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 $z(\ensuremath{\zeta},t)$ that maps the upper-half \ensuremath{\zeta} plane into the exterior of a dendrite. In this paper, we specifically examine surface energy effects on the singularities of $z(\ensuremath{\zeta},t)$ in the lower-half \ensuremath{\zeta} plane, as they move toward the real axis from below. Until the time when any of the singularities of the corresponding zero-surface-energy solution or a surface-energy-generated daughter singularity cluster comes very close to the real axis, the leading-order outer solution is the zero-surface-energy solution in a strip of the lower-half complex that includes the real axis (i.e., the interface). There is an inner region around each singularity of the zero-surface-energy solution where surface energy plays a dominant role. However, the scalings in such an inner region, and hence the equation itself, must be modified when such singularities are very close to the real axis. The relative ordering of anisotropy, surface energy, and singularity strength strongly influences the form of the inner equations and hence their solutions. A singularity with initial strength weaker than some critical value is dissipated over a fast time scale by surface energy effects, leaving no trace of the initial singularity. This cutoff in singularity strength limits the size and growth rate of the interfacial disturbances that singularities generate. Also, the variation of time scale over which surface energy acts, due to differing singularity strengths in an ensemble, is shown to account for a $|y{|}^{1/2}$ coarsening rate for some intermediate range of distances, |y|, from the dendrite tip. As in the case of the isotropic Hele-Shaw problem [S. Tanveer, Philos. Trans. R. Soc. London, Ser. A 343, 155 (1993)], we find here too that each initial zero of ${z}_{\ensuremath{\zeta}}$ gives birth to a ``daughter'' singularity cluster that moves away from the zero and necessarily approaches the real axis, before dispersing. One effect of this ``daughter'' singularity cluster, if it approaches the real axis before any other singularity, is to singularly perturb a smoothly evolving zero-surface-energy solution. In addition, numerical and analytical results for a certain general class of initial conditions indicate that daughter-singularity effects necessarily prevent an interface from ever approaching the cusp implied by the corresponding zero-surface-energy solution. Finally, we find that for a set of localized distortions, the local rescaling of dependent and independent variables (i.e., on an ``inner scale'') leads to the original equations, with an effectively larger surface-energy parameter.
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