Time dependent motion of a conducting sheet of liquid in an electromagnetic field
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Abstract
The basic objective ot the dissertation was to derive and evolve the time dependent one dimensional equation of motion for an infinite sheet or viscous, incompressible, conducting liquid of finite thickness. The liquid is acted upon by a transverse spatially homogeneous magnetic field, by current from an external source, and by mechanical pressure gradients. Two simultaneous, linear, partial differential equations were derived which give the relationship between the fluid velocity v(x, t) and a "reduced" variable h'(x,t). This latter variable is proportional to the induced magnetic field h(x,t) but has the dimensions of a velocity.
To obtain as much generality as possible, it was assumed that the mechanical and electromagnetic driving terms could be expressed either in terms of Fourier series or Fourier integrals. The system equations were first solved for an arbitrary, single-frequency, complex harmonic driving function. The solution for driving functions expressible as a sum or integral of complex harmonic terms become simply the corresponding sum or integral of the solutions for the individual harmonic components.
Special attention was given to the two important cases in which the driving terms are either constant or sinusoidal. In the former case, it is shown how Hartmann's two steady state solutions depend on the induced magnetic field boundary condition. It is pointed out that Hartmann's solutions are really one-dimensional approximations of a two dimensional problem. In the latter case, the system equations are similar to those for coupled circuits and the solutions exhibit a phase lag for v and h with respect to the driving pressure gradients.
Both solutions are shown to vary in transcendental fashion with the value of the dimensionless parameter that has been labelled by the letter M in recent work by Shercliff and Murgatroyd. A dimensional analysis reveals that M^2 is the ratio between the electromagnetic and viscous forces per unit volume of the fluid, a fact that was overlooked in Lehnert's recent work.
The complete solutions of the system equations were found by making use of the Laplace transformation technique in combination with Fourier series expansions. The transient portion of the solutions involve two exponential terms. It is shown how these solutions may be simplified when they are applied to liquid metals under conditions ordinarily existing in laboratory work.
Several examples are worked out which illustrate the acceleration from root to a steady state condition or sheets of liquid mercury. It is found that the fluid velocity and induced magnetic field approach their steady state values in a smooth, monotonically increasing, exponential fashion, provided that a certain inequality exists between various parameters of the syetem. When this inequality is violated, as it may be, the approach to steady state becomes an oscillating, exponential approach.
A separate chapter is devoted to a derivation of the system equations for two-dimensional magnetohydrodynamic motion in rectangular channels. It is assumed that a charge distribution develops in the fluid. The resulting set of partial differential equations is non-linear.
Some experimental work concerning the movement of mercury electromagnetically is described. Results are reported which confirm the order of magnitude of liquid mercury motion under conditions similar to those assumed in the theoretical examples.
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Thesis (Ph.D.)--Boston University
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