Title: Is the kinetic equation for turbulent gas-particle flows ill-posed?,
Abstract: This paper is about well-posedness and realizability of the kinetic equation
for gas-particle flows and its relationship to the Generalized Langevin Model
(GLM) PDF equation. Previous analyses claim that this kinetic equation is
ill-posed, that in particular it has the properties of a backward heat equation
and as a consequence, its solutions will in the course of time exhibit
finite-time singularities. We show that the analysis leading to this conclusion
is fundamentally incorrect because it ignores the coupling between the phase
space variables in the kinetic equation and the time and particle inertia
dependence of the phase space diffusion tensor. This contributes an extra $+ve$
diffusion that always outweighs the contribution from the$-ve$ diffusion
associated with the dispersion along one of the principal axes of the phase
space diffusion tensor. This is confirmed by a numerical evaluation of analytic
solutions of these $+ve$ and $-ve$ contributions to the particle diffusion
coefficient along this principal axis. We also examine other erroneous claims
and assumptions made in previous studies that demonstrate the apparent
superiority of the GLM PDF approach over the kinetic approach. In so doing we
have drawn attention to the limitations of the GLM approach which these studies
have ignored or not properly considered, to give a more balanced appraisal of
the benefits of both PDF approaches.