Abstract
A family of classical Boussinesq system of nonlinear wave theory is presented in a form of conservative equations supplemented by Riemann solver, which is a fundamental block in the Godunov frame formulation of flow problems. The governing system simulates different physical phenomena, such as propagation of a small amplitude waves on a surface of water, or pulse wave propagation in compliant thin walled arterial systems. While the first model is for verification purpose only, pulse wave propagation through the junction of thin walled elastic branches is of primary interest. The inertial effects associated with transversal motion of the wall are introduced, affirming the dispersive nature of pulsating waves. The problem of accounting for branching and possible discontinuity of wall properties is addressed. Preliminary analysis is presented which leads to the correct jump conditions across bifurcated area. As a result, the model accurately describes the formation of transmission and reflection waves at bifurcation, including effects of discontinuities. The Riemann solver supplies the inter cell flux and junction fluxes, based on conservation of volume, momentum and energy, with account of losses associated with the flow turn angle at bifurcation. An implicit monotonic total variation diminishing (TVD) scheme, second order accurate in time and space has been applied for the analysis of solitary wave solution in bifurcated arteries. Numerical results are in a good agreement with the known analytical and numerical solutions reported elsewhere. Based on direct computational analysis the inverse solution was obtained, calculating the local elastic properties of the arterial wall, using typical diagnostic measurements. Mathematical modelling presented in this work leads to a physiological understanding and interpretation of diagnostic measurements of the wave forms of a blood pressure, flow rate and an artery wall deflection.
Highlights
Consider a broad family of Boussinesq type dispersive wave models [1] presented in a conservative form ux ( u) x auxxx b xx 0
Preliminary analysis is presented which leads to the correct jump conditions across bifurcated area
The speed - amplitude relationship for the peak of flow velocity Au, volume flow rate AQ, and pressure Ap are retrieved from equations (9), (10) for one way wave propagation
Summary
Consider a broad family of Boussinesq type dispersive wave models [1] presented in a conservative form. Korteweg de Vries or Benjamin-Bona-Mahony dispersive waves models [2,3], propagation of small amplitude waves on the surface of incompressible flow. The Riemann solver supplies the inter cell flux and junction fluxes, in case of bifurcation, upheld by conservation of volume, momentum and energy flow rates. Based on direct computational analysis the inverse solution was obtained, calculating the local elastic properties of the arterial wall using typical diagnostic measurements. Setting Aη=maxX(η(X)), AP=maxX(p(X)), Au=maxX(u(X)), and using (9) to calculate integrals in equation (11), one can find speed of a wave propagation (pressure wave velocity (PWV)) in a form.
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