Improved theoretical model of two-dimensional flow field in a severely narrow circular pipe

Abstract

Based on the two-dimensional theory of a Newtonian incompressible fluid, an improved model is proposed by combining Reynolds stresses of new disturbance factors and velocity polynomials. It is used to solve the Reynolds averaged Navier-Stokes equation for flow through a severely narrow pipe at the continuous change of the Reynolds number from laminar flow to turbulence. Both axial and radial velocity polynomials are considered in the momentum integral method. Under boundary and symmetry conditions, a first-order differential equation for a coefficient of the axial velocity with the disturbance factors is derived. Using a numerical shooting method to solve the equation, the axial distributions of pressure are obtained in the range of Reynolds numbers from 20 to 105 when the degree of stenosis equals 0.4 or 0.9. Also, under a lower Reynolds number, the velocity profiles in axial and radial directions, the streamlines at downstream and the wall shear stresses (WSS) in narrow regions are illustrated. The disturbance factors introduced can sensitively regulate the variation of inertia, pressure gradient, and viscosity term in the Reynolds averaged Navier-Stokes equation. With an increase in the Reynolds number and the parameters from 0.02 to 20 in the disturbance factors, the axial and radial velocities reverse at some narrow regions gradually, the WSS falls to below zero downstream, and the pressure drop increases in the narrow section of the pipe. It is implied that the pressure drop plays an important role in artery collapse when it is less than 40% stenosis. When the percentage of stenosis is increased to more than 40% and the Reynolds number is only 200, WSS gradually exceeds the tolerance of endothelial cells in blood vessels. The increase in pressure drop at downstream and WSS at upstream leads to the aggravation of vascular stenosis and exfoliation of the atherosclerotic plaque.