Asymptotic behavior of steady dendrite growth at far field

The asymptotic behavior of the solution to steady dendrite growth with isotropic surface tension is examined. The results show that (1) at far field (z→∞), the deviation of the nonisothermal dendrite from an Ivantsov &, where the eigenvalue ${\alpha }_1$ is within the interval (0,(1/2) and can be determined by the undercooling of the melt and (2) the asymptotic character of the dendrite at far field cannot be fully determined locally. It is also influenced by the behavior of the solution at the tip region. These results demonstrate that the conventional far-field boundary condition in which the difference between the dendrite and the Ivantsov paraboloid is vanishingly small is not correct. Moreover, the results also imply that for any finite surface tension linearization around the Ivantsov solution corresponding to zero surface tension may not be uniformly valid in the whole region (0≤z<∞). As a consequence, the conclusions drawn from the linearized system, such as the nonexistence of the steady solution and the ‘‘solvability condition theory,’’ are questionable.