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.