Abstract
We now turn to study global linear stability of the steady or “nearly” steady needle crystal solutions that were demonstrated in the last chapter and hereinafter defined as the basic states of the system. We shall first deal with the 3D axially symmetric dendrite with nonzero isotropic surface tension. The effect of anisotropy will be discussed in the next chapter. For global linear stability, one needs to investigate the evolution of infinitesimal perturbations around the basic state solutions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
M. Abramovitz, I.A. Stegun (eds.), Handbook of Mathematical Functions (Dover, New York, 1964)
R. Ananth, W.N. Gill, Dendritic growth of an elliptical paraboloid with forced convection in the melt. J. Fluid Mech. 208, 575–593 (1989)
R. Ananth, W.N. Gill, Self-consistent theory of dendritic growth with convection. J. Cryst. Growth 108, 173–189 (1991)
C.M. Bender, S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978)
P.H. Bouissou, A. Chiffaudel, B. Perrin, P. Tabling, Europhys. Lett. 13, 89 (1990)
D. Canright, S.H. Davis, Buoyancy effect of a growing, isolated dendrite. J. Cryst. Growth 114, 153–185 (1991)
Y.Q. Chen, X.X. Tang, J.J. Xu, Three-dimensional interfacial wave theory of dendritic growth: (I). Multiple variables expansion solutions. Chin. Phys. B 18 (2), 1674-1056/2009/18(02)/0671-15 (2009)
Y.Q. Chen, X.X. Tang, J.J. Xu, Three-dimensional interfacial wave theory of dendritic growth: (II). Non-axisymmetric global wave modes and selection criterion of pattern formation. Chin. Phys. B 18 (2), 1674-1056/2009/18(02)/0686-13 (2009)
R.B. Dingle, Asymptotic Expansions: Their Derivation and Interpretation (Academic, London, 1973)
P.G. Drazin, W.H. Reid, Hydrodynamic Stability (Cambridge University Press, Cambridge, 1971)
M.E. Glicksman, M.B. Koss, E.A. Winsa, Dendritic growth velocities in microgravity. Phys. Rev. Lett. 73 (4), 573–576 (1994)
M.E. Glicksman, M.B. Koss, L.T. Bushnell, J.C. Lacombe, E.A. Winsa, Dendritic growth of succinonitrile in terrestrial and microgravity conditions as a test of theory. ISIJ Int. 35 (6), 604–610 (1995)
J.C. LaCombe, M.B. Koss, J.E. Frei, C. Giummarra, A.O. Lupulecscu, M.E. Glicksman, Evidence for tip velocity oscillations in dendritic solidification. Phys. Rev. E 65, 031604 (2012)
J.S. Langer, 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, 1681–1708 (1978)
C.C. Lin, The Theory of Hydrodynamic Stability (Cambridge University Press, Cambridge, 1955)
C.C. Lin, Y.Y. Lau, On spiral waves in galaxies—a gas dynamic approach. SIMA. J. Appl. Math. 29 (2), 352–370 (1975)
C.C. Lin, Y.Y. Lau, Density wave theory of spiral structure of galaxies, in Studies in Applied Mathematics, vol. 60 (MIT Press, Cambridge, MA, 1979), pp. 97–163
M. Rabaud, Y. Couder, N. Gerard, Phys. Rev. A 37, 935 (1988)
B. Steinin, V. Shatalov, Borel–Laplace Transform and Asymptotic Theory: Introduction to Resurgent Analysis (CRC-Press, Boca Raton, 1995)
L.M. Williams, M. Muschol, X. Qian, W. Losert, H.Z. Cummins, Phys. Rev. E 48, 489–498 (1993)
J.J. Xu, Interfacial wave theory for dendritic structure of a growing needle crystal (I): local instability mechanism. Phys. Rev. A 40 (3), 1599–1608 (1989)
J.J. Xu, Interfacial wave theory for dendritic structure of a growing needle crystal (II): wave-emission mechanism at the turning point. Phys. Rev. A 40 (3), 1609–1614 (1989)
J.J. Xu, Global neutral stable state and selection condition of tip growth velocity. J. Cryst. Growth 100, 481–490 (1990)
J.J. Xu, Interfacial wave theory of solidification—dendritic pattern formation and selection of tip velocity. Phys. Rev. A15 43 (2), 930–947 (1991)
J.J. Xu, Two-dimensional dendritic growth with anisotropy of surface tension. Phys. (D) 51, 579–595 (1991)
J.J. Xu, Interfacial wave theory of two-dimensional dendritic growth with anisotropy of surface tension. Can. J. Phys. 69 (7), 789–800 (1991)
J.J. Xu, Dendritic growth from a melt in an external flow: uniformly valid asymptotic solution for the steady state. J. Fluid Mech. 263, 227–243 (1994)
J.J. Xu, D.S. Yu, Selection and resonance of dendritic growth with interference of oscillatory external sources. J. Cryst. Growth 226, 378–392 (2001)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Xu, JJ. (2017). Global Interfacial Wave Instability of Dendritic Growth from a Pure Melt. In: Interfacial Wave Theory of Pattern Formation in Solidification. Springer Series in Synergetics. Springer, Cham. https://doi.org/10.1007/978-3-319-52663-8_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-52663-8_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-52662-1
Online ISBN: 978-3-319-52663-8
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)