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

Abstract

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.