the spiral trajectory of the dust particle shown in Fig. 5.15 is the result of such
“rocket force” effects. Unfortunately, such effects are impossible to predict.
The expressions (which are rather cumbersome) for the dust grain charge, forces,
heat flux and dust material temperature variation for spherical dust grains, relevant to
the edge plasma conditions, can be found in [15].
However, all these expressions are only valid for spherical dust particles. The
dynamics of non-spherical grains that are naturally present in the fusion devices
(recall Fig. 5.3) is more complex. For solid dust particles one should treat the grain
dynamics as the motion of a rigid body:
M d
dV
!
d
dt
¼ F
!
d ,
ð5:3Þ
dL
!
d
dt
¼ K
!
d ,
ð5:4Þ
where M d and L
!
d are the grain mass and angular momentum, whereas F
!
d and K
!
d are
the force and torque acting on the grain.
In addition, one cannot describe any more grain charging with just the charge
number Z d but should consider the distribution of the charge over the grain surface,
which, in particular, results in the departure of the force acting on the grain from
Eq. (5.2). Therefore, strictly speaking, the exact analysis of the dynamics of
non-spherical grains can be performed only numerically.
Even though rather comprehensive numerical simulations of the dust dynamics in
fusion devices, which will be discussed in Sect. 5.2.2, are based on the spherical dust
particle approximation, it is important to have at least some estimate of the difference
between the dynamics of the spherical and non-spherical grains.
Under the edge-plasma-relevant conditions, the dynamics of non-spherical grains
can be analyzed by using symmetry principles. As an example, we follow [61] and
consider the dynamics of a non-spherical grain in plasma without magnetic field.
First, we assume that the angular velocity of the grain spinning, Ω
!
d , is relatively low,
so that Ω
!
d
τ ch ( 1. Next, we will also assume that the speeds of the grain and the
plasma flow are much lower than the thermal speed of the plasma ions (recall that we
consider T e ~ T i ~ T), which means that V
!
p
, V
!
d
( V Ti and Ω
!
d
ℓ d ( V Ti . Under
such assumptions, both grain charging and the forces imposed on the grain can be
considered in a quasi-stationary approximation. For the case where the properties of
the grain surface responsible for the grain-plasma interactions are homogeneous, the
directions of F
!
d and K
!
d will only depend on the directions of W
! V
!
p À V
!
d , Ω
!
d and
the orientation of the grain. Moreover, for relatively small jW
! j and jΩ
!
d j one can keep
only linear dependence of F
!
d and K
!
d on W
!
and Ω
!
d . As a result, we have
5.2 Theoretical Aspects and Numerical Simulations of Dust-Related Phenomena in. . .
101
“rocket force” effects. Unfortunately, such effects are impossible to predict.
The expressions (which are rather cumbersome) for the dust grain charge, forces,
heat flux and dust material temperature variation for spherical dust grains, relevant to
the edge plasma conditions, can be found in [15].
However, all these expressions are only valid for spherical dust particles. The
dynamics of non-spherical grains that are naturally present in the fusion devices
(recall Fig. 5.3) is more complex. For solid dust particles one should treat the grain
dynamics as the motion of a rigid body:
M d
dV
!
d
dt
¼ F
!
d ,
ð5:3Þ
dL
!
d
dt
¼ K
!
d ,
ð5:4Þ
where M d and L
!
d are the grain mass and angular momentum, whereas F
!
d and K
!
d are
the force and torque acting on the grain.
In addition, one cannot describe any more grain charging with just the charge
number Z d but should consider the distribution of the charge over the grain surface,
which, in particular, results in the departure of the force acting on the grain from
Eq. (5.2). Therefore, strictly speaking, the exact analysis of the dynamics of
non-spherical grains can be performed only numerically.
Even though rather comprehensive numerical simulations of the dust dynamics in
fusion devices, which will be discussed in Sect. 5.2.2, are based on the spherical dust
particle approximation, it is important to have at least some estimate of the difference
between the dynamics of the spherical and non-spherical grains.
Under the edge-plasma-relevant conditions, the dynamics of non-spherical grains
can be analyzed by using symmetry principles. As an example, we follow [61] and
consider the dynamics of a non-spherical grain in plasma without magnetic field.
First, we assume that the angular velocity of the grain spinning, Ω
!
d , is relatively low,
so that Ω
!
d
τ ch ( 1. Next, we will also assume that the speeds of the grain and the
plasma flow are much lower than the thermal speed of the plasma ions (recall that we
consider T e ~ T i ~ T), which means that V
!
p
, V
!
d
( V Ti and Ω
!
d
ℓ d ( V Ti . Under
such assumptions, both grain charging and the forces imposed on the grain can be
considered in a quasi-stationary approximation. For the case where the properties of
the grain surface responsible for the grain-plasma interactions are homogeneous, the
directions of F
!
d and K
!
d will only depend on the directions of W
! V
!
p À V
!
d , Ω
!
d and
the orientation of the grain. Moreover, for relatively small jW
! j and jΩ
!
d j one can keep
only linear dependence of F
!
d and K
!
d on W
!
and Ω
!
d . As a result, we have
5.2 Theoretical Aspects and Numerical Simulations of Dust-Related Phenomena in. . .
101