@Article{IJNAM-19-6,
author = {K., Barua, Amlan and Shuwang, Li and Xiaofan, Li and Perry, Leo},
title = {Asymptotic and Exact Self-Similar Evolution of a Growing Dendrite},
journal = {International Journal of Numerical Analysis and Modeling},
year = {2022},
volume = {19},
number = {6},
pages = {777--792},
abstract = {<p style="text-align: justify;">In this paper, we investigate numerically the long-time dynamics of a two-dimensional
dendritic precipitate. We focus our study on the self-similar scaling behavior of the primary
dendritic arm with profile $x∼t^{α_1}$ and $y∼t^{α_2},$ and explore the dependence of parameters $α_1$ and $α_2$ on applied driving forces of the system (e.g. applied far-field flux or strain). We consider two
dendrite forming mechanisms: the dendritic growth driven by (i) an anisotropic surface tension
and (ii) an applied strain at the far-field of the elastic matrix. We perform simulations using a
spectrally accurate boundary integral method, together with a rescaling scheme to speed up the
intrinsically slow evolution of the precipitate. The method enables us to accurately compute the
dynamics far longer times than could previously be accomplished. Comparing with the original
work on the scaling behavior $α_1 = 0.6$ and $α_2 = 0.4$ [Phys. Rev. Lett. 71(21) (1993) 3461–3464],
where a constant flux was used in a diffusion only problem, we found at long times this scaling
still serves a good estimation of the dynamics though it deviates from the asymptotic predictions
due to slow retreats of the dendrite tip at later times. In particular, we find numerically that
the tip grows self-similarly with $α_1 = 1/3$ and $α_2 = 1/3$ if the driving flux $J ∼ 1/R(t)$ where $R(t)$ is the equivalent size of the evolving precipitate. In the diffusive growth of precipitates in an
elastic media, we examine the tip of the precipitate under elastic stress, under both isotropic and
anisotropic surface tension, and find that the tip also follows a scaling law.</p>},
issn = {2617-8710},
doi = {https://doi.org/2022-IJNAM-21033},
url = {https://global-sci.com/article/82959/asymptotic-and-exact-self-similar-evolution-of-a-growing-dendrite}
}