19. Magnetohydrodynamics: An Introduction

Magnetohydrodynamics, or shortly called MHD, studies the mutual interaction of fluid flow and magnetic fields. The fluids in this case, must be electrically conducting and non-magnetic; therefore, we limit ourselves to liquid metals, hot ionized gases (plasmas) and strong electrolytes. Magnetic fields frequently used in industry for instance to heat, pump, stir and levitate liquid metals. Magnetic field is naturally maintained by the fluid motion in the earth’s core and there is also galactic magnetic field which influences the formation of stars from interstellar clouds.

Theoretically, as a result of the laws of Faraday and Ampere, the mutual interaction of magnetic field B and a velocity field u arises and also due to the Lorentz force experienced by a current-carrying body. We are going to split the process conveniently into three parts and analyze the exact form of this interaction in detail in the following sections.

i.                    In accordance with Faraday’s law of induction, the relative movement of a conducting liquid and a magnetic field develops an electromotive force (e.m.f) of order |u x B|. Generally, electrical current will be induced with current density of order σ(u x B) with σ being the electrical conductivity.

ii.                  According to Ampere’s law, the induced current will give rise to a second induced magnetic field and this will add to the original magnetic field. The change is then such that the fluid appears to ‘drag’ the magnetic field lines along with it.

iii.                The combination of imposed plus induced magnetic field interacts with the induced current density, J, to give rise to a Lorentz force per unit volume: JxB.   

Please notice that in the last two effects (ii) and (iii), the relative movement of fluid and field tends to be reduced, i.e. in effect (ii): fluids can ‘drag’ magnetic field lines; on the other hand, the magnetic fields can pull on conducting fluids (effect (iii)).

These effects can be explained by means of conventional electrodynamics. Figure 19.1 shows a wire loop which is pulled through a magnetic field.

Figure 19.1 Conventional electrodynamics: A wire loop pulled through a magnetic field.

An e.m.f. of the order |u x B| is generated as the wire pulled to the right which drives a current as shown (effect (i)). The magnetic field accompanied the induced current perturbs the original magnetic field, and the net result is that the magnetic field lines seem to be dragged along by the wire (effect (ii)). The current gives then rise to Lorentz force, J x B acting on the wire in a direction opposite to that of the motion (effect (iii)). Shortly speaking, the wire drags the field lines while the magnetic field reacts back on the wire, tending to oppose the relative movement of the two.

As we shall see later in more detail, the range to which a velocity field influences an imposed magnetic field depends on the product of: the typical velocity of the motion, the fluid’s conductivity, and the characteristic length scale l of the motion. For instance, if the wire shown in Figure 19.1 is a poor conductor, or moves very slowly, then the induced current and the associated magnetic field will be weak. Therefore, if the fluid is non-conducting or the fluid’s velocity is very small, there will be no significant induced magnetic field. In contrast, if σ or u is relatively large, then the induced magnetic field may significantly alter the imposed field.

Consider a modest current density that spread over a large area which can produce a high magnetic field, in contrast, the same current density spread over a small area induces only a weak magnetic field. It is therefore the product of σul which determines the ratio of the induced field to the applied magnetic field. In case of ideal conductors, if σul → ∞ then the combined magnetic field behaves as if it were locked into the fluid. An example for this case is in the astrophysical MHD because of the vast characteristic length scale, but not due to the high conductivity of the plasma involved. In contrast, if σul → 0, then the imposed magnetic field remains relatively unperturbed. An example of this is in the case of liquid metal MHD, where the velocity u leaves B unperturbed. Nevertheless, it should be kept in mind that effect (iii) is still strong in liquid metal MHD, such that an imposed magnetic field can significantly alter the velocity field. In the case of liquid metals, the reasonable conductivity is in the order of ${{10}^{6}}{{\Omega }^{-1}}{{m}^{-1}}$ and the velocity involved in a typical laboratory of industrial process is in the order of $1{}^{m}/{}_{s}$ which is relatively small. Hence the induced current densities are generally rather modest, i.e. a few Amps per cm², and the induced magnetic field is usually found to be very small in comparison to the imposed field. However, the imposed magnetic field is often strong enough for the Lorentz force J x B to dominate the motion of the fluid.

To summarize, the ‘freezing together’ of the magnetic field and the medium is usually strong in astrophysical MHD and weak in metallurgical MHD. However, the influence of B and u can be important in all of these situations.