The purpose of this chapter is to focus on a variety of exact results applying to the perfectly elastic incompressible Varga materials. For these materials it is shown that the governing equations for plane strain, plane stress and axially symmetric deformations, admit certain first order integrals, which together with the constraint of incompressibility, give rise to various second order problems. These second order problems are much easier to solve than the full fourth order systems, and indeed some of these lower order problems admit elegant general solutions. Accordingly, the Varga elastic materials give rise to numerous exact deformations, which include the controllable deformations known to apply to all perfectly elastic incompressible materials. However, in addition to the standard deformations, there are many exact solutions for which the corresponding physical problem is not immediately apparent. Indeed, many of the simple exact solutions display unusual and unexpected behaviour, which possibly reflects non-physical behaviour of the Varga elastic materials for extremely large strains. Alternatively, these exact results may well mirror the full consequences of nonlinear theory. This chapter summarizes a number of recent developments.

Introduction

Natural and synthetic rubbers can accurately be modelled as homogeneous, isotropic, incompressible and hyperelastic materials, and which are sometimes referred to as perfectly elastic materials. The governing partial differential equations tend to be highly nonlinear and as a consequence the determination of exact analytical solutions is not a trivial matter.

Liquid metals freeze in much the same way as water. First, snowflake-like crystals form, and as these multiply and grow a solid emerges. However, this solid can be far from homogeneous. Just as a chef preparing icecream has to beat and stir the partially solidified cream to break up the crystals and release any trapped gas, so many steelmakers have to stir partially solidified ingots to ensure a fine-grained, homogeneous product. The preferred method of stirring is electromagnetic, and has been dubbed the ‘electromagnetic teaspoon’. We shall describe this process shortly.

First, however, it is necessary to say a little about commercial casting processes.

Casting, Stirring and Metallurgy

(Leonardo da Vinci on the extraction and casting of metals)

Man has been casting metals for quite some time. Iron blades, perhaps 5000 years old, have been found in Egyptian pyramids, and by 1000 BC we find Homer mentioning the working and hardening of steel blades. Until relatively recently, all metal was cast by a batch process involving pouring the melt into closed moulds. However, today the bulk of aluminium and steel is cast in a continuous fashion, as indicated in Figure 8.1.

In his 1964 lectures on physics, R P Feynman noted that:

In this chapter we build the dam and write down the equations. Later, particularly in Chapter 7 where we discuss turbulence, we shall see how the dam bursts open.

Fluid Flow in the Absence of Lorentz Forces

In the first seven sections of this chapter we leave aside MHD and focus on fluid mechanics in the absence of the Lorentz force. We return to MHD in Section 3.8. Readers who have studied fluid mechanics before may be familiar with much of the material in Sections 3.1 to 3.7, and may wish to proceed directly to Section 3.8. The first seven sections provide a self-contained introduction to the subject, with particular emphasis on vortex dynamics, which is so important in the study of MHD.

Elementary Concepts

Different categories of fluid flow

The beginner in fluid mechanics is often bewildered by the many diverse categories of fluid flow which appear in the text books. There are entire books dedicated to such subjects as potential flow, boundary layers, turbulence, vortex dynamics and so on.

When Rm is high there is a strong influence of u on B, and so we obtain a two-way coupling between the velocity and magnetic fields. The tendency for B to be advected by u, which follows directly from Faraday's law of induction, results in a completely new phenomenon, the Alfvèn wave. It also underpins existing explanations for the origin of the earth's magnetic field and of the solar field. We discuss both of these topics below. First, however, it may be useful to comment on the organisation of this chapter.

The subject of high-Rm MHD is vast, and clearly we cannot begin to give a comprehensive coverage in only one chapter. There are many aspects to this subject, each of which could, and indeed has, filled text-books and monographs. Our aim here is merely to provide the beginner with a glimpse of some of the issues involved, offering a stepping-stone to more serious study. The subject naturally falls into three or four main categories. There is the ability of magnetic fields to support inertial waves, both Alfvèn waves and magnetostrophic waves.

Magnetohydrodynamics (MHD for short) is the study of the interaction between magnetic fields and moving, conducting fluids. In the following seven chapters we set out the fundamental laws of MHD. The discussion is restricted to incompressible flows, and we have given particular emphasis to the elucidation of physical principles rather than detailed mathematical solutions to particular problems.

We presuppose little or no background in fluid mechanics or electromagnetism, but rather develop these topics from first principles. Nor do we assume any knowledge of tensors, the use of which we restrict (more or less) to Chapter 7, in which an introduction to tensor notation is provided. We do, however, make extensive use of vector analysis and the reader is assumed to be fluent in vector calculus.

