Numerical study of flow in a constricted curved annulus: An application to flow in a catheterised artery

The flow of an incompressible Newtonian fluid in a curved annulus with a local constriction at the outer wall is investigated numerically. The three-dimensional nonlinear elliptic partial differential equations governing the flow are simplified by use of small curvature and mild constriction approximations. The simplified equations of motion, which are locally two-dimensional elliptic in nature at each cross-section, are solved numerically by means of the finite-difference method described by Collins and Dennis [Quart. Jour. Mech. Appl. Math. 28 (1975) 133-156]. Although the results are restricted to small curvature and mild constriction, these are valid for all Dean numbers D in the entire laminar flow regime. The numerical results show that, for higher values of radii ratio k, the pressure gradient, pressure drop, and frictional resistance increase considerably and they vary markedly across the constricted length. These results are used to estimate the increase in frictional resistance in an artery when a catheter is inserted into it. In the absence of constriction (delta sub 1 = O) and depending on the value of k ranging from 0*1 to 0*7, the frictional resistance increases by a factor ranging from 1*32 to 23*91 for D = 500 and 1*20 to 16*56 for D = 2000. But, in the presence of constriction (delta sub 1 = 0*1) with the same range for k, the increase in frictional resistance is by a factor ranging from 1*34 to 42*32 for D = 500 and 1-18 to 29*5 for D = 2000. In a straight annulus, the increased factor ranges from 1*74 to 32*61 for delta sub 1 = 0 and 1*78 to 58*27 for delta sub 1 =0*1 (for all Dean numbers D). [Application is made to the case of flow in a stenosed artery with an inserted catheter, but results are of general interest in piping applications.]