Jean-Frédéric Gerbeau, Claude Le Bris, and Tony Lelièvre
- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198566656
- eISBN:
- 9780191718014
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566656.001.0001
- Subject:
- Mathematics, Mathematical Physics
This text focuses on mathematical and numerical techniques for the simulation of magnetohydrodynamic phenomena, with an emphasis on the magnetohydrodynamics of liquid metals, on two-fluid flows, and ...
More
This text focuses on mathematical and numerical techniques for the simulation of magnetohydrodynamic phenomena, with an emphasis on the magnetohydrodynamics of liquid metals, on two-fluid flows, and on a prototypical industrial application. The approach is a highly mathematical one, based on the rigorous analysis of the equations at hand, and a solid numerical analysis of the discretization methods. Up-to-date techniques, both on the theoretical side and the numerical side, are introduced to deal with the nonlinearities of the multifluid magnetohydrodynamics equations. At each stage of the exposition, examples of numerical simulations are provided, first on academic test cases to illustrate the approach, next on benchmarks well documented in the professional literature, and finally on real industrial cases. The simulation of aluminium electrolysis cells is used as a guideline throughout the book to motivate the study of a particular setting of the magnetohydrodynamics equations.Less
This text focuses on mathematical and numerical techniques for the simulation of magnetohydrodynamic phenomena, with an emphasis on the magnetohydrodynamics of liquid metals, on two-fluid flows, and on a prototypical industrial application. The approach is a highly mathematical one, based on the rigorous analysis of the equations at hand, and a solid numerical analysis of the discretization methods. Up-to-date techniques, both on the theoretical side and the numerical side, are introduced to deal with the nonlinearities of the multifluid magnetohydrodynamics equations. At each stage of the exposition, examples of numerical simulations are provided, first on academic test cases to illustrate the approach, next on benchmarks well documented in the professional literature, and finally on real industrial cases. The simulation of aluminium electrolysis cells is used as a guideline throughout the book to motivate the study of a particular setting of the magnetohydrodynamics equations.
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/9780198566656.003.0004
- Subject:
- Mathematics, Mathematical Physics
This chapter deals with the theoretical aspects of multifluid magnetohydrodynamics problems. In addition to the coupling between hydrodynamics and electromagnetics examined in Chapter 2, the high ...
More
This chapter deals with the theoretical aspects of multifluid magnetohydrodynamics problems. In addition to the coupling between hydrodynamics and electromagnetics examined in Chapter 2, the high nonlinearity that now needs to be addressed is geometrical in nature. It results from the presence of one (or many) free interface(s) separating the fluids, assumed non-miscible. The mathematical analysis is substantially more intricate, and a long list of simple, however open, problems can be drawn up. Throughout the chapter, the equation that plays a crucial role is the equation of the conservation of mass. Owing to an argument based on the theory of renormalized solutions, a global-in-time existence result of weak solution is proved. The long-time behaviour of sufficiently regular solutions is also investigated.Less
This chapter deals with the theoretical aspects of multifluid magnetohydrodynamics problems. In addition to the coupling between hydrodynamics and electromagnetics examined in Chapter 2, the high nonlinearity that now needs to be addressed is geometrical in nature. It results from the presence of one (or many) free interface(s) separating the fluids, assumed non-miscible. The mathematical analysis is substantially more intricate, and a long list of simple, however open, problems can be drawn up. Throughout the chapter, the equation that plays a crucial role is the equation of the conservation of mass. Owing to an argument based on the theory of renormalized solutions, a global-in-time existence result of weak solution is proved. The long-time behaviour of sufficiently regular solutions is also investigated.
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/9780198566656.003.0001
- Subject:
- Mathematics, Mathematical Physics
This chapter presents how MHD equations can be derived from the general conservation equations for fluid mechanics coupled with the Maxwell equations modelling the electromagnetic phenomena. A ...
More
This chapter presents how MHD equations can be derived from the general conservation equations for fluid mechanics coupled with the Maxwell equations modelling the electromagnetic phenomena. A hierarchy of models is considered, from the most general one (a full time-dependent system consisting in the incompressible Navier-Stokes equations with a Lorentz body force calculated from the Maxwell equations and the Ohm's law) to the most simplified one. Depending on the physical context, one model or the other is appropriate. The most sophisticated model raises unsolved questions of existence and uniqueness, mainly related with the hyperbolic nature of the Maxwell equations, but some simpler models can be fully analysed.Less
This chapter presents how MHD equations can be derived from the general conservation equations for fluid mechanics coupled with the Maxwell equations modelling the electromagnetic phenomena. A hierarchy of models is considered, from the most general one (a full time-dependent system consisting in the incompressible Navier-Stokes equations with a Lorentz body force calculated from the Maxwell equations and the Ohm's law) to the most simplified one. Depending on the physical context, one model or the other is appropriate. The most sophisticated model raises unsolved questions of existence and uniqueness, mainly related with the hyperbolic nature of the Maxwell equations, but some simpler models can be fully analysed.
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/9780198566656.003.0005
- Subject:
- Mathematics, Mathematical Physics
This chapter deals with the discretization issues raised by multifluid magnetohydrodynamics problems. The additional difficulty compared to those in Chapter 3, is the presence of one (or many) free ...
More
This chapter deals with the discretization issues raised by multifluid magnetohydrodynamics problems. The additional difficulty compared to those in Chapter 3, is the presence of one (or many) free interface(s) separating the fluids. Numerically, one has also to resort to up-to-date techniques for the simulation of moving interfaces. In particular, the chapter presents a numerical method based on the Arbitrary Lagrangian Eulerian formulation, and lays emphasis on the stability of the time-advancing schemes. A short review of some other numerical methods to deal with moving interfaces is provided. Some numerical test cases illustrate the capabilities of the ALE method.Less
This chapter deals with the discretization issues raised by multifluid magnetohydrodynamics problems. The additional difficulty compared to those in Chapter 3, is the presence of one (or many) free interface(s) separating the fluids. Numerically, one has also to resort to up-to-date techniques for the simulation of moving interfaces. In particular, the chapter presents a numerical method based on the Arbitrary Lagrangian Eulerian formulation, and lays emphasis on the stability of the time-advancing schemes. A short review of some other numerical methods to deal with moving interfaces is provided. Some numerical test cases illustrate the capabilities of the ALE method.
