Instability and Convection in Cylindrical Geometry

Abstract

In chapter 15 we mentioned that the instability problem arises in the three electrode geometries of high symmetry when the injection is homogeneous on the injector. Chapter 15 was devoted to instabilities in the case of plane parallel electrodes. In this section we consider the two other configurations of coaxial cylinders and of concentric spheres. Given a constant value q₀ of the injected space charge at the injecting electrode, the motionless state is a solution and the instability problem is to determine under what conditions this solution destabilises. Qualitatively the instability mechanism is the same than in the case of plane parallel electrodes. Due to coulombic repulsion, the charge density decreases when going from injector to collector along a radial line and, therefore, there is a positive coupling between velocity disturbances and the subsequent charge density perturbations (the argument of §15–1 is of general nature and is not specific to the case of parallel electrodes). Again the instability parameter is the nondimensional number T and the criterion depends on the injection parameter C = q₀d²/εV.