Extending the scope of microscopic solvability: Combination of the Kruskal-Segur method with Zauderer decomposition

Abstract

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...

Selection theory of free dendritic growth in a potential flow

“…After introducing the combination of Zauderer decomposition with the Kruskal-Segur approach recently [30,31], we have now presented the method in more detail. The analytic part of the calculation has been exemplified with a fully nonlinear problem.…”

“…Approximations that were introduced in Ref. [30] for didactic reasons have been removed, rendering the full power of the method visible.…”

“…Zauderer decomposition is the step allowing reduction of nonlinear bulk equations to an interface equation and thus circumventing the necessity of an exact integral equation, available only for linear bulk equations. Reference [30], dealing with potential flow, was more or less a proof of concept, in which we copiously used additional approximations to simplify the result to an easily digestable form, allowing us to map it in the end to the flowless finite Péclet number case treated by Ben Amar [22]. The main purpose of the present paper is to remove these approximations, which renders the treatment more complex, but does not impose insurmountable obstacles.…”

Influence of short-range forces on melting along grain boundaries

“…The equivalent Green's function formulation of the moving boundary problem is more convenient for our purposes in view of a combined numerical and analytical treatment [16][17][18], as well as by the need to resolve the dynamics of the process on various scales as depicted in Fig. 1.…”

Selection Theory of Dendritic Growth with Anisotropic Diffusion

“…It will turn out later that this neglect is indeed justified. Equation (14) is inserted into the interface conditions (15) and (16). When taking the total derivative of (15), we have to differentiate along the interface; that is, / = ( / ) + ( / ).…”