Slide 82
Slide 82 text
Ritz-Galerkin discretization
23
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
vA
i
Ri + ⇢i
=
1
X
l=1
V(l)
i
· Y(l 1)(ˆ
⇢i
),
boundary velocity
v(ri) = Vi + ⌦i
⇥ ⇢i
+ vA
i
(⇢i
)
Mazur and Van Saarloos, Physica A 1982; Hess 2015; RS et al. JSTAT 2015, PRL 2016, JOSS 2020
Expansion of boundary fields in tensorial spherical harmonics
Y(l) - dimensionless, symmetric, irreducible Cartesian tensors of
rank l that form a complete, orthogonal basis on the sphere
Y(l)(ˆ
⇢) = ( 1)l⇢l+1r(l)⇢ 1
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
Y (0) = 1, Y (1)
↵
= ˆ
⇢↵, Y (2)
↵
= 3ˆ
⇢↵ ˆ
⇢ ↵
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
v↵(r) =
Z h
G↵ (r, ri)f (ri) K ↵ (ri, r)ˆ
⇢ v (ri)
i
dSi