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Dynamical Theory of Dendritic Growth in Convective Flow : Advances in Mechanics and Mathematics - Xu Jian-Jun

Dynamical Theory of Dendritic Growth in Convective Flow

Advances in Mechanics and Mathematics

Hardcover

Published: 30th April 2004
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Convective flow in the liquid phase is always present in a realistic process of freezing and melting and may significantly affect the dynamics and results of the process. The study of the interplay of growth and convection flow during the solidification has been an important subject in the broad fields of materials science, condensed matter physics, fluid physics, micro-gravity science, etc. The present book is concerned with the dynamics of free dendritic growth with convective flow in the melt. It systematically presents the results obtained in terms of a unified asymptotic approach in the framework of the interfacial wave (IFW) theory. In particular, the book explores the effect of the various types of convection flow on the selection and pattern formation of dendritic growth based on the global stability analysis. Audience: Applied mathematicians, physicists, material scientists and engineers will all find this volume of interest.

Prefacep. xi
Introductionp. 1
Interfacial Pattern Formations in Dendritic Growthp. 1
Dendritic Growth Interacting with Convective Flowp. 3
Mathematical Formulation of the General Problemp. 6
Scalingp. 6
Macroscopic Transport Equationsp. 7
Interface Conditionsp. 8
Interfacial Wave Theory of Dendritic Growth with No Convectionp. 11
Steady State of Dendritic Growth with Zero Surface Tension--Ivantsov's Solutionp. 14
The Basic State for Dendritic Growth with Nonzero Surface Tensionp. 16
Regular Perturbation Expansion of Axi-symmetric, Basic State of Dendritic Growthp. 17
O([epsilon superscript 0])p. 18
O([epsilon superscript 2])p. 18
The Asymptotic Behavior of the Regular Perturbation Expansion Solution as [xi] to [infinity]p. 21
Some Numerical Results of the Interface Shape Correctionp. 25
Global Interfacial Wave Instabilityp. 26
Three-Dimensional, Linear Perturbed States Around the Axi-symmetric Basic State of Dendritic Growthp. 28
Outer Solution in the Outer Region away from the Singular Pointsp. 30
Zeroth-Order Approximationp. 33
First-Order Approximationp. 36
Singular Point [xi subscript c] of the Outer Solution and Stokes Phenomenonp. 41
The Inner Solutions near the Singular Point [xi subscript c]p. 43
Tip Inner Solution in the Tip Regionp. 46
Global Trapped-Wave (GTW) Modes and the Quantization Conditionp. 49
The Comparison of Theoretical Predictions with Experimental Datap. 57
Steady Dendritic Growth from Melt with Convective Flowp. 63
Mathematical Formulation of Problem with Navier-Stokes Modelp. 63
Steady Viscous Flow Past a Paraboloid of Revolutionp. 69
Mathematical Formulation of the Problemp. 69
The Oseen Model Problemp. 72
Laguerre Series Representation of Solutionsp. 76
Solution of the Oseen Model and the Paradoxp. 79
The Solution of Type (I)p. 83
The Solution of Type (II)p. 84
The Paradox of Oseen Model Solutions and Its Resolutionp. 85
Appendix (A)p. 87
The Properties of Laguerre Functionsp. 87
Important Formulasp. 88
The derivation of the solution {A[subscript n], B[subscript n]} for (4.44)p. 89
The Determination of the Functions: {A[subscript n,k]([tau]), A[subscript n,k]([tau]), B[subscript n,k]([tau]), B[subscript n,k]([tau])}p. 90
Uniformly Valid Asymptotic Solution for Steady Viscous Flow past a Slender Paraboloid of Revolutionp. 92
Mathematical Formulation of the Problemp. 92
Laguerre Series Representation of Solutionsp. 93
Outer Asymptotic Expansion Solution in the Limit Re to 0p. 94
Zeroth-Order Solution of Velocity Field O(v[subscript 0]([epsilon subscript 0]))p. 95
Inner Asymptotic Expansion of the Solutionp. 96
The Zeroth-Order Inner Solutionp. 98
Matching Conditions of the Solutionsp. 99
