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

Титова, Э. А.; Alexandrov, Dmitri V.

doi:10.1088/1361-648x/ac623e

Cited by 3 publications

(6 citation statements)

References 33 publications

(43 reference statements)

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“…At a → 1 (figure 2) we might have expected the transition to the 2D case. However, in 2D with the same Péclet number p, the undercooling is larger than in 3D, and we have the opposite situation [11]. Namely, the undercooling in figure 2 becomes lower with increasing the ellipticity parameter a.…”

Section: Resultsmentioning

confidence: 66%

“…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%

“…At the dendritic surface, the heat balance condition is given by dT dw w=1 = −p. (11) In addition, the temperature tends to a constant value far away from the growing dendrite…”

Section: The Temperature Gradientmentioning

confidence: 99%

“…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]. It is worth noting that one of the important applications of the boundary integral theory is the study of various shapes of growing patterns in an undercooled liquid.…”

Section: Introductionmentioning

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: 66%

Section: Resultsmentioning

confidence: 99%

“…At the dendritic surface, the heat balance condition is given by dT dw w=1 = −p. (11) In addition, the temperature tends to a constant value far away from the growing dendrite…”

Section: The Temperature Gradientmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 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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“…(i) A numerical simulation of dendritic growth in an inclined liquid flow should be carried out, taking into account the conductive and convective heat and mass transfer on various (with respect to the flow) sides of the dendrite. Such simulations can be done using a direct solution of the heat and mass transfer problem with flow (e.g., the phase-field or enthalpy methods) [53][54][55][56], as well as using the convective boundary integral equation [57][58][59]; (ii) Intense convective currents on the upstream side of the dendrite are a potential source of morphological instability on its surface. It is therefore necessary to study the surface stability of the dendrite crystal to small morphological perturbations on its upstream side.…”

Section: Discussionmentioning

confidence: 99%

The Tip of Dendritic Crystal in an Inclined Viscous Flow

et al. 2022

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We study the flow around the tip of a dendritic crystal by an inclined stream of viscous incompressible liquid. The tip shape is chosen accordingly to recent theory[Phil. Trans. R. Soc. A 2020, 378, 20190243] confirmed by a number of experiments and computations [Phil. Trans. R. Soc. A 2021, 379, 20200326]. Our simulations have been carried out for a 0, 30, 60, and 90-degree flow slope to the dendrite axis. We show that the stream inclination has a significant effect on the hydrodynamic flow and shear stress. In particular, a transition from laminar to turbulent currents on the upstream side of the dendritic crystal may occur in an inclined hydrodynamic flow. This leads to the fact that the heat and mass transfer mechanisms on the upstream and downstream sides of a growing dendritic crystal may be different.

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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

Cited by 3 publications

References 33 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 Tip of Dendritic Crystal in an Inclined Viscous Flow

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

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