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



FB + FH

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