A slab ingot is being formed as a continuous casting by withdrawal from a mold with parallel walls. The sides of the ingot below the mold are cooled to remove heat of fusion and energy transferred to the solidification interface by superheated liquid metal in the mold. A two-region analysis is made to determine the non-uniform heat conduction from the liquid metal to the interface, and then from the interface to the cooled ingot sides. The solidification interface shape is found that is compatible with the removal of fusion energy and nonuniform heating from the liquid. The solution is obtained by two applications of a Cauchy boundary value method.

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