Abstract
In this article we study the combined effect of internal heating and time-periodic gravity modulation on thermal instability in a closely packed anisotropic porous medium, heated from below and cooled from above. The time-periodic gravity modulation, considered in this problem can be realized by vertically oscillating the porous medium. A weak non-linear stability analysis has been performed by using power series expansion in terms of the amplitude of gravity modulation, which is assumed to be small. The Nusselt number has been obtained in terms of the amplitude of convection which is governed by the non-autonomous Ginzburg–Landau equation derived for the stationary mode of convection. The effects of various parameters such as; internal Rayleigh number, amplitude and frequency of gravity modulation, thermo-mechanical anisotropies, and Vadász number on heat transport has been analyzed. It is found that the response of the convective system to the internal Rayleigh number is destabilizing. Further it is found that the heat transport can also be controlled by suitably adjusting the external parameters of the system.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Abbreviations
- A :
-
Amplitude of convection
- d :
-
Height of the fluid layer
- Da :
-
Darcy number Da = K z /d2
- g 0 :
-
Mean of gravity
- Q :
-
Internal heat source
- k c :
-
Critical wave number
- Nu :
-
Nusselt number
- p :
-
Reduced pressure
- Pr :
-
Prandtl number, \({Pr=\nu/\kappa_{{T}_{z}}}\)
- Ra :
-
Thermal Rayleigh number, \({Ra=\alpha_{T}{g}_{0}K_{z}(\Delta T){\rm d}/\nu {\kappa_{T}}_{z}}\)
- R 0c :
-
Critical Rayleigh number
- R i :
-
Internal Rayleigh number, \({R_{\rm i}=Q{\rm d}^{2}/{\kappa_{T}}_{z}}\)
- Va :
-
Vadász number, \({Va=\phi Pr/Da}\)
- ξ :
-
Mechanical anisotropy parameter ξ = K x /K z
- η :
-
Thermal anisotropy parameter \({\eta={\kappa_{T}}_{x}/{\kappa_{T}}_{z}}\)
- t :
-
Time
- T :
-
Temperature
- ΔT :
-
Temperature difference across the fluid layer
- x,y,z :
-
Space co-ordinates
- α T :
-
Coefficient of thermal expansion
- δ 2 :
-
Horizontal wave number k 2c + π 2
- δ 1 :
-
Amplitude of temperature modulation
- Ω:
-
Frequency of modulation
- \({\epsilon}\) :
-
Perturbation parameter
- γ :
-
Heat capacity ratio \({\frac{(\rho c_{p})_{m}}{(\rho c_{p})_{f}}}\)
- K :
-
Permeability tensor
- \({\kappa_{T}}\) :
-
Effective thermal diffusivity
- μ :
-
Effective dynamic viscosity of the fluid
- ν :
-
Effective kinematic viscosity, \({\left ({\frac{\mu}{\rho _{0}}}\right )}\)
- \({\phi}\) :
-
Porosity
- ψ :
-
Stream function
- ρ :
-
Fluid density
- τ :
-
Slow time \({\tau =\epsilon ^{2}t}\)
- \({\nabla^{2}}\) :
-
\({\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial z^{2}}}\)
- \({\nabla^{2}_{1}}\) :
-
\({\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial z^{2}}}\)
- \({\nabla^{2}_{\xi}}\) :
-
\({\frac{\partial^{2}}{\partial x^{2}}+\frac{1}{\xi}\frac{\partial^{2}}{\partial z^{2}}}\)
- \({\nabla^{2}_{\eta}}\) :
-
\({\eta\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial z^{2}}}\)
- b:
-
Basic state
- c:
-
Critical
- 0:
-
Reference value
- ′:
-
Perturbed quantity
- *:
-
Dimensionless quantity
- st:
-
Stationary
References
Bhadauria B.S., Anoj K., Jogendra K., Sacheti N.C., Pallath C. (2011) Natural convection in a rotating anisotropic porous layer with internal heat generation. Transp. Porous Media 90: 687–705
Bhadauria B.S. (2012) Double diffusive convection in a saturated anisotropic porous layer with internal heat source. Trans. Porous Media 92: 299–320
