  We present a mathematical formulation of a multiscale model for
solidification with convective flow in the liquid phase. The model is an
extension of the dendritic needle network approach for crystal growth in a
binary alloy. We propose a simple numerical implementation based on finite
differences and step-wise approximations of parabolic dendritic branches of
arbitrary orientation. Results of the two-dimensional model are verified
against reference benchmark solutions for steady, unsteady, and buoyant flow,
as well as steady-state dendritic growth in the diffusive regime. Simulations
of equiaxed growth under forced flow yield dendrite tip velocities within 10%
of quantitative phase-field results from the literature. Finally, we perform
illustrative simulations of polycrystalline solidification using physical
parameters for an aluminum-10wt%copper alloy. Resulting microstructures show
notable differences when taking into account natural buoyancy in comparison to
a purely diffusive transport regime. The resulting model opens new avenues for
computationally and quantitatively investigating the influence of fluid flow
and gravity-induced buoyancy upon the selection of dendritic microstructures.
Further ongoing developments include an equivalent formulation for directional
solidification conditions and the implementation of the model in three
dimensions, which is critical for quantitative comparison to experimental
measurements.
