Interdisciplinary Applied Mathematics

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a



d2U?



6 dx2


so the complete particle equation of motion is


dVi



Шр dt



— = (mpmf)gi + m,f



DU°°


Dt


2



_™±d_ (    _ a


2 dt Г* 4    10 dx2



2


-6пра1Ц — UT



i-t



-6nga2



1


-6пар V, — UT



d2U


u, d2U°° 6 dx2


d


I о J 7Tv(t — s) ds d2UT


dx2

V<- UT —


6 dx2


2


a



d2U.


Here mp and mf are the particle mass and fluid mass respectively, and the convective total derivatives are denoted by for the flow and ^ for the particle (Maxey and Riley, 1983).


The second-derivative terms are corrections due to the flow curvature. The last    term is    due    to initial    slip    velocity,    and    the    term before    the    last


one is the Basset history force, which is associated with diffusion generated at the particle surface. A more general expression valid for small but finite particle Reynolds number was derived by (Lovalenti and Brady, 1993), who considered also the slip effects and enhanced the Basset history term with a long-time correction. This equation was later extended in (Mei and Adrian, 1992), to particle Reynolds numbers up to 100, again introducing convection to the history term associated with the vorticity transport from the particle surface onto the bulk of the fluid. It is clear from this work that convective inertia is very important even at very small scales and cannot be neglected in microflows, as the numerical and experimental results of (Lomholt, 2000) have also shown.

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