# Multiscale dendritic needle network model of alloy solidification with fluid flow

## TL;DR

This paper introduces a multiscale mathematical model for alloy solidification that incorporates fluid flow and buoyancy effects, validated against benchmarks and applied to simulate realistic microstructures in aluminum-copper alloys.

## Contribution

It extends the dendritic needle network model to include fluid flow and buoyancy, with a simple finite difference implementation and validation against benchmark solutions.

## Key findings

- Dendrite tip velocities within 10% of phase-field results.
- Notable microstructure differences when including buoyancy effects.
- Model enables investigation of fluid flow influence on microstructure selection.

## Abstract

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.