<p>Despite the success of the classical sharp interface model in describing solidification, there are no analytical or numerical solutions for the solidification of a long solid cylinder growing radially from an undercooled infinite melt when both interface curvature and attachment kinetics are considered. In this work, such a solution is obtained for the early stage of solidification using the Green’s function method and the phase-field model. The appropriate Green’s function for the problem is defined, and numerical criteria for quantitatively solving the phase-field model are established. Calculations span a wide range of undercoolings and kinetic coefficients, encompassing both metallic and non-metallic materials. Under most conditions, the growth velocity initially increases, reaches a maximum, and then decreases. This peak arises from a shift of the relative importance of capillarity (decrease in curvature) to that of heat transfer (increase in thermal resistance). An electrical circuit analogy is introduced to decouple thermodynamic effects of curvature from the kinetic processes of atomic attachment and heat flow, providing a clear explanation for the observed growth behavior across all conditions. Distinct kinetic regimes are identified, revealing that attachment kinetics dominates the early stages of growth (near the critical radius), while heat transfer kinetics becomes increasingly significant over time.</p>