JOHANNESKEPLER UNIVERSITÄTLINZ Netzwerk für Forschung, Lehre und Praxis Introduction to Magnetohydrodynamics Bakkalaureatsarbeit zur Erlangung des akademischen Grades Bakkalaureus der Technischen Wissenschaften in der Studienrichtung Technische Mathematik Angefertigt am Institut für Numerische Mathematik Betreuung: DI Peter Gangl Eingereicht von: Andreas Schafelner Linz, März 2016 Johannes Kepler Universität A-4040 Linz · Altenbergerstraße 69 · Internet: http://www.jku.at · DVR 0093696 Abstract Magnetohydrodynamics denotes the study of the dynamics of electrically conducting fluids. It establishes a coupling between the Navier-Stokes equations for fluid dynamics and Maxwell’s equations for electromagnetism. The main concept behind Magnetohy- drodynamics is that magnetic fields can induce currents in a moving conductive fluid, which in turn create forces on the fluid and influence the magnetic field itself. This Bachelor thesis is concerned with the mathematical modelling of Magnetohydro- dynamics. After introducing and deriving Maxwell’s Equations and the Navier-Stokes equations, their coupling is described and the governing equations for Magnetohydro- dynamics are obtained. Eventually some applications of Magnetohydrodynamics are described, i.e., in industrial processes, geophysics and astrophysics. i Contents 1 Introduction 1 1.1 What is Magnetohydrodynamics? . .1 1.2 Notation . .1 2 Electromagnetism 4 2.1 Introduction . .4 2.1.1 Electromagnetic quantities . .4 2.2 Maxwell’s Equations . .5 2.2.1 The Ampère-Maxwell law . .5 2.2.2 The magnetic Gauss law . .5 2.2.3 Faraday’s law . .6 2.2.4 Gauss’s law for electric fields . .6 2.2.5 Constitutive equations and material laws . .7 2.2.6 The Lorentz force . .8 2.2.7 The conservation of charge . .8 2.2.8 Summary of Maxwell’s Equations . .9 2.3 Simplifications for MHD . .9 2.3.1 Maxwell’s Equations, constitutive laws and the Laplace force . .9 2.3.2 The induction equation . 12 3 Fluid dynamics 14 3.1 Introduction . 14 3.2 Representation of a flow . 14 3.2.1 Lagrangian representation . 14 3.2.2 Eulerian representation . 15 3.3 The transport theorem . 16 3.4 The continutiy equation . 17 3.5 Equations of motion . 18 3.6 The Navier-Stokes equations . 19 3.6.1 Non-dimensionalisation and the Reynolds number Re ...... 21 4 Magnetohydrodynamics (MHD) 22 4.1 The governing equations . 22 4.2 Non-dimensionalisation . 23 ii CONTENTS iii 4.2.1 Summary . 24 4.3 Diffusion and Convection . 24 4.3.1 Diffusion of the magnetic field . 25 4.3.2 Convection of the magnetic field . 28 5 Applications of MHD 30 5.1 Simplifications for low Rem ........................ 30 5.2 MHD generators and pumps . 31 5.2.1 Theory of Hartmann layers . 31 5.2.2 Applications of Hartmann layers . 34 6 Conclusion 36 Chapter 1 Introduction 1.1 What is Magnetohydrodynamics? In nature as in industrial processes, we can observe magnetic fields influencing the behaviour of fluids and flows. In the metallurgical industry, magnetic fields are used to stir, pump, levitate and heat liquid metals. The earth’s magnetic field, protecting the surface from deadly radiation, is generated by the motion of the earth’s liquid core. Sunspots and solar flares are generated by the solar magnetic field and the galactic magnetic field which influences the formation of stars from interstellar gas clouds. We use the word Magnetohydrodynamics (MHD) for all of these phenomena, where the magnetic field B and the velocity field u are coupled, given there is an electrically conducting and non-magnetic fluid, e.g. liquid metals, hot ionised gases (plasmas) or strong electrolytes. The magnetic field can induce currents into such a moving fluid and this creates forces acting on the fluid and altering the magnetic field itself. 1.2 Notation In this short section of the thesis all facts about notation are presented. First of all, if not specified otherwise, a normal letter x represents a scalar value, whereas a bold T letter x = (x1, x2, x3) indicates a three dimensional vector. This holds for variables, constants and functions. Functions are either of static or dynamic nature. Static functions depend only on space coordinates, e.g., T (x) gives the temperature in the point x, whereas dynamic functions additionally have a time-dependent component t, e.g., T (x, t) gives the temperature in the point x at time t. The symbol · represents the euclidean inner product and × stands for the cross product. With | . | we denote either the absolute value of a scalar or the euclidean norm for