  Starting from coupled fluid-kinetic equations for the modeling of laden
flows, we derive relevant viscous corrections to be added to asymptotic
hydrodynamic systems, by means of Chapman-Enskog expansions and analyse the
shock profile structure for such limiting systems. Our main findings can be
summarized as follows. Firstly, we consider simplified models, which are
intended to reproduce the main difficulties and features of more intricate
systems. However, while they are more easily accessible to analysis, such
toy-models should be considered with caution since they might lose many
important structural properties of the more realistic systems. Secondly, shock
profiles can be identified also in such a case, which can be proven to be
stable at least in the regime of small amplitude shocks. Last, but not least,
regarding at the temperature of the mixture flow as a parameter of the problem,
we show that the zero-temperature model admits viscous shock profiles.
Numerical results indicate that a similar conclusion should apply in the regime
of small positive temperatures.
