Analytical solution for pulsatile axial flow velocity waveforms in curved elastic tubes

An analytical solution for pulsatile axial flow velocity waveforms in curved elastic tubes is presented. The result is obtained by exact solution of linearized Navier-Stokes and tube motion equations in a toroidal coordinate system. Fourier analysis is used to divide the flow into constant and oscillatory components which are separately considered. The solution is used to investigate the effects of curvature on volumetric axial velocity flow waveforms, as would be measured by Doppler ultrasound techniques. In typical human arteries, the greatest effects of curvature on the volumetric axial flow are exerted on the constant component and at low values of the frequency parameter for the oscillatory components. Here, the magnitude and phase angle of oscillatory flow in the curved tube, relative to that in the straight tube, differ by maximum values of 1.2% and 0.15 rad, respectively. However, constant flow may vary by as much as 60% at high Dean numbers. The solution is presented in a form similar to Womersley´s solution for the straight elastic tube and may, thus, be incorporated into a transmission-line analog model. These models are frequently used to investigate axial flow velocity variations in mammalian circulatory systems and this work offers a tool which may extend these models to incorporate the effects of curvature.