Skip to main content
SpringerLink
Log in

Energy stability of modulation driven porous convection with magnetic field

  • Published:
Meccanica Aims and scope Submit manuscript

Abstract

Free convective onset in a bottom heated porous medium under the influence of transverse magnetic field is studied within the framework of nonlinear theory. Simple harmonic time modulation is imposed on either the bounding surface temperature or the gravitational field acting on the medium. The Darcy–Lapwood–Brinkman model of flow through porous media is employed with two basic types of physical boundary conditions. The generalized energy approach coupled with the variational principles is followed. Higher order Galerkin method is then used to determine the unconditional and sharp stability limits for arbitrary values of modulational parameters. The increase in the Hartmann number is found to inhibit instability, thereby delaying the onset of two dimensional rolls. Moreover the initiation of instability mechanism can be well adjusted through a good knowledge of the parameters involved.

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

Similar content being viewed by others

Abbreviations

a :

Thermal modulation amplitude

\(A_m, B_m\) :

Adjustable functions

\(\mathbf {B}\) :

Applied magnetic field

\(c_p\) :

Specific heat

d :

Porous medium depth

Da :

Darcy number

E :

Disturbance energy

\(g_0\) :

Reference acceleration level

H :

Darcy-Hartmann number

Ha :

Hartmann number

\(\mathbf {j}\) :

Electric current density

k :

Permeability

\(\mathbf {k}\) :

Vertical unit vector

K :

Thermal conductivity

p :

Pressure

\(p_b\) :

Basic state pressure

\(p\prime\) :

Perturbed pressure

Pr :

Prandtl number

\(\mathbf {q}\) :

Filtration velocity, (u, v, w)

\(\mathbf {q_b}\) :

Basic state velocity

\(\mathbf {q\prime }\) :

Perturbed velocity

R :

Darcy–Rayleigh number

Ra :

Rayleigh number

t :

Time

T :

Temperature

\(T_b\) :

Basic state temperature

\(T_r\) :

Reference temperature

V :

Volume of porous media

x, y, z :

Space coordinates

\(\alpha\) :

Wavenumber, \(\sqrt{\alpha _x^2+\alpha _y^2}\)

\(\alpha _x\) :

Wavenumber in the x-direction

\(\alpha _y\) :

Wavenumber in the y-direction

\(\rho\) :

Fluid density

\(\theta\) :

Perturbed temperature

\(\vartheta\) :

Porosity

\(\nu\) :

Kinematic viscosity

\(\mu\) :

Coefficient of thermal expansion

\(\epsilon\) :

Gravity modulation amplitude

\(\kappa\) :

Thermal diffusivity

\(\phi\) :

Phase angle

\(\lambda\) :

Coupling parameter

\(\delta\) :

Variation of a quantity

\(\psi\) :

Lagrange multiplier

\(\sigma\) :

Electric conductivity

\(\varGamma\) :

Space of admissible functions

\(\varOmega\) :

Modulation frequency

\(\omega\) :

Non-dimensional modulation frequency

\(\zeta\) :

Isoperimetric constant

\(\varDelta T\) :

Temperature difference

D :

\(\partial /\partial z\)

\(\nabla ^{2}\) :

Laplacian operator

b :

Basic state

cr :

Critical value

L :

Linear

N :

Nonlinear

\('\) :

Perturbed quantity

References

  1. Chandrasekhar S (1961) Hydrodynamic and Hydromagnetic Stability. Dover Publications, New York

    MATH  Google Scholar 

  2. Finlayson BA (1970) Convective instability of ferromagnetic fluids. J Fluid Mech 40:753–767. https://doi.org/10.1017/S0022112070000423

    Article  MATH  Google Scholar 

  3. Nield DA, Bejan A (2013) Convection in porous media. Springer, New York

    Book  Google Scholar 

  4. Venezian G (1969) Effect of modulation on the onset of thermal convection. J Fluid Mech 35:243–254. https://doi.org/10.1017/S0022112069001091

    Article  MATH  Google Scholar 

  5. Gresho PM, Sani RL (1970) The effects of gravity modulation on the stability of a heated fluid layer. J Fluid Mech 40:783–806. https://doi.org/10.1017/S0022112070000447

    Article  MATH  Google Scholar 

  6. Saravanan S, Meenasaranya M (2017) Nonlinear stability of modulated Horton-Rogers-Lapwood problem. Int J Eng Sci 120:71–81. https://doi.org/10.1016/j.ijengsci.2017.06.027

    Article  Google Scholar 

  7. Series RW, Hurle DTJ (1991) The use of magnetic fields in semiconductor crystal growth. J Cryst Growth 113:305–328. https://doi.org/10.1016/0022-0248(91)90036-5

    Article  Google Scholar 

  8. Siddheshwar PG, Pranesh S (1999) Effect of temperature/gravity modulation on the onset of magneto-convection in weak electrically conducting fluids with internal angular momentum. J Magn Magn Mater 192:159–176. https://doi.org/10.1016/S0304-8853(98)00384-9

