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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

Quan Wang 1,

1. 

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

Received  June 2013 Revised  October 2013 Published  February 2014

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 & 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).  Google Scholar

[2]

D. Biskamp, Nonlinear Magnetohydrodynamics, Cambridge University Press, Cambridge, 1993. doi: 10.1017/CBO9780511599965.  Google Scholar

[3]

P. Drazin and W. Reid, Hydrodynamic Stability, Cambridge University Press, 1981.  Google Scholar

[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.  Google Scholar

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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  Google Scholar

[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.  Google Scholar

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K. Kirchg$\ddota$ssner, Bifurcation in nonlinear hydrodynamic stability, SIAM Rev., 17 (1975), 652-683. doi: 10.1137/1017072.  Google Scholar

[8]

R. Moreau, Magnetohydrodynamics, Kluwer Academic Publishers, Dordrecht, 1990.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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  Google Scholar

[13]

T. Ma and S. Wang, Stability and Bifurcation of Nolinear Evolution Equations, Science Press, Beijing, 2007. Google Scholar

[14]

R. V. Polovin and V. P. Demutskii, Fundamentals of Magnetohydrodynamics, Consultants Bureau, New York, 1990. Google Scholar

[15]

G. I. Taylor, Stability of a viscous liquid contained between two rotating cyinders,, Phil. Trans. Roy. Soc. Lond. A, 223 (): 289.   Google Scholar

[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.  Google Scholar

show all references

References:
[1]

S. Chandrasekhar, Hydrodynamic and hydromagnetic Stability, The International Series of Monographs on Physics Clarendon Press, Oxford, 1961. xix+654 pp. (16 plates).  Google Scholar

[2]

D. Biskamp, Nonlinear Magnetohydrodynamics, Cambridge University Press, Cambridge, 1993. doi: 10.1017/CBO9780511599965.  Google Scholar

[3]

P. Drazin and W. Reid, Hydrodynamic Stability, Cambridge University Press, 1981.  Google Scholar

[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.  Google Scholar

[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  Google Scholar

[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.  Google Scholar

[7]

K. Kirchg$\ddota$ssner, Bifurcation in nonlinear hydrodynamic stability, SIAM Rev., 17 (1975), 652-683. doi: 10.1137/1017072.  Google Scholar

[8]

R. Moreau, Magnetohydrodynamics, Kluwer Academic Publishers, Dordrecht, 1990.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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  Google Scholar

[13]

T. Ma and S. Wang, Stability and Bifurcation of Nolinear Evolution Equations, Science Press, Beijing, 2007. Google Scholar

[14]

R. V. Polovin and V. P. Demutskii, Fundamentals of Magnetohydrodynamics, Consultants Bureau, New York, 1990. Google Scholar

[15]

G. I. Taylor, Stability of a viscous liquid contained between two rotating cyinders,, Phil. Trans. Roy. Soc. Lond. A, 223 (): 289.   Google Scholar

[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.  Google Scholar

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