A modified Orr–Sommerfeld equation for axisymmetric stability of particulate pipe flow

A two-fluid model is used to study linear stability of circular Poiseuille–Couette flow of a particle-laden viscous fluid to axisymmetric disturbances. A modified circular Orr–Sommerfeld equation of Saffman type with a complex-form effective mean velocity is derived for non-uniform axisymmetric distribution of dispersed particles. In the two limiting cases of uniform distribution of particles of extremely small or large Stokes number, explicit formulas for the effective Reynolds number ratio of the particulate fluid to the clear fluid are given for finite particle volume fraction and finite particle-to-fluid density ratio, which imply that Poiseuille flow of a particulate fluid in a circular pipe with uniform distribution of particles of extremely small or large Stokes number is always linearly stable. For an annular circular pipe, on the other hand, the implications of the derived formulas to linear stability are discussed for particulate sliding Couette flow and pressure gradient-driven particulate Poiseuille flow with uniform distribution of particles of extremely small or large Stokes number. The derived results could be used to study axisymmetric stability of circular particulate pipe flows which are not well addressed in existing literature.