Comment on "A Prediction of Particle Behavior via the Basset-Boussinesq-Oseen Equation"

Abstract

N a recent Note, Catalano1 has given an analytical solution of the Basset-Boussinesq-Oseen (BBO) equation. This solution was applied to the interesting problem of the discrepancy between theory and laser velocimetry applications (LDA) measurements in the case of unstationary flows arising especially in Raleigh-Benard convection or at the onset of turbulence in a circular Couette flow. This discrepancy is observed mainly for high frequencies. In the following we would like to comment on the author's assumptions: Accepting, as Catalano did, the BBO equation in the form given by Tchen2 (for a thorough discussion of this topic, we refer to Reek's recent paper),5 we do not agree that the Basset term can be neglected. For example, we look at the experiments cited by Catalano6'7: Taking into account the properties of liquid helium, it turns out that Hinze's4 conditions for the neglect of the Basset term are not fulfilled for marker particles with a diameter less 10 \im. Also, it should be mentioned that Catalano neglected the mass addition term, which represents the force to accelerate the virtual mass of the particle relative to the fluid. In extending Catalano's analytical investigation, we present a solution of the complete BBO equation. We use a standard method3 for solving the inhomogeneous Volterraintegral equation of second kind. Our solution accounts for the Basset term describing the time behavior of the flow in the neighborhood of the particles. We proceeded in this way because we believe that, in unstationary flows of the type considered, near the onset of turbulence, the Bassset term cannot be neglected. The resultant closed-form solution for the particle motion should be examined for the case of Rayleigh-Benard flow as well. The nomenclature we use is the same as Catalano's. At this opportunity we also want to make some corrections of Eqs. (2), (3), (6), (7), (15), (17), (18), (20), and (21) of Ref. 1. Further, we would like to mention that Catalano did not remark that the lift forces [see Eqs. (4) and (5)] are directed perpendicular to the relative velocity vector up—uF. So the inclusion of the lift force is rather questionable in the obviously one-dimensional case presented.1