On stability of particulate flow in a pipe of non-circular cross section

Abstract

Linear stability of particulate flow of a viscous fluid with dispersed solid particles in a pipe of arbitrarily shaped uniform cross section is formulated using a novel two-fluid model. It is shown that the mathematical equations for linear stability of the particulate flow in a pipe of arbitrarily shaped uniform cross section are identical to that for the pipe flow of a clear fluid (without particles) with the same cross section provided that the original (real-form) mean velocity for the clear fluid is replaced by a complex-form effective mean velocity of Saffman type derived in the present work. In particular, in the two limiting cases of particles of extremely small or large Stokes number, the complex-form effective mean velocity reduces to a real-form proportional to the original mean velocity, and simple formulas are given for the critical Reynolds number ratio of the particulate fluid to the clear fluid under otherwise identical conditions. As examples, the derived formulas are applied to discuss qualitatively the critical Reynolds number for linear stability of particulate flow in a pipe of elliptical or rectangular cross section.