The effects of inertia on the stability of the convective flow in inclined particle settlers

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Abstract
A linear stability analysis of the base‐state convective flows, which occur in inclined particle settlers and which were studied in a companion paper by Shaqfeh and Acrivos [Phys. Fluids 2 9, 3935 (1986)], is described. The analysis considers the spatial growth of small two‐dimensional disturbances over the entire range of the parameters θ (the angle of inclination of the vessel), ω̂ (the wave frequency), R̂ (a stability Reynolds number), and ξ1/6 (the parameter governing inertial effects in the base flow). Numerical solutions to the relevant Orr–Sommerfeld equations for the wave growth rates and other important quantities are presented over this entire range. It is shown that, for very large values of R̂, these solutions approach asymptotically those of a certain set of Rayleigh equations which are also derived and solved. The results demonstrate that the base flows become most unstable when ξ1/6≊O(1), primarily owing to the development of an inviscid instability. This instability is shown to have a number of interesting characteristics including the fact that (a) it is insensitive to the inclination of the vessel, and (b) it tends to destabilize higher frequencies. In addition, it is demonstrated that, although it is very unstable for ξ1/6≊O(1), the flow restabilizes as ξ1/6→∞, as predicted by Prasad [J. Fluid. Mech. 1 5 0, 417 (1985)]. The analysis, therefore, bridges the gap between previous asymptotic results and, at the same time, resolves the existing paradox between the experimental observations of Schaflinger [Int. J. Multiphase Flow 1 1, 189 (1985)] and theoretical predictions. Moreover, it provides a theoretical basis for predicting the vigorous instabilities which are witnessed in practice.