A compressible fluid particle model based on the Voronoi ...

How can we incorporate in the <strong>model</strong> the dissipative dynamics in order to
keep as close as possible to the continuum Navier Stokes equations?
dv ∂v
= ∑ dt ∂R
i i
j j
dV
∂v
m = p +
i
j
i
dt j ∂Ri
j
dS
i Ti dt
dR
dt
i
Irreversible dynamics
=
= 0 +
V
i
-A e
∑
?
i.
V
j
?
We requiere expressions for the second derivatives,
ij
ij
dv
= v∇⋅v
dt
dv
2 ⎛ d −2
⎞
m =− v∇P+ ηv ∇ ∇ v+ ⎜η + ζ⎟v
∇(∇
∇(∇⋅v)
dt
⎝ d ⎠
dS 2
T = κv∇∇ ∇ ∇T+ 2 ηv ∇ ∇v: ∇ ∇ v+ ζv(∇
(∇
(∇⋅v)
dt
⎣ ⎦ ∑ v
⎣ ⎦
⎡ f⎤ ⎣
∇∇
∇∇
⎦
≈− ∑
f
11 ⎡ f ⎤ =− A ⎡ ij ij f ⎤
⎣
=− ∇i
⎦ ∑ ijeij e i
v ⎣ ⎦
Divergence
⎡ theorem ∇∇ ∇∇f⎤ A ⎡ ∇
∇f⎤
fij
i i
i j i j
ij ij
i
1
vi j
Aij
ee ij ij
Rij
ij
≈
2
R
ij
e
ij