Dendritic crystal growth for weak undercooling. II. Surface energy effectson nonlinear evolution

We extend the previous work of Kunka, Foster, and Tanveer [Phys. Rev. E 56, 3068 (1997)] by incorporating small but nonzero surface energy effects in the nonlinear dynamics of a conformal mapping function z(zeta,t) that maps the upper-half zeta plane into the exterior of a dendrite. In this paper, w e specifically examine surface energy effects on the singularities of z(zet a, t) in the lower-half zeta plane, as they move toward the real axis from below. Until the time when any of the singularities of the corresponding ze ro-surface-energy solution or a surface-energy-generated daughter singulari ty cluster comes very close to the real axis, the leading-order outer solut ion is the zero-surface-energy solution in a strip of the lower-half comple x that includes the real axis (i.e., the interface). There is an inner regi on around each singularity of the zero-surface-energy solution where surfac e energy plays a dominant role. However, the scalings in such an inner regi on, and hence the equation itself, must be modified when such singularities are very close to the real axis. The relative ordering of anisotropy, surf ace energy, and singularity strength strongly influences the form of the in ner equations and hence their solutions. A singularity with initial strengt h weaker than some critical value is dissipated over a fast time scale by s urface energy effects, leaving no trace of the initial singularity. This cu toff in singularity strength limits the size and growth rate of the interfa cial disturbances that singularities generate. Also, the variation of time scale over which surface energy acts, due to differing singularity strength s in an ensemble, is shown to account for a \y\(1/2) coarsening rate for so me intermediate range of distances, \y\, from the dendrite tip. As in the c ase of the isotropic Hele-Shaw problem [S. Tanveer, Philos. Trans. R. Sec. London, Ser. A 343, 155 (1993)], we find here too that each initial zero of z(zeta) gives birth to a "daughter" singularity cluster that moves away fr om the zero and necessarily approaches the real axis, before dispersing. On e effect of this "daughter" singularity cluster, if it approaches the real axis before any other singularity, is to singularly perturb a smoothly evol ving zero-surface-energy solution. In addition, numerical and analytical re sults fora certain general class of initial conditions indicate that daught er-singularity effects necessarily prevent an interface from ever approachi ng the cusp implied by the corresponding zero-surface-energy solution. Fina lly, we find that for a set of localized distortions, the local rescaling o f dependent and independent variables (i.e., on an "inner scale"). leads to the original equations, with an effectively larger surface-energy paramete r. [S1063-651X(99)04501-8].