Abstract An analysis is presented for steady free-convection flow past a semi-infinite vertical flat plate at large Grashof numbers. If it is assumed that the wall temperature varies as a power of the distance from the leading edge of the plate, then the governing equations can be reduced to a set of ordinary differential equations by the use of a similarity variable. Numerical and asymptotic solutions of these equations are given. The unsteady approach to these solutions are also investigated by considering the impulsive heating of the plate. If the temperature increases along the length of the plate, numerical solutions are presented which match the large- and small-time solutions. However, no matching of these limiting solutions has been achieved where the temperature decreases along the length of the plate. An asymptotic solution, which is valid at large values of time, is also given. For all the temperature distributions at the plate that are considered in this paper the disturbance from the leading edge of the plate travels fastest within the boundary layer. The unsolved problem, in which the temperature is impulsively increased to a constant value, is a special case of the problem considered here.