We consider the steady flow between two totally reflecting walls at different temperatures
with a temperature difference
. Since
, the moment equations read
The closure moments for the solution are determined from the asymptotic solution
for the model at hand of
In the
Navier-Stokes description,
contains no contribution to
, and from equations ... we find the heat layer solution
Comparing the temperature profiles of numerical simulations for the nonlinear and the BGK system (
Figure 3) we find that the BGK profile is considerably flatter than that of the nonlinear case. The latter one is close to the constant gradient profile (solid line) linearly connecting the wall temperatures. An explanation for th the difference is given by comparing the closure moments
for both models (
Figure 4). In
Figure 4(a) we find a small almost constant contribution (maybe weakly dependent on the temperature) in the nonlinear case. Zooming into
Figure 4(b) confirms this. However, for BGK we find a distinct, almost constant gradient over the whole field of calculation. An explanation for this is again provided by the trace description which allows to draw a rough picture pointing out the differences between nonlinear collision operator and BGK model. A full discussion of the trace solutions for both models in the fluid dynamic limit is provided in [
5]. We do not repeat the tedious calculations of [
5] but explain by a short argument why the BGK system does not provide the correct temperature gradient in the fluid dynamic limit. Suppose
is the steady solution of the balance equation for the BGK system,
The first term on the left hand side represents in the limit a finite heat flux modeled by some term of the form
. For small
we may use the approximation (65) for
shortly denoted as
. Furthermore,
yields an unknown contribution
which comes out as a solvability condition for the above equation,
Its solution is
and
Thus under the above assumptions
A is singular in the limit
which is not realistic. The only way out is that
c and with this
B are
-dependent and vanish in the limit. This explains
qualitatively the flattened curve for the temperature profile of the BGK model. In the limit it is supposed to yield a constant temperature profile.