The boundary integral theory for slow and rapid curved solid/liquid interfaces propagating into binary systems

The present article is focused on the shapes of dendritic tips occurring in undercooled binary systems in the absence of convection. A circular/globular shape appears in limiting cases of small and large Péclet numbers. A parabolic/paraboloidal shape describes the tip regions of dendrites whereas a fractional power law defines a shape behind their tips in the case of low/moderate Péclet number. The parabolic/paraboloidal and fractional power law shapes are sewed together in the present work to describe the dendritic shape in a broader region adjacent to the dendritic tip. Such a generalized law is in good agreement with the parabolic/paraboloidal and fractional power laws of dendritic shapes. A special case of the angled dendrite is considered and analysed in addition. The obtained results are compared with previous experimental data and the results of numerical simulations on dendritic growth. This article is part of the theme issue ‘Patterns in soft and biological matters’.

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“…Note that if a = −1/2, the thermal integral (A 9) becomes ) which is identical to expressions (30) in [27] and (5.1) in [28].…”

Section: The Thermal and Concentration Integrals I Tmentioning

confidence: 62%

The shape of dendritic tips

Alexandrov

Galenko

2020

Phil. Trans. R. Soc. A.

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“…In one approach, interface-fitted meshes are used. Examples include finite element methods (e.g., [1]), boundary integral (e.g., [2]) and boundary element (e.g., [3]) methods. Because of the challenges associated with generating interface-fitted meshes for complex geometries, especially in three-dimensions, and because in many applications the complex geometry evolves in time, which would require a new mesh to be generated at each time step, embedded domain methods have been developed as an alternative approach.…”

Section: Introductionmentioning

confidence: 99%

Higher-order accurate diffuse-domain methods for partial differential equations with Dirichlet boundary conditions in complex, evolving geometries

Guo

Lowengrub

2020

Journal of Computational Physics

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The diffuse-domain, or smoothed boundary, method is an attractive approach for solving partial differential equations in complex geometries because of its simplicity and flexibility. In this method the complex geometry is embedded into a larger, regular domain. The original PDE is reformulated using a smoothed characteristic function of the complex domain and source terms are introduced to approximate the boundary conditions. The reformulated equation, which is independent of the dimension and domain geometry, can be solved by standard numerical methods and the same solver can be used for any domain geometry. A challenge is making the method higher-order accurate. For Dirichlet boundary conditions, which we focus on here, current implementations demonstrate a wide range in their accuracy but generally the methods yield at best first order accuracy in , the parameter that characterizes the width of the region over which the characteristic function is smoothed. Typically, ∝ h, the grid size. Here, we analyze the diffuse-domain PDEs using matched asymptotic expansions and explain the observed behaviors. Our analysis also identifies simple modifications to the diffuse-domain PDEs that yield higher-order accuracy in , e.g., O( 2 ) in the L 2 norm and O( p ) with 1.5 ≤ p ≤ 2 in the L ∞ norm. Our analytic results are confirmed numerically in stationary and moving domains where the level set method is used to capture the dynamics of the domain boundary and to construct the smoothed characteristic function.

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“…it can contain the convective and conductive fluxes. Such a theory containing the mixed type of heat and mass transfer at the dendritic boundary can be developed on the basis of the present analysis and previous theory summarized in [36][37][38][39]. This subject represents a new research direction for future investigations.…”

Section: Resultsmentioning

confidence: 71%

On the Theory of Stable Mode of Dendritic Growth in the Presence of Convective Heat and Mass Transfer Boundary Conditions

2020

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The dendritic form is one of the most common forms of crystals growing from supercooled melts and supersaturated solutions. In recent decades, an analytical theory has been developed that describes a stable dendrite growth mode under the conditions of a conductive heat and mass transfer process. However, in experiments, the growth of dendritic crystals is often observed under the conditions of convective fluid flow. In the present work, the theory of the growth of dendritic crystals is developed taking into account the convective mechanism of heat and mass transfer at the crystal-melt interface. A stable mode of dendritic growth in the case of intense convective flows near the steady-state growing dendritic tip is analyzed. The selection theory determining a stable growth mode in the vicinity of parabolic solutions as well as the undercooling balance condition are used to find the dendrite tip velocity and its tip diameter as functions of the melt undercooling. It is shown that the theoretical predictions in the case of convective boundary conditions are in agreement with experimental data for small undercoolings. In addition, the convective and conductive heat and mass transfer mechanisms near the growing dendritic surfaces are compared. Our calculations show that the convective boundary conditions essentially influence the stable mode of dendritic growth.

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The boundary integral theory for slow and rapid curved solid/liquid interfaces propagating into binary systems

Cited by 68 publications

References 93 publications

The shape of dendritic tips

The shape of dendritic tips

Higher-order accurate diffuse-domain methods for partial differential equations with Dirichlet boundary conditions in complex, evolving geometries

On the Theory of Stable Mode of Dendritic Growth in the Presence of Convective Heat and Mass Transfer Boundary Conditions

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