The present paper is concerned with  steady eutectic  growth in  the directional solidification of a binary mixture by using the two-dimensional model with the isotropic surface tension. We consider the case that Peclet number ε is small and the segregation coefficient κ is closed to the unit. The Peclet number ε is defined as the ratio of  the half of primary spacing and mass diffusion length. By using multiple variable expansion and matched asymptotic method and exploring the singularity existing at the triple point, we obtain  the global uniformly valid asymptotic solution for the steady state in the whole physical space. It is shown that the steady state solution  contains two free parameters, the tilted angle parameter, which determines the orientation of the liquid open angle at the triple point and the Peclet number, which measures the primary spacing. Hence, given growth conditions the primary space may not be uniquely selected. The results show that the system has a boundary layer with the thickness of O(ε^1/2), where the slope and curvature of the interface are drastically large, hence the effect of isotropic surface tension pays a significant role and the interface has a noticeable non-uniform undercooling temperature. Our results lead to a self-consistent theory for steady lamellar eutectic growth, and raise the question on the validity of the hypothesis proposed by Jackson-Hunt, that the steady state with the minimum average undercooling at the interface is uniquely selected. The comparisons between the theoretical predictions and the available experimental observations have been made, which show quite good quantitative agreements.

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