Taylor-Prandtl modification of Reynolds analogy for heat

Taylor-Prandtl modification of Reynolds analogy for heat transfer and mass transfer 

The original Reynolds analogy involves a number of simplifying assumptions which are justifiable only in a limited range of conditions. Thus it was assumed that fluid was transferred from outside the boundary layer to the surface without mixing with the intervening fluid, that it was brought to rest at the surface, and that theirnal equilibrium was established. Various modifications have been made to this simple theory to take account of the existence of the laminar sub-layer and the buffer layer close to the surface. 

Taylor and Prandtu - allowed for the existence of the laminar sub-layer but ignored the existence of the buffer layer in their treatment and assumed that the simple Reynolds analogy was applicable to the transfer of heat and momentum from the main stream to the edge of the laminar sub-layer of thickness 8b- Transfer through the laminar sub-layer was then presumed to be attributable solely to molecular motion. 

The rate of transfer of heat by conduction through the laminar sub-layer from a surface of area A is given by  

The rate of transfer of momentum is equal to the shearing force and therefore  

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