# Thermoelectric magnetohydrodynamics in dendritic solidification

Kao, Andrew (2010) *Thermoelectric magnetohydrodynamics in dendritic solidification.* PhD thesis, University of Greenwich.

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## Abstract

The focus of this work is to investigate the effects of applying an external magnetic field to a solidifying liquid metal melt. The principle is that thermoelectric currents that are naturally inherent to solidification processes will interact with this magnetic field, resulting in a Lorentz force. This force will exist in a microscopic region in the vicinity of the solidification front, generating microscopic fluid flow in the liquid region which can significantly effect the mechanism of dendritic growth. The work contained in this thesis provides an initial insight into the complex behaviour of this process, through the use of numerical models.

To model the soldification dynamics, an enthalpy based model for dendritic growth in a supercooled melt is used in 2-dimensions and extended into 3-dimensions. The dendrite is defined as being equiaxed in nature and, for purely diffusion driven growth, numerical calculations show a good agreement with other methods under similar growth parameters. To investigate the effects of fluid dynamics, dendritic growth is tested under forced convection conditions and significant morphological changes occur. The incident tip velocity is increased and the downstream tip velocity is decreased; in agreement with many other authors investigating similar situations.

In the presence of a magnetic field the Lorentz force will form in planes perpendicular to the direction of the magnetic field. Due to the morphology and anisotropy of the surface temperature, the nature of the flow is dependent on the relative orientation of the magnetic field and the crystallographic orientation of the lattice. Using a low

magnetic field strength approximation, thus removing the non-linear and resistive terms in Navier-Stokes equation, the resulting fluid velocity is arbitrarily small so that convective transport is negated. At some time, when the morphological features of a dendrite are apparent,

steady state simulations show the flow fields that exist with different orientations of the magnetic field. The results are compared to an analytic solution for the Lorentz force, which is described by reducing the morphology of a dendrite to a sphere and assuming that the surface temperature is equivalent to the anisotropy in the surface

energy.

When the thermoelectric currents are large and the magnetic field strength is substantial the convective transport, non-linear and resistive terms become significant. The problem is purely 3-dimensional and it is shown that classical 2-dimensional boundary conditions lead to stagnant conditions. A 2-dimensional quasi 3-dimensional approximation is proposed and, with the magnetic field orientated in the

(001) direction, the effect of heat and solute redistribution through convection on the crystal morphology is modelled. Two significant morphological changes occur; the first is a deflection of the dendrite tip and the second is the initiation of secondary branching into the incident flow. The deflection is caused by circulations at the tips of the dendrite; the circulations continuously provide a region of higher free energy on the incident side while lowering it on the other. The net effect is a bias of growth in the direction of incident flow. The increase in secondary branching, in a similar fashion to the deflection,

is caused by both a circulation at the tip and also a global circulation around the entire dendrite, destabilising the incident interface and initiating secondary growth. To qualify the quasi 3-dimensional approximation, a moving mesh technique is developed that tracks a single tip of 3-dimensional growth and the similar morphological features

are observed in comparison to the quasi 3-dimensional case.

Finally a discussion into possible extensions of this work is proposed and preliminary results for grain growth in the presence of a magnetic field are given.

Item Type: | Thesis (PhD) |
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Additional Information: | uk.bl.ethos.525138 |

Uncontrolled Keywords: | Lorentz force, electricity, magnetism, electromagnetism, solidification |

Subjects: | Q Science > QC Physics |

School / Department / Research Groups: | School of Computing & Mathematical Sciences School of Computing & Mathematical Sciences > Centre for Numerical Modelling & Process Analysis > Computational Science & Engineering Group |

Related URLs: | |

Last Modified: | 29 Aug 2012 20:02 |

URI: | http://gala.gre.ac.uk/id/eprint/6365 |

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