flow, the VDF is a drifting Maxwellian 1.5 source radii from the centre of the source
(middle panel right). However, the more tenuous flow (middle panel left) is distinctly
non-spherically symmetric in velocity-space. Clearly, at this distance, the flow on the
left has still not reached equilibrium. By the time the gas has reached 6.7 source radii
from the source (lower panels), the low production rate source (left) shows an
elongated, almost conical, VDF. This is indicating that we have already entered
the free molecular flow regime. For the higher production rate source at 6.7 source
radii, the VDF is close to spherical but with a little elongation indicating that the
transition to free molecular flow is beginning. This is consistent with Fig. 3.19.
This description shows that the sizes of the Knudsen and equilibrium regions (r Kn
and r Eq in Fig. 3.20) are strongly dependent upon the production rate. r Kn can
become very small for high production rate cases while for low production rate
cases, the size of the equilibrium flow regime can go to zero so that the gas emission
can transit from a Knudsen regime to a free molecular flow without ever
reaching LTE.
We now look in detail at the gas expansion using these three regimes.
3.4.2 The Knudsen Number
In gas dynamics, the macroscopic model describes gas as a continuous medium and
uses terms such as velocity, pressure and temperature to describe the flow. The
mathematical equations used to describe a continuum flow are the Navier-Stokes
equations for viscous fluids. The simpler Euler equations can be used if the flow is
that of an ideal fluid where viscosity and thermal conductivity can be ignored. The
microscopic model, on the other hand, describes the gas in terms of discrete
molecules and follows their individual behaviour as a result of interactions such as
collisions. In the microscopic model, the equation to be solved is the Boltzmann
equation.
The microscopic and macroscopic approaches have advantages and disadvantages. As we shall see, the microscopic model can, in principle, be used for all
applications but numerical issues arise when local densities become too high. The
macroscopic approach is, in general, faster but the continuum description begins to
break down when the gradients in the macroscopic variables becomes too steep. If
the scale lengths of changes are of the same order as the mean free path, λ MFP , in the
gas then the continuum equations become inaccurate and LTE can no longer be
assumed. As we have seen above, the expansion of the gas implies that this accuracy
limit will be reached at some distance from the nucleus source even if the production
rate is high. A rough estimate for its distance from the nucleus, r HD , can be obtained
using the approximation
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3 Gas Emissions Near the Nucleus
(middle panel right). However, the more tenuous flow (middle panel left) is distinctly
non-spherically symmetric in velocity-space. Clearly, at this distance, the flow on the
left has still not reached equilibrium. By the time the gas has reached 6.7 source radii
from the source (lower panels), the low production rate source (left) shows an
elongated, almost conical, VDF. This is indicating that we have already entered
the free molecular flow regime. For the higher production rate source at 6.7 source
radii, the VDF is close to spherical but with a little elongation indicating that the
transition to free molecular flow is beginning. This is consistent with Fig. 3.19.
This description shows that the sizes of the Knudsen and equilibrium regions (r Kn
and r Eq in Fig. 3.20) are strongly dependent upon the production rate. r Kn can
become very small for high production rate cases while for low production rate
cases, the size of the equilibrium flow regime can go to zero so that the gas emission
can transit from a Knudsen regime to a free molecular flow without ever
reaching LTE.
We now look in detail at the gas expansion using these three regimes.
3.4.2 The Knudsen Number
In gas dynamics, the macroscopic model describes gas as a continuous medium and
uses terms such as velocity, pressure and temperature to describe the flow. The
mathematical equations used to describe a continuum flow are the Navier-Stokes
equations for viscous fluids. The simpler Euler equations can be used if the flow is
that of an ideal fluid where viscosity and thermal conductivity can be ignored. The
microscopic model, on the other hand, describes the gas in terms of discrete
molecules and follows their individual behaviour as a result of interactions such as
collisions. In the microscopic model, the equation to be solved is the Boltzmann
equation.
The microscopic and macroscopic approaches have advantages and disadvantages. As we shall see, the microscopic model can, in principle, be used for all
applications but numerical issues arise when local densities become too high. The
macroscopic approach is, in general, faster but the continuum description begins to
break down when the gradients in the macroscopic variables becomes too steep. If
the scale lengths of changes are of the same order as the mean free path, λ MFP , in the
gas then the continuum equations become inaccurate and LTE can no longer be
assumed. As we have seen above, the expansion of the gas implies that this accuracy
limit will be reached at some distance from the nucleus source even if the production
rate is high. A rough estimate for its distance from the nucleus, r HD , can be obtained
using the approximation
214
3 Gas Emissions Near the Nucleus