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We study scaling laws characterizing the inter-diffusive zone of two miscible fluids flowing side by side in a Y-shape laminar micromixer using the lattice Boltzmann method. The lattice Boltzmann method solves the coupled 3D hydrodynamics and mass transfer equations and incorporates intrinsic features of 3D flows related to this problem. We observe the different power law regimes occurring at the center of the channel and close to the top/bottom wall. The extent of the inter-diffusive zone scales as square root of the axial distance at the center of the channel. At the top/bottom wall, we find an exponent 1/3 at early stages of mixing as observed in the experiments of Ismagilov and coworkers [Appl. Phys. Lett. 76,2376 (2000)]. At a larger distance from the entrance, however, the scaling exponent close to the walls changes to 1/2. For a channel with infinite aspect ratio (width/height), a criterion for this cross over can be given using the homogenization of the tracer concentration field along the shortest dimension (height) of the channel [J.-B. Salmon et al J. Appl. Phys.101, 074902 (2007)]. We analyze the effect of finite aspect ratio and volumetric flow rate on the extent of diffusive broadening. Interestingly, we find the same scaling laws regardless of the channel's aspect ratio. However, the point at which the exponent 1/3 characterizing the broadening at the top/bottom wall reverts to the normal diffusive behavior downstream varies with the aspect ratio.