Flows and Chemical Reactions in an Electromagnetic Field

By Roger Prud'homme

This book - a sequel of previous publications ‘Flows and Chemical Reactions’, ‘Chemical Reactions Flows in Homogeneous Mixtures’ and ‘Chemical Reactions and Flows in Heterogeneous Mixtures’ - is devoted to flows with chemical reactions in the electromagnetic field.

The first part, entitled basic equations, consists of four chapters. The first chapter provides an overview of the equations of electromagnetism in Minkowski spacetime. This presentation is extended to balance equations, first in homogeneous media unpolarized in the second chapter and homogeneous fluid medium polarized in the third chapter. Chapter four is devoted to heterogeneous media in the presence of electromagnetic field. Balance equations at interfaces therein.

The second part of this volume is entitled applications. It also includes four chapters. Chapter five provides a study of the action of fields on fire. Chapter six deals with a typical application for the Peltier effect, chapter seven is devoted to metal-plasma interaction, especially in the Langmuir probe and finally Chapter Eight deals with the propulsion Hall effect.

Are given in appendix supplements the laws of balance with electromagnetic field and described the methodology for establishing one-dimensional equations for flow comprising active walls as is the case in some Hall effect thrusters.

• Language: English
• Publisher: Wiley
• Release date: Oct 30, 2014
• ISBN: 9781119054313

Book preview

Part 1

Introduction

In this first part of this volume, Chapters 1-3 recap the fundamental equations for homogeneous media, mainly in the case of mixtures of conductive fluids in which chemical reactions do take place.

In Chapter 1, we look at the general principles which govern the establishment of the equations of electromagnetism in the case of a simple medium. These equations are expressed in the Minkowski space and then transferred into the usual three-dimensional space.¹ The quantities used in the four-dimensional space are the tensors of the electromagnetic field and the current 4-vector, the momentum-energy tensor. These quantities will also be presented in Chapter 4, where we shall establish the interfacial equations. First, we shall consider non-polarized media, followed by polarized media.

Reactive mixtures involved additional quantities such as the velocity 4-vectors associated with the chemical species and the force 4-vectors. The balance equations for conductive reactive fluid mixtures are first established in Chapter 2 in the absence of polarization.

Chapter 3 is dedicated to the case of conductive reactive fluid mixtures in the presence of electrical and magnetic polarization.

Additional information about the homogeneous balances with electromagnetic fields is presented in the Appendix. The primary objective is to establish the constitutive relations of these conductive homogeneous media, which we can only do if we specify the type of medium in which we are interested: metal, and then homogeneous plasma. We will need these constitutive equations in Chapter 4, when we look at interfaces, because on both sides of these interfaces, we have homogeneous media.

The balance equations of the electromagnetic field give us the two groups of Maxwell equations. The mass balances are established for the species and the mixture; that of the energy-momentum leads us to the equations of conservation of momentum and energy.

The case of non-polarized media is studied first followed by that of polarized media for which the constitutive equations are deduced from linearized TIP.² Attention is drawn to the ambiguity of the definition of certain quantities in a polarized medium. This difficulty of definition arises particularly when we wish to separate the true electromagnetic effects from the mass effects.³

¹ This way of working is not the only way. It is perfectly possible to study electromagnetism without operating in timespace in the domain of non-relativistic velocities (on this subject, see the remark made by Groot and Mazur [GRO 63, p. 376]; also see [CAB 70, ERI 90]). The method used here to establish the basic equations is, however, fairly conventional (for instance, see [LAN 69, LAN 82, GRO 69b, SAN 68]). In addition, it will enable us to seamlessly introduce interfacial heterogeneities in Chapter 3.

² TIP: thermodynamics of irreversible processes.

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1

Relativistic Considerations

To begin with, here, we shall present the basic principles and the expressions of the classic quantities, such as the proper time and the universal velocity in the Minkowski timespace. The law of dynamics of the material point is then stated.

The expressions involved in continua, such as the electromagnetic field tensors and that of the electromagnetic momentum-energy, are presented in the case of media with a single component. The Maxwell equations are written, as are the balances of the electrical charge and the electrical momentum-energy in a polarized or non-polarized medium.

1.1. Recap of electromagnetics and mechanics in special relativity

We first recap the equations of electromagnetism, considering them to be deduced from the balance of various tensorial quantities in timespace [EIN 05, LAN 82]. Thus, the formulation is relativistic, but we believe this simplifies the reasoning process. The drawback is that the conventional balance equations in Aerothermochemistry are not relativistic. Hence, at first glance, this presentation seems non-homogeneous. In reality, though, it is not so at all: here, the homogeneity stems from a unique presentation of the balance equation, relativistic or otherwise [GRO 69a, GRO 69b]. However, we shall limit ourselves to the relativity restricted to Galilean systems.¹

Let us recap some of the basic principles of our developments.

PRINCIPLE OF RELATIVITY.— All of the laws of nature are identical in all Galilean frames of reference; it follows that the equation of a law retains its form in time and space when we change the inertial frame of reference. The rate of propagation of the interactions is the same in all inertial frames of reference.

GALILEO’s PRINCIPLE OF RELATIVITY.— The rate of propagation of the interactions is infinite.

EINSTEIN’s PRINCIPLE OF RELATIVITY.— The rate of propagation of the interactions is constant and equal to the celerity of light c.

This principle leads us to work in timespace.²

1.1.1. Minkowski timespace

A point M of spacetime is represented by a complex vector 4M, with the associated column matrix:

where c is the celerity of light in a vacuum in the absence of a field. Here, the coordinates are indicated in a Galilean frame of