  A theoretical model framework of spherical symmetry is presented for a
composite astrophysical system of two polytropic fluids coupled together by
gravity to explore large-scale shocks and flow dynamics in clusters of galaxies
or in globular clusters. The existence of such large-scale shocks in clusters
of galaxies as inferred by high-resolution X-ray and radio imaging observations
implies large-scale systematic flows that are beyond usual static models for
clusters of galaxies. Here, we explore self-similar two-fluid flow solutions
with shocks for a hot polytropic gas flow in a cluster of galaxies in the
presence of a massive dark matter (DM) flow after the initiation of a
gravitational core collapse or a central AGN activity or a large-scale merging
process. In particular, the possibility of DM shocks or sharp jumps of mass
density and of velocity dispersion in dark matter halo is discussed and such DM
shocks might be detectable through gravitational lensing effects. To examine
various plausible scenarios for clusters of galaxies, we describe three
possible classes of shock flows within our model framework for different types
of temperature, density and flow speed profiles. Depending upon sensible model
parameters and shock locations, the hot ICM and DM halo may have various
combinations of asymptotic behaviours of outflow, breeze, inflow, contraction
or static envelopes at large radii at a given time. We refer to asymptotic
outflows of hot ICM at large radii as the galaxy cluster wind. As a result of
such galaxy cluster winds and simultaneous contractions of DM halo during the
course of galaxy cluster evolution, there would be less hot ICM within clusters
of galaxies as compared to the average baryon fraction in the Universe.