vectors. Throughout the thesis the following differential operators will occur: 1 CHAPTER 1. INTRODUCTION 2 Definition 1.1. Let u : R3 × [0,T ] → R3 and f : R3 × [0,T ] → R be sufficiently smooth. Then the gradient ∇, the divergence ∇·, the curl ∇× and the Laplacian ∆ are defined as ∂f ∂x1 ∂u2 − ∂u3 ∂f ∂x3 ∂x2 ∇f = ∂u ∂u ∂x2 ∇ × u = − 3 + 1 ∂f ∂x1 ∂x3 ∂u ∂u ∂x3 2 − 1 2 2 2 ∂x1 ∂x2 ∂ f ∂ f ∂ f ∆f = 2 + 2 + 2 ∆u1 ∂x1 ∂x2 ∂x3 ∆u = ∆u2 ∂u1 ∂u2 ∂u3 ∇ · u = + + ∆u3 ∂x1 ∂x2 ∂x3 Remark 1.2. As we can see in Definition 1.1, the differential operators are only using the partial derivatives of the space coordinates, but not the time coordinate. Usually the operators are defined for every variable of a function. Apart from differential operators, we will encounter different types of integrals, which are also summarized in the following definition. Definition 1.3. Let u : R3 × [0,T ] → R3 be a sufficiently smooth vector field. We define the line integral as u · dl := u(φ(τ)) · φ0(τ) dτ, (1.1) ˆC ˆI with φ : I → R3 a parametrisation of the curve C ⊂ R3 and I = [a, b] ⊂ R. Furthermore we define the surface integral ∂ψ ∂ψ ! u · ~n dS := u(ψ(σ, τ)) · × d(σ, τ), (1.2) ˆS ˆK ∂σ ∂τ where ψ : K → R3 is a parametrisation of the surface S ⊂ R3 and K ⊂ R2 is a reference domain. The volume integral is defined as u dx = u(x1, x2, x3, t) d(x1, x2, x3), (1.3) ˆV ˆV where V ⊂ R3 is a volume. With these definitions, we can formulate two important theorems, which relate the different types of integrals to each other. Theorem 1.4 (Stokes’ Theorem). Let S be a surface in R3, parametrised by ϕ : M → R3, where M is a superset of a normal range K in R2, with ϕ(K) = S. Let ϕ ∈ C2(M, R3) and ∂S be the boundary curve of S. Let u ∈ C1(ϕ(M), R3). Then it holds that ∇ × u · ~n dS = u · dl. (1.4) ˆS ˆ∂S CHAPTER 1. INTRODUCTION 3 Proof. See Thm. 8.50 in [2]. Theorem 1.5 (Divergence Theorem). Let V be a sufficiently smooth subset of R3 and let ∂V its boundary. Let M ⊇ V and u ∈ C1(M, R3). Then it holds that u · ~n dS = ∇ · u dx. (1.5) ˆ∂V ˆV Proof. See Thm. 8.58 and Remark 8.59 in [2] Chapter 2 Electromagnetism - Maxwell’s Equations 2.1 Introduction 2.1.1 Electromagnetic quantities Throughout this thesis, we will deal with a set of physical quantities to describe elec- tromagnetic processes. These are summarized in Table 2.1. However, some quantities need further explanation. The electric field E = Es + Ei consists of two parts, the electrostatic field Es and the induced electric field Ei which have a similar effect but a different structure. The electric current density J = J c + J i consists of the conduct current density J c and the induced current density J i. The magnetisation M quanti- fies the strength of a permanent magnet and the electric polarisation P is its electric counterpart. Variable Physical quantity Unit E electric field intensity [V/m] D electric flux density [As/m2] H magnetic field density [A/m] B magnetic flux density [V s/m2] J electric current density [A/m2] 2 ρc electric charge density [As/m ] M magnetisation [V s/m2] P electric polarisation [As/m2] Table 2.1: Overview of electromagnetic quantities 4 CHAPTER 2. ELECTROMAGNETISM 5 2.2 Maxwell’s Equations We continue now by deriving Maxwell’s Equations. We start with the integral form of each equation and will then extract the differential form. Both forms are commonly used, but for the following chapters, the latter will be used. The content of this section is based on [3] and [4]. 2.2.1 The Ampère-Maxwell law Ampère’s law, or more correctly Ampère-Maxwell’s law describes the relation of a magnetic field and an electric current. In terms of mathematics, we can write the relation as ∂D H · dl = (J + ) · ~n dS, (2.1) ˛C ˆS ∂t where C is a closed curve bounding a surface S. Now applying Stokes’ theorem to the left hand side of (2.1) gives us ∂D ∇ × H · ~n dS = (J + ) · ~n dS, ˆS ˆS ∂t and as this holds for any surface S, we obtain ∂D ∇ × H = J + . (2.2) ∂t We call (2.2) the Ampère-Maxwell law in differential form. In the static case, (2.2) reduces to ∇ × H = J, which is commonly known as Ampère’s law. 2.2.2 The magnetic Gauss law Gauss’s law for magnetic fields gives information about the magnetic flux. It states that the total magnetic flux through any closed surface is zero [3]. We can write this in terms of mathematics as B · ~n dS = 0, (2.3) ˛S with S a closed surface surrounding a volume V. Using the divergence theorem we obtain 0 = B · ~n dS = ∇ · B dx, ˛S ˆV which holds for arbitrary volumes V with closed surface S.