Skip to main content
SpringerLink
Log in

Chaotic Convection in a Porous Medium Under Temperature Modulation

  • Published:
Transport in Porous Media Aims and scope Submit manuscript

Abstract

The work proposed by Vadász et al. (Transp Porous Media 103:279–294, 2014) motivated us to take up the problem of chaotic convection under temperature modulation for study. The analysis of buoyancy driven convection for moderate Prandtl number in a fluid saturated porous layer heated from below and subject to temperature modulation is presented. It has been investigated that a better combination of values of \(\Omega , \delta \) and scaled Rayleigh number \(Ra\) provides a way for chaos. It is found that the temperature modulation (suitable choice of frequency \(\Omega \), amplitude \(\delta \) along with scaled Rayleigh number \(Ra\)) of the boundaries is to enhance the behaviour of the chaotic motion. The lower boundary plate modulation is similar to the gravity modulation given by Vadász et al. (Transp Porous Media 103:279–294, 2014) and Bhadauria and Kiran (Int J Heat Mass Transf 84:610–624, 2015). It is also found that heat transfer results conform the results of Bhadauria and Kiran (Transp Porous Med 100:279–295, 2013, Int J Heat Mass Transf 77:843–851, 2014a, Int Commun Heat Mass Transf 58:166–175, 2014b).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Abbreviations

\(a\) :

Wave number

\(\delta \) :

Amplitude of temperature modulation

\(d\) :

Depth of the fluid layer

\(g\) :

Acceleration due to gravity

\(p\) :

Reduced pressure

\(Pr\) :

Prandtl number, \(Pr=\nu /\kappa _{T}\)

\(Da\) :

Darcy number, \(Da=K/d^{2}\)

\(R\) :

Thermal Rayleigh–Darcy number, \(R=\frac{\beta _{T} g \Delta T d K}{\nu \kappa _{T}}\)

\(Ra\) :

Scaled Rayleigh number, \(Ra=\frac{R}{\pi ^2\theta ^2}\)

\(Va\) :

Vadász number, \(Va=\delta _1 Pr/Da\)

\(Vas\) :

Scaled Vadász number, \(Vas=\frac{Va\gamma }{\pi ^2}\)

\(T\) :

Temperature

\(\Delta T\) :

Temperature difference across the porous layer

\(t\) :

Time

\((x,z)\) :

Horizontal and vertical coordinates

\(\alpha _T\) :

Coefficient of thermal expansion

\(\kappa _{T}\) :

Effective thermal diffusivity

\(K\) :

Permeability

\(\Omega \) :

Frequency of modulation

\(\mu \) :

Dynamic viscosity of the fluid

\(\delta _1\) :

Porosity

\(\nu \) :

Kinematic viscosity, \(\left( {\frac{\mu }{\rho _{0}}} \right) \)

\(\rho \) :

Fluid density

\(\psi \) :

Stream function

\(\tau \) :

Time (dimensionless)

\(\phi \) :

Phase angle

\(T^{'}\) :

Perturbed temperature

\(\nabla ^{2}\) :

\(\frac{\partial ^{2}}{\partial x^{2}}+\frac{\partial ^{2}}{\partial y^{2}}+\frac{\partial ^{2}}{\partial z^{2}}\)

\(b\) :

Basic state

0:

Reference value

\('\) :

Perturbed quantity

\(*\) :

Dimensionless quantity

References

  • Adomian, G.: A review of the decomposition method in applied mathematics. J. Math. Anal. Appl. 135, 501–544 (1988)

    Article  Google Scholar 

  • Adomian, G.: Solving frontier problems in physics: the decomposition method. Kluwer Academic Publishers, Dordrecht (1994)

    Book  Google Scholar 

  • Bejan, A.: Convection heat transfer, 2nd edn. Wiley, New York (1995)

    Google Scholar 

  • Bardana, G., Knoblochb, E., Mojtabia, A., Khallouf, A.: Natural double diffusive convection with vibration. Fluid Dyn. Res. 28, 159–187 (2001)

    Article  Google Scholar 

  • Bardan, G., Mojtabi, A.: On the Horton–Rogers–Lapwood convective instability with vibration: onset of convection. Phys. Fluids 12, 2723–2731 (2000)

