Abstract
The problem of convection in a variable gravity field with magnetic field effect is studied using methods of linear instability theory and non-linear energy theory. Then, the accuracies of both the linear instability and global non-linear energy stability thresholds are tested using a three-dimensional simulation. The strong stabilizing effect of gravity field and magnetic field is shown. Moreover, the results support the assertion that the linear theory is very accurate in predicting the onset of convective motion, and thus regions of stability.
Similar content being viewed by others
References
Chandrasekhar, S.: Hydrodynamic and hydromagnetic stability. Dover, New York (1981)
Chen, C.F., Chen, F.: Onset of salt finger convection in a gravity gradient. Phys. Fluids A 4, 451–452 (1992)
Davis, C., Carpenter, P.W.: A novel velocity–vorticity formulation of the Navier–Stokes equations with application to boundary layer disturbance evolution. J. Comp. Phys. 172, 119–165 (2001)
Daube, O.: Resolution of the 2D Navier–Stokes equations in velocity-vorticity form by means of an influence matrix technique. J. Comput. Phys. 103, 402–414 (1992)
Fasel, H.: Investigation of the stability of boundary layers by a finite-difference model of the Navier-Stokes equations. J. Fluid Mech. 78, 355–383 (1976)
Fabrizio, M., Morro, A.: Electromagnetism of continuous media. Oxford University Press, Oxford (2003)
Galdi, G.P.: Nonlinear stability of the magnetic Bénard problem via a generalized energy method. Arch. Rational Mech. Anal. 87, 167–186 (1985)
Galdi, G.P., Straughan, B.: Exchange of stabilities, symmetry and nonlinear stability. Arch. Rational Mech. Anal. 89, 211–228 (1985)
Guevremont, G., Habashi, W.G., Hafez, M.M.: Finite element solution of the Navier-Stokes equations by a velocity–vorticity method. Int. J. Numer. Meth. Fluids 10, 461–475 (1990)
Guj, G., Stella, F.: A vorticity–velocity method for the numerical solution of 3D incompressible flows. J. Comput. Phys. 106, 286–298 (1993)
Harfash, A.J.: Magnetic effect on instability and nonlinear stability of double diffusive convection in a reacting fluid. Contin. Mech. Thermodyn. 25, 89–106 (2013)
Harfash, A.J.: Continuous dependence on the coefficients for double diffusive convection in Darcy flow with Magnetic field effect. Anal. Math. Phys. 3, 163–181 (2013)
Harfash, A.J.: Three dimensions simulation for the problem of a layer of non-Boussinesq fluid heated internally with prescribed heat flux on the lower boundary and constant temperature upper surface. Int. J. Eng. Sci. 74, 91–102 (2014)
Harfash, A.J.: Structural stability for convection models in a reacting porous medium with magnetic field effect. Ricerche mat. (2014). doi:10.1007/s11587-013-0152-x
Harfash, A.J.: Three-dimensional simulations for convection in a porous medium with internal heat source and variable gravity effects. Transp. Porous Media 101, 281–297 (2014)
Harfash, A.J.: Three dimensional simulation of radiation induced convection. Appl. Math. Comput. 227, 92–101 (2014)
Harfash, A.J.: Three-dimensional simulations for convection problem in anisotropic porous media with nonhomogeneous porosity, thermal diffusivity, and variable gravity effects, Transp. Porous Media, doi:10.1007/s11242-013-0260-9.
Harfash, A.J.: Three dimensional simulations for penetrative convection in a porous medium with internal heat sources, To appear in Acta Mechanica Sinica (2014).
Harfash, A.J., Hill, A.A.: Simulation of three dimensional double-diffusive throughflow in internally heated anisotropic porous media. Int. J. Heat Mass Trans. 72, 609–615 (2014)
Harfash, A.J., Straughan, B.: Magnetic effect on instability and nonlinear stability in a reacting fluid. Meccanica 47, 1849–1857 (2012)
Landau, L., Lifshitz, E., Pitaevskii, L.: Electrodynamics of continuous media. Pergamon, London (1984)
Mallinson, G.D., de Vahl Davis, G.: Three-dimensional natural convection in a box: a numerical study. J. Fluid Mech. 83, 1–31 (1977)
Napolitano, M., Catalano, L.A.: A multigrid solver for the vorticity–velocity Navier–Stokes equations. Int. J. Numer. Meth. Fluids 13, 49–59 (1993)
Ni, J., Beckerma, C., Smith, T.F.: Effect of an electromagnetic field on natural convection in porous medium. Fundam. Heat Transf. Electromagn. Electrost. Acoust. Field ASME HTD (1993).
Nield, D.A., Bejan, D.: Convection in porous media, 4th edn. Springer, NewYork (2013)
Padula, M.: Non-linear energy stability for the compressible Benard problem. Boll. Cln. Mar. Ital. B 5, 581–602 (1986)
Pradhan, G.K., Samal, P.C.: Thermal stability of a fluid layer under variable body forces. J. Math. Anal. Appl. 122, 487–495 (1987)
Rionero, S., Mulone, G.: Nonlinear stability analysis of the magnetic Bénard problem through the Lyapunov direct method. Arch. Rational Mech. Anal. 103, 347–368 (1988)
Rionero, S.: Metodi variazionali per la stabilitá asintotica in media in magnetoidrodinamica. Ann. Matem. Pura Appl. 78, 339–364 (1968)
Roberts, P.H.: An introduction to magnetohydrodynamics. Longman, London (1967)
Spiegel, E.A.: Convective instability in a compressible atmosphere. I. Asrrophys. J. 139, 1068–1090 (1964)
Straughan, B.: Convection in a variable gravity field. J. Math. Anal. Appl. 140, 467–475 (1989)
Straughan, B.: The energy method, stability, and nonlinear convection. Series in Applied Mathematical Sciences, vol. 91, second edition. Springer, NewYork (2004).
Straughan, B., Harfash, A.J.: Instability in Poiseuille flow in a porous medium with slip boundary conditions. Microfluid Nanofluid 15, 109–115 (2013)
Wong, K.L., Baker, A.J.: A 3D incompressible Navier-Stokes velocity-vorticity weak form finite element algorithm. Int. J. Numer. Meth. Fluids 38, 99–123 (2002)
Acknowledgments
This work was supported by a scholarship from the Iraqi ministry of higher education and scientific research. The author would like to thank Prof. B. Straughan for his guidance. Also, the author would like to thank anonymous referees for their comments that have led to improvements in the manuscript.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Harfash, A.J. Convection in a Porous Medium with Variable Gravity Field and Magnetic Field Effects. Transp Porous Med 103, 361–379 (2014). https://doi.org/10.1007/s11242-014-0305-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11242-014-0305-8