Solidification in the Presence of Natural Convection in a Hele-Shaw Cell - Selected Topics of Computational and Experimental Fluid Mechanics - page 360

1 Introduction
The phenomenon of solidification is of great importance in many industrial processes
like casting, refrigeration, crystal growth and others. Frequently, in these processes,
the large temperature gradients and the unavoidable presence of gravity results in
a convective motion that interacts with the change of phase. In turn, the geometry
modification of the volume occupied by the fluid is decisive for the dynamic charac-
teristics of the convective motion making this a two-way coupled phenomenon. The
systematic study of heat transfer during solidification started in the nineteenth cen-
tury with the analysis of the displacement of a solidification front by J. Stefan in the
context of ice formation in the polar seas. In the classical Stefan problem, the growth
of the solid takes place in absence of fluid motion, and it considers a one dimensional
system composed of the liquid and solid regions and the interface, whose position
is determined by the heat exchange between the two phases. An important result of
this theory indicates that the displacement of the solidification front is proportional
to the square root of time. Many refinements and generalizations of this model that
consider more realistic physical situations are now available in the literature. See, for
instance, Langlois ( 1985 ). Specifically, a two dimensional model of the solidification
front indicates that a straight front is unstable and develops a wavy shape whose crests
evolve to form cusps that are interpreted as the precursors of fingers and dendrites
(Davis 1990 ). Presently, it is recognized that the solidification process is extremely
complex, and many studies have focused in the description of the microstructure of
the newly formed solid. A topic that has received much attention is the formation of a
semisolid region that forms between the solid and the liquid regions which has been
named the “mushy layer” and that has its own complex dynamics. For instance, the
solidification of an aqueous ammonium chloride solution confined in a Hele-Shaw
was studied by Chen ( 1995 ) to clarify the role played by the local convection in
the formation of channels devoid of solid or “chimneys” in the solidified material.
An approximate estimation of the convective velocity near the solidification front is
given, but no attempt is made to describe the velocity distribution or its time depen-
dence. In the presence of a body force, the thermal gradients that take place due to
the latent heat released at the solidification front induce buoyancy-driven convec-
tion that greatly affects the interfacial patterns, i.e. the solidification microstructures
that are present in the solidified material (Rosenberger 1979 ; Worster 1997 ). The
emphasis of the present study is not on the analysis of the solid structure or in the
liquid-solid transition layer but in the fluid motion. This phenomenon is strongly
dependent on the geometry of the container, and thus our observations refer to a
quasi two-dimensional motion in the direction parallel to the Hele-Shaw plates.