  Details of the explosion mechanism of core-collapse supernovae (CCSNe) are
not yet fully understood. There is now an increasing number of successful
examples of reproducing explosions in the first-principles simulations, which
have shown a slow increase of explosion energy. However, it was recently
pointed out that the growth rates of the explosion energy of these simulations
are insufficient to produce enough $^{56}$Ni mass to account for observations.
We refer to this issue as the `nickel mass problem' (Ni problem, hereafter) in
this paper. The neutrino-driven wind is suggested as one of the most promising
candidates for the solution to the Ni problem in previous literature, but a
multi-dimensional simulation for this is computationally too expensive to allow
long-term investigations. In this paper, we first built a consistent model of
the neutrino-driven wind with an accretion flow onto a protoneutron star (PNS),
by connecting a steady-state solution of the neutrino-driven wind and a
phenomenological mass accretion model. Comparing the results of our model with
the results of first-principles simulations, we find that the total ejectable
amount of the neutrino-driven wind is roughly determined within $\sim$ 1 sec
from the onset of the explosion and the supplementable amount at a late phase
($t_e \gtrsim 1$ sec) remains $M_\mathrm{ej} \lesssim 0.01M_\odot$ at most. Our
conclusion is that it is difficult to solve the Ni problem, by continuous
injection of $^{56}$Ni by the neutrino-driven wind. We suggest that the total
amount of synthesized $^{56}$Ni can be estimated robustly if simulations are
followed up to $\sim 2$ seconds.

High Energy Astrophysical Phenomena