Abstract
The paper is concerned with the onset of bifurcations in fluid mixtures. The instability of thermal conduction state in a rotating fluid layer heated and salted from below, is analyzed. The Taylor number threshold for the transition to Hopf bifurcations, in simple closed form, is obtained. The basic property of each coefficient of the spectrum equation to drive instability and type of bifurcation, is shown and applied.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Dover, New York (1981)
Drazin, P., Reid, W.: Hydrodynamic Stability. Cambridge University Press, Cambridge (1981)
Joseph, D.D.: Stability of fluid motions I. II. Springer, New York (1992)
Straughan B.: The Energy Method, Stability, and Nonlinear Convection 2\(^{nd}\) edn, Applied Mathematics Science, 91, Springer, Berlin (2004)
Nield, D.A., Bejan, A.: Convection in Porous Media. Springer, Berlin (2017)
Flavin, J., Rionero, S.: Qualitative estimates for partial differential equations. An introduction. CRC Press, Boca Raton (1996)
Rionero, S.: Soret effects on the onset of convection in rotating porous layers via the auxiliary system method. Ricerche Mat. 62(2), 183–208 (2013)
Rionero, S.: Absence of subcritical instabilities and global nonlinear stability for porous ternary diffusive-convective fluid mixtures. Phys. Fluids 24, 104101 (2012)
Rionero, S.: Multicomponent diffusive-convective fluid motions in porous layers: ultimately boundedness, absence of subcritical instabilities, and global nonlinear stability for any number of salts. Phys. Fluids 25, 054104 (2013)
Rionero, S.: Heat and mass transfer by convection in multicomponent Navier-Stokes mixture: absence of subcritical instabilities and global nonlinear stability via the Auxiliary System Method. Rend. Lincei Mat. Appl. 25, 1–44 (2014)
Capone, F., Rionero, S.: Porous MHD Convection: stabilizing effect of magnetic field and bifurcation analysis. Ricerche Mat. 65, 163–186 (2016)
Capone, F., De Luca, R.: Porous MHD Convection: effect of Vadasz Inertia term. Transp. Porous Med. 118, 519–536 (2017)
Rionero, S.: Hopf bifurcation in dynamical systems. Ricerche Mat. (2019). https://doi.org/10.1007/s11587-019-00440-4
Rionero, S.: Hopf bifurcations and global nonlinear \(L^2-\)energy stability in thermal MHD. Rend. Lincei Mat. Appl. 30, 881–905 (2019)
De Luca, R., Rionero, S.: Dynamic of rotating fluid layers: \(L^2-\)absorbing sets and onset of convection. Acta Mech. 228, 4025–4037 (2017)
De Luca, R., Rionero, S.: Steady and oscillatory convection in rotating fluid layers heated and salted from below. Int. J. Non-Linear Mech. 78, 121–130 (2016)
Gantmaker, F.R.: The Theory of Matrices, vol. 1-2. AMS (Chelsea Publishing), Salt Lake (2000)
Gantmaker, F.R.: Lectures in Analytical Mechanics. MIR Publisher, Moscoww (1975)
Rionero, S.: Cold convection in porous layers salted from above. Meccanica 49(9), 2061–2068 (2014)
Acknowledgements
This paper has been performed under the auspices of the G.N.F.M. of INdAM.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflicts of interest
The corresponding author states that there is no conflict of interest.
Additional information
This paper is dedicated to Prof. G. Toscani and Prof. M. Sugiyama on the occasion of their 70th birthday.
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
Rionero, S. Hopf bifurcations in quaternary dynamical systems of rotating thermofluid mixtures, driven by spectrum characteristic coefficients. Ricerche mat 70, 331–346 (2021). https://doi.org/10.1007/s11587-020-00514-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11587-020-00514-8