TRAPPED MODE FREQUENCIES EMBEDDED IN THE CONTINUOUS SPECTRUM

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Abstract
Experiments have been performed by Gross (1) in which a layer of insulating oil is confined between horizontal conducting planes, spaced at a distance d, and is heated from above and cooled from below. When a vertical electric field E of sufficient strength is applied across the layer, a tessellated pattern of motions is observed which, despite the fact that buoyancy is here stabilizing, is strikingly similar to that characteristic of Bénard convection. These motions cannot be explained by the “free surface” and “interfacial” effects investigated by Malkus and Veronis (2) since the layer is homogeneous in composition and is in contact with rigid boundaries at its surfaces. Gross suggested that they might be set up via the variation of dielectric constant of the fluid with temperature. This mechanism is described below and its governing equations are derived. An approximation analogous to the Boussinesq approximation of Bénard convection is developed. It is shown that the behaviour of the system is decided by the values of two non-dimensional parameters; P =ν/κ, the Prandtl number (where ν is kinematic viscosity and κ is thermal diffusivity), and L, a parameter analogous to the Rayleigh number R of Bénard convection and defined by where V is the potential difference between the planes, ξ the difference in ε in the fluid in contact with them, and ρ is the density. It is shown that if L exceeds a critical value, Lo, of 2129, independent of P, the layer is indeed convectionally unstable to motions whose horizontal semi-wavelength is 0.974 times the depth of the layer. The values of L attained by Gross in his experiments appear, however, to be only about 10−7Lo. An alternative possibility, that the convection is due to free charge conducted into the layer, is examined briefly, and is also found wanting.

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