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

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Abstract
The heat transfer to constant-property laminar boundary layer with power-function variations of free stream velocity ( u 1 = c x m ) ({u_1} = c{x^m}) and of temperature difference between wall and free stream ( T 0 − T 1 = b x n ) ({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 h can be approximately given in the form \[ h x / k ( u 1 x / v ) 1 / 2 = 1 ( 2 − β ) 1 / 2 [ Γ ( 2 / 3 ) 3 2 / 3 Γ ( 4 / 3 ) { 1 2 + n ( 2 − β ) } 1 / 3 ( σ α ) 1 / 3 − β 10 α ] , \frac {{hx/k}}{{{{\left ( {{u_1}x/v} \right )}^{1/2}}}} = \frac {1}{{{{\left ( {2 - \beta } \right )}^{1/2}}}}\left [ {\frac {{\Gamma \left ( {2/3} \right )}}{{{3^{2/3}}\Gamma \left ( {4/3} \right )}}{{\left \{ {\frac {1}{2} + n\left ( {2 - \beta } \right )} \right \}}^{1/3}}{{\left ( {\sigma \alpha } \right )}^{1/3}} - \frac {\beta }{{10\alpha }}} \right ], \] where β = 2 m / ( m + 1 ) \beta = 2m/(m + 1) , α \alpha is the non-dimensional velocity gradient at the wall (usually expressed as α = f ( 0 ) \alpha = f(0) ), σ \sigma is the Prandtl number, k k is the thermal conductivity, and v v is the kinematic viscosity.