Thermal instability in a non-Darcy porous medium saturated with a nanofluid and with a convective boundary condition
© Shaw and Sibanda; licensee Springer 2013
Received: 10 April 2013
Accepted: 5 August 2013
Published: 20 August 2013
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.
Keywordsthroughflow convective boundary condition oscillatory convection linear stability analysis
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 . Nanoparticles also have medical applications including cancer therapy and nano-drug delivery (Shaw and Murthy ). Buongiorno  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. , Malik et al. , Nadeem et al.  and the review by Das et al. .
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 ), sodium oxide-silicon dioxide glass melt flows (Guloyan ), reactive polymer flows in heterogeneous porous media , electrochemical generation of elemental bromine in porous electrode systems (Qi and Savinell ).
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  studied the Blasius flow over a flat plate with a convection thermal boundary condition. Ishak  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 , Yao et al. .
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. ). Instability of nanofluids in natural convection was studied by Tzou . Tzou  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  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  and Nield and Kuznetsov [23–26].
In this study, we extend the work by Nield and Kuznetsov  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
where 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, is the density of nanoparticles, β is the coefficient of thermal expansion, is effective viscosity, μ is viscosity and is electric conductivity. In the energy equation, is the heat capacity of the fluid in the porous medium, is the heat capacity of nanoparticles and is thermal conductivity. In the equation of continuity for nanoparticles, is the Brownian diffusion coefficient, given by the Einstein-Stokes equation and is the thermophoretic diffusion coefficient of nanoparticles.
We note here that the parameter Rm is a measure of the basic static pressure gradient.
2.1 Basic solution
the same results were obtained by Nield and Kuznetsov .
2.2 Perturbation solution
3 Normal modes and stability analysis
For the non-trivial solution, the determinant of the augmented matrix is equal to zero, s is a dimensionless growth factor. We put , where ω is real and is the dimensionless frequency.
3.1 Stationary convection
We calculate the corresponding critical Rayleigh number using the above critical value of α for stationary convection.
3.2 Oscillatory convection
The critical Rayleigh number 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 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 .
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.
The authors wish to thank the University of KwaZulu-Natal for financial support.
- 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. 66
- Shaw S, Murthy PVSN, Sibanda P: Magnetic drug targeting in permeable microvessel. Microvasc. Res. 2013, 85: 77-85.View Article
- Buongiorno J: Convective transport in nanofluids. J. Heat Transf. 2006, 128: 240-250. 10.1115/1.2150834View Article
- 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.
- 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.
- 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.
- Das SK, Choi SUS, Yu W, Pradeep T: Nanofluids: Science and Technology. Wiley, New York; 2007.View Article
- Takahashi K, Alkire R: Mass transfer in gas-sparged porous electrodes. Chem. Eng. Commun. 1985, 38: 209-227. 10.1080/00986448508911307View Article
- Guloyan YA: Chemical reactions between components in the production of glass-forming melt. Glass Ceram. 2003, 60: 233-235. 10.1023/A:1027395310680View Article
- 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 Article
- 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 Article
- 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 Article
- 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 ArticleMATH
- 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 Article
- 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 ArticleMATH
- 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 Article
- Tzou DY: Instability of nanofluids in natural convection. J. Heat Transf. 2008., 130: Article ID 072401
- 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 ArticleMATH
- 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 ArticleMATH
- 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 Article
- 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 Article
- 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 Article
- 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 ArticleMATH
- 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 074503
- 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 Article
- 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 Article
- 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 Article
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.