The subjects covered in Part A are:

1. qualitative overview of MHD

2. governing equations of electrodynamics

3. governing equations of fluid mechanics

4. kinematics of MHD: advection and diffusion of a magnetic field

5. at low magnetic Reynolds' number

6. at high magnetic Reynolds' number

7. turbulence at low and high magnetic Reynolds' numbers

One point is worth emphasising from the outset. The governing equations of MHD consist simply of Newton's laws of motion and the pre-Maxwell form of the laws of electrodynamics. The reader is likely to be familiar with elements of both sets of laws and many of the phenomena associated with them.

What is MHD?

Magnetic fields influence many natural and man-made flows. They are routinely used in industry to heat, pump, stir and levitate liquid metals. There is the terrestrial magnetic field which is maintained by fluid motion in the earth's core, the solar magnetic field which generates sunspots and solar flares, and the galactic magnetic field which is thought to influence the formation of stars from interstellar clouds. The study of these flows is called magnetohydrodynamics (MHD). Formally, MHD is concerned with the mutual interaction of fluid flow and magnetic fields. The fluids in question must be electrically conducting and non-magnetic, which limits us to liquid metals, hot ionised gases (plasmas) and strong electrolytes.

The mutual interaction of a magnetic field, B, and a velocity field, u, arises partially as a result of the laws of Faraday and Ampère, and partially because of the Lorentz force experienced by a current-carrying body. The exact form of this interaction is analysed in detail in the following chapters, but perhaps it is worth stating now, without any form of proof, the nature of this coupling.

The History of Electrometallurgy

There were two revolutions in the application of electricity to industrial metallurgy. The first, which occurred towards the end of the nineteenth century, was a direct consequence of Faraday's discoveries. The second took place around eighty years later. We start with Faraday.

The discovery of electromagnetic induction revolutionised almost all of 19th century industry, and none more so than the metallurgical industries. Until 1854, aluminium could be produced from alumina only in small batches by various chemical means. The arrival of the dynamo transformed everything, sweeping aside those inefficient, chemical processes. At last it was possible to produce aluminium continuously by electrolysis. Robert Bunsen (he of the ‘burner’ fame) was the first to experiment with this method in 1854. By the 1880s the technique had been refined into a process which is little changed today (Figure I.1).

In the steel industry, electric furnaces for melting and alloying iron began to appear around 1900. There were two types: arc-furnaces and induction furnaces (Figure 1.2).

Prefaces are rarely inspiring and, one suspects, seldom read. They generally consist of a dry, factual account of the content of the book, its intended readership and the names of those who assisted in its preparation. There are, of course, exceptions, of which Den Hartog's preface to a text on mechanics is amongst the wittiest. Musing whimsically on the futility of prefaces in general, and on the inevitable demise of those who, like Heaviside, use them to settle old scores, Den Hartog's preface contains barely a single relevant fact. Only in the final paragraph does he touch on more conventional matters with the observation that he has ‘placed no deliberate errors in the book, but he has lived long enough to be quite familiar with his own imperfections’.

We, for our part, shall stay with a more conventional format. This work is more of a text than a monograph. Part A (the larger part of the book) is intended to serve as an introductory text for (advanced) undergraduate and post-graduate students in physics, applied mathematics and engineering. Part B, on the other hand, is more of a research monograph and we hope that it will serve as a useful reference for professional researchers in industry and academia. We have at all times attempted to use the appropriate level of mathematics required to expose the underlying phenomena. Too much mathematics can, in our opinion, obscure the interesting physics and needlessly frighten the student.

When an electric current is made to pass through a liquid-metal pool it causes the metal to pinch in on itself. That is to say, like-signed currents attract one another, and so each part of the pool is attracted to every other part. When the current is perfectly uniform, the only effect is to pressurise the liquid. However, often the current is non-uniform; for example, it may spread radially outwards from a small electrode placed at the surface of the pool. In such cases the radial pinch force will also be non-uniform, being largest at places where the current density is highest (near the electrode). The (irrotational) pressure force, – ∇p, is then unable to balance the (rotational) Lorentz force. Motion results, with the fluid flowing inward in regions of high current density and returning through regions of small current.

Perhaps the first systematic experimental investigation of the ‘pinch effect’ in current-carrying melts was that of E F Northrup who, in 1907, injected current into pools of mercury held in a variety of different configurations. It should be noted, however, that industrial metallurgists have been routinely passing large currents through liquid metals since 1886, when Hall and Hèroult first developed the aluminium reduction cell and von Siemens designed the first electric-arc furnace. One of the many descendants of the electric-arc furnace is vacuum-arc remelting (VAR).