By Jian-Jun Xu
ISBN-10: 3642804357
ISBN-13: 9783642804359
ISBN-10: 3642804373
ISBN-13: 9783642804373
The stabiltiy mechanisms of a curved entrance and the trend formation in dendrite development and viscous fingering were basic topics within the components of condensed-matter physics, fabrics technological know-how, crystal development, and fluid mechanics for approximately part a century. This ebook reports interfacial instability and development formation in dynamic platforms clear of the equilibrium country. specifically, it bargains with the 2 well known prototype platforms: dendrite progress in solidification and viscous fingering in Hele--Shaw circulate. It elucidates the major difficulties step-by-step and systematically derives their mathematical ideas, at the foundation of the newly validated interfacial wave concept. ultimately, it conscientiously examines those effects via comparisons with the to be had experimental effects. The unified, asymptotic process defined during this publication may be worthwhile for research of development formation phenomena taking place in a much wider category of inhomogeneous dynamic structures. the implications on worldwide balance and choice mechanisms of development formation given during this publication can be of specific curiosity to researchers. For the readers who're now not conversant in this quickly turning out to be interdisciplinary box, this publication might be a stimulating and important introduction.
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Extra resources for Interfacial Wave Theory of Pattern Formation: Selection of Dendritic Growth and Viscous Fingering in Hele-Shaw Flow
Example text
Sekerka, "Stability of a Planar Interface During Solidification of a Dilute Binary Alloy", J. Appl. Phys. 35, pp. 444-451, (1964). 10 G. · Nash and M. E. Glicksman, "Capillarity-limited Steady-State Dendritic Growth 1. Theoretical Development", Acta Metall. 22, pp. 1283-1299, (1974). 31 25 M. E. Glicksman, R. J. Schaefer and J. D. Ayers, "High-Confidence Measurement of Solid/Liquid Surface Energy in a Pure Material", Philosophical Magazine 32, pp. 725-743, (1975). M. E. Glicksman, R. J. Schaefer and J.
18) is not applicable in the long-wavelength range ko = O(e). 26), becomes important. They must be taken into account in the leading order approximation. 18) is not valid in the extremely short-wavelength range ko = O( ~) either. For any given small e > 0, as ko » 1, we have 10'01 « leO'll and Igol « leg11. Therefore, in order to study the behaviors of solution in the ranges k « 1 and k » 1, one needs to make different asymptotic expansions. In what follows, we shall look into these cases separately.
32 2. G 2 )h- = O. 26) Hereby, it is seen that as € --t 0, in the leading order approximation, both the convective and unsteady heat transfer terms in the bulk are negligible, whereas the Gibbs-Thomson effect at the interface is important. 26), one can successively derive each order of approximation. Zeroth-order approximation solutions. 27) o. 28) o. 29) ax~ + ay~ To = 0 , and the boundary conditions: 1. As y+ --t 00, To 2. As y+ --t --t -00, Tso 3. 31 ) kO-a a (To - Tso) y+ + (loho = O. 33) Aso(x, y)eix ++ y + = ho = Doe ix + .
Interfacial Wave Theory of Pattern Formation: Selection of Dendritic Growth and Viscous Fingering in Hele-Shaw Flow by Jian-Jun Xu
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