Abstract
In this article, we study double-diffusive convection in a horizontal porous medium saturated by a nanofluid, for the case when the base fluid of the nanofluid is itself a binary fluid such as salty water. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis, while the Darcy model is used for the porous medium. The thermal energy equations include the diffusion and cross-diffusion terms. The linear stability is studied using normal mode technique and for non-linear analysis, a minimal representation of the truncated Fourier series analysis involving only two terms has been used. For linear theory analysis, critical Rayleigh number has been obtained, while non-linear analysis has been done in terms of the Nusselt numbers.
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Abbreviations
- C :
-
Solute concentration
- D B :
-
Brownian diffusion coefficient
- D T :
-
Thermophoretic diffusion coefficient
- d :
-
Dimensional layer depth
- k T :
-
Effective thermal conductivity of porous medium
- k m :
-
Thermal diffusivity of porous medium
- K :
-
Permeability
- Le :
-
Thermo-solutal Lewis number
- Ln :
-
Thermo-nanofluid Lewis number
- N A :
-
Modified diffusivity ratio
- N B :
-
Modified particle-density increment
- N CT :
-
Soret parameter
- N TC :
-
Dufour parameter
- p :
-
Pressure
- g :
-
Gravitational acceleration
- Ra :
-
Thermal Rayleigh-Darcy number
- Rm :
-
Basic density Rayleigh number
- Rn :
-
Nanoparticle concentration Rayleigh number
- Rs :
-
Solutal Rayleigh number
- t :
-
Time
- T :
-
Nanofluid temperature
- T c :
-
Temperature at the upper wall
- T h :
-
Temperature at the lower wall
- v :
-
Nanofluid velocity
- (x, y, z):
-
Cartesian coordinates
- β C :
-
Solutal volumetric coefficient
- β T :
-
Thermal volumetric coefficient
- ε :
-
Porosity
- μ :
-
Viscosity of the fluid
- ρ f :
-
Fluid density
- ρ p :
-
Nanoparticle mass density
- γ :
-
Thermal capacity ratio
- \({\phi }\) :
-
Nanoparticle volume fraction
- ψ :
-
Stream function
- α :
-
Wave number
- ω :
-
Frequency of oscillations
- b:
-
Basic solution
- f:
-
Fluid
- p:
-
Particle
- *:
-
Dimensional variable
- ′:
-
Perturbation variable
- \({\nabla^2}\) :
-
\({\displaystyle\frac{\partial^2}{\partial x^2} + \displaystyle\frac{\partial^2}{\partial y^2} + \displaystyle\frac{\partial^2}{\partial z^2}}\) .
- \({\nabla_1^2}\) :
-
\({\displaystyle\frac{\partial^2}{\partial x^2} + \displaystyle\frac{\partial^2}{\partial z^2}}\) .
References
Agarwal S., Bhadauria B.S.: Natural convection in a nanofluid saturated rotating porous layer with thermal non equilibrium model. Transp. Porous Media 2(1), 53–64 (2011)
Agarwal S., Bhadauria B.S., Siddheshwar P.G.: Thermal instability of a nanofluid saturating a rotating anisotropic porous medium. Spec. Top. Rev. Porous Media Begell House Publ. 2(1), 53–64 (2011)
Bhadauria B.S.: Double diffusive convection in a porous medium with modulated temperature on the boundaries. Transp. Porous Media 70, 191–211 (2007a)
Bhadauria B.S.: Fluid convection in a rotating porous layer under modulated temperature on the boundaries. Transp. Porous Media 67(2), 297–315 (2007b)
Bhadauria B.S.: Effect of temperature modulation on Darcy convection in a rotating porous medium. J. Porous Media 11(4), 361–375 (2008)
Bhadauria B.S., Agarwal S.: Natural convection in a nanofluid saturated rotating porous layer: a nonlinear study. Transp. Porous Media 87(2), 585–602 (2011a)
Bhadauria B.S., Agarwal S.: Convective transport in a nanofluid saturated porous layer with thermal non equilibrium model. Transp. Porous Media 88(1), 107–131 (2011b)
Bhadauria B.S., Agarwal S., Kumar A.: Non-linear two-dimensional convection in a nanofluid saturated porous medium. Transp. Porous Media 90(2), 605–625 (2011)
Buongiorno J.: Convective transport in nanofluids. ASME J Heat Transfer 128, 240–250 (2006)
Buongiorno, J., Hu, W.: Nanofluid coolant for advanced nuclear power plants. Paper No. 5705. In: Proceedings of ICAPP’05, Seoul, 15–19 May 2005
Chandrashekhar S.: Hydrodynamic and hydromagnetic stability. Oxford University Press, Oxford (1961)
Choi, S.: Enhancing thermal conductivity of fluids with nanoparticles. In: Signier, D.A., Wang, H.P. (eds.) Development and applications of Non-Newtonian flows, ASME FED, vol. 231/MD vol. 66, pp. 99–105 (1995)
Choi S.: Nanofluid Technology: Current Status and Future Research. Energy Technology Division, Argonne National Laboratory, Argonne (1999)
Das S.K., Putra N., Thiesen P., Roetzel W.: Temperature dependence of thermal conductivity enhancement for nanofluids. ASME J. Heat Transf. 125, 567–574 (2003)
Drazin P.G., Reid D.H.: Hydrodynamic stability. Cambridge University Press, Cambridge (1981)
