On the heat transferred to the air surrounding a semi-infinite inclined hot plate

Abstract

AbstractAn asymptotic analysis of laminar free convection in a boundary layer over an isothermal semi-infinite flat plate inclined at some angle to the vertical has been performed. Existing analytical solutions show no difference in the heat-transfer rate between the upper and lower surfaces of the plate, contrary to observations. To investigate this, higher-order perturbations of the non-dimensional temperature, velocity and pressure across the boundary layer were computed and found to show only small variations from first-order perturbations previously reported. Unexpectedly, third-order perturbations of all functions were found to be identical to those of the vertical plate, indicating that differences in temperature between both sides of the plate are limited to exceedingly small terms of order ${x}^{- 9/ 4} $ or smaller, $x$ being the distance from the leading edge, non-dimensionalized by the buoyancy length scale. Dominant differences between heat-transfer rates on the upper and lower surfaces were therefore concluded to be due to near-leading-edge effects. In applying an integral form of the conservation equations to the near-leading-edge region, it was found that, up to terms of order unity in $x$, the total heat-exchange rate for the inclined plate is identical to that of the vertical plate, so that the heat-transfer gain on one side balances exactly the loss occurring on the other. This simplification allowed determination of an upper bound for differences in heat-transfer rates between the upper and lower sides, even though complete determination of the differences would require a numerical integration of the full Navier–Stokes equations near the leading edge of the plate.