Near Field Asymptotic Formulation for Dendritic Growth Due to Buoyancy

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1 Advances in Theoretical and Applied Mathematics. ISSN Volume 11 Number 4 16 pp Research India Publications Near Field Asymptotic Formulation for Dendritic Growth Due to Buoyancy Tat Leung Yee Department of Mathematics and Information Technology The Education University of Hong Kong Tai Po New Territories Hong Kong. tlyee@eduhk.hk Abstract The paper is devoted to the asymptotic analysis for a mathematical model of physical dendritic crystal growth problem. A theoretical framework governing the flow field and the temperature field under the effect of buoyancy-driven convection in the tip region of the dendritic growth is studied. By assuming the asymptotic expansions as the gravity parameter Gr the governing equations and the corresponding interface boundary conditions for the leading-order and the first-order expansions have been formulated. AMS subject classification: 3E15 35Q3 35Q79. Keywords: Asymptotic expansion dendritic growth buoyancy effect. 1. Introduction This paper is concerned with the single dendritic growth [1] [7] under the effect of buoyancy-driven convection. We neglect the effects due to the density change during phase transition and the external flow in the far field. The dendrite is subject to the small gravity which means the gravity parameter Gr is negligible. We also assume that the surface tension is negligible. Therefore we only need to consider the temperature field in liquid melt. The governing equations are: Pr D 1 ζ = η ξ ζ 1 η ξη D 1 = ξ + η ζ 1 ξ ηζ + ζ η η ξ η ξ ξ η η ζ ξ η ζ Gr ξη η T η ξ T ξ + ξ T η

2 438 Tat Leung Yee 1 T = η ξ T ξ ηt + 1 η η ξη η T ξ ξ T. 3 η We attempt to study the near field asymptotic expansion solutions of the flow field and the temperature field T under the assumption that the gravitational acceleration is negligible. With such consideration the mathematical formulation follows from assuming the asymptotic expansions in the small gravity parameter Gr. Note that the entire physical space can be divided into two regions such as the near field and the far field. The inner and the outer asymptotic solutions should be determined independently. We may assume that there exists an intermediate region in which both inner and outer asymptotic solutions are valid. We believe that the solutions can be matched in the intermediate region and the matched solution should be a globally valid asymptotic solution in the whole physical domain. In Sect. we shall first present the mathematical formulation of the physical problem. Asymptotic analysis has been performed to derive the governing equations and the corresponding boundary conditions. In Sect. 3 we shall derive the zero-th order and the first-order approximations of the PDE systems by assuming the asymptotic expansions of the flow field and the temperature field in the tip region of the dendritic growth.. Mathematical formulation We shall assume that the interface shape ηs = OGr 1 so it could be written in the form where ˆηs = O1. The inner solutions are given by ηs = Gr 1 ˆη s 4 ˆ ξ ˆη Gr = ξηgr ˆζξ ˆη Gr = ζξηgr ˆTξˆη Gr = TξηGr 5 where we have introduced the new inner variables ξ ˆη such that ˆη = η Gr 1 = O1. 6 From 1 3 the inner solutions are subject to the following governing equations: ˆη 1ˆη ˆη ˆ = Gr ξ 1 ξ ξ ˆ Gr ξ ˆζ Gr ˆη ˆζ 7

3 Near Field Asymptotic Formulation for Dendritic Growth 439 Pr ˆη 1ˆη { ˆζ = Gr Pr ˆη ξ 1 ξ ˆη + 1ˆη + Gr η ξ ˆζ ξ ˆη ˆζ ˆη 1 ˆ ˆζ η ξ ˆη ξ ˆη ˆ ˆζ ˆη ξ ˆη ˆT = Gr 1 η ξ ˆη ˆζ ξ + ˆζ η ξ ˆη Gr ξ ˆη ˆT T ˆη ξ ξ + 1 ξ ˆ ˆT ˆη ξ ˆ ˆT ξ ˆη with the interface boundary conditions at ˆη =ˆηs ξ ˆ ξ ˆη ˆ ˆη ˆT + Gr η ξ ˆη ˆT Gr3 T ξ ξ ˆT ξ ˆη ˆT ˆη 8 9 ˆT = 1 ˆT ˆη Gr ˆη s ˆT ξ + Gr η ξ ˆη s +ˆη s = 11 ˆ ξ + Gr η4 ξ ˆη s +ˆη s ˆ ˆη + Gr η4 ξ ˆηs = Gr η 4 ξ ˆη sξ ˆη s +ˆη s 1 ˆ ˆη + Gr η4 ξ ˆηs Gr ˆη s ˆ ξ + Gr η4 ξ ˆη s 13 + Gr η 4 ξ ˆη sgr ˆηs ˆη s ξ =. 3. Asymptotic analysis in the near field η = O Gr 1 Now we consider the following inner region asymptotic expansions in the limit Gr : { ˆ = Gr ˆ ξ ˆη + Gr ˆ 1 ξ ˆη + ˆζ {ˆζ = Gr α ξ ˆη + Gr ˆζ 1 ξ ˆη + { 14 ˆT = Gr ˆT ξ ˆη + Gr ˆT 1 ξ ˆη + ηˆ s = ĥ ξ + Gr ĥ 1 ξ +. It could be seen that α = for the less degenerate case. In order to obtain the inner region solution we substitute the expansions 14 into the governing equations Then we are able to derive each order of the inner region expansions successively.

