Thickness data, which span a wider time (t) range than previous data, are presented for the liquid phase epitaxial growth of GaAs and Al0.4Ga0.6As by the near equilibrium cooling technique. It is demonstrated that for longer times the layer thickness data are consistent with t32 diffusion limited growth, while for short times there is a relatively rapid initial growth. This rapid initial data is followed by a weak time dependence which may be attributed to a flow mechanism in the melt produced by movement of the slider and driven by supersaturation.