Flow in a tube with a small side branch

Abstract

Atherosclerotic lesions in mammalian blood vessels show a definite spatial pattern, and it has been proposed that lesions occur preferentially in regions with a low wall shear stress. Near the entrance to an intercostal artery, lesions on the wall of the aorta occur, initially at least, downstream of the entrance. We model the flow in this region by a linear shear flow along a flat wall (the wall of the main tube/aorta) past an infinitely deep circular hole perpendicular to the wall (the side-branch/intercostal artery), with fluid being sucked into the hole. By assuming Stokes flow, the three-dimensional model problem is reduced to two independent problems on a two-dimensional domain. By addition, the solutions for any particular side-branch flow rate. We find that the wall shear stress in the main tube is elevated upstream of the side-branch entrance, and downstream as well is the side-branch flow rate is small. However, if the flow rate in the side branch is large enough, there will be regions of both elevated and reduced wall shear stress in the main tube downstream of the side-branch entrance, including a stagnation point. The wall shear is lower downsteam than upstream unless there is no net flow into the side branch.The solutions given apply to the case with flow out of as well as into the hole. Also, the asymptotic structure for the flow in the hole when there is no net flow into the hole, and the analysis of three-dimensional flow near a sharp corner, are given.