    Article  Google Scholar 

  • Bhadauria, B.S.: Thermal modulation of Rayleigh-Bénard convection in a sparsely packed porous medium. J. Porous Media 10, 175–188 (2007a)

    Article  Google Scholar 

  • Bhadauria, B.S.: Fluid convection in a rotating porous layer under modulated temperature on the boundaries. Transp. Porous Media 67, 297–315 (2007b)

    Article  Google Scholar 

  • Bhadauria, B.S.: Double diffusive convection in a porous medium with modulated temperature on the boundaries. Transp. Porous Media 70, 191–211 (2007c)

    Article  Google Scholar 

  • Bhadauria, B.S., Sherani, A.: Onset of Darcy-convection in a magnetic fluid saturated porous medium subject to temperature modulation of the boundaries. Transp. Porous Media 73, 349–368 (2008)

    Article  Google Scholar 

  • Bhadauria, B.S., Srivastava, A.K.: Magneto-double diffusive convection in an electrically conducting-fluidsaturated porous medium with temperature modulation of the boundaries. Int. J. Heat Mass Transf. 53, 2530–2538 (2010)

    Article  Google Scholar 

  • Bhadauria, B.S., Siddheshwar, P.G., Kumar, J., Suthar, O.P.: Non-linear stability analysis of temperature/gravity modulated Rayleigh-Bénard convection in a porous medium. Transp. Porous Media 92, 633–647 (2012)

    Article  Google Scholar 

  • Bhadauria, B.S., Hashim, I., Siddheshwar, P.G.: Effects of time-periodic thermal boundary conditions and internal heating on heat transport in a porous medium. Transp. Porous Media 97, 185–200 (2013)

    Article  Google Scholar 

  • Bhadauria, B.S., Kiran, P.: Heat transport in an anisotropic porous medium saturated with variable viscosity liquid under temperature modulation. Transp. Porous Media 100, 279–295 (2013)

    Article  Google Scholar 

  • Bhadauria, B.S., Kiran, P.: Weakly nonlinear oscillatory convection in a viscoelastic fluid saturating porous medium under temperature modulation. Int. J. Heat Mass Transf. 77, 843–851 (2014a)

    Article  Google Scholar 

  • Bhadauria, B.S., Kiran, P.: Heat and mass transfer for oscillatory convection in a binary viscoelastic fluid layer subjected to temperature modulation at the boundaries. Int. Commun. Heat Mass Transf. 58, 166–175 (2014b)

    Article  Google Scholar 

  • Bhadauria, B.S., Kiran, P.: Weak nonlinear oscillatory convection in a viscoelastic fluid-saturated porous medium under gravity modulation. Transp. Porous Media 104(3), 451–467 (2014c)

    Article  Google Scholar 

  • Bhadauria, B.S., Kiran, P.: Chaotic and oscillatory magneto-convection in a binary viscoelastic fluid under G-jitter. Int. J Heat Mass Transf. 84, 610–624 (2015)

    Article  Google Scholar 

  • Chen, G.R., Ueta, T.: Yet another chaotic attractor. Int. J. Bifurc. Chaos 9, 1465–1466 (1999)

    Article  Google Scholar 

  • Chhuon, B., Caltagirone, J.P.: Stability of a horizontal porous layer with timewise periodic boundary conditions. ASME J. Heat Transf. 101, 244–248 (1979)

    Article  Google Scholar 

  • Davis, S.H.: The stability of time periodic flows. Annu. Rev. Fluid Mech. 8, 57–74 (1976)

    Article  Google Scholar 

  • Feki, M.: An adaptive feedback control of linearizable chaotic systems. Chaos Solitons Fractals 15, 883–890 (2003)

    Article  Google Scholar 

  • Gershuni, G.Z., Zhukhovitskii, E.M.: On parametric excitation of convective instability. J. Appl. Math. Mech. 27, 779–783 (1963)

    Google Scholar 

  • Gresho, P.M., Sani, R.L.: The effects of gravity modulation on the stability of a heated fluid layer. J. Fluid Mech. 40, 783–806 (1970)

