Boundary Layer Treatment of the Porous Channel Flow with Wall Regression

Abstract

This work extends a sequence of studies devoted to the analysis of the laminar flow in porous channels with retracting walls. This problem was originally used to model slab propellant grain regression by Zhou and Majdalani (Zhou, C., and Majdalani, J., “Improved Mean Flow Solution for Slab Rocket Motors with Regressing Walls,” Journal of Propulsion and Power, Vol. 18, No. 3, 2002, pp. 703711. doi: 10.2514/2.5987). After identifying a subtle endpoint singularity that affects the former solution in its third derivative, a slow variable is introduced to capture the rapid variations in the channel’s core. The core refers to the midsection plane where the shear layer is displaced due to hard blowing at the walls. Then using matched-asymptotic expansions with logarithmic corrections, a composite solution is developed following successive integrations that start with the third derivative. In the process, the inner correction is retrieved from the fourth-order equation governing the symmetric injection-driven flow near the core. The resulting approximation is expressed in terms of generalized hypergeometric functions and is confirmed using numerics and limiting process verifications. The composite solution is shown to outperform the former, outer solution, as the core is approached or as the injection Reynolds number is increased. Without undermining the practicality of the former solution outside the thin core region, the development of a matched-asymptotic approximation enables us to suppress singular terms, thus ensuring a uniformly valid outcome down to the fourth derivative.