Dynamical Theory of Dendritic Growth in Convective Flow

by Xu, Jian-Jun

• ISBN13: 9781402079245
• ISBN10: 1402079249
• Edition: 2nd
• Format: Hardcover
• Copyright: 2004-05-01
• Publisher: Kluwer Academic Pub

Summary

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.

Author Biography

Jian-Jun Xu: Department of Mathematics and Statistics, McGill University, Montreal, Quebec, Canada

Table of Contents

Preface	xi
1. INTRODUCTION	1	(10)
1 Interfacial Pattern Formations in Dendritic Growth	1	(2)
2 Dendritic Growth Interacting with Convective Flow	3	(2)
3 Mathematical Formulation of the General Problem	5	(6)
3.1 Scaling	6	(1)
3.2 Macroscopic Transport Equations	7	(1)
3.3 Interface Conditions	8	(3)
2. INTERFACIAL WAVE THEORY OF DENDRITIC GROWTH WITH NO CONVECTION	11	(52)
1 Steady State of Dendritic Growth with Zero Surface Tension-Ivantsov's Solution	14	(2)
2 The Basic State for Dendritic Growth with Nonzero Surface Tension	16	(1)
3 Regular Perturbation Expansion of Axi-symmetric, Basic State of Dendritic Growth	17	(9)
3.1 Omicron(epsilon0)	18	(1)
3.2 Omicron(epsilon2)	18	(3)
3.3 The Asymptotic Behavior of the Regular Perturbation Expansion Solution as ξ-> infinity	21	(4)
3.4 Some Numerical Results of the Interface Shape Correction	25	(1)
4 Global Interfacial Wave Instability	26	(2)
5 Three-Dimensional, Linear Perturbed States Around the Axi-symmetric Basic State of Dendritic Growth	28	(2)
6 Outer Solution in the Outer Region away from the Singular Points	30	(13)
6.1 Zeroth-Order Approximation	33	(3)
6.2 First-Order Approximation	36	(5)
6.3 Singular Point ξc of the Outer Solution and Stokes Phenomenon	41	(2)
7 The Inner Solutions near the Singular Point ξc	43	(3)
8 Tip Inner Solution in the Tip Region	46	(3)
9 Global Trapped-Wave (GTW) Modes and the Quantization Condition	49	(8)
10 The Comparison of Theoretical Predictions with Experimental Data	57	(6)
3. STEADY DENDRITIC GROWTH FROM MELT WITH CONVECTIVE FLOW	63	(6)
1 Mathematical Formulation of Problem with Navier-Stokes Model	63	(6)
4. STEADY VISCOUS FLOW PAST A PARABOLOID OF REVOLUTION	69	(44)
1 Mathematical Formulation of the Problem	69	(3)
2 The Oseen Model Problem	72	(20)
2.1 Laguerre Series Representation of Solutions	76	(3)
2.2 Solution of the Oseen Model and the Paradox	79	(4)
2.3 The Solution of Type (I)	83	(1)
2.4 The Solution of Type (II)	84	(2)
2.5 The Paradox of Oseen Model Solutions and Its Resolution	86	(2)
2.6 Appendix (A)	88	(1)
2.6.1 The Properties of Laguerre Functions	88	(1)
2.6.2 Important Formulas	89	(1)
2.6.3 The derivation of the solution {Αn, Βn} for (4.44)	90	(1)
2.6.4 The Determination of the Functions: {Αn,k(Τ), &Âlpha;n,k(Τ), Βn,k(Τ), Βn,k(Τ)}	91	(1)
3 Uniformly Valid Asymptotic Solution for Steady Viscous Flow past a Slender Paraboloid of Revolution	92	(21)
3.1 Mathematical Formulation of the Problem	93	(1)
3.2 Laguerre Series Representation of Solutions	93	(1)
3.3 Outer Asymptotic Expansion Solution in the Limit Re -> 0	94	(1)
3.3.1 Zeroth-Order Solution of Velocity Field Omicron(upsilon0(Epsilon0))	95	(1)
3.4 Inner Asymptotic Expansion of the Solution	96	(3)
3.4.1 The Zeroth-Order Inner Solution	99	(1)
3.5 Matching Conditions of the Solutions	100	(3)
3.6 Skin Friction at the Surface of a Paraboloid	103	(4)
3.7 Appendix (B)	107	(1)
3.7.1 Asymptotic behavior of the outer solution Ψ0 in the limit Τ-> 0	107	(2)
3.7.2 Determination of the special outer solution Ψ0	109	(4)
5. ASYMPTOTIC SOLUTION OF DENDRITIC GROWTH IN EXTERNAL FLOW (I)	113	(18)
1 Mathematical Formulation of the Problem	114	(4)
