A steady, two-dimensional cellular convection modifies the morphological instability
of a binary alloy that undergoes directional solidification. When the convection
wavelength is far longer than that of the morphological cells, the behaviour of the
moving front is described by a slow, spatial–temporal dynamics obtained through
a multiple-scale analysis. The resulting system has a parametric-excitation structure
in space, with complex parameters characterizing the interactions between flow,
solute diffusion, and rejection. The convection in general stabilizes two-dimensional
disturbances, but destabilizes three-dimensional disturbances. When the flow is weak,
the morphological instability is incommensurate with the flow wavelength, but as the
flow gets stronger, the instability becomes quantized and forced to fit into the flow
box. At large flow strength the instability is localized, confined in narrow envelopes. In
this case the solutions are discrete eigenstates in an unbounded space. Their stability
boundaries and asymptotics are obtained by a WKB analysis. The weakly nonlinear
interaction is delivered through the Lyapunov–Schmidt method.
Directional solidification of a binary alloy into a
cellular convective flow: localized morphologies