Response of blood flow through an artery under stenotic conditions

This paper presents an analytical study on the behavoiur of blood flow in an artery having a stenosis. This is basically formulated through the use of a suitable mathematical model. The arterial segment under consideration is simulated by an anisotropically elastic cylindrical tube filled with a viscous incompressible fluid representing blood. The analysis is carried out for an artery with mild local narrowing in its lumen forming a stenosis. Particular emphasis has been paid to the effect of the surrounding connective tissues on the motion of the arterial wall. Blood is treated as a Newtonian fluid. The analysis is restricted to propagation of small amplitude harmonic waves, generated due to the flow of blood whose wave length is large compared to the radius of the arterial segment. The effect of the shape of stenosis on the resistance to blood flow has been well illustrated quantitatively through numerical computations of the resulting expressions. A quantitative analysis is also made for the variation of the phase velocity, as well as the velocity of wave propagation and the flow rate, in order to illustrate the applicability of the model.     

Pressure wave propagation in liquid-filled tubes of viscoelastic material

The propagation of small-amplitude waves in a thick-walled long viscoelastic tube of variable cross-section, filled with a viscous incompressible fluid, is considered with account for wave reflection at the tube end in application to arterial pulse wave propagation. A solution is obtained in the form of expansions in a small parameter. The effect of the coefficient of wave reflection at the tube end and the wall material parameters on the fluid volume flow-rate and the tube wall displacement is investigated. It is shown that the volume flow-rate phase spectrum characteristics depend only slightly on the wall properties and can be used in clinical diagnostics for finding the reflection coefficient from pressure and flow-rate records.     

Propagation of pressure waves in two-phase flow

We deal with a pressure wave of finite amplitude propagating in a gas and liquid medium or in the fluid in an elastic tube. We study the effects of pipe elasticity on the propagation velocity of the pressure wave. Pressure waves of finite amplitude progressing in the two-phase flow are treated considering the void fraction change due to pressure rise. The propagation velocity of the two-phase shock wave is also investigated, and the behavior of the reflection of the pressure wave at the rigid wall is analyzed and compared to that in a pure gas or liquid. The results are compared to experimental data of a pressure wave propagating in the two-phase flow in a vertical shock tube.     

Effect of elastic wall motion on oscillatory flow of micropolar fluid in an annular tube

Summary Oscillatory flow of a micropolar fluid in an annular tube is investigated. The outer wall of the tube is taken to be elastic and the variation in the diameter of the elastic wall due to pulsatile nature of pressure gradient is assumed to be small. The wall motion is governed by a tube law. The nonlinear equations governing the fluid flow and the tube law are solved using perturbation analysis. The steady-streaming phenomenon due to the interaction of convected inertia with viscous effects is studied. The analysis, is carried out for zero mean flow rate. It presents the effects of the elastic nature of the wall combined with micropolar fluid parameters on the mean pressure gradient and wall shear stress for different catheter sizes and frequency parameters. It is found that the effect of micropolarity is of considerable importance for small steady-streaming Reynolds number. Also, it is observed that the relationship between mean pressure gradient and the flow rate depends on the amplitude of the diameter variation, flow rate waveforms and the phase difference between them.     

WAVE PROPAGATION IN A SYSTEM OF COAXIAL TUBES FILLED WITH INCOMPRESSIBLE MEDIA: A MODEL OF PULSE TRANSMISSION IN THE INTRACRANIAL ARTERIES

The propagation of harmonic waves through a system formed of coaxial tubes filled with incompressible continua is considered as a model of arterial pulse propagation in the craniospinal cavity. The inner tube represents a blood vessel and is modelled as a thin-walled membrane shell. The outer tube is assumed to be rigid to account for the constraint imposed on the vessels by the skull and the vertebrae. We consider two models: in the first model the annulus between the tubes is filled with fluid; in the second model the annulus is filled with a viscoelastic solid separated from the tubes by thin layers of fluid. In both models, the elastic tube is filled with fluid. The motion of the fluid is described by the linearized form of the Navier–Stokes equations, and the motion of the solid by classical elasticity theory. The results show that the wave speed in the system is lower than that for a fluid-filled elastic tube free of any constraint. This is due to the stresses generated to satisfy the condition that the volume in the system has to be conserved. However, the effect of the constraint weakens as the radius of the outer tube is increased, and it should be insignificant for the typical physiological parameter range.     

Oscillatory flow of second grade fluid in cylindrical tube

The unsteady oscillatory flow of an incompressible second grade fluid in a cylindrical tube with large wall suction is studied analytically. Flow in the tube is due to uniform suction at the permeable walls, and the oscillations in the velocity field are due to small amplitude time harmonic pressure waves. The physical quantities of interest are the velocity field, the amplitude of oscillation, and the penetration depth of the oscillatory wave. The analytical solution of the governing boundary value problem is obtained, and the effects of second grade fluid parameters are analyzed and discussed.     

Weakly non-linear waves in a fluid-filled elastic tube with variable prestretch

In the present work, by utilizing the non-linear equations of motion of an incompressible, isotropic thin elastic tube subjected to a variable prestretch both in the axial and the radial directions and the approximate equations of motion of an incompressible inviscid fluid, which is assumed to be a model for blood, we studied the propagation of weakly non-linear waves in such a medium, in the long wave approximation. Employing the reductive perturbation method we obtained the variable coefficient KdV equation as the evolution equation. By seeking a travelling wave solution to this evolution equation, we observed that the wave speed is variable in the axial coordinate and it decreases for increasing circumferential stretch (or radius). Such a result seems to be plausible from physical considerations.     

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研究了埋置于弹性地基内充液压力管道中非线性波的传播. 假设管壁是线弹 性的,地基反力采用Winkler线性地基模型,管中流体为不可压缩理想流体. 假定系统初始 处于内压为$P_0$的静力平衡状态,动态的位移场及内压和流速的变化是叠加在静 力平衡状态上的扰动. 基于质量守恒和动量定理,建立了管壁和流体耦合作用的非 线性运动方程组; 进而用约化摄动法, 在长波近似情况下得到了KdV方程,表征 着系统有孤立波解.     

爆破振动波叠加数值预测方法

根据场地地震波的传播叠加原理,以实测单炮孔爆破振动波形为基础,考虑预测点位置与各炮孔 的相对位置关系,并按照实际起爆网路设计的各炮孔起爆时差和实测的地震波传播速度等参数,计算获得预 测点的爆破振动波形。不仅可以预测爆破振动速度峰值,而且可以预测完整的振动波形,并可获知爆破振动 持续时间及主振频率分布范围。根据现场应用数码电子雷管的深孔爆破实验,该方法计算的预测波形与实测 波形相当吻合,计算结果可靠性较好,可以在实际工程中推广使用。     

Low-frequency wave propagation in an elastic plate floating on a two-layered fluid

For some technical applications related to the ice–sea interaction, it is necessary to predict waveguide properties of elastic plates floating on a relatively thin layer of water, which has a non-uniform density distribution across its depth. The issue of particular concern is propagation of low-frequency waves in such a coupled waveguide. In the present paper, a stratified fluid is modelled as two homogeneous, inviscid and incompressible layers with slightly different densities. The lighter layer of fresh water lies on top of the heavier layer of salty water. The former one produces fluid loading at the pre-stressed plate, whereas the latter one is bounded by the sea bottom. The classical asymptotic methods are employed to identify significant regimes of wave motion in such a three-component waveguide. Dispersion diagrams obtained from approximate dispersion relations are compared with their exact counterparts. The phenomena of veering and generation of waves with zero group velocity induced by pre-stress are identified and quantified.     

Modeling of the phase lag causing fluidelastic instability in a parallel triangular tube array

Fluidelastic instability is considered a critical flow induced vibration mechanism in tube and shell heat exchangers. It is believed that a finite time lag between tube vibration and fluid response is essential to predict the phenomenon. However, the physical nature of this time lag is not fully understood. This paper presents a fundamental study of this time delay using a parallel triangular tube array with a pitch ratio of 1.54. A computational fluid dynamics (CFD) model was developed and validated experimentally in an attempt to investigate the interaction between tube vibrations and flow perturbations at lower reduced velocities U_r=1–6 and Reynolds numbers Re=2000–12 000. The numerical predictions of the phase lag are in reasonable agreement with the experimental measurements for the range of reduced velocities U_g/fd=6–7. It was found that there are two propagation mechanisms; the first is associated with the acoustic wave propagation at low reduced velocities, U_r<2, and the second mechanism for higher reduced velocities is associated with the vorticity shedding and convection. An empirical model of the two mechanisms is developed and the phase lag predictions are in reasonable agreement with the experimental and numerical measurements. The developed phase lag model is then coupled with the semi-analytical model of Lever and Weaver to predict the fluidelastic stability threshold. Improved predictions of the stability boundaries for the parallel triangular array were achieved. In addition, the present study has explained why fluidelastic instability does not occur below some threshold reduced velocity.     

Non-linear waves in a fluid-filled inhomogeneous elastic tube with variable radius

In the present work, by employing the non-linear equations of motion of an incompressible, inhomogeneous, isotropic and prestressed thin elastic tube with variable radius and the approximate equations of an inviscid fluid, which is assumed to be a model for blood, we studied the propagation of non-linear waves in such a medium, in the longwave approximation. Utilizing the reductive perturbation method we obtained the variable coefficient Korteweg–de Vries (KdV) equation as the evolution equation. By seeking a progressive wave type of solution to this evolution equation, we observed that the wave speed decreases for increasing radius and shear modulus, while it increases for decreasing inner radius and the shear modulus.     

Effective equations describing the flow of a viscous incompressible fluid through a long elastic tubeEquations efficaces décrivant l'écoulement d'un fluide visqueux incompressible dans un tuyau long et élastique

We study the flow of a viscous incompressible fluid through a long and narrow elastic tube whose walls are modeled by the Navier equations for a curved, linearly elastic membrane. The flow is governed by a given small time dependent pressure drop between the inlet and the outlet boundary, giving rise to creeping flow modeled by the Stokes equations. By employing asymptotic analysis in thin, elastic, domains we obtain the reduced equations which correspond to a Biot type viscoelastic equation for the effective pressure and the effective displacement. The approximation is rigorously justified by obtaining the error estimates for the velocity, pressure and displacement. Applications of the model problem include blood flow in small arteries. We recover the well-known Law of Laplace and provide a new, improved model when shear modulus of the vessel wall is not negligible. To cite this article: S. ?ani?, A. Mikeli?, C. R. Mecanique 330 (2002) 661–666.     

弯曲动脉壁非线性弹性力学性质的理论分析

血管壁非线性力学特性是探索心血管中的血液流动规律及脉搏波传播现象的一个重要前提．在血管力学研究中,直管动脉壁的本构方程已有相当深人的研究,唯独象主动脉弓那样的弯管动脉壁本构方程;至今还没有建立一个理论模型．文中在已有的直管动脉壁本构方程研究基础上,提出一个理论方法来分析弯曲动脉壁的非线性弹性力学性质,在弯曲动脉壁被模拟为均质、正交各向异性、不可压缩材料的前提下,作者从理论上建立了一个表达弯管动脉壁非线性弹性性质的三维ｅ指数型本构方程文中还探讨了弯曲动脉壁内的残余应变分析．     

A two-dimensional effective model describing fluid–structure interaction in blood flow: analysis, simulation and experimental validation

We derive a closed system of effective equations describing a time-dependent flow of a viscous incompressible Newtonian fluid through a long and narrow elastic tube. The 3D axially symmetric incompressible Navier–Stokes equations are used to model the flow. Two models are used to describe the tube wall: the linear membrane shell model and the linearly elastic membrane and the curved, linearly elastic Koiter shell model. We study the behavior of the coupled fluid–structure interaction problem in the limit when the ratio between the radius and the length of the tube, , tends to zero. We obtain the reduced equations that are of Biot type with memory. An interesting feature of the reduced equations is that the memory term explicitly captures the viscoelastic nature of the coupled problem. Our model provides significant improvement over the standard 1D approximations of the fluid–structure interaction problem, all of which assume an ad hoc closure assumption for the velocity profile. We performed experimental validation of the reduced model using a mock circulatory flow loop assembled at the Cardiovascular Research Laboratory at the Texas Heart Institute. Experimental results show excellent agreement with the numerically calculated solution. Major applications include blood flow through large human arteries. To cite this article: S. Čanić et al., C. R. Mecanique 333 (2005).     

The fluid hopkinson bar

Experiments in which pressure pulses are propagated in a column of fluid held in a stiff tube are described. A parameter, η, which characterizes the tube stiffness has been defined and a one-dimensional model of the wave propagation which includes dissipation both in the volume of the fluid and at the wall of the containing tube has been developed. It is found that dissipation at the wall dominates dissipation in the fluid volume for pulse lengths long compared to the tube radius. The experimental results delineate practical limits on the ratio of pulse length to tube radius for which the wave propagation can be characterized as one-dimensional. The validity of a one-dimensional representation of pulse transmission and reflection at a solid-fluid interface is also evaluated with the aid of experimental results. Finally, the dissipation model in combination with the experimental results leads to a simple expression for pressure pulse attenuation in terms of a nondimensional physical parameter, Ξ, and tube radius.     

Effect of the rotation,magnetic field and initial stress on peristaltic motion of micropolar fluid

In this work, the effect of magnetic field, rotation and initial stress on peristaltic motion of micropolar fluid in a circular cylindrical flexible tube with viscoelastic or elastic wall properties has been considered. Runge–Kutta technique are used. Runge–Kutta method is developed to solve the governing equations of motion resulting from a perturbation technique for small values of amplitude ratio. The time mean axial velocity profiles are presented for the case of free pumping and analyzed to observe the influence of wall properties, magnetic field, rotation and initial stress for various values of micropolar fluid parameters. In the case of viscoelastic wall, the effect of viscous damping on mean flow reversal at the boundary is seen. The numerical results of the time mean velocity profile are discussed in detail for homogeneous fluid under the effect of wall properties, magnetic field, initial stress and rotation for different cases by figures. The results indicate that the effect of wall properties, rotation, initial stress and magnetic field are very pronounced. Numerical results are given and illustrated graphically.     

One-Dimensional Interaction of a Cylindrical Unloading Wave with a Moving Elastic–Plastic Boundary

The one-dimensional dynamic problem of the theory of large elastic–plastic deformations is considered for the interaction of an unloading wave with an elastic–plastic boundary. It is shown that before the occurrence of the unloading wave, the increasing pressure gradient leads to quasistatic deformation of the elasti©viscoplastic material filling the round tube, which is retained in the tube due to friction on its wall, resulting in the formation of near-wall viscoplastic flow and an elastic core. The unloading wave is initiated at the moment of the onset of slippage of the material along the inner wall of the tube. Calculations were conducted using the ray method of constructing approximate solutions behind strong discontinuity surfaces, and ray expansions of the solutions behind the cylindrical surfaces of discontinuities were obtained.     

Quasi-periodic waves and quasi-solitary waves in stratified fluid of slowly varying depth

The nonlinear waves in a stratified fluid of slowly varying depth are investigated in this paper. The model considered here consists of a two-layer incompressible constantdensity inviscid fluid confined by a slightly uneven bottom and a horizontal rigid wall. The Korteweg-de Vries (KdV) equation with varying coefficients is derived with the aid of the reductive perturbation method. By using the method of multiple scales, the approximate solutions of this equation are obtained. It is found that the unevenness of bottom may lead to the generation of so-called quasi-periodic waves and quasisolitary waves, whose periods, propagation velocities and wave profiles vary slowly. The relations of the period of quasiperiodic waves and of the amplitude, propagation velocity of quasi-solitary waves varying with the depth of fluid are also presented. The models with two horizontal rigid walls or single-layer fluid can be regarded as particular cases of those in this paper.Project Supported by National Natural Science Foundation of China.