Heat transfer to a quadratic shear profile

An operational method is used to obtain an exact solution for the heat transfer from a surface on which the temperature is prescribed to a stream in which the velocity profile is given by \[ u = (\mu_{e}\rho_w)^{-1}\left[\tau_w\eta + \left(\frac{1}{2\rho_w}\frac{dp}{dx}\right)\eta^2\right], \] where \[ \eta = \int_0^y \rho dy \] and other symbols have their usual meanings. The solution is expanded for small and large values of a dimensionless parameter proportional to , where \[ \alpha = \sigma\rho_w\tau_w\frac{l^2}{\mu^2_w},\quad\beta = \bigg(\frac{\sigma\rho_w}{\mu^2_w}\bigg)\frac{dp}{dx}l^3, \] and expansions for small and large values of are given. Over the whole range both are adequately represented for experimental purposes by the equation \[ c_1\alpha + \frac{c_2\beta}{Nu} = Nu^3, \] which has a form suggested by consideration of the integral approximation of Curle (1962). The experimental application of the results to both laminar and turbulent flows is discussed.