Stability of a vertical Couette flow in the presence of settling particles

Abstract

We consider the linear stability of a particle-laden vertical Couette flow between two flat plates. The flow is described in the framework of a two-fluid approach with a negligibly small volume fraction of the dispersed phase (the “dusty-gas model”). The carrier-fluid velocity profile in the main flow is modified by the presence of settling particles distributed non-uniformly between the plates. Two types of particle number density distributions are considered, namely, a single layer with a maximum at the flow center line and two symmetric layers located between the center line and the walls. Linearized governing equations are reduced to a modified Orr–Sommerfeld equation for the amplitude of a disturbance in the form of a normal mode. On the basis of a parametric study of the eigenvalues, it is shown that, in contrast to the case of two-phase Couette flow with zero particle velocity slip, the vertical disperse flow is unstable over a wide range of governing parameters, including small Reynolds numbers. In flows with both kinds of particle concentration profiles considered, there are two types of growing modes. A gravitational mode is triggered at low Reynolds numbers and typically small wavenumbers, while a shear mode with the wavelength of the order of the channel width and larger time-amplification rates is triggered at large Reynolds numbers. It is found that both modes are amplified with an increase in the particle mass loading or the ratio of the Reynolds number to squared Froude number, determining the carrier-fluid velocity profile. The results of the study can be used to control the stability of various technological processes involving particle-laden flows.