Mathematical Methods for the

This comprehensive text focuses on mathematical and numerical techniques for the simulation of magnetohydrodynamic phenomena, with an emphasis laid on the magnetohydrodynamics of liquid metals, and on a prototypical industrial application. Aimed at research mathematicians, engineers, and physicists, as well as those working in industry, and starting from a good understanding of the physics at play, the approach is a highly mathematical one, based on the rigorous analysis of the equations at hand, and a solid numerical analysis to found the simulations. 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, whenever possible, on real industrial cases.

1 The magnetohydrodynamics equations

1

(20)

1.1 The general fluid equations

1

(7)

1.1.1 The conservation equations

1

(5)

1.1.2 Boundary and initial conditions

6

(1)

1.1.3 Steady-state equations

7

(1)

1.2 The electromagnetic description

8

(3)

1.3 The MHD coupling

11

(3)

1.3.1 The general MHD system

11

(2)

1.3.2 A commonly used simplified MHD coupling

13

(1)

1.3.3 The density-dependent case

14

(1)

1.4 Other MHD models

14

(1)

1.5 The MHD system considered in the sequel

15

(3)

1.6 Non-dimensionalized equations

18

(3)

2 Mathematical analysis of one-fluid problems

21

(62)

2.1 Mathematical results on the incompressible homogeneous Navier–Stokes equations

24

(32)

2.1.1 Some basics

25

(14)

2.1.2 The illustrative example of the two-dimensional case

39

(10)

2.1.3 The three-dimensional hydrodynamic case

49

(5)

2.1.4 Related issues

54

(2)

2.2 Mathematical results on the one-fluid MHD equations

56

(27)

2.2.1 A brief overview of the literature

57

(1)

2.2.2 Mathematical analysis

58

(5)

2.2.3 Back to the hyperbolic system

63

(1)

2.2.4 Stationary problems

64

(10)

2.2.5 A hybrid problem

74

(6)

2.2.6 Other MHD models and formulations

80

(3)

3 Numerical approximation of one-fluid problems

83

(62)

3.1 A general framework for problems with constraints

83

(14)

3.1.1 A model problem: the Stokes equations

84

(1)

3.1.2 Abstract framework for a linear problem

85

(3)

3.1.3 Application to the Stokes problem

88

(1)

3.1.4 The inf–sup condition

88

(3)

3.1.5 The mixed Galerkin method

91

(1)

3.1.6 Algebraic aspects

92

(1)

3.1.7 Mixed finite element for the Stokes problem

93

(2)

3.1.8 Extension to nonlinear problems

95

(2)

3.2 A glance at stabilized finite elements

97

(3)

3.3 Mixed formulations of the stationary MHD equations

100

(15)

3.3.1 A formulation for convex polyhedra and regular domains

101

(7)

3.3.2 A formulation for non-convex polyhedra

108

(7)

3.4 Mixed finite elements for MHD

115

(4)

3.4.1 Mixed finite elements on convex polyhedra and regular domains

116

(1)

3.4.2 Mixed finite elements on non-convex polyhedra

117

(2)

3.5 Stabilized finite elements for MHD

119

(8)

3.6 Solution strategy and algebraic aspects

127

(5)

3.6.1 Fully coupled iterations for stationary problems

127

(2)

3.6.2 Decoupled iterations for stationary problems

129

(2)

3.6.3 Fully coupled iterations for transient problems

131

(1)

3.6.4 MHD versus Navier–Stokes solvers

131

(1)

3.7 Examples of test cases and simulations

132

(8)

3.7.1 Hartmann flows

133

(3)

3.7.2 A fluid carrying current in the presence of a magnetic field

136

(1)

3.7.3 Convergence of nonlinear algorithms

137

(3)

3.8 About the boundary conditions

140

(5)

3.8.1 First set of boundary conditions

142

(1)

3.8.2 Second set of boundary conditions

143

(1)

3.8.3 Practical implementation of the boundary conditions

144

(1)

4 Mathematical analysis of two-fluid problems

145

(40)

4.1 The difficulties of the non-homogeneous case

146

(8)

4.1.1 A formal mathematical argument

147

(1)

4.1.2 The major ingredient

148

(5)

4.1.3 Short overview of the state of the art for the hydrodynamic case

153

(1)

4.2 Weak solutions of the multifluid MHD system

154

(14)

4.2.1 Mathematical setting of the equations

154

(2)

4.2.2 Existence of a weak solution

156

(12)

4.3 On the long-time behavior

168

(17)

4.3.1 The nonlinear hydrodynamics case

169

(10)

4.3.2 A detour by linearized models

179

(4)

4.3.3 The MHD case

183

(2)

5 Numerical simulation of two-fluid problems

185

(48)

5.1 Numerical approximations in the ALE formulation

186

(34)

5.1.1 Weak ALE formulation

187

(11)

5.1.2 Time and space discretization

198

(6)

5.1.3 Geometric conservation law, stability and conservation properties

204

(11)

5.1.4 Surface tension effects

215

(5)

5.2 Other approaches

220

(4)

5.2.1 Fixed-mesh methods

221

(2)

5.2.2 Moving-mesh methods

223

(1)

5.3 Example of test cases and simulations

224

(9)

5.3.1 A benchmark problem with a free surface

224

(1)

5.3.2 On the discrete mass conservation

225

(2)

5.3.3 An MHD experiment with a free surface and a free interface

227

(6)

6 MHD models for one industrial application

233

(54)

6.1 Presentation of aluminum electrolysis

233

(10)

6.1.1 The electrolysis process

233

(3)

6.1.2 Questions of stability and efficiency of the cell

236

(2)

6.1.3 The magnetohydrodynamic modeling of the cell

238

(2)

6.1.4 Qualitative interpretation of MHD instability and the Sele criterion

240

(3)

6.2 Linearized approaches

243

(29)

6.2.1 Basic assumptions

244

(4)

6.2.2 A prototypical derivation of a linearized system

248

(8)

6.2.3 Analysis of a single Fourier mode and dispersion relations

256

(3)

6.2.4 Taking into account the boundaries: analysis of coupling of standing plane waves

259

(8)

6.2.5 Numerical computations of the eigenvalues around a precomputed stationary state

267

(5)

6.3 A nonlinear approach

272

(12)

6.3.1 Generalities about linear versus nonlinear approaches

272

(2)

6.3.2 Some experiments on realistic cells

274

(4)

6.3.3 Metal pad roll instabilities

278

(4)

6.3.4 Spectral analysis

282

(2)

6.4 Other nonlinear approaches and conclusions

284

(3)

References

287

(16)

Index

303

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Mathematical Methods for the Magnetohydrodynamics of Liquid Metals

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