Eastman J.A., Choi S.U.S., Yu W., Thompson L.J.: Anomalously increased effective thermal conductivities of ethylene glycol-based nanofluids containing copper nanoparticles. Appl. Phys. Lett. 78, 718–720 (2001)
Eastman J.A., Choi S.U.S., Yu W., Thompson L.J.: Thermal transport in nanofluids. Annu. Rev. Matter Res. 34, 219–246 (2004)
Horton W., Rogers F.T.: Convection currents in a porous medium. J. Appl. Phys. 16, 367–370 (1945)
Keblinski P., Cahil D.G.: Comments on model for heat conduction in nanofluids. Phy. Rev. Lett. 95, 209401 (2005)
Kim J., Kang Y.T., Choi C.K.: Analysis of convective instability and heat transfer characteristics of nanofluids. Phys. Fluids 16, 2395–2401 (2004)
Kim J., Choi C.K., Kang Y.T., Kim M.G.: Effects of thermodiffusion and nanoparticles on convective instabilities in binary nanofluids. Nanoscale Microscale Thermophys. Eng. 10, 29–39 (2006)
Kim J., Kang Y.T., Choi C.K.: Analysis of convective instability and heat transfer characteristics of nanofluids. Int. J. Refrig. 30, 323–328 (2007)
Kleinstreuer C., Li J., Koo J.: Microfluidics of nano-drug delivery. Int. J. Heat Mass Transf. 51, 5590–5597 (2008)
Kuznetsov A.V.: Thermal nonequilibrium forced convection in porous Media. In: Ingham, D.B., Pop, I. (eds) Transport Phenomenon in Porous Media, pp. 103–130. Pergamon, Oxford (1998)
Kuznetsov A.V., Nield D.A.: Thermal instability in a porous medium layer saturated by a nanofluid: Brinkman model. Transp. Porous Media 81, 409–422 (2010a)
Kuznetsov A.V., Nield D.A.: Effect of local thermal non-equilibrium on the onset of convection in porous medium layer saturated by a nanofluid. Transp. Porous Media 83, 425–436 (2010b)
Kuznetsov A.V., Nield D.A.: Natural convective boundary-layer flow of a nanofluid past a vertical plate. Int. J. Therm. Sci. 49, 243–247 (2010c)
Kuznetsov A.V., Nield D.A.: The onset of double-diffusive nanofluid convection in a layer of a saturated porous medium. Transp. Porous Media 85, 941–951 (2010d)
Kuznetsov A.V., Nield D.A.: Double-diffusive natural convective boundary-layer flow of a nanofluid past a vertical plate. Int. J. Therm. Sci. 50, 712–717 (2011)
Lapwood E.R.: Convection of a fluid in a porous medium. Proc. Camb. Phil. Soc. 44, 508–521 (1948)
Malashetty M.S.: Anisotropic thermo convective effects on the onset of double diffusive convection in a porous medium. Int. J. Heat Mass Transf. 36, 2397–2401 (1993)
Masuda H., Ebata A., Teramae K., Hishinuma N.: Alteration of thermal conductivity and viscosity of liquid by dispersing ultra fine particles. Netsu Bussei 7, 227–233 (1993)
Murray B.T., Chen C.F.: Double diffusive convection in a porous medium. J. Fluid Mech. 201, 147–166 (1989)
Nield D.A.: Onset of thermohaline convection in a porous medium. Water Resour. Res. 4, 553–560 (1968)
Nield D.A., Bejan A.: Convection in Porous Media. 3rd edn. Springer, New York (2006)
Nield D.A., Kuznetsov A.V.: Thermal instability in a porous medium layer saturated by nonofluid. Int. J. Heat Mass Transf. 52, 5796–5801 (2009a)
Nield D.A., Kuznetsov A.V.: The Cheng-Minkowycz problem for natural convective boundary-layer flow in a porous medium saturated by a nanofluid. Int. J. Heat Mass Transf. 52, 5792–5795 (2009b)
Nield D.A., Kuznetsov A.V.: The effect of local thermal nonequilibrium on the onset of convection in a nanofluid. J. Heat Transf. 132, 052405 (2010a)
Nield D.A., Kuznetsov A.V.: The onset of convection in a horizontal nanofluid layer of finite depth. Eur. J. Mech. B 29, 217–223 (2010b)
Rudraiah N., Malashetty M.S.: The influence of coupled molecular diffusion on the double diffusive convection in a porous medium. ASME J. Heat Transf. 108, 872–876 (1986)
Tzou D.Y.: Instability of nanofluids in natural convection. ASME J. Heat Transf. 130, 072401 (2008a)
Tzou D.Y.: Thermal instability of nanofluids in natural convection. Int. J. Heat Mass Transf. 51, 2967–2979 (2008b)
Vadasz, P.: Nanofluids suspensions: possible explanations for the apparent enhanced effective thermal conductivity, ASME paper #HT2005-72258. In: Proceedings of 2005 ASME Summer Heat Transfer Conference, San Francisco, 17–22 July 2005
Vadasz P.: Heat conduction in nanofluid suspensions. ASME J. Heat Transf. 128, 465–477 (2006)
Vafai K.: Handbook of Porous Media. Taylor and Francis, London (2005)
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Agarwal, S., Sacheti, N.C., Chandran, P. et al. Non-linear Convective Transport in a Binary Nanofluid Saturated Porous Layer. Transp Porous Med 93, 29–49 (2012). https://doi.org/10.1007/s11242-012-9942-y
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DOI: https://doi.org/10.1007/s11242-012-9942-y