Tip splitting without interfacial tension and dendritic growth patterns arising from molecular anisotropy

Dynamical systems as models for physical processes

“…Random fractals, on the other hand, may very well exist. Computer generated random fractals (from algorithms or physically-based models) resemble natural objects quite well [1,2,10,11]. However, for reasons similar to those discussed previously, fractals and scaling may not be as abundant as claimed, and their existence may be restricted to a certain range of scales depending on the "connectivity" of the generating physical system to other systems [12].…”

“…The actual growth and shape of a snowflake is subjected to Darcy's law [111 and to environmental effects (noise). Snowflakes have been simulated by a statistical mechanical model [11] that incorporates a parameter producing the six-fold anisotropy of natural flakes, Darcy's law, and random motion of water molecules. Such simulations, unlike the Koch snowflake (Figure 1), which is an exact fractal, produce flakes that are random fractals having a dimension of about 1.5, and display a striking resemblance to natural snowflakes [22].…”

A boundary integral approach to unstable solidification

“…The large-scale shape of the resulting solid -for example the diameter of a snowflake-is typically a reproducible function of a few physical parameters such as the undercooling, but the detailed small-scale structure has a complex but organized appearance. Currently many workers [5,6,24,67,66,58,59,60,85,7,91] are interested in this problem of pattern formation.…”

“…Meiron [73] has exhibited similar tip-splitting instabilities in an approximate steady-state model. Nittmann & Stanley [85] also discuss connections between tip-splitting, sidebranching and anisotropy, in a more general context. This computation failed at about t = 5.4, due to exponent overflow.…”

“…Numerical methods have been successfully applied to steady-state equations [72,91] and to equations for smoothed interfaces [13,34]. Only a few methods have been proposed to solve the full time-dependent problem [21,94,99], and it seems that none have been capable of computing the interface accurately far into the nonlinear regime, where interesting phenomena like sidebranching [53], tip-splitting [85], and formation of cellular fronts [66] are presumed to occur.…”