Inertial Effects on Linear and Locally Linear Flows

The dynamics of a suspension of particles in a gas may be characterized by the coexistence of a nearly equilibrated incompressible component (the gas) with a highly compressible component, which is often very far from equilibrium (the particle phase). The governing equations for such a system are complicated due to the coupling between the two phases, and have been successfully solved in only a limited number of cases. In the first part of this paper the wide range of so-called “linear flows” (where the host gas velocity field varies linearly with position) for which the particle velocity field may be obtained analytically is introduced. Because the family of linear flows includes stagnation point flows, Couette flows, and solid body rotations, relatively wide conclusions may be reached on the particle behavior, with important implications for the aerodynamic separation of aerosols or heavy molecules. In particular, it is seen that sufficiently far downstream from particle injection points the particle velocity becomes independent of initial conditions and is a function of the local fluid properties only. This remarkable feature (“normal behavior”) always occurs for solid body rotations or parallel flows, but not for decelerated flows. For these, there is a critical value of the ratio between the particle relaxation time τ and the flow deceleration time to ω^?1, above which the “normal solutions” break down and the particle motion is influenced also by its past history rather than by local fluid properties only. In the second part, a phenomenological approach (constitutive law) is proposed for solving particulate problems. The particle velocity is given at every point as a function of the local host gas velocity and velocity gradient tensor, using the normal solutions obtained previously as if the flow were locally linear. The corresponding results are compared with numerical calculations (easily performed in the limit of small particle mass fraction) for two different nonlinear stagnation-type flows, and the agreement is quite good in both cases up to values of ωτ near critical. If more extensive testing of the proposed “locally linear flow approximation” proves successful, its use will simplify significantly the treatment of the flow of dusty gases.