Abstract
The effects of cross-diffusion on linear and weak nonlinear stability of double diffusive convection in an electrically conducting horizontal fluid layer with an imposed vertical magnetic field are investigated. The criterion for the onset of stationary and oscillatory convection is obtained analytically by performing the linear instability analysis. Several noteworthy departures from those of doubly diffusive fluid systems are unveiled under certain parametric conditions. It is shown that (i) disconnected closed convex oscillatory neutral curve separated from the stationary neutral curve exists requiring three critical thermal Rayleigh numbers to completely specify the linear instability criteria instead of a usual single critical value, (ii) an electrically conducting fluid layer in the presence of magnetic field can be destabilized by stable solute concentration gradient, and (iii) a doubly diffusive conducting fluid layer can be destabilized in the presence of magnetic field. It is demonstrated that small variations in the off-diagonal elements enforce discrepancies in the instability criteria. A weak nonlinear stationary stability analysis has been performed using a perturbation method and a cubic Landau equation is derived and the stability of bifurcating equilibrium solution is discussed. It is found that subcritical bifurcation occurs depending on the choices of governing parameters.
Similar content being viewed by others
References
Turner, J.: Double-diffusive phenomena. Annu. Rev. Fluid Mech. 6, 37–54 (1974)
Huppert, H.E., Turner, J.S.: Double-diffusive convection. J. Fluid Mech. 106, 299–329 (1981)
Platten, J.K., Legros, J.C.: Convection in Liquids. Springer, Berlin (2012)
Mortimer, R.G., Eyring, H.: Elementary transition state theory of the Soret and Dufour effects. Proc. Natl. Acad. Sci. 77, 1728–1731 (1980)
Knobloch, E.: Convection in binary fluids. Phys. Fluids 23, 1918–1920 (1980)
McDougall, T.J.: Double-diffusive convection caused by coupled molecular diffusion. J. Fluid Mech. 126, 379–397 (1983)
Terrones, G., Chen, C.F.: Convective stability of gravity-modulated doubly cross-diffusive fluid layers. J. Fluid Mech. 255, 301–321 (1993)
Straughan, B., Hutter, K.: A priori bounds and structural stability for double-diffusive convection incorporating the Soret effect. Proc. R. Soc. A 455, 767–777 (1999)
Malashetty, M.S., Gaikwad, S.N., Swamy, M.: An analytical study of linear and non-linear double diffusive convection with Soret effect in couple stress liquids. Int. J. Therm. Sci. 45, 897–907 (2006)
Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Oxford University Press, London (1961)
Weiss, N.O.: Convection in an imposed magnetic field, part 1: the development of nonlinear convection. J. Fluid Mech. 108, 247–272 (1981)
Knobloch, E., Weiss, N.O., Da Costa, L.N.: Oscillatory and steady convections in a magnetic field. J. Fluid Mech. 113, 153–186 (1981)
Proctor, M.R.E., Weiss, N.O.: Magnetoconvection. Rep. Prog. Phys. 45, 1317–1379 (1982)
Rudraiah, N., Kumudini, V., Unno, W.: Theory of nonlinear magnetoconvection and its application to solar convection problems. I, II. Publ. Astron. Soc. Jpn. 37, 183–233 (1985)
Hughes, D.W., Weiss, N.O.: Double-diffusive convection with two stabilizing gradients: strange consequences of magnetic buoyancy. J. Fluid Mech. 301, 383–406 (1995)
Tagare, S.G., Rameshwar, Y.: Magnetoconvection in rotating stars. Astrophys. Space Sci. 284, 983–999 (2003)
Güray, E., Tarman, H.I.: Thermal convection in the presence of a vertical magnetic field. Acta Mech. 194, 33–46 (2007)
Harfash, J., Straughan, B.: Magnetic effect on instability and nonlinear stability in a reacting fluid. Meccanica 47, 1849–1857 (2012)
Reza, M., Gupta, A.S.: Magnetohydrodynamic thermal instability in a conducting fluid layer with throughflow. Int. J. Non-Linear Mech. 47, 616–625 (2012)
Stein, R.F.: Magneto-convection. Philos. Trans. R. Soc. A 370, 3070–3087 (2012)
Lortz, D.: A stability criterion for steady finite amplitude convection with an external magnetic field. J. Fluid Mech. 23, 113–128 (1965)
Malkus, W.V.R., Veronis, G.: Finite amplitude cellular convection. J. Fluid Mech. 4, 225–260 (1958)
Rudraiah, N., Shivakumara, I.S.: Double-diffusive convection with an imposed magnetic field. Int. J. Heat Mass Trans. 27, 1825–1836 (1984)
Rudraiah, N., Shivakumara, I.S.: Effect of magnetic field on the stability of a doubly-diffusive conducting fluid layer. Vignana Bharathi (Bangalore Univ. J.) 10, 25–47 (1987)
Rudraiah, N.: Double-diffusive magnetoconvection. Pramana J. Phys. 27, 233–266 (1986)
Shivakumara, I.S.: Further results on double diffusive magnetoconvection. In: Sachdev, P.L., Venkatachalappa, M. (eds.) Recent Advances in Fluid Mechanics, pp. 233–248. Gordon and Breach Science Publishers, New York (1995)
Prakash, J., Bala, R., Vaid, K.: On the characterization of magnetohydrodynamic triple diffusive convection. J. Magn. Magn. Mater. 377, 378–385 (2015)
Prakash, J., Kumar, R., Lata, K.: Onset of convection in a multicomponent fluid layer in the presence of a uniform magnetic field. J. Appl. Mech. Tech. Phys. 58, 36–46 (2017)
Naveen Kumar, S., Shivakumara, I.S., Shankar, B.M.: Linear and weakly nonlinear double-diffusive magnetoconvection in a non-Newtonian fluid layer. Microgravity Science and Technology, pp. 1–18 (2020)
Veronis, G.: Effect of a stabilizing gradient of solute on thermal convection. J. Fluid Mech. 34, 315–336 (1968)
Rajagopal, K.R., Ruzicka, M., Srinivas, A.R.: On the Oberbeck-Boussinesq approximation. Math. Model Methods Appl. Sci. 16, 1157–1167 (1996)
Rosenblat, S.: Thermal convection in a viscoelastic liquid. J. Non-Newton. Fluid Mech. 21, 201–223 (1986)
Raghunatha, K.R., Shivakumara, I.S., Swamy, M.S.: Effect of cross-diffusion on the stability of a triple-diffusive Oldroyd-B fluid layer. Z. Angew. Math. Phys. 70, 100–121 (2019)
Knobloch, E., Weiss, N.O., Da Costa, L.N.: Oscillatory and steady convections in a magnetic field. J. Fluid Mech. 113, 153–186 (1981)
Nagata, W., Thomas, J.W.: Bifurcation in doubly-diffusive systems I. Equilibrium solutions. SIAM J. Math. Anal. 17, 91–113 (1986)
Acknowledgements
The author (KRR) gratefully acknowledge the support offered by the “Davangere University”, India, under the Grant No: DU/HRM/2020-21/6045 for this research work. The authors thank the reviewer for the constructive comments and useful suggestions which helped in improving the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Shivakumara, I.S., Raghunatha, K.R., Savitha, M.N. et al. Implication of cross-diffusion on the stability of double diffusive convection in an imposed magnetic field. Z. Angew. Math. Phys. 72, 117 (2021). https://doi.org/10.1007/s00033-021-01544-4
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00033-021-01544-4