2 Laguerre Series Representation of Solutions	118	(1)
3 Asymptotic Expansion Form of the Solution as Epsilon0->0	119	(12)
3.1 Leading-Order Solutions of Flow Field	120	(1)
3.2 Zeroth-Order Solution of Temperature Field Omicron(1)	121	(1)
3.3 First Order Solution of Temperature Field Omicron(Epsilon0)	122	(9)
6. ASYMPTOTIC SOLUTION OF DENDRITIC GROWTH IN EXTERNAL FLOW (II)	131	(26)
1 Laguerre Series Representation of Solutions	134	(1)
2 Asymptotic Expansion Forms of the Solution for the Flow Field	135	(4)
2.1 Outer Expansion Form of the Solution	135	(1)
2.2 Inner Expansion Form of the Solution	136	(3)
3 Leading-Order Asymptotic Solutions of Flow Field	139	(7)
3.1 Zeroth-Order Outer Solution of the Velocity Field	139	(1)
3.2 First Sequence of Inner Solutions of the Velocity Field	140	(1)
3.3 Second Sequence of Inner Solutions of the Velocity Field	141	(1)
3.4 Matching Conditions for Leading-Order Solutions of the Flow Field	141	(5)
4 Asymptotic Expansion Solution of the Temperature Field	146	(7)
4.1 First Sequence of Solutions of the Temperature Field	147	(2)
4.2 Second Sequence of Solutions of the Temperature Field	149	(4)
5 A Brief Summary	153	(4)
7. STEADY DENDRITIC GROWTH WITH NATURAL CONVECTION (I)	157	(20)
1 Mathematical Formulation of The Problem	158	(4)
2 Laguerre Series Representation of Solutions	162	(1)
3 Asymptotic Expansion Solution with Small Buoyancy Effect	162	(14)
3.1 Zeroth-Order Solution of the Temperature Field Omicron(1)	164	(1)
3.2 Zeroth-Order Solution of the Velocity Field Omicron(Epsilon0)	165	(6)
3.3 First-Order Solution of the Temperature Field Omicron(Epsilon0)	171	(5)
4 Summary	176	(1)
8. STEADY DENDRITIC GROWTH WITH NATURAL CONVECTION (II)	177	(20)
1 Laguerre Series Representation and Asymptotic Forms of Solutions	178	(4)
1.1 Laguerre Series Representation of the Solution	178	(1)
1.2 Outer Expansion Form of the Solution	178	(1)
1.3 Inner Expansion Form of the Solution	179	(3)
2 Leading-Order Asymptotic Expansion Solutions	182	(7)
2.1 Leading-Order Asymptotic Expansion Solution of the Temperature Field	182	(3)
2.2 Leading-Order Inner Solutions of the Velocity Field Omicron(Epsilon2)	185	(1)
2.3 Leading-Order Outer Solutions of the Velocity Field Omicron(upsilon0)(Epsilon2))	186	(2)
2.4 Matching Conditions for the Leading Order Solutions of the Flow Field	188	(1)
3 First-Order Asymptotic Expansion Solutions	189	(4)
3.1 First-Order Asymptotic Solution for the Temperature Field	189	(4)
4 Summary of the Results	193	(4)
9. STABILITY AND SELECTION OF DENDRITIC GROWTH WITH CONVECTIVE FLOW	197	(36)
1 Basic Steady State Solution	198	(5)
1.1 Convection Flow Field Induced by Uniform External Flow	198	(1)
1.2 Convection Flow Field Induced by Buoyancy Effect	199	(1)
1.3 Convection Motion Induced by Density Change During Phase Transition	200	(1)
1.4 More General Steady State Solutions with Nearly Paraboloid Interface	201	(2)
2 Linear Perturbed System around the Basic Steady State Solution	203	(3)
3 Outer Expansion Solution	206	(11)
3.1 Zeroth-Order Multiple Variables Expansion (MVE) Solutions	210	(4)
3.2 First-Order Approximation	214	(3)
4 Stability Criterion and Selection Condition of Tip Velocity	217	(1)
5 Some Special Cases	218	(9)
5.1 Convection Motion Induced by Uniform External Flow with Pr >> 1	218	(4)
5.2 Convection Motion Induced by Buoyancy Effect with Pr >> 1	222	(4)
5.3 Convection Motion Induced by Density Change During Phase Transition	226	(1)
6 A Summary	227	(6)
10. CONCLUDING REMARK	233	(2)
REFERENCE	235	(4)
INDEX	239