An FVE-FAC APPROACH TO DETERMINING THERMOCAPILLARY
                          EFFECTS ON WELD POOL SHAPE

                     David Canright and Van Emden Henson
                              Mathematics Dept.
                 Naval Postgraduate School Monterey, CA 93943

                                   Abstract

Several practical materials processes, e.g., welding, float-zone purification,
and Czochralski crystal growth, involve a pool of molten metal with a free
surface, with strong temperature gradients along the surface.  In some cases,
the resulting thermo capillary flow is vigorous enough to convect heat toward
the edges of the pool, increasing the driving force in a sort of positive
feedback.  In this work we examine this mechanism and its effect on the
solid-liquid interface through a model problem:  a half space of pure
substance with concentrated axisymmetric surface heating, where surface
tension is strong enough to keep the liquid free surface flat.  The numerical
method proposed for this problem utilizes a finite volume element (FVE)
discretization in cylindrical coordinates.  Because of the axisymmetric nature
of the model problem, the control volumes used are torroidal prisms, formed by
taking a polygonal cross-section in the (r; z) plane and sweeping it
completely around the z-axis.  Conservation of energy (in the solid), and
conservation of energy , momentum, and mass (in the liquid) are enforced
globally by integrating these quantities and enforcing conservation over each
control volume.  Judicious application of the Divergence Theorem and Stokes'
Theorem, combined with a Crank-Nicolson time-stepping scheme leads to an
implicit algebraic system to be solved at each time step.  It is known that
near the boundary of the pool, that is, near the solid-liquid interface, the
full conduction-convection solution will require extremely fine length scales
to resolve the physical behavior of the system.  Furthermore, this boundary
moves as a function of time.  Accordingly, we develop the foundation of an
adaptive refinement scheme based on the principles of Fast Adaptive Composite
Grid methods (FAC).  An interesting sidelight to this work is the development
of an automated scheme, using the symbolic processor MAPLE, for computing the
values of the volume and area integrals that make up the FVE stencils.