Far East Journal of Applied Mathematics

The Far East Journal of Applied Mathematics publishes original research papers and survey articles in applied mathematics, covering topics such as nonlinear dynamics, approximation theory, and mathematical modeling. It encourages papers focusing on algorithm development.

Submit Article

ANALYTICAL STUDY OF BUOYANCY-DRIVEN CONVECTION EFFECTS ON DENDRITIC GROWTH AND HEAT TRANSFER

Authors

  • Tat Leung Yee

Keywords:

dendritic growth, buoyancy effects, matched asymptotic solutions, flow-thermal coupling, morphological instability

DOI:

https://doi.org/10.17654/0972096025009

Abstract

This paper presents a matched asymptotic analysis of dendritic crystal growth in an undercooled melt affected by buoyancy-driven convection. Focusing on small Grashof numbers ($\mathrm{Gr} \to 0$), we derive asymptotic solutions for the flow and temperature fields by dividing the domain into near-field and far-field regions and matching solutions in between. The analysis explicitly incorporates buoyancy effects, revealing how gravity alters flow patterns, enhances heat transfer, and can cause interface instability. This extended model broadens classical dendritic growth theories by including gravitational influences, with implications for materials processing and geological phenomena.

Received: July 1, 2025
Accepted: August 4, 2025

References

[1] D. V. Alexandrov and P. Galenko, Dendrite growth under forced convection: Analysis methods and experimental tests, Phys. Usp. 57 (2014), 771.

https://doi.org/10.3367/UFNe.0184.201408b.0833.

[2] R. Ananth and W. N. Gill, Self-consistent theory of dendritic growth with convection, J. Crystal Growth 108 (1991), 173-189.

https://doi.org/10.1016/0022-0248(91)90365-C.

[3] C. Beckermann, H. J. Diepers, I. Steinbach, A. Karma and X. Tong, Modeling melt convection in phase-field simulations of solidification, J. Comput. Phys. 154 (1999), 468-496. https://doi.org/10.1006/jcph.1999.6323.

[4] M. W. Chen, X. F. Wang, Z. D. Wang and J. X. Xie, Dendritic growth from a binary system in an external flow: Steady state solution with zero surface tension, J. Cryst. Growth 310 (2008), 655-664.

https://doi.org/10.1016/j.jcrysgro.2007.10.085.

[5] S. K. Das and W. N. Gill, Forced convection heat and momentum transfer to dendritic structures (parabolic cylinders and paraboloids of revolution), Int. J. Heat Mass Transf. 27 (1984), 1345-1356.

https://doi.org/10.1016/0017-9310(84)90062-0.

[6] L. C. Felgueroso and R. Juanes, A phase-field model of two-phase Hele-Shaw flow, J. Fluid Mech. 758 (2014), 522-552. https://doi.org/10.1017/jfm.2014.512.

[7] S. C. Huang and M. E. Glicksman, Overview 12: Fundamentals of dendritic solidification - I. Steady-state tip growth, Acta. Metall. 29 (1981), 701-715.

https://doi.org/10.1016/0001-6160(81)90115-2.

[8] G. P. Ivantsov, Temperature field around a spherical, cylindrical and needle-shaped crystal, growing in a pre-cooled melt, Dokl. Akad. Nauk SSSR 58 (1947), 567-569.

[9] D. A. Kessler, J. Koplik and H. Levine, Pattern selection in fingered growth phenomena, Adv. Phys. 37 (1988), 255-339.

https://doi.org/10.1080/00018738800101379.

[10] M. D. Kruskal and H. Segur, Asymptotics beyond all orders in a Model of Crystal growth, Stud. Appl. Math. 85 (1991), 129-181.

https://doi.org/10.1002/sapm1991852129.

[11] P. Pelce and Y. Pomeau, Dendrites in the small undercooling limit, Dyn. Curved Fronts (1988), 327-340. https://doi.org/10.1038/s41524-024-01245-2.

[12] D. A. Saville and P. J. Beaghton, Growth of needle-shaped crystals in the presence of convection, Phys. Rev. A 37 (1988), 3423-3430.

https://doi.org/10.1103/PhysRevA.37.3423.

[13] I. Steinbach, Pattern formation in constrained dendritic growth with solutal buoyancy, Acta. Mater 57 (2009), 2640-2645.

https://doi.org/10.1016/j.actamat.2009.02.004.

[14] J. J. Xu, Dendritic growth from a melt in an external flow: uniformly valid asymptotic solution for the steady state, J. Fluid Mech. 263 (1994), 227-244. https://doi.org/10.1017/S002211209400409X.

[15] J. J. Xu and J. Schimizu, Asymptotic theory of disc-like crystal growth: (II): interfacial instability and pattern formation at early stage of growth, Commun. Pure Appl. Anal. 3 (2004), 527-543. https://doi.org/10.3934/cpaa.2004.3.527.

[16] J. J. Xu, Interfacial Wave Theory of Pattern Formation in Solidification -Dendrites, Fingers, Cells and Free Boundaries, Springer Cham, 2017.

[17] T. L. Yee, Asymptotic theory for dendritic solidification with oscillatory sources, Global J. Pure Appl. Math. 10 (2014), 401-416.

Downloads

Published

2025-08-25

Issue

Vol. 118 No. 2 (2025)

Section

Articles

License

Copyright (c) 2025 PUSHPA PUBLISHING HOUSE, PRAYAGRAJ, INDIA

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

____________________________

Licensing Terms:

This work is published by Pushpa Publishing House and is subject to the following conditions:

Attribution: You must credit Pushpa Publishing House as the original publisher. Include the publication title and author(s) if applicable.

No Derivatives: Modifying the work or creating derivative works is not permitted without prior written permission.

For more information or permissions beyond the scope of this license, contact Pushpa Publishing House.

How to Cite

ANALYTICAL STUDY OF BUOYANCY-DRIVEN CONVECTION EFFECTS ON DENDRITIC GROWTH AND HEAT TRANSFER. (2025). Far East Journal of Applied Mathematics, 118(2), 155-176. https://doi.org/10.17654/0972096025009

Similar Articles

1-10 of 25

You may also start an advanced similarity search for this article.