# On convection and stability of some welding and solidification processes

Sharpe, Michael Anthony (2000) On convection and stability of some welding and solidification processes. University of Southampton, Faculty of Mathematical Studies, Doctoral Thesis , 236pp.

## Description/Abstract

In this thesis a variety of problems are considered, the first of which is associated with the
welding process. Experiments indicate that the shape of a weld pool is influenced by convection
in the liquid metal. In recent years it has been shown that this convection is crucially
affected by the Marangoni (or thermocapillary) force. Recently Craine and Belgrove ([30],
[7]) have developed a two-dimensional, axisymmetric model which includes the Marangoni
force on the free surface of a semi-infinite region of liquid steel when a point source of current
and heat is incident on the free surface. An asymptotic solution to this problem is obtained in
this thesis, and the surface tension gradient with respect to temperature, dj/dT, a parameter
which is crucial to the magnitude of the Marangoni force, is found to affect every coefficient
in the leading and first order asymptotic expansions. In various theoretical and experimental
models purely poloidal flow bifurcates to a rotating flow. To investigate this possibility for
our flow a linear stability analysis is performed on a numerically obtained poloidal solution
for the flow and temperature distribution in a hemisphere (a model first derived in [7]). For
the azimuthal stability mode m = 0 the equation governing the linear stability of the rotating
motion is found to decouple from the corresponding poloidal equations. The poloidal and
azimuthal stability equations both become unstable at different critical currents dependent
on the sign and magnitude of d^/dT. An investigation of the eigenvectors indicates the onset
of instability near to the point source. For the upper modes instability occurs only when
m = 1 and in a very small region of parameter space.
In the second part of this thesis a freezing sphere problem with flow is used to compare
a sharp interface Stefan model and a diffuse interface phase-field model. Firstly a Stefan
model that includes a disparity between the density of the solid and liquid phases is derived
and solved numerically. This model is compared with a recent phase-field model with flow,
derived by Anderson et al. in [2]. In this thesis the one-dimensional isotropic version of
Anderson's model is obtained in spherical polar coordinates and using certain simplifications
when the dimensionless thickness of the interface £5 is vanishingly small a leading order
asymptotic expression reproduces the Stefan model with flow. The phase-field model is
subsequently modified and solved numerically, and the results are compared with the sharp
interface model. Close agreement is observed between these models when es < 0.01.

Item Type: Thesis (Doctoral)
Subjects: Q Science > QA Mathematics
T Technology > TA Engineering (General). Civil engineering (General)
Divisions : University Structure - Pre August 2011 > School of Mathematics
ePrint ID: 50624
Accepted Date and Publication Date:
Status