Analytical Scaling Investigation of Free Convection Flow of a Hot Fluid Over a Cooler Vertical Isothermal Wall

Natural convection boundary layer flow of a hot fluid past a cooler semi-infinite plate is analyzed using the method of scale analysis. This kind of flow is a gravity opposing flow. For Prandtl numbers greater than one, there exist two separate boundary layers, namely, the thermal and viscous boundary layers. At moderate Prandtl numbers, these two layers are almost identical but become distinct at higher Prandtl numbers. This is particularly so for heavy oils whose Prandtl number can be as high as 100 000. As the Prandtl number increases, direct numerical integration of the governing equations become inaccurate because there is a very small parameter multiplying the highest derivative. This small parameter is the inverse of the root of the Prandtl number. The method of matched asymptotic expansion is therefore used to compute temperatures, velocities, Nusselt number and skin friction coefficient at moderate and high Prandtl numbers. Five terms inner expansion were matched with five terms outer expansion at the edge of the thermal boundary layer. Results obtained showed that for Prandtl number of up to 100000, both the Nusselt number and skin friction coefficient approached different asymptotes. Also, in all cases considered in this study, the skin friction coefficient and Nusselt numbers for the cold plate immersed in hot fluid are greater than for the case of hot plate immersed in cold fluid when they were compared.