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Details of the explosion mechanism of core-collapse supernovae (CCSNe) arenot yet fully understood. There is now an increasing number of successfulexamples of reproducing explosions in the first-principles simulations, whichhave shown a slow increase of explosion energy. However, it was recentlypointed out that the growth rates of the explosion energy of these simulationsare insufficient to produce enough $^{56}$Ni mass to account for observations.We refer to this issue as the `nickel mass problem' (Ni problem, hereafter) inthis paper. The neutrino-driven wind is suggested as one of the most promisingcandidates for the solution to the Ni problem in previous literature, but amulti-dimensional simulation for this is computationally too expensive to allowlong-term investigations. In this paper, we first built a consistent model ofthe neutrino-driven wind with an accretion flow onto a protoneutron star (PNS),by connecting a steady-state solution of the neutrino-driven wind and aphenomenological mass accretion model. Comparing the results of our model withthe results of first-principles simulations, we find that the total ejectableamount of the neutrino-driven wind is roughly determined within $\sim$ 1 secfrom 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. Ourconclusion is that it is difficult to solve the Ni problem, by continuousinjection of $^{56}$Ni by the neutrino-driven wind. We suggest that the totalamount of synthesized $^{56}$Ni can be estimated robustly if simulations arefollowed up to $\sim 2$ seconds.

A Consistent Modeling of Neutrino-driven Wind with Accretion Flow onto a Protoneutron Star and its Implications for $^{56}$Ni Production