    Article  Google Scholar 

  • Govender, S.: Weak non-linear analysis of convection in a gravity modulated porous layer. Transp. Porous Media 60, 33–42 (2005)

    Article  Google Scholar 

  • Govender, S.: stability of gravity driven convection in a cylindrical porous layer subjected to vibration. Transp. Porous Media 63, 489–502 (2006)

    Article  Google Scholar 

  • Govender, S.: Vadasz number influence on vibration in a rotating porous layer placed far away from the axis of rotation. ASME J. Heat Transf. 132, 112601 (2010)

    Article  Google Scholar 

  • Horton, C.W., Rogers, F.T.: Convection currents in a porous medium. J. Appl. Phys. 16, 367–370 (1945)

    Article  Google Scholar 

  • Ingham, D.B., Pop, I. (eds.): Transport Phenomena in Porous Media. Oxford, Pergamon (1998)

  • Lapwood, E.R.: Convection of a fluid in a Porous medium. Proc. Camb. Philos. Soc. 44, 508–521 (1948)

    Article  Google Scholar 

  • Lorenz, E.N.: Deterministic non-periodic flow. J. Atmos. Sci. 20, 130–141 (1963)

    Article  Google Scholar 

  • Lü, J.H., Chen, G.R.: A new chaotic attractor coined. Int. J. Bifur. Chaos 12, 659–661 (2002)

    Article  Google Scholar 

  • Magyari, E.: The butterfly effect in a porous slab. Transp. Porous Media 84(3), 711–715 (2010)

    Article  Google Scholar 

  • Malashetty, M.S., Wadi, V.S.: Rayleigh-Bénard convection subject to time dependent wall temperature in a fluid saturated porous layer. Fluid Dyn. Res. 24, 293–308 (1999)

    Article  Google Scholar 

  • Mojtabi, M.C.C., Razi, Y.P., Maliwan, K., Mojtabi, A.: Influence of vibration on Soret-driven convection in porous media. Numer. Heat Transf. 46, 981–993 (2004)

    Article  Google Scholar 

  • Narayana, M., Gaikwad, S.N., Sibanda, P., Malge, R.E.: Double diffusive magneto-convection in viscoelastic fluids. Int. J. Heat Mass Transf. 67, 194–201 (2013a)

    Article  Google Scholar 

  • Narayana, M., Sibanda, P., Siddheshwar, P.G., Jayalatha, G.: Linear and nonlinear stability analysis of binary viscoelastic fluid convection. Appl. Math. Model. 37, 8162–8178 (2013b)

    Article  Google Scholar 

  • Nield, D.A., Bejan, A.: Convection in porous media, 4th edn. Springer, New York (2013)

    Book  Google Scholar 

  • Peng, C.-C., Chen Chaos, C.-L.: Robust chaotic control of Lorenz system by backstepping design. Solitons Fractals 37, 598–608 (2008)

    Article  Google Scholar 

  • Poincaré, J.H.: (1892, 1893, 1899) Les méthodes nouvelles de la méchanique céleste. Paris

  • Poincaré, J.H.: Sur le probléme des trois corps et les équations de la dynamique. Acta Math. 13, 01–279 (1890)

    Article  Google Scholar 

  • Rudraiah, N., Malashetty, M.S.: Effect of modulation on the onset of convection in a sparsely packed porous layer. ASME J. Heat Transf. 122, 685–689 (1990)

    Article  Google Scholar 

  • Rössler, O.E.: An equation for continuous chaos. Phys. Lett. A 57, 397–398 (1976)

    Article  Google Scholar 

  • Sheu, L.J., Tam, L.M., Chen, J.H., Chen, H.K., Lin, K.T., Kang, Y.: Chaotic convection of viscoelastic fluids in porous media. Chaos Solitons Fractals 37, 113–124 (2008)

    Article  Google Scholar 

  • Siddheshwar, P.G., Bhadauria, B.S., Srivastava, A.: An analytical study of nonlinear double diffusive convection in a porous medium with temperature modulation/gravity modulation. Transp. Porous Media 91, 585–604 (2012)

    Article  Google Scholar 

  • Sparrow, C.: The Lorenz equations: bifurcations, chaos and strange attractors. Springer, New York (1982)

