The Basset, Boussinesq, Oseen, and Tchen equation

Writing the fundamental law of dynamics to describe the movement of the particle requires a balance of forces exerted on the particle. These forces are gravity on one hand and the hydrodynamic forces exerted by the fluid flow on the particle on the other hand. The particle disturbs the velocity field of the flow in its 

1 A creeping flow is a flow for which nonlinear ferms are neglecfed in fhe Navier-Stokes equations, and the pressure gradient balances out the viscous terms. The book by Happel and Brenner Low Reynolds Number Hydrodynamics, Kluwer Academic Publishers, 1991) discusses these questions. 

In the following we resume the notations used by Maxey and Riley (1983) and write the Basset, Boussinesq, Oseen, and Tchen (BBOT) equations in a form that is quasi-identical to that of Maxey and Riley. This formulation is interesting insofar as it highlights the relative movement of the particle with respect to the fluid. The three components of the particle s relative velocity with respect to the fluid are obtained by solving the following differential equations  

The differences with the equations obtained by Maxey and Riley are the corrections brought in by Anton (1987, 1988). An in-depth discussion of these equations can be found in the book by Michaehdes.  

The notations appearing in [16.5] have already been introduced, with the exception of the two time derivatives. Indeed, equation [16.5] distinguishes between  

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