Asymptotic expansion of the full nonlocal solidification problem

We analyze the shape z(x) of two-dimensional needle crystals far away from the tip and find that in general the deviation Δz away from the Ivantsov solution has an asymptotic behavior of the form Δz∼$x^{\mathrm{-}\alpha }$, with α a noninteger exponent. For the asymptotic behavior, the regime where the Péclet number p is less than (1/2) and the one where p is larger than (1/2) are distinct. For p>(1/2), the exponent is calculated explicitly, while for pα. These results differ from those used in earlier numerical and analytical studies of two-dimensional dendritic growth.