    Book  Google Scholar 

  • Srivastava, A., Bhadauria, B.S., Siddheshwar, P.G., Hashim, I.: Heat transport in an anisotropic porous medium saturated with variable viscosity liquid under G-jitter and internal heating effects. Transp. Porous Media 99, 359–376 (2013)

    Article  Google Scholar 

  • Straughan, B.: A sharp nonlinear stability threshold in rotating porous convection. Proc. R. Soc. Lond. A8(457), 87–93 (2001)

    Article  Google Scholar 

  • Vadász, P.: Analytical prediction of the transition to chaos in Lorenz equations. Appl. Math. Lett. 23, 503–507 (2010)

    Article  Google Scholar 

  • Vadász, P.: Local and global solutions for transitions to chaos and hysteresis in a porous layer heated from below. Transp. Porous Media 37, 213–245 (1999a)

    Article  Google Scholar 

  • Vadász, P.: Subcritical transitions to chaos and hysteresis in a fluid layer heated from below. Int. J. Heat Mass Transf. 43, 705–724 (1999b)

    Article  Google Scholar 

  • Vadász, P., Olek, S.: Transitions and chaos for free convection in a rotating porous layer. Int. J. Heat Mass Transf. 41(11), 1417–1435 (1998)

    Article  Google Scholar 

  • Vadász, P., Olek, S.: Weak turbulence and chaos for low Prandtl number gravity driven convection in porous media. Transp. Porous Media 37, 69–91 (1999)

    Article  Google Scholar 

  • Vadász, P., Olek, S.: Route to chaos for moderate Prandtl number convection in a porous layer heated form below. Transp. Porous Media 41, 211–239 (2000)

    Article  Google Scholar 

  • Vadász, J.J., Meyer, J.P., Saneshan, G.: Chaotic and periodic natural convection for moderate and high Prandtl numbers in a porous layer subject to vibrations. Transp. Porous Media 103, 279–294 (2014)

    Article  Google Scholar 

  • Vafai, K. (ed.): Handbook of porous media. Marcel Dekker, New York (2000)

    Google Scholar 

  • Venezian, G.: Effect of modulation on the onset of thermal convection. J. Fluid Mech. 35, 243–254 (1969)

    Article  Google Scholar 

  • Wang, Y., Singer, J., Bau, H.H.: Controlling chaos in a thermal convection loop. J. Fluid Mech. 237, 479–498 (1992)

    Article  Google Scholar 

  • Yau, H.T., Chen, C.K., Chen, C.L.: Sliding mode control of chaotic systems with uncertainties. Int. J. Bifurc. Chaos 10, 1139–1147 (2000)

    Google Scholar 

  • Yau, H.T., Chen, C.L.: Chaos control of Lorenz systems using adaptive controller with input saturation. Chaos Solitons Fractals 34, 1567–1574 (2007)

    Article  Google Scholar 

  • Yuen, P., Bau, H.H.: Rendering a subcritical Hopf bifurcation supercritical. J. Fluid Mech. 317, 91–109 (1996)

    Article  Google Scholar 

Download references

Acknowledgments

This work was done during the lien sanctioned to the author B.S. Bhadauria by Banaras Hindu University, Varanasi, India, to work as Professor of Mathematics at Department of Applied Mathematics, School for Physical Sciences, Babasaheb Bhimrao Ambedkar Central University, Lucknow, India, during 06.07.2011 to 03.07.2014. Author Palle Kiran acknowledges the financial assistance from Babasaheb Bhimrao Ambedkar Central University, Lucknow, India, as a research fellowship.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, School for Physical Sciences, Babasaheb Bhimrao Ambedkar University, Lucknow, 226025, India

    Palle Kiran

  2. Department of Mathematics, Faculty of Sciences, Banaras Hindu University, Varanasi, 221005, India

    B. S. Bhadauria

Authors
  1. Palle Kiran
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. S. Bhadauria
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. S. Bhadauria.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kiran, P., Bhadauria, B.S. Chaotic Convection in a Porous Medium Under Temperature Modulation. Transp Porous Med 107, 745–763 (2015). https://doi.org/10.1007/s11242-015-0465-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11242-015-0465-1

Keywords

Search

Navigation