Interfacial Wave Theory of Pattern Formation: Selection of Dendritic Growth and Viscous Fingering in Hele-Shaw Flow
The stability mechanisms of a curved front and the selection of pattern formation in dendrite growth and viscous fingering have been long-standing fundamental subjects in the areas of condensed matter physics, material science, crystal growth and fluid mechanics. This book studies the interfacial stability and pattern formation in dynamic systems away from the equilibrium state. In particular, it deals with the two prominent prototype systems: dendrite growth in solidification and viscous fingering in Hele--Shaw flow. It elucidates the key problems step by step and systematically derives their mathematical solutions on the basis of the newly established interfacial wave theory. Finally, it carefully examines these results by comparisons with the available experimental results.
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anisotropy anisotropy of surface anti-Stokes line asymptotic expansion asymptotic solution binary mixture boundary conditions coefficients constant critical number Crystal Growth dendrite growth derive dimensionless dispersion relation eigenvalues exact solution experimental data far-field condition first-order approximation formula function Gibbs-Thomson condition global GTW modes Hele-Shaw cell inner equation inner solution instability mechanism interface conditions interface shape Interfacial Wave Theory isotropic surface tension Ivantsov solution J. J. Xu leading order approximation linear liquid phase mode solutions MVE solution Nash-Glicksman needle crystal neutral curve neutral mode nose bubble obtain outer region outer solution pattern formation Peclet number perturbation expansion Phys problem pure melt quantization condition regular perturbation expansion root condition S-modes shown in Fig singular points solid phase steady needle solution Stokes lines surface tension temperature thermodynamic equilibrium three-dimensional tip velocity triple point viscous fingering wave number zero surface tension zeroth-order approximation ZST solution