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

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Complete Voronoi dynamics dRi () t = Vi dt dV ⎛ cij e ⎞ ij Aij ⎡ i ⎛ ⎛ 2 ⎞ ⎞ ⎤ m i = ∑Aij⎜ − ⎟Pij + − η ij + η 1 − + ζ ( ij⋅ ij) ij dt ⎜ j i Rij 2 ⎟ ∑ ⎢ v ≠ j R ⎜ ⎜ ⎟ ij d ⎟ e v e ⎥ ⎝ ⎠ ⎣ ⎝ ⎝ ⎠ ⎠ ⎦ dS A i ij ⎡ 1 2 1⎛ ⎛ 2 ⎞ ⎞ 2⎤ = 0 + ∑ ⎢− κ Tij + η vij + η⎜1 − ⎟+ ζ ( ij ⋅ ij ) ⎥ dt j Rij 2 2 ⎜ d ⎟ e v ⎣ ⎝ ⎝ ⎠ ⎠ ⎦ REVERSIBLE IRREVERSIBLE
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 ≥
Page 1 and 2: A compressible <st
Page 3 and 4: Outline 1. Motivacion: Lagrangian t
Page 5 and 6: SPH Discretization of continuum hyd
Page 7 and 8: Mesoscopic fluid <
Page 11 and 12: m i dVi dt This dynamics can be und
Page 13 and 14: Voronoi Ri Aij Rij Rj cij eij lim
Page 17: Spatial derivatives SDPD/Voronoi
Page 21 and 22: Including thermal fluctuations ⎛
Page 23 and 24: y Comparing the performance of SDPD
Page 25 and 26: Stationary velocity profiles Disord
Page 29: Conclusions -The discrete m
Page 33 and 34: 2d implementation Structure of the
Page 35 and 36: OLD TOPOLOGY Fik Fil v1 i j v2 Time
Page 37 and 38: Properties of the non dissipative V
Page 39: SDPD Overlapping Input theory s=3 s

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