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On the Gunn effect and other physical examples of perturbed conservation equations

Published online by Cambridge University Press:  29 March 2006

J. D. Murray
J. D. Murray
Affiliation:
Courant Institute, New York University Present address: Mathematical Institute, University of Oxford.
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Abstract

Many situations of practical importance both in fluid mechanics and elsewhere are governed by perturbed forms of conservation laws. Generally the perturbations are in the nature of positive dissipation terms in the sense that any initial disturbance from a uniform state ultimately decays to that state. Diverse examples of these are discussed briefly.

A situation in which the perturbation results naturally in a negative dissipation term, in the sense that initial disturbances grow, although not necessarily indefinitely, arises in what has been accepted for a model for the Gunn (1963) effect and other so-called bulk negative resistance effects in semiconductors. The Gunn effect, which is of immense importance in electron-device technology (comparable with transistors), is the appearance of coherent microwave current oscillations in the crystals of a suitable semiconductor, in particular Gallium Arsenide, when they are subjected to a large electric field generally of the order of several kilovolts per centimetre. It now seems to be accepted that the effect is a consequence of the negative resistance (that is the electron drift speed decreases with increasing electric field) properties of the semiconductor crystal.

A typical model with negative resistance properties is described in detail, the resulting perturbed (both singularly and otherwise) non-linear conservation equations ((2.19) and (2.21)) are studied for practical situations of interest and the physical implications discussed in the light of experimental facts. Particular care is given to the shocks or discontinuities that must appear in the solutions when the diffusion is zero. As a result of these comparisons with experiment a simpler model is suggested which should suffice for a large number of practical situations and various quantitative features of this model are given.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 44 , Issue 2 , 11 November 1970 , pp. 315 - 346
Copyright
© 1970 Cambridge University Press

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