  We compute non-linear corrections to the matter power spectrum taking the
time- and scale-dependent free-streaming length of neutrinos into account. We
adopt a hybrid scheme that matches the full Boltzmann hierarchy to an effective
two-fluid description at an intermediate redshift. The non-linearities in the
neutrino component are taken into account by using an extension of the
time-flow framework. We point out that this remedies a spurious behaviour that
occurs when neglecting non-linear terms for neutrinos. This behaviour is
related to how efficiently short modes decouple from long modes and can be
traced back to the violation of momentum conservation if neutrinos are treated
linearly. Furthermore, we compare our results at next to leading order to
various other methods and quantify the accuracy of the fluid description. Due
to the correct decoupling behaviour of short modes, the two-fluid scheme is a
suitable starting point to compute higher orders in perturbations or for
resummation methods.
