Steady flow through a slender tapered vessel with a partially permeable wall and closed end

We study the flow of a viscous fluid through a slender tapered tube whose radius may reduce to zero.The vessel is closed at the end, so that the flow is made possible owing to the fact thata portionof the tube wall is permeable.The smallness of the tube aspect ratio is exploited using an upscaling technique leading to adegenerate differential equation for pressure. Solutions are found either in explicit form or as power series expansions. This class of flows may represent, though in a largely approximated way, the blood flow though a coronary artery.