Semisimilar solutions for unsteady free-convective boundary-layer flow on a vertical flat plate

The analysis of unsteady free convection has classically been made difficult because of the singularities which occur in the governing boundary-layer equations, and because anomalies often occur which are related to the occurrence of these singularities. In the present paper a semisimilar analysis of unsteady free convection in the vicinity of a vertical flat plate is presented, wherein a number of possible wall-temperature variations with time and position are derived. Unique scalings are formulated for the semisimilar equations that aid in the numerical solutions and in the physical interpretation of the results. These scalings collapse the infinite time and position coordinates into a finite region, and present the semisimilar problem in a format bounded by similarity equations. Solutions are carried out which indicate the occurrence of overshoots in the temperature profiles and heat transfer for a variety of conditions. Also, concepts such as the ‘limit-of-pure-conduction’ and ‘leading-edge penetration distance’ are shown to require special interpretation under variable wall-temperature conditions.