofluid layer.
*Int. J. Heat Fluid Flow*2011, 32: 771-776. 10.1016/j.ijheatfluidflow.2011.03.010View ArticleGoogle Scholar - Nield DA, Kuznetsov AV: The effect of vertical through flow on thermal instability in a porous medium layer saturated by a nanofluid.
*Transp. Porous Media*2011, 87: 765-775. 10.1007/s11242-011-9717-xMathSciNetView ArticleGoogle Scholar - Chand R, Rana GC: On the onset of thermal convection in rotating nanofluid layer saturating a Darcy-Brinkman porous medium.
*Int. J. Heat Mass Transf.*2012, 55: 5417-5424. 10.1016/j.ijheatmasstransfer.2012.04.043View ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.