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

2
m i
Ex ( ) = ∑ + ε ( mi, Si, vi)
i 2
V
POTENTIALS
∂Ex
( )
Sx () =∑Si
i ∂
⎛ ⎞
∂x
∂x
⎜ ⎟
d
⎛ ⎞
⎜
R
i ⎟ ⎛ ⎞⎜
⎟ ⎛ ⎞⎛⎞
⎜ dt ⎟ ⎜ ⎟ vk
δij
0 ⎜
∂
− p ⎟ ⎜ ⎟⎜⎟
k 0 0
d
⎜
0 1
⎜ ⎟ ⎟ ∑ ⎜
0
⎟⎜0
V
⎜ k ∂R
i
j ⎟
⎟
VV SV
⎜m= ⎜ δ 0 ⎟ ⎜ i ⎟ ∑ −1
ij
0 ⎜ ⎟+
∑ 0 M ij M ⎟ ij ⎜0⎟ ⎜ dt ⎟ j ⎜ ⎟ mj
j ⎜ VS SS ⎟
0
⎜ V ⎟ j
⎜ ⎟
⎜
0 M ij M ij 1
dS ⎟ ⎜ 0 0 ⎟⎜
i
T ⎟ ⎜ ⎟⎜⎟
⎜ ⎟ j
⎜ ⎟ ⎜ ⎟ ⎜ ⎟
dt
⎝ ⎠⎜ ⎟ ⎝ ⎜
⎠⎝⎟
⎜ ⎟
⎜ ⎟ ⎝
⎠
⎠
⎝ ⎠
T
Lx ( ) =− Lx ( )
M( x) = M ( x)
≥0
OPERATORS
S( x)
A SV ij ⎡ ⎛ ⎛d−2⎞ ⎞ ⎤
M ij = ⎢− η vij + ( ij ij ) ij
R
⎜η⎜ ⎟+
ζ
ij
d
⎟ e ⋅v
e ⎥
∂S(
x)
⎣ ⎝ ⎝ ⎠ ⎠ ∂E(
x⎦)
L(
x)
⋅ = 0 PROPERTIES M ( x)
⋅ = 0
∂x
A VS ij ⎡ ⎛ ⎛d−2⎞ ⎞ ∂x
T T ⎤
M ij = ⎢− η v ij + η⎜ ⎟+
ζ ( ij ⋅ ij ) ij ⎥
R
⎜
ij
d
⎟ e v e
The <strong>model</strong> has a STRUCTURE ⎣ known ⎝ ⎝ as GENERIC
⎠ ⎠ ⎦
dEx
( )
Assure the
A SS
ij A ⎡ ij η 2 ⎛η( d − 2)
ζ ⎞ ⎤ 2
M ij =−
First
κ
Law
= 0
thermodynamical
Tij
+ ⎢ vdt ij + ⎜ Grmela + & ⎟Öttinger
( eij
⋅vij
) ⎥
Rij Rij
⎢⎣2 ⎝ 2d2 consistency
dSx
( ) Phys. Rev. ⎠ E 1997
⎥
Second Law
⎦
0
dt ≥