Scaling laws for two-dimensional dendritic crystal growth in a narrow channel

We investigate analytically and computationally the dynamics of 2D needle crystal growth from the melt in a narrow channel. Our analytical theory predicts that, in the low supersaturation limit, the growth velocity $V$ decreases in time $t$ as a power law $V\sim t^{-2\mathrm{/}3}$, which we validate by phase-field and dendritic-needle-network simulations. Simulations further reveal that, above a critical channel width $\mathrm{\Lambda }\approx 5l_D$, where $l_D$ the diffusion length, needle crystals grow with a constant $V<V_s$, where $V_s$ is the free-growth needle crystal velocity, and approaches $V_s$ in the limit $\mathrm{\Lambda }\gg l_D$.