Bhattacharya S.P., Jena S.K. (1984) Thermal instability of a horizontal layer of micropolar fluid with heat source. Proc. Indian Acad. Sci. (Math. Sci.) 93(1): 13–26
Epherre, J.F.: Crit’ere d’apparition de la convection naturalle dans une couche poreuse anisotrope. Rev. Gén. Therm. 168, 949–950 (1975) (English translation, Int. Chem. Engng. 17, 615–616 (1977))
Govender S. (2004) Stability of convection in a gravity modulated porous layer heated from below. Transp. Porous Media 57(1): 113–123
Govender S. (2005) Linear stability and convection in a gravity modulated porous layer heated from below-transition from synchronous to subharmonic solutions. Transp. Porous Media 59(2): 227–238
Govender S. (2005) Weak non-linear analysis of convection in a gravity modulated porous layer. Transp. Porous Media 60: 33–42
Haajizadeh M., Ozguc A.F., Tien C.L. (1984) Natural convection in a vertical porous enclosure with internal heat generation. Int. J. Heat Mass Transf. 27: 1893–1902
Herron Isom H. (2001) Onset of convection in a porous medium with internal heat source and variable gravity. Int. J. Eng. Sci. 39: 201–208
Ingham D.B., Pop I. (1998) Transport phenomena in porous media. Pergamon, Oxford
Ingham D.B., Pop I. (2005) Transport Phenomena in Porous Media, vol. III. Elsevier, Oxford
Joshi M.V., Gaitonde U.N., Mitra S.K. (2006) Analytical study of natural convection in a cavity with volumetric heat generation. ASME J. Heat Transf. 128: 176–182
Khalili A., Huettel M. (2002) Effects of throughflow and internal heat generation on convective instabilities in an anisotropic porous layer. J. Porous Media 5(3): 64–75
Kuznetsov A.V., Nield D.A. (2008) The effects of combined horizontal and vertical heterogenity on the onset of convection in a porous medium: double diffusive case. Transp. Porous Media 72: 157–170
Kuznetsov A.V. (2005) The onset of bioconvection in a suspension of negatively geotactic microorganisms with high-frequency vertical vibration. Int. Comm. Heat Mass Transf. 32: 1119–1127
Kuznetsov A.V. (2006) Linear stability analysis of the effect of vertical vibration on bioconvection in a horizontal porous layer of finite depth. J. Porous Media 9: 597–608
Kuznetsov A.V. (2006) Investigation of the onset of bioconvection in a suspension of oxytactic microorganisms subjected to high frequency vertical vibration. Theor. Comput. Fluid Dynam. 20: 73–87
Lapwood E.R. (1948) Convection of a fluid in porous medium. Proc. Camb. Phil. Soc. 44: 508–521
Malashetty M.S., Padmavathi V. (1997) Effect of gravity modulation on the onset of convection in a fluid and porous layer. Int. J. Eng. Sci. 35: 829–839
Malashetty M.S., Basavaraj D. (2002) Rayleigh–Bénard convection subject to time-dependent wall temperature/ gravity in a fluid saturated anisotropic porous medium. Heat Mass Transf. 38: 551–563
Malashetty M.S., Swamy M. (2011) Effect of gravity modulation on the onset of thermal convection in rotating fluid and porous layer. Phys. Fluids 23(6): 064108
Nield D.A., Bejan A. (2006) Convection in Porous Media. Springer, New York
Nield D.A., Kuznetsov A.V. (2007) The effects of combined horizontal and vertical heterogeneity and anisotropy on the onset of convection in a porous medium. Int. J. Thermal Sci. 46: 1211–1218
Parthiban C., Patil P.R. (1997) Thermal instability in an anisotropic porous medium with internal heat source and inclined temperature gradient. Int. Comm. Heat Mass Transf. 24(7): 1049–1058
Rajagopal K.R., Saccomandib G., Vergoric L. (2010) A systematic approximation for the equations governing convection-diffusion in a porous medium. Nonlinear Anal. Real World Appl. 11(4): 2366–2375
