Formal Derivation and Stability Analysis of Boundary Layer Models in MHD

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Derivation of the dimensionless MHD equations

We first remind here the classical derivation of the dimensional MHD system, following for instance [24]. We will notably discuss the appropriate boundary conditions. We will then put the equations in dimensionless form, to lay the ground for the following boundary layer developments.
The aim of the MHD system is to describe the motion of a viscous incompressible conducting fluid. We denote by the three dimensional bounded domain occupied by the fluid. The governing equations for the fluid velocity u and pressure p are the classical Navier-Stokes equations, with constant density and kinematic viscosity coefficient : Btu 􀀀 pu rqu 􀀀 1 rp u j b in ; div u 0 in ; u|B 0: (2.1) With regards to the purely hydrodynamic case, the difference lies in the addition of the Lorentz force, at the right-hand side of the momentum equation. It is given by the cross product of the current density j and the magnetic field b. Hence, to complete the description, one needs to couple this Navier-Stokes dynamic to the one for the magnetic and electric fields b and e. Therefore, we write the Maxwell equations, in which we neglect the displacement currents: Btb 􀀀 curl e 0 in R3.

Weak solutions of the MHD system

The system (MHD) derived in the previous section is used to describe various magnetohydrodynamic flows, notably in the context of dynamo theory ([24]). Without further assumptions on the magnetic field in the insulator 1, it is mathematically and numerically challenging, as it couples the dynamics of the fluid in the bounded domain to the dynamics of the magnetic field in the whole space R3. This creates some difficulties in the well-posedness analysis, for which we could find no references. We will therefore address this issue before turning to the boundary layer analysis. We focus in this section on weak solutions of Leray type. Further remarks on smooth solutions will be made in the next section. For simplicity, we take all parameters Re, Rm and S equal to unity.

Additional remarks on the well-posedness of (MHD)

We devoted most of this chapter to the construction of Leray type solutions to the system (MHD). We tried to emphasize the issues and tools associated with the treatment of the different domains occupied by u and b, that are and R3. Such issues and tools were not met in the treatment of the other MHD type systems found in the literature. Let us stress that our analysis extends also without difficulty to the two-dimensional case, with the usual changes. Note that in the 2d setting, the non-linearity curl b b has to be replaced by pcurl bq bK pcurl bq b2 b1  , where the 2d curl is the scalar operator defined by curl b B1b2 B2b1. Similarly, the non-linearity curl pu bq has to be replaced by rKpu bKq.
It would be interesting to check if the usual results for Navier-Stokes subsist in this MHD setting: uniqueness of the Leray type solutions and global regularity in 2d, local existence of strong solutions in 3d. Nevertheless, we do not push further the analysis here: in the next sections, we will concentrate on a refined asymptotic analysis of the system, focusing on MHD boundary layer stability. Notably, in the next chapter, we will consider a simplified setting in which:
the external magnetic field is assumed to be constant: b e in c, where e is a unit vector.
the fluid domain is a half-space: R3.

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Well-posedness for the mixed Prandtl/Shercliff Equations

We will, in this paragraph, re-use the notations pu »; v »; b »; ! »q to distinguish the approximate system from the original one. We remember that all the calculations above were performed with these quantities. Existence: The existence of a solution to the system (4.5) is a standard consequence of the uniform estimates established on the approximate system (4.11). From such estimates and the Gronwall lemma, one can easily show that there exists a time T ¡ 0 and some M ¡ 0 such that for all t P r0; minpT; T »qs (where T » is the maximal time of existence of the approximate solution), one has.

Table of contents :

1 Introduction 
1.1 The Euler and Navier Stokes models
1.2 The high Reynolds number limit
1.3 The derivation of the Prandtl system
1.4 The regularity of the Prandtl solution
1.5 Well-posedness in a weighted Sobolev space for a monotone initial profile
1.6 Conclusion
1.7 Presentation of the results
2 Preliminaries on the MHD system 
2.1 Derivation of the dimensionless MHD equations
2.2 Weak solutions of the MHD system
2.2.1 Variational formulation and existence result
2.2.2 Sketch of proof
2.3 Additional remarks on the well-posedness of (MHD)
3 Formal Derivation and Stability Analysis of Boundary Layer Models in MHD 
3.1 Introduction
3.2 Derivation of MHD layers
3.2.1 Layers under a transverse magnetic field
3.2.2 Layers in a tangent magnetic field.
3.2.3 Summary of the formal derivation
3.3 Linear Stability
3.3.1 Mixed Prandtl/Hartmann regime
3.3.2 Mixed Prandtl/Shercliff regime
3.3.3 Fully nonlinear MHD layer
3.4 Conclusion
4 Analysis of the nonlinear models 
4.1 Mixed Prandtl/Hartmann regime
4.2 Mixed Prandtl/Shercliff regime
4.2.1 Regularized system
4.2.2 Uniform estimates
4.2.3 Well-posedness for the mixed Prandtl/Shercliff Equations
4.3 Fully nonlinear MHD layer
4.3.1 The linearised zero viscosity case
5 Numerical Study 
5.1 Mixed Prandtl/Shercliff system
5.2 Simulation of the stability mechanism and comparison

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