  Deformations of the Reeb flow of a Sasakian manifold as transversely K\"ahler
flows may not admit compatible Sasakian metrics anymore. We show that the
triviality of the (0,2)-component of the basic Euler class characterizes the
existence of compatible Sasakian metrics for given small deformations of the
Reeb flow as transversely holomorphic Riemannian flows. We also prove a
Kodaira-Akizuki-Nakano type vanishing theorem for basic Dolbeault cohomology of
homologically orientable transversely K\"ahler foliations. As a consequence of
these results, we show that any small deformations of the Reeb flow of a
positive Sasakian manifold admit compatible Sasakian metrics.
