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 (₁=0) and depending on the value of k ranging from 01 to 07, the frictional resistance increases by a factor ranging from 132 to 2391 for D=500 and 120 to 1656 for D=2000. But, in the presence of constriction (₁ = 01) with the same range for k, the increase in frictional resistance is by a factor ranging from 134 to 4232 for D=500 and 118 to 295 for D=2000. In a straight annulus, the increased factor ranges from 174 to 3261 for ₁=0 and 178 to 5827 for ₁ = 01 (for all Dean numbers D).