Skin Friction at the Surface of a Paraboloidp. 103
Appendix (B)p. 107
Asymptotic behavior of the outer solution [psi subscript 0] in the limit [tau] to 0p. 107
Determination of the special outer solution [psi]*[subscript 0]p. 109
Asymptotic Solution of Dendritic Growth in External Flow (I)p. 113
Mathematical Formulation of the Problemp. 114
Laguerre Series Representation of Solutionsp. 118
Asymptotic Expansion Form of the Solution as [epsilon subscript 0] to 0p. 119
Leading-Order Solutions of Flow Fieldp. 120
Zeroth-Order Solution of Temperature Field O(1)p. 121
First Order Solution of Temperature Field O([epsilon subscript 0])p. 122
Asymptotic Solution of Dendritic Growth in External Flow (II)p. 131
Laguerre Series Representation of Solutionsp. 134
Asymptotic Expansion Forms of the Solution for the Flow Fieldp. 135
Outer Expansion Form of the Solutionp. 135
Inner Expansion Form of the Solutionp. 136
Leading-Order Asymptotic Solutions of Flow Fieldp. 139
Zeroth-Order Outer Solution of the Velocity Fieldp. 139
First Sequence of Inner Solutions of the Velocity Fieldp. 140
Second Sequence of Inner Solutions of the Velocity Fieldp. 141
Matching Conditions for Leading-Order Solutions of the Flow Fieldp. 141
Asymptotic Expansion Solution of the Temperature Fieldp. 146
First Sequence of Solutions of the Temperature Fieldp. 147
Second Sequence of Solutions of the Temperature Fieldp. 149
A Brief Summaryp. 153
Steady Dendritic Growth with Natural Convection (I)p. 157
Mathematical Formulation of The Problemp. 158
Laguerre Series Representation of Solutionsp. 162
Asymptotic Expansion Solution with Small Buoyancy Effectp. 162
Zeroth-Order Solution of the Temperature Field O(1)p. 164
Zeroth-Order Solution of the Velocity Field O([epsilon subscript 0])p. 165
First-Order Solution of the Temperature Field O([epsilon subscript 0])p. 171
Summaryp. 176
Steady Dendritic Growth with Natural Convection (II)p. 177
Laguerre Series Representation and Asymptotic Forms of Solutionsp. 178
Laguerre Series Representation of the Solutionp. 178
Outer Expansion Form of the Solutionp. 178
Inner Expansion Form of the Solutionp. 179
Leading-Order Asymptotic Expansion Solutionsp. 182
Leading-Order Asymptotic Expansion Solution of the Temperature Fieldp. 182
Leading-Order Inner Solutions of the Velocity Field O([epsilon superscript 2 subscript 2])p. 185
Leading-Order Outer Solutions of the Velocity Field O(v[subscript 0]([epsilon subscript 2]))p. 186
Matching Conditions for the Leading Order Solutions of the Flow Fieldp. 188
First-Order Asymptotic Expansion Solutionsp. 189
First-Order Asymptotic Solution for the Temperature Fieldp. 189
Summary of the Resultsp. 193
Stability and Selection of Dendritic Growth with Convective Flowp. 197
Basic Steady State Solutionp. 198
Convection Flow Field Induced by Uniform External Flowp. 198
Convection Flow Field Induced by Buoyancy Effectp. 199
Convection Motion Induced by Density Change During Phase Transitionp. 200
More General Steady State Solutions with Nearly Paraboloid Interfacep. 201
Linear Perturbed System around the Basic Steady State Solutionp. 203
Outer Expansion Solutionp. 206
Zeroth-Order Multiple Variables Expansion (MVE) Solutionsp. 210
First-Order Approximationp. 214
Stability Criterion and Selection Condition of Tip Velocityp. 217
Some Special Casesp. 218
Convection Motion Induced by Uniform External Flow with Pr [double greater-than sign] 1p. 218
Convection Motion Induced by Buoyancy Effect with Pr [double greater-than sign] 1p. 222
Convection Motion Induced by Density Change During Phase Transitionp. 226
A Summaryp. 227
Concluding Remarkp. 231
Referencesp. 237
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9781402079245
ISBN-10: 1402079249
Series: Advances in Mechanics and Mathematics
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 256
Published: 30th April 2004
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.5 x 15.5  x 1.91
Weight (kg): 1.19
Edition Number: 2
Edition Type: Revised