The prime example of a relevant industrial process in which fluid flow dynamics impacts desired process specifications is given by the Czochralski (CZ) crystal growth process. Melt flow in the crucible induces a variety of transport phenomena with profound effects on the mass, momentum, and heat transfer during the process. To provide novel insight into the transport phenomena associated with the CZ process melt flow, the approach of discerning the mixing templates by identifying Lagrangian coherent structures (LCS) is explored. In particular, the LCS have been extracted to identify transport features in the melt during the crystal growth. The finite-element method has been employed to simulate realistic axisymmetric melt flow accounting for physically relevant geometries (including the melt-crystal and melt-ambient interfaces) for identification of the LCS. These structures are known to represent dynamically driven material surfaces, and they provide insight into mixing properties of the melt flow that influence the characteristics of the grown crystal.