*Jean-Frédéric Gerbeau, Claude Le Bris, and Tony Lelièvre*

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198566656
- eISBN:
- 9780191718014
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566656.003.0006
- Subject:
- Mathematics, Mathematical Physics

This chapter is entirely devoted to one industrial application, the simulation of the industrial production of aluminium in electrolytic cells. The simulation of this specific problem has indeed ...
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This chapter is entirely devoted to one industrial application, the simulation of the industrial production of aluminium in electrolytic cells. The simulation of this specific problem has indeed motivated the whole scientific strategy described in the first five chapters. It serves as an illustration of the efficiency of the approach presented throughout this book. A schematic description of the problem is as follows. An electric current of huge intensity runs downwards through two horizontal layers of incompressible non-miscible conducting fluids. Owing to the magnetohydrodynamics coupling, the interface between the fluids moves, and, in view of the very high intensity of the electric current, the system is very sensitive to instabilities. The industrial challenge is to model, understand, and control these instabilities. Numerical simulation of nonlinear systems can help to reach such a goal. Other techniques (such as a stability analysis for the linearized system) are also employed, and are overviewed for comparison.Less

This chapter is entirely devoted to one industrial application, the simulation of the industrial production of aluminium in electrolytic cells. The simulation of this specific problem has indeed motivated the whole scientific strategy described in the first five chapters. It serves as an illustration of the efficiency of the approach presented throughout this book. A schematic description of the problem is as follows. An electric current of huge intensity runs downwards through two horizontal layers of incompressible non-miscible conducting fluids. Owing to the magnetohydrodynamics coupling, the interface between the fluids moves, and, in view of the very high intensity of the electric current, the system is very sensitive to instabilities. The industrial challenge is to model, understand, and control these instabilities. Numerical simulation of nonlinear systems can help to reach such a goal. Other techniques (such as a stability analysis for the linearized system) are also employed, and are overviewed for comparison.

*Gary A. Glatzmaier*

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691141725
- eISBN:
- 9781400848904
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691141725.003.0003
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

This chapter describes a linear stability analysis (that is, solving for the critical Rayleigh number Ra and mode) that allows readers to check their linear codes against the analytic solution. For ...
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This chapter describes a linear stability analysis (that is, solving for the critical Rayleigh number Ra and mode) that allows readers to check their linear codes against the analytic solution. For this linear analysis, each Fourier mode n can be considered a separate and independent problem. The question that needs to be addressed now is under what conditions—that is, what values of Ra, Prandtl number Pr, and aspect ratio a—will the amplitude of the linear solution grow with time for a given mode n. This is a linear stability problem. The chapter first introduces the linear equations before discussing the linear code and explaining how to find the critical Rayleigh number; in other words, the value of Ra for a and Pr that gives a solution that neither grows nor decays with time. It also shows how the linear stability problem can be solved using an analytic approach.Less

This chapter describes a linear stability analysis (that is, solving for the critical Rayleigh number Ra and mode) that allows readers to check their linear codes against the analytic solution. For this linear analysis, each Fourier mode *n* can be considered a separate and independent problem. The question that needs to be addressed now is under what conditions—that is, what values of Ra, Prandtl number Pr, and aspect ratio *a*—will the amplitude of the linear solution grow with time for a given mode *n*. This is a linear stability problem. The chapter first introduces the linear equations before discussing the linear code and explaining how to find the critical Rayleigh number; in other words, the value of Ra for *a* and Pr that gives a solution that neither grows nor decays with time. It also shows how the linear stability problem can be solved using an analytic approach.

*Mark Newman*

- Published in print:
- 2018
- Published Online:
- October 2018
- ISBN:
- 9780198805090
- eISBN:
- 9780191843235
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198805090.003.0017
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

An introduction to the theory of dynamical systems on networks. This chapter starts with a short introduction to classical (non-network) dynamical systems theory, including linear stability analysis, ...
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An introduction to the theory of dynamical systems on networks. This chapter starts with a short introduction to classical (non-network) dynamical systems theory, including linear stability analysis, fixed points, and limit cycles. Dynamical systems on networks are introduced, focusing initially on systems with only one variable per node and progressing to multi-variable systems. Linear stability analysis is developed in detail, leading to master stability conditions and the connection between stability and the spectral properties of networks. The chapter ends with a discussion of synchronization phenomena, the stability of limit cycles, and master stability conditions for synchronization.Less

An introduction to the theory of dynamical systems on networks. This chapter starts with a short introduction to classical (non-network) dynamical systems theory, including linear stability analysis, fixed points, and limit cycles. Dynamical systems on networks are introduced, focusing initially on systems with only one variable per node and progressing to multi-variable systems. Linear stability analysis is developed in detail, leading to master stability conditions and the connection between stability and the spectral properties of networks. The chapter ends with a discussion of synchronization phenomena, the stability of limit cycles, and master stability conditions for synchronization.