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Volume 19, Issue 6
Asymptotic and Exact Self-Similar Evolution of a Growing Dendrite

Amlan K. Barua, Shuwang Li, Xiaofan Li & Perry Leo

Int. J. Numer. Anal. Mod., 19 (2022), pp. 777-792.

Published online: 2022-09

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  • Abstract

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.

  • AMS Subject Headings

52B10, 65D18, 68U05, 68U07

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-19-777, author = {K. Barua , AmlanLi , ShuwangLi , Xiaofan and Leo , Perry}, 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 = {

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.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/21033.html} }
TY - JOUR T1 - Asymptotic and Exact Self-Similar Evolution of a Growing Dendrite AU - K. Barua , Amlan AU - Li , Shuwang AU - Li , Xiaofan AU - Leo , Perry JO - International Journal of Numerical Analysis and Modeling VL - 6 SP - 777 EP - 792 PY - 2022 DA - 2022/09 SN - 19 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/21033.html KW - Moving boundary problems, self-similar, dendrite growth, boundary integral equations. AB -

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.

Amlan K. Barua, Shuwang Li, Xiaofan Li & Perry Leo. (2022). Asymptotic and Exact Self-Similar Evolution of a Growing Dendrite. International Journal of Numerical Analysis and Modeling. 19 (6). 777-792. doi:
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