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

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.
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. Kirchgässner, 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, 289-343. Also in Sci, Paper4, 34-85.[125,127,129,130]

[16]

W. Velte, Stabilität 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.

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).

[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. Kirchgässner, 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, 289-343. Also in Sci, Paper4, 34-85.[125,127,129,130]

[16]

W. Velte, Stabilität 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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