The boundary integral equation for curved solid/liquid interfaces propagating into a binary liquid with convection

Титова, Э. А.; Alexandrov, Dmitri V.

doi:10.1088/1751-8121/ac463e

Cited by 10 publications

(18 citation statements)

References 26 publications

(58 reference statements)

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“…Note that a = 0 corresponds to the previously studied case of the paraboloid of revolution [10,11]. The theory under consideration has a limiting transition at a → 0 to this simpler case of the convective BIE.…”

Section: Resultsmentioning

confidence: 99%

“…In the case of a paraboloid of revolution, it is possible to pass to curvilinear orthogonal coordinates reflecting the symmetry of the surface and isolate the integral of the Green function concerning variables x 1 , y 1 , t 1 in the convective term, which simplifies the case substantially [10].…”

Section: The Modelmentioning

confidence: 99%

“…The convective boundary integral containing the solute concentration C i at the solid/liquid interface takes the form [8,10] (1…”

Section: Thermo-chemical Dendrite Growthmentioning

confidence: 99%

“…This was first done by Saville and Beaghton [9] in the framework of solving a thermal problem with convection. Their method was then extended to describe the evolution of curvilinear crystallization fronts in binary undercooled liquids by Titova and Alexandrov [10]. Based on this theory, we recently showed that the perfect fluid hydrodynamic model can be used instead of the viscous flow model in the case of solidification of an undercooled metal [11].…”

Section: Introductionmentioning

confidence: 99%

“…are the tip diameters of the flat parabolas, −1 < a < 1, and D T is the thermal diffusivity. In dimensionless variables, the convective boundary integral for a stationary moving interface has the form[10] …”

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confidence: 99%

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Convective boundary integral equation: the case of a non-axisymmetric dendrite with a forced viscous flow

Титова¹,

Alexandrov²

2022

J. Phys. A: Math. Theor.

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The paper is devoted to obtaining an integral solution to the problem of convective heat transfer in the vicinity of the tip of a stationary growing nonaxisymmetric dendrite. The boundary integral equation for an elliptical paraboloid growing in a viscous forced flow is solved using the Green function technique. The total undercooling at the dendrite tip is found for single-component and binary melts, which is a function of the P\'eclet, Reynolds, and Prandtl numbers as well as the ellipticity parameter. Also, we demonstrate that these parameters substantially influence the total undercooling. We show that the increase of fluid flow and ellipticity of the crystal tip allows it to grow faster at fixed undercooling and average tip diameter. The 3D nonaxisymmetric theory under consideration is verified with previous solutions constructed by Ananth and Gill (Ananth and Gill 1989 {\it J. Fluid Mech.} {\bf 208} 575--593) for elliptic paraboloid and Alexandrov and Galenko (Alexandrov and Galenko 2021 {\it Phil. Trans. R. Soc. A} {\bf 379} 20200325) for a parabolid of revolution and a parabolic cylinder with a forced flow. The method developed can be used for the stationary growth of arbitrary patterns in the presence of convective flow.

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Section: Resultsmentioning

confidence: 99%

Section: The Modelmentioning

confidence: 99%

“…The convective boundary integral containing the solute concentration C i at the solid/liquid interface takes the form [8,10] (1…”

Section: Thermo-chemical Dendrite Growthmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Convective boundary integral equation: the case of a non-axisymmetric dendrite with a forced viscous flow

Титова¹,

Alexandrov²

2022

J. Phys. A: Math. Theor.

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The boundary integral equation describing a curvilinear solid/liquid interface with nonlinear phase transition temperature

Титова

Ivanov

Toropova

2023

Math Methods in App Sciences

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The boundary integral equation (BIE) describes the dynamics of a curved crystallization front separating liquid melt and solid material. We derive a generalized BIE for the thermal‐concentration problem taking into account the nonlinear dependence of the crystallization temperature on solute concentration and the kinetics of atomic attachment at the interface. This equation determines the evolution of the interface function and the equation for crystallization driving force ‐ the melt undercooling at the crystal surface. Our calculations carried out for a dendritic vertex in the form of a paraboloid of revolution have shown that the growth rate of the dendritic tip and its diameter substantially depend on the nonlinear effects under study. In particular, the velocity and diameter of the dendrite tip respectively become greater and narrower with increasing deviation of the liquidus equation from the linear relationship. Also, the dendrite tip velocity can be significantly affected by variations in the exponent of atomic kinetics.

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Analysis of the boundary integral equation for the growth of a parabolic/paraboloidal dendrite with convection

Титова

Alexandrov

2022

J. Phys.: Condens. Matter

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The growth of a parabolic/paraboloidal dendrite streamlined by viscous and potential flows in an undercooled one-component melt is analyzed using the boundary integral equation. The total melt undercooling is found as a function of the P\'eclet, Reynolds, and Prandtl numbers in two- and three-dimensional cases. The solution obtained coincides with the modified Ivantsov solution known from previous theories of crystal growth. Varying P\'eclet and Reynolds numbers we show that the melt undercooling practically coincides in cases of viscous and potential flows for a small Prandtl number, which is typical for metals. In cases of water solutions and non-metallic alloys, the Prandtl number is not small enough and the melt undercooling is substantially different for viscous and potential flows. In other words, a simpler potential flow hydrodynamic model can be used instead of a more complicated viscous flow model when studying the solidification of undercooled metals with convection.

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The boundary integral equation for curved solid/liquid interfaces propagating into a binary liquid with convection

Cited by 10 publications

References 26 publications

Convective boundary integral equation: the case of a non-axisymmetric dendrite with a forced viscous flow

Convective boundary integral equation: the case of a non-axisymmetric dendrite with a forced viscous flow

The boundary integral equation describing a curvilinear solid/liquid interface with nonlinear phase transition temperature

Analysis of the boundary integral equation for the growth of a parabolic/paraboloidal dendrite with convection

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