(526b) Analysis of Melt Flow of Bridgman Process Using Lagrangian Coherent Structures

Bridgman process is one of the commonly used methods for the
production of melt-grown crystals. In the inverted Bridgman method, the lower
zone of the furnace with temperatures above the melting point of the material
is separated from the upper zone with a temperature below melting point by an
adiabatic baffle. The crucible containing the solid material is lowered into
the hot zone and after temperature stabilization, the
crucible is raised slowly into the upper zone, so that the crystal grows from
the melt.

In this process, thermal convection plays an important role
by affecting heat and mass transfer. It has been demonstrated that during the
formation of a crystal from its melt, mixing properties of the melt flow affect
transport phenomena. If complete mixing in the melt does not occur, a gradient
of impurities concentration exists in the melt near the crystal interface.

In the framework of dynamical systems theory, it is well-known that there exist special flow patterns in a fluid
domain which are the skeletons of observed tracer patterns and govern transport
structure and mechanics of the flow. These patterns are referred to as coherent
structures and when captured in terms of quantities derived from particle
trajectories, they are called Lagrangian coherent structures (LCS). Recently,
effective computational algorithms have been developed for the identification
of these structures based on the eigenvalues and eigenvectors of the
finite-time deformation tensor.

In this work, we study the kinematics of mixing of the melt
flow in Bridgman process by computing the LCS in the melt domain. To this end,
velocity field is obtained based on the mixed finite element resolution of the
coupled governing equations including Navier-Stokes,
continuity and energy equations. Having velocity data, fluid element
trajectories are obtained numerically and used for the computation of
deformation tensor at each point in the melt domain, which finally reveals the
LCS. Providing a novel insight into the transport of the flow, LCS can be
considered as stable and unstable manifolds in the phase space of the dynamical
systems. Because all of these manifolds are invariant, meaning that fluid
particles do not cross them, the fluid that is trapped in the intersection of
these manifolds (lobes) is confined to remain in them as time evolves.
Therefore, the motion of lobes in terms of entrainment and detrainment helps
explain the transport and mixing processes.

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