F d
ð Þ α ¼ Φ
W
ð Þ
αβ W β þ Φ
Ω
ð Þ
αβ Ω d
ð Þ β ,
ð5:5Þ
K d
ð Þ α ¼ T
W
ð Þ
αβ W β þ T
Ω
ð Þ
αβ Ω d
ð Þ β ,
ð5:6Þ
where the tensors Φ
...
ð Þ
αβ and T
...
ð Þ
αβ are determined only by the shape of the grain and
the plasma parameters. For a general case, these tensors can only be calculated
numerically. However, for the grain having rotational symmetry around some axis,
the structure of the tensors Φ
...
ð Þ
αβ and T
...
ð Þ
αβ can be found from geometrical arguments
[61]. Indeed, for this case, the spatial orientation of the grain can be characterized by
a dimensionless vector D
!
. Then, taking into account that F
!
d , D
!
, and W
!
are vectors,
whereas L
!
d , K
!
d , and Ω
!
d are pseudo-vectors, the most general form of the equations
of motion of the grain can be written as follows
M d
dV
!
d
dt
¼ Φ
W
ð Þ
1 W
! þ Φ
W
ð Þ
2 D
!
D
! Á W
!
þ Φ
Ω
ð Þ
Ω
!
d  D
!
,
ð5:7Þ
dL
!
d
dt
¼ T
W
ð Þ W
! Â D
!
þ T
Ω
ð Þ
1 Ω
!
d þ T
Ω
ð Þ
2 D
!
D
! Á Ω
!
d
,
ð5:8Þ
where (L d ) α ¼ I αβ (Ω d ) β and I αβ ¼ I 0 δ αβ + I 1 D α D β is the inertia tensor of the grain,
where δ αβ is the Kronecker delta and I 0 and I 1 describe the components of the inertia
tensor. The scalars Φ
W
ð Þ
1
, Φ
W
ð Þ
2
, Φ
(Ω) , T
(W) , T
Ω
ð Þ
1 , and T
Ω
ð Þ
2
are determined by
particular properties of the grain material and shape and, for the grain shape not
too far from spherical, can be estimated as Φ
W
ð Þ
1 ∼ Φ
W
ð Þ
2 ∼ b F d ¼ ς drag πℓ
2
d M i n i V Ti ,
Φ
Ω
ð Þ
∼ T
W
ð Þ
∼ b F d ℓ d and T
Ω
ð Þ
1 ∼ T
Ω
ð Þ
2 ∼ b F d ℓ
2
d .
As we see from Eq. (5.7), the force acting on a non-spherical grain is no longer
aligned with the direction of the relative velocity W
!
, unlike the case of a spherical
grain in Eq. (5.2). The departure is due to both the grain orientation and grain
spinning (correspondingly the second and third terms on the right-hand side, RHS,
of Eq. (5.7)). We notice that the second term was also derived in [62] from the direct
calculation of the force acting on the grain for the combined Coulomb and dipole
grain-plasma interaction potentials.
For the case V
!
p
)jV
!
d j, jΩ
!
d j ℓ d , Eqs. (5.7) and (5.8) can be simplified and we
have
M d
dV
!
d
dt
¼ Φ
W
ð Þ
1 V
!
p þ Φ
W
ð Þ
2 D
!
D
! Á V
!
p
,
ð5:9Þ
dL
!
d
dt
¼ T
W
ð Þ V
!
p  D
!
,
ð5:10Þ
102
5 Dust in Fusion Plasmas
ð Þ α ¼ Φ
W
ð Þ
αβ W β þ Φ
Ω
ð Þ
αβ Ω d
ð Þ β ,
ð5:5Þ
K d
ð Þ α ¼ T
W
ð Þ
αβ W β þ T
Ω
ð Þ
αβ Ω d
ð Þ β ,
ð5:6Þ
where the tensors Φ
...
ð Þ
αβ and T
...
ð Þ
αβ are determined only by the shape of the grain and
the plasma parameters. For a general case, these tensors can only be calculated
numerically. However, for the grain having rotational symmetry around some axis,
the structure of the tensors Φ
...
ð Þ
αβ and T
...
ð Þ
αβ can be found from geometrical arguments
[61]. Indeed, for this case, the spatial orientation of the grain can be characterized by
a dimensionless vector D
!
. Then, taking into account that F
!
d , D
!
, and W
!
are vectors,
whereas L
!
d , K
!
d , and Ω
!
d are pseudo-vectors, the most general form of the equations
of motion of the grain can be written as follows
M d
dV
!
d
dt
¼ Φ
W
ð Þ
1 W
! þ Φ
W
ð Þ
2 D
!
D
! Á W
!
þ Φ
Ω
ð Þ
Ω
!
d  D
!
,
ð5:7Þ
dL
!
d
dt
¼ T
W
ð Þ W
! Â D
!
þ T
Ω
ð Þ
1 Ω
!
d þ T
Ω
ð Þ
2 D
!
D
! Á Ω
!
d
,
ð5:8Þ
where (L d ) α ¼ I αβ (Ω d ) β and I αβ ¼ I 0 δ αβ + I 1 D α D β is the inertia tensor of the grain,
where δ αβ is the Kronecker delta and I 0 and I 1 describe the components of the inertia
tensor. The scalars Φ
W
ð Þ
1
, Φ
W
ð Þ
2
, Φ
(Ω) , T
(W) , T
Ω
ð Þ
1 , and T
Ω
ð Þ
2
are determined by
particular properties of the grain material and shape and, for the grain shape not
too far from spherical, can be estimated as Φ
W
ð Þ
1 ∼ Φ
W
ð Þ
2 ∼ b F d ¼ ς drag πℓ
2
d M i n i V Ti ,
Φ
Ω
ð Þ
∼ T
W
ð Þ
∼ b F d ℓ d and T
Ω
ð Þ
1 ∼ T
Ω
ð Þ
2 ∼ b F d ℓ
2
d .
As we see from Eq. (5.7), the force acting on a non-spherical grain is no longer
aligned with the direction of the relative velocity W
!
, unlike the case of a spherical
grain in Eq. (5.2). The departure is due to both the grain orientation and grain
spinning (correspondingly the second and third terms on the right-hand side, RHS,
of Eq. (5.7)). We notice that the second term was also derived in [62] from the direct
calculation of the force acting on the grain for the combined Coulomb and dipole
grain-plasma interaction potentials.
For the case V
!
p
)jV
!
d j, jΩ
!
d j ℓ d , Eqs. (5.7) and (5.8) can be simplified and we
have
M d
dV
!
d
dt
¼ Φ
W
ð Þ
1 V
!
p þ Φ
W
ð Þ
2 D
!
D
! Á V
!
p
,
ð5:9Þ
dL
!
d
dt
¼ T
W
ð Þ V
!
p  D
!
,
ð5:10Þ
102
5 Dust in Fusion Plasmas