Stability and bifurcation of a viscous incompressible plasma fluid contained between two concentric rotating cylinders

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March 2014, 19(2): 543-563. doi: 10.3934/dcdsb.2014.19.543

Stability and bifurcation of a viscous incompressible plasma fluid contained between two concentric rotating cylinders

1.	Department of Mathematics, Sichuan University, Chengdu, Sichuan 610021, China

Received June 2013 Revised October 2013 Published February 2014

Abstract
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In this paper, our objective is to apply the attractor bifurcation theory to study the stability and bifurcation of a viscous incompressible plasma fluid contained between two concentric rotating cylinders. We get a dimensionless parameter $T$ which can describe the stability and bifurcation of the plasma fluid through calculation. When $T$ is smaller than a critical number $T_0$, the plasma fluid is stable. When $T$ crosses the critical number $T_0$, the plasma fluid becomes unstable and will generate a new magnetic field which has an interesting structure.

Keywords: Plasma fluid, narrow-gap issue, instability, structure., attractor bifurcation.

Mathematics Subject Classification: Primary: 76W05, 76U05, 82D10; Secondary: 37N20, 37G30, 37G3.

Citation: Quan Wang. Stability and bifurcation of a viscous incompressible plasma fluid contained between two concentric rotating cylinders. Discrete and Continuous Dynamical Systems - B, 2014, 19 (2) : 543-563. doi: 10.3934/dcdsb.2014.19.543

References:

[1]	S. Chandrasekhar, Hydrodynamic and hydromagnetic Stability, The International Series of Monographs on Physics Clarendon Press, Oxford, 1961. xix+654 pp. (16 plates).
[2]	D. Biskamp, Nonlinear Magnetohydrodynamics, Cambridge University Press, Cambridge, 1993. doi: 10.1017/CBO9780511599965.
[3]	P. Drazin and W. Reid, Hydrodynamic Stability, Cambridge University Press, 1981.
[4]	C. Foias, O. Manley and R. Temam, Attractors for the Bénard problem: existence and physical bounds on their fractal dimension, Nonlinear Anal, 11 (1987), no.8, 939-967. doi: 10.1016/0362-546X(87)90061-7.
[5]	D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981. iv+348 pp. ISBN: 3-540-10557-3
[6]	V. I. Iudovich, Secondary flows and fluid instability between rotating cylinders, Prikl. Mat. Meh., 30 688-698 (Russian); translated as J. Appl. Math. Mech. 30 (1966), 822-833. doi: 10.1016/0021-8928(66)90033-5.
[7]	K. Kirchg$\ddota$ssner, Bifurcation in nonlinear hydrodynamic stability, SIAM Rev., 17 (1975), 652-683. doi: 10.1137/1017072.
[8]	R. Moreau, Magnetohydrodynamics, Kluwer Academic Publishers, Dordrecht, 1990.
[9]	T. Ma and S. Wang, Structural classification and stability of divergence-free vector fields, Phys. D, 171 (2002), 107-126. doi: 10.1016/S0167-2789(02)00587-0.
[10]	T. Ma and S. Wang, Stability and bifurcation of the Taylor problem, Arch. Ration. Mech. Anal., 181 (2006), 149-176. doi: 10.1007/s00205-006-0415-8.
[11]	T. Ma and S. Wang, Bifurcation Theory and Applications, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, 53. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. xiv+375 pp. ISBN: 981-256-287-7 doi: 10.1142/9789812701152.
[12]	T. Ma and S. Wang, Geometric theory of incompressible flows with applications to fluid dynamics, Mathematical Surveys and Monographs, 119. American Mathematical Society, Providence, RI, 2005. x+234 pp. ISBN: 0-8218-3693-5
[13]	T. Ma and S. Wang, Stability and Bifurcation of Nolinear Evolution Equations, Science Press, Beijing, 2007.
[14]	R. V. Polovin and V. P. Demutskii, Fundamentals of Magnetohydrodynamics, Consultants Bureau, New York, 1990.
[15]	G. I. Taylor, Stability of a viscous liquid contained between two rotating cyinders,, Phil. Trans. Roy. Soc. Lond. A, 223 (): 289.
[16]	W. Velte, Stabilit$\ddota$t and verzweigung station$\ddotarer$ l$\ddot{0}$sungen der davier-stokeschen gleichungen beim Taylorproblem, Arch. Ration. Mech. Anal., 22 (1966), 1-14. doi: 10.1007/BF00281240.

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References:

[1]	S. Chandrasekhar, Hydrodynamic and hydromagnetic Stability, The International Series of Monographs on Physics Clarendon Press, Oxford, 1961. xix+654 pp. (16 plates).
[2]	D. Biskamp, Nonlinear Magnetohydrodynamics, Cambridge University Press, Cambridge, 1993. doi: 10.1017/CBO9780511599965.
[3]	P. Drazin and W. Reid, Hydrodynamic Stability, Cambridge University Press, 1981.
[4]	C. Foias, O. Manley and R. Temam, Attractors for the Bénard problem: existence and physical bounds on their fractal dimension, Nonlinear Anal, 11 (1987), no.8, 939-967. doi: 10.1016/0362-546X(87)90061-7.
[5]	D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981. iv+348 pp. ISBN: 3-540-10557-3
[6]	V. I. Iudovich, Secondary flows and fluid instability between rotating cylinders, Prikl. Mat. Meh., 30 688-698 (Russian); translated as J. Appl. Math. Mech. 30 (1966), 822-833. doi: 10.1016/0021-8928(66)90033-5.
[7]	K. Kirchg$\ddota$ssner, Bifurcation in nonlinear hydrodynamic stability, SIAM Rev., 17 (1975), 652-683. doi: 10.1137/1017072.
[8]	R. Moreau, Magnetohydrodynamics, Kluwer Academic Publishers, Dordrecht, 1990.
[9]	T. Ma and S. Wang, Structural classification and stability of divergence-free vector fields, Phys. D, 171 (2002), 107-126. doi: 10.1016/S0167-2789(02)00587-0.
[10]	T. Ma and S. Wang, Stability and bifurcation of the Taylor problem, Arch. Ration. Mech. Anal., 181 (2006), 149-176. doi: 10.1007/s00205-006-0415-8.
[11]	T. Ma and S. Wang, Bifurcation Theory and Applications, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, 53. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. xiv+375 pp. ISBN: 981-256-287-7 doi: 10.1142/9789812701152.
[12]	T. Ma and S. Wang, Geometric theory of incompressible flows with applications to fluid dynamics, Mathematical Surveys and Monographs, 119. American Mathematical Society, Providence, RI, 2005. x+234 pp. ISBN: 0-8218-3693-5
[13]	T. Ma and S. Wang, Stability and Bifurcation of Nolinear Evolution Equations, Science Press, Beijing, 2007.
[14]	R. V. Polovin and V. P. Demutskii, Fundamentals of Magnetohydrodynamics, Consultants Bureau, New York, 1990.
[15]	G. I. Taylor, Stability of a viscous liquid contained between two rotating cyinders,, Phil. Trans. Roy. Soc. Lond. A, 223 (): 289.
[16]	W. Velte, Stabilit$\ddota$t and verzweigung station$\ddotarer$ l$\ddot{0}$sungen der davier-stokeschen gleichungen beim Taylorproblem, Arch. Ration. Mech. Anal., 22 (1966), 1-14. doi: 10.1007/BF00281240.

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