  Self-similar shock solutions in spherically symmetric polytropic gas flows
are constructed and analyzed in contexts of proto-star formation processes.
Among other possible solutions, we model a similarity shock across the sonic
critical curve with an inner free-fall core collapse and a simultaneous outer
expansion of the extended envelope; the separation or stagnation surface
between these two flow zones travels outwards in a self-similar manner at a
variable speed. One readily obtains accretion shock solutions. Semi-complete
self-similar solutions across the sonic critical curve either once or twice
without shocks can also be constructed. Features of star formation clouds of
our polytropic model include the mass density scaling in the outer flow zone
$\rho\propto r^{-2/(2-\gamma)}$, the temperature scalings of the inner flow
zone $T\propto r^{-3(\gamma-1)/2}$ and of the outer flow zone $T\propto
r^{-2(\gamma-1)/(2-\gamma)}$, and the variable central mass accretion rate
$\dot{M}=k^{3/2}t^{(3-3\gamma)}m_0/G$ where $\gamma$ is the polytropic index,
$k$ is a constant, $m_0$ is the core mass, and $G$ is the gravitational
constant. Spectral line profiles characteristic of the `envelope expansion with
core collapse' (EECC) shock solutions are expected. Random magnetic field
permeated in a partially ionized cloud can be incorporated into this
theoretical polytropic model framework. We discuss briefly our results in
context of the oft-observed starless B335 cloud system as an example.
