New analytic solutions for wave propagation in flexible, tapered vessels with reference to mammalian arteries

Abstract

Novel, closed-form, analytic solutions for the pressure and velocity fields are derived for the linear problem of wave propagation inside a tapered flexible vessel of conical shape. It is shown that pressure and velocity can be written in terms of Bessel functions of orders 1/3 and 4/3 respectively. An expression is also derived that quantifies the effect of the cone angle on the wave propagation velocity. The analytic solutions are general and valid for tube variations at any length scale in relation to the wavelength of the wave. In other words, the requirement that the changes in vessel properties with distance should take place over a length scale large compared to the wavelength of the wave, is not employed or needed. This is the basic condition for the application of WKB theory to tapered vessels. However, this condition is not satisfied in pressure pulses propagating in mammalian arteries. The general expressions derived in this paper are directly applicable to the cardiovascular system of mammals. It is further shown that the presented solution naturally tends to the asymptotic WKB solution when the assumptions of the theory are applied to the general expressions. An explicit formula is provided for the time-averaged energy flux of the wave that shows clearly the effect of the continuous reflection of the wave from the vessel wall. Viscous effects are incorporated by coupling the derived analytic solution with the radial velocity profile of Womersley. The results are compared with full nonlinear fluid-structure interaction simulations and very good agreement is found (maximum differences are ∼1 % and 1.6 % for area-averaged pressure and velocity respectively, and 4-6 % for local velocity values).

Verification of the coupled‐momentum method with Womersley's Deformable Wall analytical solution

“…Constants H and Q represent the imposed oscillatory pressure or flow, respectively. Applying equations (5), (6), the dimensional solutions of the system (A16) at fixed frequency ω are obtained, see Table 2. Since solutions are complex, only the real parts of velocities, pressure and displacements are taken.…”

“…Wave attenuation and dispersion are also observed within the cardiovascular system. Numerous mathematical formulations have been developed to represent these complex physical phenomena, usually describing oscillatory flow in an idealized tube, rigid or elastic [3][4][5][6][7][8] . One of such formulations is given by Womersley's analytical velocity profile for oscillatory flow in rigid tubes 9 .…”

Wave propagation in stenotic vessels; theoretical analysis and comparison between 3D and 1D fluid–structure-interaction models

“…The central aim of the paper is to derive, apply and validate a linear 1D model for a stenotic vessel in terms of space-frequency (as opposed to spacetime) variables. The model is based on analytical solutions of pressure and velocity waveforms in elastic, tapered vessels that were derived by Papadakis (2011). These solutions were found to match very well with 2D fluid-structure interaction (FSI) results.…”

“…It was shown in Papadakis (2011) that if the system of equations ( 1) is formulated in the frequency domain, i.e. we assume a solution of the form,…”

Method of characteristics for transient, spherical flows

“…(1.1) and ( 1.2 ) and, in practice, accurate solutions are possible only with undesirably small grid sizes. Analytical solutions for transient flow in a tapered tube section can be found in [21,22] , where the continuous variation of the cross-sectional area leads to continuous reflection of the propagating waves in the positive and negative axial directions.…”