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

Abstract

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 non-axisymmetric 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 Péclet, 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 non-axisymmetric theory under consideration is verified with previous solutions constructed by Ananth and Gill (1989 J. Fluid Mech. 208 575–593) for elliptic paraboloid and Alexandrov and Galenko (2021 Phil. Trans. R. Soc. A 379 20200325) for a paraboloid 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.