An analysis is presented of the steady states of two-dimensional convection near threshold in a laterally finite container with aspect ratio 2L 9 1. It is shown that the allowed wavevectors which can occur in the bulk of the container are reduced from a band Jqj-[(R-Ro)/R,li in the laterally infinite system to a band 1q1-(R-R,)/R, in a system with sidewalls (R is the Rayleigh number and R, its critical value in the infinite system). The analysis involves an expansion of the hydrodynamic equations in the small parameter [(R-RO)/RO]i, and leads to amplitude equations with boundary conditions, which generalize to higher order those previously obtained by Newel1 & Whitehead and Segel. The precise values of the allowed wavevectors depend on the Prandtl number of the fluid and the thermal properties of the sidewalls. For certain values of these parameters all the allowed wavevectors are less than the critical value q,. The applicability of the results to convection in a rectangular container is briefly discussed.
Models the positive column in strongly electronegative gases, where recombination is the dominant mechanism of charge loss in the volume. The pressure range is taken to be such that mobility governs particle motion. It is related to earlier work in gases like oxygen, where detachment is the dominant loss mechanism and there are many similarities with the case where the attachment/ionization rate ratio P was <1, treated by Daniels, Franklin and Snell (1990). It is shown that P is necessarily less than unity and that it is not physically possible to have a discharge where electron-ion recombination is the only loss mechanism. The structure found is that of a central positive ion-negative ion core surrounded by a conventional electron-positive ion plasma with a sharp transition region of fractional thickness given by the parameter l1/2 where l is a normalized ratio of ion recombination rate beta i and ionization rate nu i given by beta ine0/ nu i where ne0 is the central electron density. Expressions are found for the ion plasma dimensions and the eigenvalue lambda , which relates the ionization rate, discharge radius, ion mobility and electron temperature, and comparison is made with the work of Volynets et al. (1993). The physical characteristics of such discharges are explained by the treatment given for the first time since the problem was formulated in 1949.
It is shown that, in the flow of a viscous wall layer past a relatively steep obstacle at the wall, the Goldstein (1948) singularity generated in the classical boundary-layer approach to separation is removable in a physically sensible fashion. The removal is effected by means of a sequence of local double structures, the last of which arises just beyond separation owing to the occurrence of a further singularity which is also removable and describes the necessary complete breakaway of the viscous layer from the wall. The novel forms of the local pressure–displacement relations are the key elements allowing the solution to retain physical reality throughout. Beyond the breakaway the reattachment process takes place only at a relatively large distance downstream, before the motion returns to its original uniform shear form. The present flow configuration, the first we know of where Goldstein's singularity proves to be removable, has important applications in both internal and external flows at high Reynolds numbers and these are also discussed.
The theory of the positive column at moderate pressures in electron-attaching gases is examined for small attachment rates relative to the ionisation rate. It is shown that the negative (and positive) ions are confined to a channel in the centre of the discharge, by both computational and analytical approximation methods. Approximate expressions for the channel width and the eigenvalue which determines the electron temperature are derived.
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