    Article  Google Scholar 

  9. Li BQ (2001) Stability of modulated gravity induced thermal convection in magnetic fields. Phys Rev E 63:041508. https://doi.org/10.1103/PhysRevE.63.041508

    Article  Google Scholar 

  10. Bhadauria BS (2006) Time-periodic heating of Rayleigh-Benard convection in a vertical magnetic field. Phys Scr 73:296–302. https://doi.org/10.1088/0031-8949/73/3/010

    Article  MATH  Google Scholar 

  11. Bhadauria BS (2008) Combined effect of temperature modulation and magnetic field on the onset of convection in an electrically conducting-fluid-saturated porous medium. J Heat Transfer 130:052601. https://doi.org/10.1115/1.2885871

    Article  Google Scholar 

  12. Siddheshwar PG, Bhadauria BS, Mishra P, Srivastava AK (2012) Study of heat transport by stationary magneto-convection in a Newtonian liquid under temperature or gravity modulation using Ginzburg-Landau model. Int J of Non Linear Mech 47:418–425. https://doi.org/10.1016/j.ijnonlinmec.2011.06.006

    Article  Google Scholar 

  13. Basak A (2019) The fate of self-aligned rolls in gravity modulated magnetoconvection. Phys Lett A 383:1466–1472. https://doi.org/10.1016/j.physleta.2019.01.058

    Article  MathSciNet  Google Scholar 

  14. Hazra S, Kumar K, Mitra S (2020) Rayleigh - Benard magnetoconvection with temperature modulation. Proc Royal Soc A 476:1–11. https://doi.org/10.1098/rspa.2020.0289

    Article  MATH  Google Scholar 

  15. Joseph DD (1966) Nonlinear stability of the Boussinesq equations by the method of energy. Arch Rat Mech Anal 22:163–184. https://doi.org/10.1007/BF00266474

    Article  MathSciNet  Google Scholar 

  16. Straughan B (2004) The energy method, stability and nonlinear convection. Springer-Verlag, New York

    Book  Google Scholar 

  17. Theofilis V (2011) Global linear instability. Annu Rev Fluid Mech 43:319–352. https://doi.org/10.1146/annurev-fluid-122109-160705

    Article  MathSciNet  MATH  Google Scholar 

  18. Pierrehumbert RT, Widnall SE (1982) The two- and three-dimensional instabilities of a spatially periodic shear layer. J Fluid Mech 114:59–82. https://doi.org/10.1017/S0022112082000044

    Article  MATH  Google Scholar 

  19. Eriksson LE, Rizzi A (1985) Computer-aided analysis of the convergence to steady state of discrete approximations to the Euler equations. J Comput Phys 57:90–128. https://doi.org/10.1016/0021-9991(85)90054-3

    Article  MathSciNet  MATH  Google Scholar 

  20. Chomaz JM, Huerre P, Redekopp LG (1988) Bifurcations to local and global modes in spatially developing flows. Phys Rev Lett 60:25–28. https://doi.org/10.1103/PhysRevLett.60.25

    Article  Google Scholar 

  21. Jerome JJS, Chomaz JM, Huerre P (1992) Transient growth in Rayleigh-Benard-Poiseuille/Couette convection. Phys Fluids 24:044103. https://doi.org/10.1063/1.4704642

    Article  Google Scholar 

  22. Saravanan S, Brindha D (2008) A note on the universal stability of convective flow with variable viscosity. Math Methods Appl Sci 31:769–773. https://doi.org/10.1002/mma.940

    Article  MathSciNet  MATH  Google Scholar 

  23. Flavin JN, Rionero S (1999) Nonlinear stability for a thermofluid in a vertical porous slab. Contin Mech Thermodyn 11:173–179. https://doi.org/10.1007/s001610050109

    Article  MathSciNet  MATH  Google Scholar 

  24. Saravanan S (2009) Centrifugal acceleration induced convection in a magnetic fluid saturated anisotropic rotating porous medium. Transp Porous Media 77:79–86. https://doi.org/10.1007/s11242-008-9263-3

    Article  Google Scholar 

Download references

Acknowledgements

The second author thanks DST, India for providing financial assistance in the form of INSPIRE Fellowship. This work was supported by UGC, India through DRS Special Assistance Programme in Differential Equations and Fluid Dynamics.

Author information

Authors and Affiliations

  1. Centre for Differential Equations and Fluid Dynamics, Department of Mathematics, Bharathiar University, Coimbatore, Tamil Nadu, 641046, India

    S. Saravanan & M. Meenasaranya

Authors
  1. S. Saravanan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Meenasaranya
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Saravanan.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Saravanan, S., Meenasaranya, M. Energy stability of modulation driven porous convection with magnetic field. Meccanica 56, 2777–2788 (2021). https://doi.org/10.1007/s11012-021-01420-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11012-021-01420-5

Keywords

Search

Navigation