Steady and oscillating axisymmetric laminar flows are determined by a finite-difference solution of the vorticity and continuity equations for an incompressible fluid contained in a straight concertina-shaped tube far from its ends. In steady flow the size of the wall corrugations is varied as well as the Reynolds number of the flow. In unsteady flow one tube geometry is studied, and the parameters varied are the Reynolds number, the ratio of the mean volume flow rate to its amplitude, and the frequency of oscillation. The analysis produces streamlines, particle paths and the pressure difference across a length of the tube. The resistance to the flow is determined in terms of an equivalent cylindrical tube diameter.
In steady flow the onset of flow separation and the growth of the separated region of flow is determined. The equivalent diameter is found to be principally a function of the product of Reynolds number and the non-dimensional pressure difference. This product depends on the height of the wall corrugations and less strongly on Reynolds number and the length of the corrugations. Resistance increases with increasing height of the corrugations. Comparison is made with other computational and experimental values of the pressure difference.
In unsteady flow the mean velocity to amplitude ratio has little effect except on the particle paths. The flow pattern is found to be governed by the Stokes number (radius × (2π/(kinematic viscosity × period))½) and the Reynolds number. There is a region of quasi-steady flow at the time of zero acceleration at maximum flow, but unsteady flow in between. The mixing produced by radial convection is restricted to the outer parts of the tube where the wall is corrugated. In oscillating flow the resistance relative to a cylindrical tube decreases as frequency and Reynolds number increase.
In the medical application of the work the concern is whether sustained stagnant regions occur in the corrugations and whether there is a large change in resistance relative to a cylindrical tube. This part of the investigation was made with an arterial waveform which contained six harmonics. It is found that there are no regions of stagnant fluid in the range of parameters considered. The difference between the variation with the flow parameters of the resistance of the corrugated tube and of a cylindrical tube was found not to be large.