We use a combined phase-field/lattice-Boltzmann scheme [D. Medvedev, K. Kassner, Phys. Rev. E 72, 056703 (2005)] to simulate non-facetted crystal growth from an undercooled melt in external flows. Selected growth parameters are determined numerically.For growth patterns at moderate to high undercooling and relatively large anisotropy, the values of the tip radius and selection parameter plotted as a function of the Péclet number fall approximately on single curves. Hence, it may be argued that a parallel flow changes the selected tip radius and growth velocity solely by modifying (increasing) the Péclet number. This has interesting implications for the availability of current selection theories as predictors of growth characteristics under flow.At smaller anisotropy, a modification of the morphology diagram in the plane undercooling versus anisotropy is observed. The transition line from dendrites to doublons is shifted in favour of dendritic patterns, which become faster than doublons as the flow speed is increased, thus rendering the basin of attraction of dendritic structures larger.For small anisotropy and Prandtl number, we find oscillations of the tip velocity in the presence of flow. On increasing the fluid viscosity or decreasing the flow velocity, we observe a reduction in the amplitude of these oscillations.

Abstract. -Successful applications of the Kruskal-Segur approach to interfacial pattern formation have remained limited due to the necessity of an integral formulation of the problem. This excludes nonlinear bulk equations, rendering convection intractable. Combining the method with Zauderer's asymptotic decomposition scheme, we are able to strongly extend its scope of applicability and solve selection problems based on free boundary formulations in terms of partial differential equations alone. To demonstrate the technique, we give the first analytic solution of the problem of velocity selection for dendritic growth in a forced potential flow.The fundamental equations describing the growth of a crystal into its undercooled melt are very difficult to solve, if surface tension effects are accounted for, even when restriced to the simplest case of merely diffusive heat transport. On the other hand, the capillary length d 0 describing these effects is typically very small in comparison with other length scales of the problem such as the sizes of growing patterns or the diffusion length. Therefore, it was a natural step to first look for solutions with d 0 set equal to zero. This simplified problem was solved exactly by Ivantsov [1] who showed that the crystal can grow in the shape of a parabola in 2D or a paraboloid in 3D. A major drawback of these solutions is that they constitute a whole continuum for any given undercooling: the mathematics fixes only the Péclet number P c = V ρ/D, where V is the growth velocity of the crystal, ρ the tip radius of the parabolic needle, and D the thermal diffusion coefficient. Hence, only the product of velocity and length scale is determined, but neither of the two quantities separately. In experiments, a given undercooling leads to both a welldefined growth velocity and a well-defined tip radius of the needle crystal, which after developing side branches is called a dendrite. This situation became known as the selection problem of diffusion-limited dendritic growth and is was not solved until some twenty years ago [2][3][4][5], with the advent of microscopic solvability theory.Because the theory was mathematically complex and not very intuitive, it failed to enjoy unanimous appraisal. Moreover, its success in explaining experiments remained controversial to some extent [6]. It has been emphasized by Tanveer [7] that even small fluid flows in the melt might account for changes in the theoretically predicted scalings as the problem is structurally unstable. Hence, selection theory should be extended to nondiffusive transport such as convection. To our knowledge, the only approach to solvability theory available so far for models with convection is due to Bouissou and Pelcé (BP) [8]. Their method relies on a linearized solvability condition, which prevents it from becoming exact in the limit of vanishing d 0 . Also, it has been shown [9] that nonlinearity may be crucial in problems involving multiple parameters. Hence, a method would be more than desirable that takes nonlinear sol...

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