Memory driven instability in a diffusion process

B.R. Duffy, M. Grinfeld, P. Freitas

Research output: Contribution to journalArticle

19 Citations (Scopus)

Abstract

We consider the ndimensional version of a model proposed by Olmstead et al. [SIAM J. Appl. Math., 46 (1986), pp. 171--188] for the flow of a non-Newtonian fluid in the presence of memory. We prove the existence of a global attractor and obtain conditions for the existence of a Lyapunov functional, which allows us to give a full description of this attractor in a certain region of the parameter space in the bistable case. We then study the stability and bifurcation of stationary solutions and, in particular, prove that for certain values of the parameters it is not possible to stabilize the flow by increasing a Rayleigh-type number. The existence of periodic and homoclinic orbits is also shown by studying the Bogdanov--Takens singularity obtained from a center manifold reduction.
LanguageEnglish
Pages1090-1106
Number of pages16
JournalSIAM Journal on Mathematical Analysis
Volume33
Issue number5
DOIs
Publication statusPublished - 2002

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Diffusion Process
Orbits
Data storage equipment
Fluids
Center Manifold Reduction
Non-Newtonian Fluid
Homoclinic Orbit
Lyapunov Functional
Global Attractor
Stationary Solutions
Rayleigh
Periodic Orbits
Parameter Space
Attractor
Bifurcation
Singularity
Model

Keywords

  • parabolic systems
  • memory effects
  • non-Newtonian fluids
  • memory

Cite this

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Memory driven instability in a diffusion process. / Duffy, B.R.; Grinfeld, M.; Freitas, P.

In: SIAM Journal on Mathematical Analysis, Vol. 33, No. 5, 2002, p. 1090-1106.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Memory driven instability in a diffusion process

AU - Duffy, B.R.

AU - Grinfeld, M.

AU - Freitas, P.

PY - 2002

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AB - We consider the ndimensional version of a model proposed by Olmstead et al. [SIAM J. Appl. Math., 46 (1986), pp. 171--188] for the flow of a non-Newtonian fluid in the presence of memory. We prove the existence of a global attractor and obtain conditions for the existence of a Lyapunov functional, which allows us to give a full description of this attractor in a certain region of the parameter space in the bistable case. We then study the stability and bifurcation of stationary solutions and, in particular, prove that for certain values of the parameters it is not possible to stabilize the flow by increasing a Rayleigh-type number. The existence of periodic and homoclinic orbits is also shown by studying the Bogdanov--Takens singularity obtained from a center manifold reduction.

KW - parabolic systems

KW - memory effects

KW - non-Newtonian fluids

KW - memory

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