Consider flow over a flat plate where the surface of the plate has a temperature distribution. Here we are solving the situation for the unheated starting length but once the plate is heated, the steady-state temperature distribution obeys:

\[T_{s}=T_{\infty}+b x\]

The heat flux for an arbitrary wall temperature distribution is given by:

\[q_{s}^{\prime \prime}=\int_{0}^{x} h\left(x_{o}, x\right)\left(\frac{d T_{s}}{d x_{o}}\right) d x_{o}+\sum_{i=1}^{n} h\left(x_{o i}, x\right) \Delta T_{s i}\]

where \(\Delta T_{s i}\) represents any abrupt jumps in surface temperature. The idea here is that we break the calculation up into a series of unheated starting length problems and the sum them up to get the final heat flux.

a. Calculate the heat flux for the linear temperature ramp. You will encounter an integral of the form

\[\int_{0}^{x} Z^{m-1}(1-Z)^{n-1} d Z=\beta_{x}(m, n)\]

which is the definition of the incomplete beta function. Tabulated values can be found in [27] or [28].

b. Determine an expression for the average Nusselt number. This expression need not be in closed form or totally evaluated.