Abstract
Lyapunov direct method has been used to study the nonlinear stability of laminar flow between two parallel planes in the presence of a coplanar magnetic field for streamwise perturbations with stress-free boundary planes. Two Lyapunov functions are defined. By means of the first, it is proved that the transverse components of the perturbations decay unconditionally and asymptotically to zero for all Reynolds numbers and magnetic Reynolds numbers. By means of the second, it is showed that the other components of the perturbations decay conditionally and exponentially to zero for all Reynolds numbers and the magnetic Reynolds numbers below \(\frac{\pi ^2}{2M}\), where M is the maximum of the absolute value of the velocity field of the laminar flow.
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References
Serrin, J.: On the Stability of Viscous Fluid Motions. Arch. Ration. Mech. Anal. 3, 1–13 (1959)
Joseph, D.D.: Nonlinear Stability of the Boussinesq Equations by the Method of Energy. Arch. Ration. Mech. Anal. 22, 163–184 (1966)
Rionero, S.: Sulla stabilità asintotica in media in magnetoidrodinamica. Ann. Mat. Pura Appl. 76, 75–92 (1967)
Galdi, G.P., Straughan, B.: A Nonlinear Analysis of the Stabilizing Effect of Rotation in the Bénard Problem. Proc. R. Soc. Lond. A 402, 257–283 (1985)
Straughan, B.: The Energy Method, Stability and Nonlinear Convection, Applied Mathematic Sciences, 91, 2nd edn. Springer, New York (2004)
Rionero, S., Mulone, G.: On the Nonlinear Stability of Parallel Shear Flows. Continuum Mech. Thermodyn. 3, 1–11 (1991)
Mulone, G., Rionero, S.: On the Nonlinear Stability of the Magnetic Bénard Problem with Rotation. ZAMM 73(1), 35–45 (1993)
Kaiser, R., Xu, L.X.: Nonlinear Stability of the Rotating Bénard Problem, the Case \(Pr=1\). Nonlinear Differ. Equ. Appl. 5, 283–307 (1998)
Lombardo, S., Mulone, G., Trovato, M.: Nonlinear Stability in Reaction-Diffusion Systems via Optimal Lyapunov Functions. J. Math. Anal. Appl. 342, 461–476 (2008)
Mulone, G., Salemi, F.: On the Nonlinear Stability of Laminar Flow between Parallel Planes in the Presence of a Coplanar Magnetic Field. Ricerche Math. XLI– Supplemento, 209–225 (1992)
Xu, L., Lan, W.: On the nonlinear stability of parallel shear flow in the presence of a coplanar magnetic field. Nonlinear Anal. 95, 93–98 (2014)
Stuart, J.T.: On the Stability of Viscous Flow between Parallel Planes in the Presence of a Coplanar Magnetic Field. Proc. R. Soc. Lond. A 221, 189–206 (1954)
Nardini, R.: Su un caso particolare di stabilità in media della magnetoidrodinamica. Ann. Mat. Pura Appl. 84, 95–100 (1970)
Maiellaro, M.: Su due casi particolari di stabilità asintotica esponenziale in media in magnetoidrodinamica. Atti Sem. Mat. Fis. Modena 20, 105–114 (1971)
Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Oxford University Press, Oxford (1961)
Hoffmann, N.P., Busse, F.H.: Instability of Shear Flow between Two Coaxial Differentially Rotating Cones. Phys. Fluids 11(6), 1676–1678 (1999)
Schmitt, B.J., von Wahl, W.: Decomposition of Solenoidal Fields into Poloidal Fields, Toroidal Fields and Mean Flow. Applications to the Boussinesq Equations. The Navier-Stokes Equations II-Theory and Numerical Methods. In: Solonnikov, S.A. (ed.) Lecture Notes in Mathematics, vol. 1530, pp. 291–305. Springer, Berlin, Heidelberg, New York
von Wahl, W.: Necessary and Sufficient Conditions for the Stability of Flows of Incompressible Viscous Fluids. Arch. Ration. Mech. Anal. 126(2), 103–129 (1994)
Falsaperla, P., Giacobbe, A., Lombardo, S., Mulone, G.: Laminar hydromagnetic flows in an inclined heated layer. Atti della Accademia Peloritana dei Pericolanti 94(1), A5-1–A5-15 (2016)
Falsaperla, P., Giacobbe, A., Lombardo, S., Mulone, G.: Stability of hydromagnetic lamina flows in an inclined heated layer Ricerche di Matematica (2016). doi:10.1007/s11587-016-0290-z
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Xu, L., Lan, W. On the Nonlinear Stability of Plane Parallel Shear Flow in a Coplanar Magnetic Field. J. Math. Fluid Mech. 19, 613–622 (2017). https://doi.org/10.1007/s00021-016-0298-x
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DOI: https://doi.org/10.1007/s00021-016-0298-x