American Institute of Mathematical Sciences

Advanced
July  2011, 16(1): 173-188. doi: 10.3934/dcdsb.2011.16.173

Dynamic phase transition for binary systems in cylindrical geometry

I-Liang Chern 1, and  Chun-Hsiung Hsia 2,

1. 

Department of Mathematics and Taidar Institute of Mathematical Science, National Taiwan University, Taipei, 10617, Taiwan
2. 

Department of Mathematics and Taidar Institute of Mthematical Science, National Taiwan University, Taipei, 10617, Taiwan

Received  February 2010 Revised  August 2010 Published  April 2011

In this article, we present a dynamic phase transition and stability analysis for the Cahn-Hilliard equations in cylindrical geometry. Two types of phase transitions (the continuous type and the jump type) are determined explicitly in terms of relevant physical and geometric parameters.
Keywords: binary system., bifurcation, Phase transition, stability.
Mathematics Subject Classification: 37G3.
Citation: I-Liang Chern, Chun-Hsiung Hsia. Dynamic phase transition for binary systems in cylindrical geometry. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 173-188. doi: 10.3934/dcdsb.2011.16.173
References:
[1]

Stephen J. Blundell and Katherine M. Blundell, "Concepts in Thermal Physics,", Oxford University Press, (2008).   Google Scholar

[2]

C. M. Elliott and H. Garcke, Diffusional phase transitions in multicomponent systems with a concentration dependent mobility matrix,, Physica D, 109 (1997), 242.  doi: 10.1016/S0167-2789(97)00066-3.  Google Scholar

[3]

J. E. Hilliard, Spinodal decomposition,, in Phase Transformations, (1970), 497.   Google Scholar

[4]

T. Ma and S. Wang, "Bifurcation Theory and Applications,", vol. \textbf{53} of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, 53 ().   Google Scholar

[5]

T. Ma and S. Wang, Dynamic phase transition theory in PVT systems,, Indiana Univ. Math. J., 57 (2008), 2861.  doi: 10.1512/iumj.2008.57.3630.  Google Scholar

[6]

T. Ma and S. Wang, Cahn-Hilliard equations and phase transition dynamics for binary systems,, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 741.  doi: 10.3934/dcdsb.2009.11.741.  Google Scholar

[7]

T. Ma and S. Wang, Phase separation of binary systems,, Phys. Rev. A., 388 (2009), 4811.   Google Scholar

[8]

J. Moser, A rapidly convergent iteration method and nonlinear partial differential equation. I,, Ann. Sc. Norm. Super. Pisa, 20 (1966), 265.   Google Scholar

[9]

L. E. Reichl, "A Modern Course in Statistical Physics," (Second ed.), A Wiley-Interscinece Publication. New York: John Wiley & Sons Inc. 1998., (1998).   Google Scholar

show all references

References:
[1]

Stephen J. Blundell and Katherine M. Blundell, "Concepts in Thermal Physics,", Oxford University Press, (2008).   Google Scholar

[2]

C. M. Elliott and H. Garcke, Diffusional phase transitions in multicomponent systems with a concentration dependent mobility matrix,, Physica D, 109 (1997), 242.  doi: 10.1016/S0167-2789(97)00066-3.  Google Scholar

[3]

J. E. Hilliard, Spinodal decomposition,, in Phase Transformations, (1970), 497.   Google Scholar

[4]

T. Ma and S. Wang, "Bifurcation Theory and Applications,", vol. \textbf{53} of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, 53 ().   Google Scholar

[5]

T. Ma and S. Wang, Dynamic phase transition theory in PVT systems,, Indiana Univ. Math. J., 57 (2008), 2861.  doi: 10.1512/iumj.2008.57.3630.  Google Scholar

[6]

T. Ma and S. Wang, Cahn-Hilliard equations and phase transition dynamics for binary systems,, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 741.  doi: 10.3934/dcdsb.2009.11.741.  Google Scholar

[7]

T. Ma and S. Wang, Phase separation of binary systems,, Phys. Rev. A., 388 (2009), 4811.   Google Scholar

[8]

J. Moser, A rapidly convergent iteration method and nonlinear partial differential equation. I,, Ann. Sc. Norm. Super. Pisa, 20 (1966), 265.   Google Scholar

[9]

L. E. Reichl, "A Modern Course in Statistical Physics," (Second ed.), A Wiley-Interscinece Publication. New York: John Wiley & Sons Inc. 1998., (1998).   Google Scholar

[1]

Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084
[2]

Tian Ma, Shouhong Wang. Topological phase transition III: Solar surface eruptions and sunspots. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 501-514. doi: 10.3934/dcdsb.2020350
[3]

Tomáš Smejkal, Jiří Mikyška, Jaromír Kukal. Comparison of modern heuristics on solving the phase stability testing problem. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1161-1180. doi: 10.3934/dcdss.2020227
[4]

Chao Xing, Jiaojiao Pan, Hong Luo. Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food. Communications on Pure & Applied Analysis, 2021, 20 (1) : 427-448. doi: 10.3934/cpaa.2020275
[5]

Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467
[6]

Xianyong Chen, Weihua Jiang. Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021013
[7]

Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292
[8]

Pan Zheng. Asymptotic stability in a chemotaxis-competition system with indirect signal production. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1207-1223. doi: 10.3934/dcds.2020315
[9]

Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, , () : -. doi: 10.3934/era.2021003
[10]

Akbar Mahmoodi Rishakani, Seyed Mojtaba Dehnavi, Mohmmadreza Mirzaee Shamsabad, Nasour Bagheri. Cryptographic properties of cyclic binary matrices. Advances in Mathematics of Communications, 2021, 15 (2) : 311-327. doi: 10.3934/amc.2020068
[11]

Izumi Takagi, Conghui Zhang. Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020400
[12]

Yangjian Sun, Changjian Liu. The Poincaré bifurcation of a SD oscillator. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1565-1577. doi: 10.3934/dcdsb.2020173
[13]

Maicon Sônego. Stable transition layers in an unbalanced bistable equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020370
[14]

Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344
[15]

Helmut Abels, Johannes Kampmann. Existence of weak solutions for a sharp interface model for phase separation on biological membranes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 331-351. doi: 10.3934/dcdss.2020325
[16]

Lin Shi, Dingshi Li, Kening Lu. Limiting behavior of unstable manifolds for spdes in varying phase spaces. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021020
[17]

Andrea Giorgini, Roger Temam, Xuan-Truong Vu. The Navier-Stokes-Cahn-Hilliard equations for mildly compressible binary fluid mixtures. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 337-366. doi: 10.3934/dcdsb.2020141
[18]

Yuyuan Ouyang, Trevor Squires. Some worst-case datasets of deterministic first-order methods for solving binary logistic regression. Inverse Problems & Imaging, 2021, 15 (1) : 63-77. doi: 10.3934/ipi.2020047
[19]

Joel Kübler, Tobias Weth. Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3629-3656. doi: 10.3934/dcds.2020032
[20]

Chihiro Aida, Chao-Nien Chen, Kousuke Kuto, Hirokazu Ninomiya. Bifurcation from infinity with applications to reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3031-3055. doi: 10.3934/dcds.2020053

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (45)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences