Asymptotic solutions for a time-dependent, axisymmetric directional solidification system

The interface shape, and the thermal and solutal profiles are analytically evaluated as functions of time for various heater temperature profiles, heater translation rates, and material properties of the system. Due to the inclusion of curvature effects on the melting point, the shape of the interface is described by a superposition of a Bessel function onto a parabolic profile. The parabolic profile is set by a combination of axial and radial diffusive transport of heat while its Bessel function correction is determined by a prescribed contact angle between the interface and the crucible wall. The model predicts an oscillatory interfacial pattern in the case of undercooling thus signifying an onset of a morphological instability.