  Effective field theory of dark matter fluid on large scales predicts the
presence of viscosity of the order of $10^{-6} H_0 M_P^2$. It has been shown
that this magnitude of viscosities can resolve the discordance between large
scale structure observations and Planck CMB data in the $\sigma_8$-$\Omega_m^0$
and $H_0$-$\Omega_m^0$ parameters space. Massive neutrinos suppresses the
matter power spectrum on the small length scales similar to the viscosities. We
show that by including the effective viscosity, which arises from summing over
non linear perturbations at small length scales, severely constrains the
cosmological bound on neutrino masses. Under a joint analysis of Planck CMB and
different large scale observation data, we find that upper bound on the sum of
the neutrino masses at 2-$\sigma$ level, decreases from $\sum m_\nu \le
0.396\,$eV (normal hierarchy) and $\sum m_\nu \le 0.378 \,$eV (inverted
hierarchy) to $\sum m_\nu \le 0.267\,$eV (normal hierarchy) and $\sum m_\nu \le
0.146\,$eV (inverted hierarchy) when the effective viscosities are included.
