Stagnation-Point Heat Transfer Near the Continuum Limit

Abstract

H EAT and momentum transfer at the stagnation point is a problem of theoretical and practical interest. Solution of the Navier–Stokes equations at the stagnation point is one of oldest known solutions to the Navier–Stokes equations [1,2] and is closely related to boundary-layer flow [3]. Once the fluid flow is computed, the heat transfer can be computed in both 2-dimensional [4] and axisymmetric [5] geometries. The importance of stagnation-point heat transfer in problems such as atmospheric reentry [6] and other rarefied hypersonic flows [7] make estimating the heat transfer a problem of practical engineering interest. Initial attempts to solve stagnation-point flow and boundary-layer flow with a slip boundary condition using perturbation methods [8] suggested that the slip condition would not affect shear stress or heat transfer. Amore complete thermal analysis partially contradicted this result, suggesting that heat transfer in a laminar boundary layer decreased in the presence of a slip boundary condition [9]. The apparent lack of a change in shear stress due to the slip condition led to the conclusion that the terms added by the slip boundary condition were smaller than the discarded second-order terms in the boundary-layer equations [10]. This led to the conclusion that slip could be ignored in both laminar boundary-layer and stagnationpoint flows. These conclusions were challenged by numerical results, including solution of the linearized Boltzmann equation for stagnation-point flow [11], solution of stagnation-point flow with slip [12], and solution of the Blasius boundary-layer equations with slip flow that incorporated the loss of self-similarity [13]. All of these analyses showed decreased shear stress and boundary-layer thickness. When heat transfer was incorporated in the boundarylayer analysis, the heat transfer decreased from the equilibrium values. The present work extends previous analysis of the fluid flow and heat transfer in the presence of a slip boundary condition [12,14] to cover rarefied flow, in which the temperature jump and slip boundary conditions are coupled. This analysis provides an estimate for change in heat transfer due to rarefied-flow effects for a range of Knudsen and Prandlt numbers for both monatomic and diatomic gases.