A Theory of Fluid Flow in Compliant Tubes

Barnard, Andries; Hunt, W.A.; Timlake, William P.; Varley, E.

doi:10.1016/s0006-3495(66)86690-0

Cited by 197 publications

(116 citation statements)

References 3 publications

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“…One-dimensional (1D) models for the description of blood flow in a compliant vessel where the only space coordinate is the one associated with the vessel axis may provide a good trade-off among the different requirements. They have been introduced almost 250 years ago by L. Euler [53], and then rediscovered in the second half of the XX century in [14] -see also [94,95]. The construction of these models is the result of two steps.…”

Section: The 1d Modelmentioning

confidence: 99%

Geometric multiscale modeling of the cardiovascular system, between theory and practice

Quarteroni

Veneziani

Vergara

2016

Computer Methods in Applied Mechanics and Engineering

169

157

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This review paper addresses the so called geometric multiscale approach for the numerical simulation of blood flow problems, from its origin (that we can collocate in the second half of '90s) to our days. By this approach the blood fluid-dynamics in the whole circulatory system is described mathematically by means of heterogeneous featuring different degree of detail and different geometric dimension that interact together through appropriate interface coupling conditions.Our review starts with the introduction of the stand-alone problems, namely the 3D fluidstructure interaction problem, its reduced representation by means of 1D models, and the so-called lumped parameters (aka 0D) models, where only the dependence on time survives. We then address specific methods for stand-alone 3D models when the available boundary data are not enough to ensure the mathematical well posedness. These so-called "defective problems" naturally arise in practical applications of clinical relevance but also because of the interface coupling of heterogeneous problems that are generated by the geometric multiscale process. We also describe specific issues related to the boundary treatment of reduced models, particularly relevant to the geometric multiscale coupling. Next, we detail the most popular numerical algorithms for the solution of the coupled problems. Finally, we review some of the most representative works -from different research groups -which addressed the geometric multiscale approach in the past years.A proper treatment of the different scales relevant to the hemodynamics and their interplay is essential for the accuracy of numerical simulations and eventually for their clinical impact. This paper aims at providing a state-of-the-art picture of these topics, where the gap between theory and practice demands rigorous mathematical models to be reliably filled.

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Section: The 1d Modelmentioning

confidence: 99%

Geometric multiscale modeling of the cardiovascular system, between theory and practice

Quarteroni

Veneziani

Vergara

2016

Computer Methods in Applied Mechanics and Engineering

169

157

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“…An example of the former is the widely used one-dimensional (1D) Navier-Stokes flow model in deformable tubes [1][2][3][4][5][6], while an example of the latter is the attempt of Vajravelu et al [7] to model the flow of Herschel-Bulkley fluids in elastic tubes. Other attempts have also been made to deal with more special cases of non-Newtonian flow using mainly numerical methods [8,9].…”

Section: Introductionmentioning

confidence: 99%

The flow of Newtonian and power law fluids in elastic tubes

Sochi

2014

International Journal of Non-Linear Mechanics

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We derive analytical expressions for the flow of Newtonian and power law fluids in elastic circularly-symmetric tubes based on a lubrication approximation where the flow velocity profile at each cross section is assumed to have its axially-dependent characteristic shape for the given rheology and cross sectional size. Two pressure-area constitutive elastic relations for the tube elastic response are used in these derivations. We demonstrate the validity of the derived equations by observing qualitatively correct trends in general and quantitatively valid asymptotic convergence to limiting cases. The Newtonian formulae are compared to similar formulae derived previously from a one-dimensional version of the Navier-Stokes equations.

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“…We will perform asymptotic reduction of the equations in non-dimensional variables by ignoring the terms of order 2 and smaller, where is the ratio of the width vs the length of the channel. Essentially this is what was done in Reference [8].…”

Section: The Derivation Of the Model Equationsmentioning

confidence: 83%

“…The simplicity of the model makes it useful in fast real-time computations when quick answers are needed in the cases when the geometry of the patient's vessel can be approximated 1. We present a rigorous derivation of the equations which are obtained using asymptotic analysis of the incompressible Navier-Stokes equations in narrow, long channels; see Reference [8] for basic reference. 2.…”

Section: Introductionmentioning

confidence: 99%

Mathematical analysis of the quasilinear effects in a hyperbolic model blood flow through compliant axi‐symmetric vessels

Čanić

Kim

2003

Math Methods in App Sciences

142

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SUMMARYIn this paper, we present a mathematical analysis of the quasilinear e ects arising in a hyperbolic system of partial di erential equations modelling blood ow through large compliant vessels. The equations are derived using asymptotic reduction of the incompressible Navier-Stokes equations in narrow, long channels.To guarantee strict hyperbolicity we ÿrst derive the estimates on the initial and boundary data which imply strict hyperbolicity in the region of smooth ow. We then prove a general theorem which provides conditions under which an initial-boundary value problem for a quasilinear hyperbolic system admits a smooth solution. Using this result we show that pulsatile ow boundary data always give rise to shock formation (high gradients in the velocity and inner vessel radius). We estimate the time and the location of the ÿrst shock formation and show that in a healthy individual, shocks form well outside the physiologically interesting region (2:8 m downstream from the inlet boundary). In the end we present a study of the in uence of vessel tapering on shock formation. We obtain a surprising result: vessel tapering postpones shock formation. We provide an explanation for why this is the case.

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A Theory of Fluid Flow in Compliant Tubes

Cited by 197 publications

References 3 publications

Geometric multiscale modeling of the cardiovascular system, between theory and practice

Geometric multiscale modeling of the cardiovascular system, between theory and practice

The flow of Newtonian and power law fluids in elastic tubes

Mathematical analysis of the quasilinear effects in a hyperbolic model blood flow through compliant axi‐symmetric vessels

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