In our previous work, we obtained the uniformly valid asymptotic solution of a cylindrical rod eutectic. In order to further study the critical point of the stable growth of a rod eutectic, we have considered the unsteady growth of a rod eutectic on the basis of the steady solution of the rod eutectic. Based on the experimental system of rod eutectic growth, combined with solidification thermodynamics and kinetics, the unsteady mathematical model of the rod eutectic was established. We used the asymptotic analysis method to seek the analytical solution of the mathematical model and used the nonlinear stability analysis theory to analyze the analytical solution and establish the corresponding disturbance model. We obtained the analytic form of the global mode solution and the corresponding quantization conditions and find that there is a stable growth mode, namely the mode (ST-mode), for rod eutectic growth; when ε<εST0, the rod eutectic growth is unstable, when ε>εST0, the rod eutectic growth is stable and when ε=εST0, the rod eutectic growth is of a neutral stability. The critical eutectic spacing of succinonitrile(D)camphor (SCN-DC) predicted by us is smaller than that predicted by Jackson–Hunt, which is consistent with the corresponding experimental data. Finally, we found that the critical eutectic spacing and stable region of rod eutectic growth changed little with the temperature gradient.