The minimum total energy (MTE) criteria for determining the minimum wetting rate and the minimum thickness of an isothermal, thin liquid film flowing down a vertical adiabatic surface are examined. An analytical expression of the profile of a stable liquid rivulet and of the two-dimensional velocity distribution in the rivulet developed using the Ritz method are incorporated into the MTE criterion to improve its predictions of the minimum wetting rate (MWR) and the corresponding minimum liquid film thickness (MLFT). The present predictions of the MWR and MLFT are in good agreement, to within +/-10-20%, with experimental data for water and glycerol-water mixtures and with the values calculated from the MWR measurements, respectively. The accuracy of the present predictions is demonstrated for three conditions, namely: (a) incipient breakup of liquid film, (b) formation of a stable dry patch, and (c) rewetting of a dry surface. For the latter the advancing contact angle is substituted for the equilibrium contact angle in the developed analytical expressions for the MLFT and corresponding MWR.