Rodionov et al. (2002) addressed the multi-fluid approach for the dust and
showed that the mean free path for dust-dust collisions can be assumed to be infinite.
Hence, the dust distribution function follows the Collisionless Boltzmann Equation
(Eq. 3.91) which is a significant simplification. But the individual dust particles sizes
cannot be grouped together because different particles sizes emitted from the same
position, at the same time will follow different streamlines and this is not the
behaviour of a single species fluid. Hence, a separate set of fluid equations are
needed for each particle size (and clearly some grouping is still required to discretize
the problem). In addition, the interaction between the gas and dust fluids results in
additional source and sink terms for the momentum and energy budgets. Hence, we
arrive at
∂ρ d,i
∂t
þ ∇ ρ d,i u d,i
À
Á ¼ Q d,i
ð4:104Þ
for the continuity equation for the dust with similar modifications for the momentum
and energy equations where the external forces and energy changes should now
include those arising from the gas-dust interaction. These adaptations can be made to
both the Euler system of equations (Rauer 2010) and the Navier-Stokes system
(Rodionov et al. 2002).
The main issue with this approach is in defining the details of the interactions.
There are several uncertainties. For example, the gas is not merely slowed by the
momentum transfer to the dust but it can also be heated by the dust. The expansion of
the gas implies that it is always colder than the dust but during a collision it is in
contact with the dust and can therefore receive thermal energy through that contact.
This is not an entirely moot point because the gas obtains the energy to expand from
its internal degrees of freedom and that energy might be replenished by interaction
with the dust giving the gas more energy to increase its outflow (terminal) velocity.
The dust also emits thermal radiation in the infrared which the gas can absorb.
There is also the question of whether the dust provides an additional gas source
which needs to be included in the fluid equations. Rodionov et al. (2002) included
this in their model based on work by Crifo (1995). There are a vast number of
possibilities and relatively little reliable data to constrain the problem.
We have seen that column densities of dust follow a 1/r law in the case of forcefree radial outflow. If the scattering cross-section within a column is unmodified by
sublimation then the observed scattered brightness from the dust should also follow
1/r and it was indeed shown at 1P/Halley that, for distances from 100 km to
~2000 km, this was an excellent approximation. In other words, the particle crosssections remained constant in this distance range. But nothing could be said at
distances closer than 100 km because of the influence of other effects
(e.g. non-point source geometry, see below). The observations of the unusual
103P/Hartley 2 by EPOXI revealed water ice particles close to the nucleus. This
suggests that in some cases, sublimation from ice particles can provide a significant
4.7 The Influence of Drag on the Equations of Motion for the Dust
335
showed that the mean free path for dust-dust collisions can be assumed to be infinite.
Hence, the dust distribution function follows the Collisionless Boltzmann Equation
(Eq. 3.91) which is a significant simplification. But the individual dust particles sizes
cannot be grouped together because different particles sizes emitted from the same
position, at the same time will follow different streamlines and this is not the
behaviour of a single species fluid. Hence, a separate set of fluid equations are
needed for each particle size (and clearly some grouping is still required to discretize
the problem). In addition, the interaction between the gas and dust fluids results in
additional source and sink terms for the momentum and energy budgets. Hence, we
arrive at
∂ρ d,i
∂t
þ ∇ ρ d,i u d,i
À
Á ¼ Q d,i
ð4:104Þ
for the continuity equation for the dust with similar modifications for the momentum
and energy equations where the external forces and energy changes should now
include those arising from the gas-dust interaction. These adaptations can be made to
both the Euler system of equations (Rauer 2010) and the Navier-Stokes system
(Rodionov et al. 2002).
The main issue with this approach is in defining the details of the interactions.
There are several uncertainties. For example, the gas is not merely slowed by the
momentum transfer to the dust but it can also be heated by the dust. The expansion of
the gas implies that it is always colder than the dust but during a collision it is in
contact with the dust and can therefore receive thermal energy through that contact.
This is not an entirely moot point because the gas obtains the energy to expand from
its internal degrees of freedom and that energy might be replenished by interaction
with the dust giving the gas more energy to increase its outflow (terminal) velocity.
The dust also emits thermal radiation in the infrared which the gas can absorb.
There is also the question of whether the dust provides an additional gas source
which needs to be included in the fluid equations. Rodionov et al. (2002) included
this in their model based on work by Crifo (1995). There are a vast number of
possibilities and relatively little reliable data to constrain the problem.
We have seen that column densities of dust follow a 1/r law in the case of forcefree radial outflow. If the scattering cross-section within a column is unmodified by
sublimation then the observed scattered brightness from the dust should also follow
1/r and it was indeed shown at 1P/Halley that, for distances from 100 km to
~2000 km, this was an excellent approximation. In other words, the particle crosssections remained constant in this distance range. But nothing could be said at
distances closer than 100 km because of the influence of other effects
(e.g. non-point source geometry, see below). The observations of the unusual
103P/Hartley 2 by EPOXI revealed water ice particles close to the nucleus. This
suggests that in some cases, sublimation from ice particles can provide a significant
4.7 The Influence of Drag on the Equations of Motion for the Dust
335