American Institute of Mathematical Sciences

Advanced
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, (1961).   Google Scholar

[2]

D. Biskamp, Nonlinear Magnetohydrodynamics,, Cambridge University Press, (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), 939.  doi: 10.1016/0362-546X(87)90061-7.  Google Scholar

[5]

D. Henry, Geometric theory of semilinear parabolic equations,, Lecture Notes in Mathematics, (1981), 3.   Google Scholar

[6]

V. I. Iudovich, Secondary flows and fluid instability between rotating cylinders,, Prikl. Mat. Meh., 30 (1966), 688.  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.  doi: 10.1137/1017072.  Google Scholar

[8]

R. Moreau, Magnetohydrodynamics,, Kluwer Academic Publishers, (1990).   Google Scholar

[9]

T. Ma and S. Wang, Structural classification and stability of divergence-free vector fields,, Phys. D, 171 (2002), 107.  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.  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, (2005), 981.  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, (2005), 0.   Google Scholar

[13]

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

[14]

R. V. Polovin and V. P. Demutskii, Fundamentals of Magnetohydrodynamics,, Consultants Bureau, (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.  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, (1961).   Google Scholar

[2]

D. Biskamp, Nonlinear Magnetohydrodynamics,, Cambridge University Press, (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), 939.  doi: 10.1016/0362-546X(87)90061-7.  Google Scholar

[5]

D. Henry, Geometric theory of semilinear parabolic equations,, Lecture Notes in Mathematics, (1981), 3.   Google Scholar

[6]

V. I. Iudovich, Secondary flows and fluid instability between rotating cylinders,, Prikl. Mat. Meh., 30 (1966), 688.  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.  doi: 10.1137/1017072.  Google Scholar

[8]

R. Moreau, Magnetohydrodynamics,, Kluwer Academic Publishers, (1990).   Google Scholar

[9]

T. Ma and S. Wang, Structural classification and stability of divergence-free vector fields,, Phys. D, 171 (2002), 107.  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.  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, (2005), 981.  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, (2005), 0.   Google Scholar

[13]

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

[14]

R. V. Polovin and V. P. Demutskii, Fundamentals of Magnetohydrodynamics,, Consultants Bureau, (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.  doi: 10.1007/BF00281240.  Google Scholar

[1]

Philipp Fuchs, Ansgar Jüngel, Max von Renesse. On the Lagrangian structure of quantum fluid models. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1375-1396. doi: 10.3934/dcds.2014.34.1375
[2]

I. D. Chueshov, Iryna Ryzhkova. A global attractor for a fluid--plate interaction model. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1635-1656. doi: 10.3934/cpaa.2013.12.1635
[3]

Eugenio Aulisa, Akif Ibragimov, Emine Yasemen Kaya-Cekin. Fluid structure interaction problem with changing thickness beam and slightly compressible fluid. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1133-1148. doi: 10.3934/dcdss.2014.7.1133
[4]

Qiang Du, M. D. Gunzburger, L. S. Hou, J. Lee. Analysis of a linear fluid-structure interaction problem. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 633-650. doi: 10.3934/dcds.2003.9.633
[5]

Yaodan Huang, Zhengce Zhang, Bei Hu. Bifurcation from stability to instability for a free boundary tumor model with angiogenesis. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2473-2510. doi: 10.3934/dcds.2019105
[6]

M. Bulíček, F. Ettwein, P. Kaplický, Dalibor Pražák. The dimension of the attractor for the 3D flow of a non-Newtonian fluid. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1503-1520. doi: 10.3934/cpaa.2009.8.1503
[7]

T. Caraballo, J. A. Langa, J. Valero. Structure of the pullback attractor for a non-autonomous scalar differential inclusion. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 979-994. doi: 10.3934/dcdss.2016037
[8]

Tomás Caraballo, David Cheban. On the structure of the global attractor for non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2012, 11 (2) : 809-828. doi: 10.3934/cpaa.2012.11.809
[9]

Thai Son Doan, Martin Rasmussen, Peter E. Kloeden. The mean-square dichotomy spectrum and a bifurcation to a mean-square attractor. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 875-887. doi: 10.3934/dcdsb.2015.20.875
[10]

Tian Ma, Shouhong Wang. Attractor bifurcation theory and its applications to Rayleigh-Bénard convection. Communications on Pure & Applied Analysis, 2003, 2 (4) : 591-599. doi: 10.3934/cpaa.2003.2.591
[11]

Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. A stability estimate for fluid structure interaction problem with non-linear beam. Conference Publications, 2009, 2009 (Special) : 424-432. doi: 10.3934/proc.2009.2009.424
[12]

Grégoire Allaire, Alessandro Ferriero. Homogenization and long time asymptotic of a fluid-structure interaction problem. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 199-220. doi: 10.3934/dcdsb.2008.9.199
[13]

Serge Nicaise, Cristina Pignotti. Asymptotic analysis of a simple model of fluid-structure interaction. Networks & Heterogeneous Media, 2008, 3 (4) : 787-813. doi: 10.3934/nhm.2008.3.787
[14]

Igor Kukavica, Amjad Tuffaha. Solutions to a fluid-structure interaction free boundary problem. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1355-1389. doi: 10.3934/dcds.2012.32.1355
[15]

Jean-Jérôme Casanova. Existence of time-periodic strong solutions to a fluid–structure system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3291-3313. doi: 10.3934/dcds.2019136
[16]

Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. FLUID STRUCTURE INTERACTION PROBLEM WITH CHANGING THICKNESS NON-LINEAR BEAM Fluid structure interaction problem with changing thickness non-linear beam. Conference Publications, 2011, 2011 (Special) : 813-823. doi: 10.3934/proc.2011.2011.813
[17]

George Avalos, Roberto Triggiani. Fluid-structure interaction with and without internal dissipation of the structure: A contrast study in stability. Evolution Equations & Control Theory, 2013, 2 (4) : 563-598. doi: 10.3934/eect.2013.2.563
[18]

Rachel Kuske, Peter Borowski. Survival of subthreshold oscillations: The interplay of noise, bifurcation structure, and return mechanism. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 873-895. doi: 10.3934/dcdss.2009.2.873
[19]

Kousuke Kuto, Tohru Tsujikawa. Bifurcation structure of steady-states for bistable equations with nonlocal constraint. Conference Publications, 2013, 2013 (special) : 467-476. doi: 10.3934/proc.2013.2013.467
[20]

Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (26)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences