Spelman Tamsin A, Onah Ifeanyi S, MacTaggart David, Stewart Peter S
Sainsbury Laboratory, University of Cambridge, 47 Bateman Street, Cambridge CB2 1LR, UK.
School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8SQ, UK.
R Soc Open Sci. 2024 Jul 17;11(7):232000. doi: 10.1098/rsos.232000. eCollection 2024 Jul.
The theory of small-amplitude waves propagating across a blood vessel junction has been well established with linear analysis. In this study, we consider the propagation of large-amplitude, nonlinear waves (i.e. shocks and rarefactions) through a junction from a parent vessel into two (identical) daughter vessels using a combination of three approaches: numerical computations using a Godunov method with patching across the junction, analysis of a nonlinear Riemann problem in the neighbourhood of the junction and an analytical theory which extends the linear analysis to the following order in amplitude. A unified picture emerges: an abrupt (prescribed) increase in pressure at the inlet to the parent vessel generates a propagating shock wave along the parent vessel which interacts with the junction. For modest driving, this shock wave divides into propagating shock waves along the two daughter vessels and reflects a rarefaction wave back towards the inlet. However, for larger driving the reflected rarefaction wave becomes transcritical, generating an additional shock wave. Just beyond criticality this new shock wave has zero speed, pinned to the junction, but for further increases in driving this additional shock divides into two new propagating shock waves in the daughter vessels.
通过线性分析，小振幅波在血管连接处传播的理论已经相当完善。在本研究中，我们采用三种方法相结合的方式，研究大振幅非线性波（即激波和稀疏波）从母血管通过连接处进入两条（相同的）子血管的传播情况：使用戈东诺夫方法并在连接处进行拼接的数值计算、对连接处附近非线性黎曼问题的分析以及将线性分析扩展到振幅下一阶的解析理论。由此呈现出一幅统一的图景：母血管入口处压力的突然（规定的）增加会沿母血管产生一个传播的激波，该激波与连接处相互作用。对于适度的驱动，这个激波会沿着两条子血管分裂为传播的激波，并向入口反射一个稀疏波。然而，对于更大的驱动，反射的稀疏波会变成跨临界的，从而产生一个额外的激波。就在临界值附近，这个新激波速度为零，固定在连接处，但随着驱动的进一步增加，这个额外的激波会在子血管中分裂为两个新的传播激波。
