  Massive neutrinos are well-known to cause a characteristic suppression in the
growth of structures at scales below the neutrino free-streaming length. A
detailed understanding of this suppression is essential in the era of precision
cosmology we are entering into, enabling us to better constrain the total
neutrino mass and possibly probe (beyond)-$\Lambda$CDM cosmological model(s).
Instead of the usual N-body simulation or Boltzmann solver, in this article we
consider a two-fluid framework at the linear scales, where the neutrino fluid
perturbations are coupled to the CDM (+ baryon) fluid via gravity at redshifts
of interest. Treating the neutrino mass fraction $f_\nu$ as a perturbative
parameter, we find solutions to the system with redshift-dependent neutrino
free-streaming length in $\Lambda$CDM background via two separate approaches.
The perturbative scale-dependent solution is shown to be in excellent agreement
with numerical solution of the two-fluid equations valid to all orders in
$f_{\nu}$, and also agrees with results from {\texttt{CLASS}} to a good
accuracy. We further generalize the framework to incorporate different evolving
dark energy backgrounds and found sub-percent level differences in the
suppression, all of which lie within the observational uncertainty of BOSS-like
surveys. We also present a brief discussion on the prospects of the current
analysis in the context of upcoming missions.