4 44 Tat Leung Yee The zero-order approximation solution is subject to the following governing equations: ˆη 1ˆη ˆ + ξ ˆζ = 15 ˆη ˆη 1ˆη ˆζ = 16 ˆη ˆη + 1ˆη ˆT = 17 ˆη with the interface conditions at ˆη = ĥ near field ˆT = 18 ˆT ˆη + η ξĥ + ĥ = 19 ˆ ξ + η4 ξĥ + ĥ ˆ ˆη + η4 ξ ĥ = η 4 ξĥ ξĥ + ĥ ˆ ˆη =. 1 From the condition 19 we may assume that the solution ˆT only depends on the variable ˆη namely ˆT = ˆT ˆη when ĥ is supposed to be a constant. The constant ĥ could be normalized to be 1 by properly choosing the parameter η which is related to the under-cooling temperature T. Now the free boundary value problem 15 1 can be easily solved. The solution is as follows: ˆζ ξ ˆη = a ξ ˆη + b ξ + c ˆ ξ ˆη = ξ {ξ a 8 ˆη4 + b ˆη ln ˆη + d ˆη + e c + ˆη ln ˆη + d 1 ˆη + e 1 ˆT ˆη = a 1 ln ˆη + b 1 where a b c d e a 1 b 1 d 1 and e 1 are some undetermined constants. We can express b d e d 1 and e 1 in terms of the constants a and c by applying the boundary conditions 18-1 at ˆη = 1 we eventually obtain the zero-order inner solution ˆζ ξ ˆη = a ξ ˆη + c ˆ ξ ˆη = ξ [ a ξ ˆη 4 ˆη c ˆη ln ˆη ˆη + 1 ] 3 8 ˆT ˆη = η ln ˆη.

5 Near Field Asymptotic Formulation for Dendritic Growth 441 The zero-order solution contains arbitrary constants a and c. The solution satisfy the near field conditions on the interface but fail to satisfy the far field boundary conditions at ˆη. The inner asymptotic expansion solution is not valid in the far field. In fact we should consider a different asymptotic expansion solution which could satisfy all the far field boundary conditions. This solution is called the outer expansion solution. The inner and the outer expansion solutions would be matched in an intermediate region while a and c could then be determined by matching. We will show this step by step in the followings. Note also that the leading order inner solution 3 do not have singularity at the tip ξ =. So at this stage we do not need to consider the asymptotic solution in the near tip region. The first-order approximation solution is subject to the following governing equations: ˆη 1ˆη { ˆ 1 + ξ ˆζ 1 = ˆη ξ 1 ˆ ˆη ˆζ 4 ξ ξ Pr ˆη 1ˆη ˆη ˆη + 1ˆη ˆη with the interface conditions at ˆη = ĥ ˆζ 1 = Pr ξ 1 ˆζ + η ξ ˆζ ξ ξ ξ ˆη ˆζ ˆη + ˆζ η ξ ˆη ξ ˆ ξ ˆη ˆ ˆη 1 ˆ ˆζ η ξ ˆη ξ ˆη ˆ ˆζ ˆη ξ ˆT 1 = ξ + 1 ˆT + η ξ ˆT ξ ξ ξ ˆη ˆT ˆη + 1 ˆ ˆT η ξ ˆη ˆη ξ ˆ ˆT ξ ˆη 5 6 ˆT 1 = 7 ˆT 1 ˆη ˆT ĥ ξ + η ξĥ 1 + ĥ 1 = 8 ˆ 1 ξ + η4 ξĥ ĥ 1 + ĥ ˆ 1 ˆη + η4 ξ ĥ 1 + ĥ ˆ 1 ˆη + η4 ξ ĥ 9 = η 4 ξ ξĥ ĥ 1 + ĥ ĥ 1 ˆ 1 ˆη + η4 ξ ĥ 1 ĥ ˆ ξ + η4 ξĥ + η 4 ξĥĥ η4 ξ ĥ 1 =. 3

6 44 Tat Leung Yee 4. Conclusion In [8] [1] we have introduced the method of matched asymptotic expansions and successfully applied it to the physical problem of dendritic growth under the effect of convection motion induced by an oscillating external force. The asymptotic formulation was based on the assumption of the small Reynolds number. In this paper we undertake the further study and has an investigation on the dendritic growth under the effect of buoyancy-driven convection. We use the asymptotic approach for the mathematical formulation as the gravity parameter Gr. Our analytical framework will provide a basis to resolve the yet unsolved problems of selection of tip velocity of dendrite and the formation of pattern on the interface under the effect of buoyant flow at the near-tip region of the dendrite. References [1] R. Ananth and W. N. Gill Self-consistent theory of dendritic growth with convection J. Crystal Growth [] S. K. Dash and W. G. Gill Forced-convection heat and momentum-transfer to dendritic structures parabolic cylinders and paraboloids of revolution Int. J. Heat Mass Transfer [3] M. D. Kruskal and H. Segur Asymptotics beyond all orders in a Model of Crystal Growth Stud. Appl. Math [4] P. Pelce and U. Pomeau Dendrites in the small undercooling limit Stud. Appl. Math [5] D. A. Saville and P. J. Beaghton Growth of needle-shaped crystals in the presence of convection Phys. Rev. A [6] J. J. Xu Uniformly Valid Asymptotic Solution for Steady Dendrite Growth in External Flow J. Crystal Growth [7] J. J. Xu and D. S. Yu Selection and Resonance of Dendritic Growth with Interference of Oscillatory External Sources J. Crystal Growth [8] T. L. Yee Asymptotic Theory For a Model of Dendritic Solidification with Effect of an Oscillatory Source Global Journal of Pure and Applied Mathematics [9] T. L. Yee Further Results on Dendritic Growth with Forced Oscillation and Pattern Formation Advances in Theoretical and Applied Mathematics [1] T. L. Yee Effect on Temperature and Interface Shape of Dendritic Growth with Forced Oscillation International of Computational and Applied Mathematics