# Steady State of Dendrite Growth with Zero Surface Tension and Its Regular Perturbation Expansion

• Jian-Jun Xu
Part of the Springer Series in Synergetics book series (SSSYN, volume 68)

## Abstract

For zero surface tension (ε = 0) and arbitrary undercooling, the three-dimensional system (3.13)–(3.25) allows the following steady similarity solution
$$T = T_* (\eta ) = T_\infty + \frac{{\eta _0^2 }} {2}\text{e}^{\frac{{\eta _0^2 }} {2}} E_1 \left( {\frac{{\eta _0^2 \eta ^2 }} {2}} \right)$$
(4.1)
$$T_{\text{S}} = T_{\text{S*}} = T_* (1) = 0$$
$$\eta _* = 1$$
$$T_\infty = - \frac{{\eta _0^2 }} {2}\text{e}^{\frac{{\eta _0^2 }} {2}} E_1 \left( {\frac{{\eta _0^2 }} {2}} \right)$$
$$(0\underline < \,\xi \, < \,\infty ),$$
where E 1(x) is the exponential function defined as
$$E_1 (x) = \int\limits_x^\infty {\frac{{e^{ - t} }} {t}dt}$$
(4.2)

(see [4.1]). This solution was first found by Ivantsov in 1946 (cf. [4.2] and [4.3]) and is now called the Ivantsov solution.

Lution

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### References

1. 4.1
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2. 4.2
G. P. Ivantsov, “Temperature Field around a Spheroidal, Cylindrical and Acicular Crystal Growing in a Supercooled Melt”, Dokl. Akad. Nauk, SSSR. 58, No. 4, pp. 567–569, (1947).Google Scholar
3. 4.3
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4. 4.4
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6. 4.6
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7. 4.7
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8. 4.8
M. E. Glicksman, R. J. Schaefer, and J. D. Ayers, “Dendrite Growth — A Test of Theory”, Metall.Trans. 7A, pp. 1747–1759, (1976).Google Scholar
9. 4.9
J. S. Langer and H. Müller-Krumbhaar, “Theory of Dendritic Growth — I. Elements of a Stability Analysis; II. Instabilities in the Limit of Vanishing Surface Tension; III. Effects of Surface Tension”, Acta Metall. 26, pp. 1681–1708, (1978).
10. 4.10
J. S. Langer, “Instability and Pattern Formation in Crystal Growth”, Rev. Mod. Phys. 52, 1–28, (1980).
11. 4.11
J. J. Xu, “Global Asymptotic Solution for Axisymmetric Dendrite Growth with Small Undercooling”, in Structure and Dynamics of Partially Solidified System, Ed. by D.E. Loper NATO ASI Series E. No. 125, (1987), pp. 97–109.Google Scholar
12. 4.12
J. J. Xu, “Asymptotic Theory of Steady Axisymmetric Needle-like Crystal Growth”, Studies in Applied Mathematics, 82, pp. 71–91, (1990).