Higher-Order Approximations for Darcian Free Convective Flow About a Semi-Infinite Vertical Flat Plate

Higher-order effects of Darcian free convection boundary-layer flow adjacent to a semi-infinite vertical flat plate with a power law variation of wall temperature (i.e., Tˆ_w αxˆ^λ for xˆ≥0) are examined theoretically in this paper. The method of matched asymptotic expansions is used to construct inner and outer expansions. The small parameter of the perturbation series is the inverse of the square root of the Rayleigh number. The leading term in the inner expansions is taken to be the boundary layer theory with the second-order term due to the entrainment effect, and the third-order term due to the transverse pressure gradient and the streamwise heat conduction. The ordering of the term due to the leading edge effect depends on the wall temperature distribution; this term is determinate within a multiplicative constant owing to the appearance of an eigenfunction in the inner expansion. Thus, the perturbation solutions are carried out up to this term. For the case of an isothermal vertical plate (λ = 0), the second-order corrections for both the Nusselt number and the vertical velocity are zero, with the leading edge effect appearing in the third-order term. For λ>0, both the second- and third-order corrections in the Nusselt number are positive. The increase in surface heat flux is due to the fact that the higher-order effects increase the velocity parallel to the heated surface. The boundary layer theory for the prediction of the Nusselt number is shown to be quite accurate even at small Rayleigh number for 0≤λ≤1/3. The higher order effects tend to have a stronger influence on the velocity distribution than the temperature distribution. These effects become more pronounced as λ is increased from λ=1/3, or as the Rayleigh number is decreased.