On the heat transfer to constant-property laminar boundary layer with power-function free-stream velocity and wall-temperature distributions

The heat transfer to constant-property laminar boundary layer with power-function variations of free stream velocity $({u_1} = c{x^m})$ and of temperature difference between wall and free stream $({T_0} - {T_1} = b{x^n})$ is studied by means of an improved version of the WKB method developed by the author. It is found that the local heat-transfer coefficient $h$ can be approximately given in the form \[ \frac {{hx/k}}{{{{\left ( {{u_1}x/v} \right )}^{1/2}}}} = \frac {1}{{{{\left ( {2 - \beta } \right )}^{1/2}}}}\left [ {\frac {{\Gamma \left ( {2/3} \right )}}{{{3^{2/3}}\Gamma \left ( {4/3} \right )}}{{\left \{ {\frac {1}{2} + n\left ( {2 - \beta } \right )} \right \}}^{1/3}}{{\left ( {\sigma \alpha } \right )}^{1/3}} - \frac {\beta }{{10\alpha }}} \right ],\] where $\beta = 2m/(m + 1)$, $\alpha$ is the non-dimensional velocity gradient at the wall (usually expressed as $\alpha = f”(0)$), $\sigma$ is the Prandtl number, $k$ is the thermal conductivity, and $v$ is the kinematic viscosity.