Scaling laws of free dendritic growth in a forced Oseen flow

Abstract

We use the method presented in M von Kurnatowski et al (2013 Phys. Rev. E 87 042405) to solve the nonlinear problem of free dendritic growth in an Oseen flow. The growth process is assumed to be limited by thermal transport via diffusion and convection. A singular perturbation expansion is treated to lowest nontrivial order in the framework of asymptotic decomposition. The resulting complex integro-differential equation is solved using an elaborate numerical method. The approximate scaling laws and for the growth velocity and the tip radius of curvature of the dendrite, respectively, are found as a function of the forced flow velocity. The results are compared to those by Pelcé and Bouissou, constituting the only other attempt so far to treat the problem analytically.

Thermo-solutal and kinetic modes of stable dendritic growth with different symmetries of crystalline anisotropy in the presence of convection

“…A rigorous method allowing one to treat the selection theory with nonlinear equations of motion has been developed by Kassner et al for dendrite growth in a forced potential flow [94] and under Oseen flow [95]. Indeed, the Nash-Glicksman integral formulation of the pattern formation problem [93] does not allow the analysis of nonlinear bulk equations that render, for instance, convection intractable.…”

“…Indeed, the Nash-Glicksman integral formulation of the pattern formation problem [93] does not allow the analysis of nonlinear bulk equations that render, for instance, convection intractable. As such, Kassner et al [94][95][96] extended the nonlinear formulation of the pattern formation problem by its re-formulation in terms of partial differential equations alone. One of the formal advantages of this extended method of Kassner et al is that it paves the way for rigorous nonlinear asymptotic analysis which might be correctly applied to the problem involving multiple parameters [92].…”

Selection Theory of Dendritic Growth with Anisotropic Diffusion

“…For given values of Δ, 2 , and , the set (13), (20) is solved numerically. For details about the numerical method, see [9,17]. Explicit results are shown for the substance CCH5 (4-pentyl-4 -cyano-trans 1,1-bicyclohexane, = 51.2 ∘ C [18], 2 = 0.06 ⋅ ⋅ ⋅ 0.18 [13], = 0.767, = 1.25 ⋅ 10 5 m 2 /s, and 0 = 5 ⋅ 10 −6 m).…”

“…with the dimensionless undercooling Δ = ( − ∞ )/ . The mathematical structure of problem (7), (8), (9), and (10) is almost the same as in the case of isotropic diffusion [2]. The difference lies in ( ) and in the function…”

“…Recently, the presence of grain boundaries was successfully included into the theory [6]. Furthermore, the selection problem was treated for convective systems, where the flow introduces additional complexity [7][8][9]. Interfacial patterns have also been observed in liquid crystals at nematic-isotropic boundaries [10,11], as well as at nematic-smectic-B boundaries [12,13].…”