# Interdisciplinary Applied Mathematics

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dui    dui

dt Jдхд

1 2 —Vp + b’VIui

Pf

+ -^АГД(х-У»;а)

pf n

(14.15)

1    дв(х-Уп;ар)

Pf^ 13 dx,

The    first    extra    source term    on    the    right    is associated    with    a monopole

contribution caused by a virtual particle that occupies a finite localized region defined by the Gaussian envelope

(x — Y»)2«

2 <72

A(x~Yn^)= (2.J2)3/2

The length    scale    a is    related    to    the    size    of the particle, i.e., radius    a,    as

follows (Maxey and Patel, 2001):

a

a = —=

This choice corresponds to a particle having velocity

Vi(t)= Ui(Y, t),

where the latter is a volume-averaged velocity obtained from the convolution

Щ =    и*(х,£)Д(х — Y,a)d3x,

and Ui(x,t) denotes the flow field computed from the Navier-Stokes equations. Note that if more particles are present, the particle interaction is already included in u(t) and therefore in the particle velocity V(t). The volume-averaged velocity ensures that

The mass of particles is conserved; see (Maxey and Dent, 1998), and

A consistent energy balance exists between the potential energy corresponding to the settling of the particle and fluid viscous dissipation; see (Maxey and Patel, 2001).

In essence, the convolution procedure represents a filtering process of the small scales, which are energetically insignificant, and thus they do not affect the particle motion. The magnitude of the monopole F™ represents the force on the fluid by particle n, and it can be computed from the equations of the motion of the particle

mp

dt