Rao Y.F., Wang B.X. (1991) Natural convection in vertical porous enclosures with internal heat generation. Int. J. Heat Mass Transf. 34: 247–252
Razi Y.P., Mojtabi I., Charrier-Mojtabi M.C. (2009) A summary of new predictive high frequency thermo-vibrational modes in porous media. Transp. Porous Media 77: 207–208
Rees D.A.S., Pop I. (2000) The effect of G-jitter on vertical free convection boundary-layer flow in porous media. Int. Comm. Heat Mass Transf. 27(3): 415–424
Rees D.A.S., Pop I. (2001) The effect of G-jitter on free convection near a stagnation point in a porous medium. Int. J. Heat Mass Transf. 44: 877–883
Rees D.A.S., Pop I. (2003) The effect of large-amplitude G-jitter vertical free convection boundary-layer flow in porous media. Int. J. Heat Mass Transf. 46: 1097–1102
Rionero S., Straughan B. (1990) Convection in a porous medium with internal heat source and variable gravity effects. Int. J. Eng. Sci. 28(6): 497–503
Saravanan S., Arunkumar A. (2010) Convective instability in a gravity modulated anisotropic thermally stable porous medium. Int. J. Engg. Sci. 48: 742–750
Saravanan S., Purusothaman A. (2009) Floquent instability of a modulated Rayleigh-Bénard problem in an anisotropic porous medium. Int. J. Therm. Sci. 48: 2085–2091
Saravanan S., Sivakumar T. (2010) Onset of filteration convection in a vibrating medium: The Brinkman model. Phys. Fluids 22: 034104
Saravanan S., Sivakumar T. (2011) Thermovibrational instability in a fluid saturated anisotropic porous medium. ASME J. Heat Transf. 133: 051601.1–051601.9
Siddhavaram V.K., Homsy G.M. (2006) The effects of gravity modulation on fluid mixing Part 1 Harmonic modulation. J. Fluid Mech. 562: 445–475
Siddheshwar P.G., Vanishree R.K., Melson A.C. (2012) Study of heat transport in Bénard-Darcy convection with g-jitter and thermomechanical anisotropy in variable viscosity liquids. Transp. Porous Media 92(2): 277–288
Strong N. (2008) Effect of vertical modulation on the onset of filtration convection. J. Math. Fluid Mech. 10: 488–502
Strong N. (2008) Double-diffusive convection in a porous layer in the presence of vibration. SIAM J. Appl. Math. 69: 1263–1276
Siddheshwar P.G., Bhadauria B.S., Srivastava Alok. (2012) An analytical study of nonlinear double diffusive convection in a porous medium under temperature/gravity modulation. Transp. Porous Media 91: 585–604
Straughan B. (2004) The Energy Method, Stability, and Nonlinear Convection—Second edition, Appl. Math. Sci. Ser.. Springer, New York
Vadász P. (1998) Coriolis effect on gravity-driven convection in a rotating porous layer heated from below. J. Fluid Mech. 376: 351–375
Vadász, P. (eds) (2008) Emerging Topics in Heat and Mass Transfer in Porous Media . Springer, New York
Vafai, K. (eds) (2000) Handbook of Porous Media. Marcel Dekker, New York
Vafai, K. (eds) (2005) Handbook of Porous Media. Taylor and Francis (CRC), Boca Raton
Vanishree R.K. (2010) Combined effect of temperature and gravity modulations on the onset of convection in an anisotropic porous medium. Int. J. Appl. Mech. Engg. 15(1): 267–291
Yang Wen-Mei. (1997) Stability of viscoelastic fluids in a modulated gravitational field. Int. J. Heat Mass Transf. 40(6): 1401–1410
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bhadauria, B.S., Hashim, I. & Siddheshwar, P.G. Study of Heat Transport in a Porous Medium Under G-jitter and Internal Heating Effects. Transp Porous Med 96, 21–37 (2013). https://doi.org/10.1007/s11242-012-0071-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11242-012-0071-4