A boundary integral approach to unstable solidification

Abstract

We consider the supercooled Stefan problem with a general anisotropic curvature-and velocity-dependent boundary c<;mdition on the moving interface. This is a well-known model for pattern formation in unstable solidification.We reformulate the problem in terms of a quasilinear history-dependent singular integral equation for the velocity of the boundary. Using this equation, we carry out a new linear stability analysis of a planar solidification front with a general boundary condition. This analysis disagrees with the classical linear stability theory, because our approach includes transient effects due to initial conditions and linearizes the boundary conditions more accurately. Our analysis exhibits the smoothing role of velocity-dependence and the destabilizing effect of anisotropy.We then present numerical methods, also based on the integral equation formulation. These methods are able to follow the evolution of a periodic solidification front far into the nonlinear regime, with 0 (M) accuracy, where M is the time step. Previous work has been limited to short times and achieved slightly less than 0 (M 112 ) accuracy. Our methods also include a new algorithm for moving curves with curvature-dependent velocity.We present numerical results obtained with these new methods. After demonstrating first-order convergence for short time spans, we compare accurate numerical results with the predictions of the new and classical linear stability theories. Our results agree very well with the new theory, and disagree with the classical theory by as much as 25%. Then we study the long-time evolution of a dendritic front. We confirm the smoothing effect of velocity-dependence numerically. Our computations exhibit the beginnings of a sidebranching instability, followed by formation of a periodic cellular front, with an anisotropic boundary condition. Tip-splitting occurs instead, in the isotropic case .

Removing the stiffness from interfacial flows with surface tension

“…We have considered other ways to enforce this condition, but we have found that just explicitly forcing these coefficients to be 1 and 0, respectively, performs the best numerically. We remark that Strain also noted this difficulty in [48], but he did not seem to employ a correction of it computationally.…”

“…See, for example, Strain [48] in the context of unstable solidification, or Goldstein and Petrich [ 20 ] in the context of integrable curve dynamics. We derive it here for purposes of completeness and illustration.…”

Convergence of a non-stiff boundary integral method for interfacial flows with surface tension

“…In these two new variables, the surface tension term has a very simple form: Sκ α = S(θ α /σ) α . This approach has been used previously by Kessler, Koplik and Levine [25] for interface evolution problems, by Strain [40] in the context of solidification, by Goldstein and Petrich [21] for integrable closed curve dynamics, and by Meiburg and Homsy [32] for Hele-Shaw flows.…”

“…Boundary integral methods are a popular choice for simulations of interfacial flows and have been used extensively to compute nonlinear surface wave [6], [19], [30], [35], [37], vortex sheet (no density difference) motion [3], [23], [26], [27], [36], [38], Rayleigh-Taylor instability [5], [35], [44], interfaces in Hele-Shaw cells [17], [24], and crystal growth and solidification [29], [40]. The advantage of boundary integral methods is that they reduce the dimension of the problem by involving quantities along the interface only [34].…”

Numerical Methods for Solidification Processes in Materials Science

“…Numerical results agreed closely with linear stability theory and predicted correct tip-splitting phenomena in the nonlinear regime. The method and numerical results are presented in (Strain, 1989).…”