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 that a portion of the tube wall is permeable. The smallness of the tube aspect ratio is exploited using an upscaling technique leading to a degenerate 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.
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