Pulsatile Flow in Tubes of Elliptic Cross Sections

Haslam, Michael C.; Zamir, M.

doi:10.1114/1.106

Cited by 45 publications

(45 citation statements)

References 6 publications

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“…With these assumptions, the equations of motion for the fluid were determined as a function of the pressure gradient in the tube, which varied as a sinusoidal function. The exact solution of the governing equations for sinusoidal flow in a vessel with an elliptical cross-section has been determined [26]. The velocity is given by the expression:…”

Section: Methodsmentioning

confidence: 99%

“…Wall shear rate ( γ ) corresponding to a fluid experiencing a sinusoidal pressure gradient is obtained through the spatial derivative of the velocity with respect to the surface normal ( n ) evaluated at the vessel wall [26]. Subsequently, the wall shear stress ( τ ) is calculated from the product of the viscosity (µ) and the wall shear rate:…”

Section: Methodsmentioning

confidence: 99%

“…The equations that govern sinusoidal flow of a viscous fluid through a straight rigid elliptical vessel [26]are the solutions of the Navier-Stokes equations [18]in an elliptical co-ordinate system (fig. 1).…”

Section: Methodsmentioning

confidence: 99%

“…A previous study [26]has demonstrated a dependence of the velocity profile on the frequency of the sinusoidal flow and the ellipticity of the vessel’s cross-section. The aim of this study was to improve the understanding of pulsatile flow in elliptical vessels through the display of idealised velocity profiles, corresponding to physiological mean flow waveforms, throughout the flow cycle.…”

Section: Introductionmentioning

confidence: 99%

“…An exact solution of the governing equations for pulsatile flow in straight rigid vessels of elliptic cross-section involves Mathieu functions [25]. A recent study [26]has overcome the inherent complexity of these functions, which had previously hindered a complete examination of the flow properties, to solve the governing equations.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Flow in Elliptical Vessels Calculated for a Physiological Waveform

Robertson

Köhler²,

Hoskins³

et al. 2001

J Vasc Res

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Understanding the nature of pulsatile flow is an important issue in haemodynamics, especially the initiation and progression of vascular disease. The geometry of a non-circular vessel was idealised to an elliptical cross-section, and the dynamic properties of the flow were calculated for a physiological waveform. The Fourier harmonics for a common carotid waveform were determined, and the velocity profile and wall shear stress were calculated from the superposition of the individual contributions from each harmonic. The effects of ellipticity on the flow pattern were found to be significant. The velocity profile along the major axis of the elliptical cross-section developed a flattened peak, which widened as the vessel became more elliptical. Wall shear stress demonstrated an angular dependence in elliptical vessels, where the point of minimum shear stress was located at the end of the major axis. Comparison with a cylindrical vessel demonstrated a 3% decrease in peak wall shear stress (τ = 2.96, N·m^–2) at the end of the major axis, and 10% in the mean wall shear stress (τ = 0.44 N· m^–2), for an elliptical vessel (Ε = 0.8). The temporal average wall shear stress, which has been associated with atherogenic sites, also displayed a minimum at the end of the major axis that decreased with more elliptical cross-sections.

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Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Flow in Elliptical Vessels Calculated for a Physiological Waveform

Robertson

Köhler²,

Hoskins³

et al. 2001

J Vasc Res

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A three‐dimensional method for morphological analysis and flow velocity estimation in microvasculature on‐a‐chip

Rota,

Possenti,

Offeddu

et al. 2023

Bioengineering & Transla Med

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Three‐dimensional (3D) imaging techniques (e.g., confocal microscopy) are commonly used to visualize in vitro models, especially microvasculature on‐a‐chip. Conversely, 3D analysis is not the standard method to extract quantitative information from those models. We developed the μVES algorithm to analyze vascularized in vitro models leveraging 3D data. It computes morphological parameters (geometry, diameter, length, tortuosity, eccentricity) and intravascular flow velocity. μVES application to microfluidic vascularized in vitro models shows that they successfully replicate functional features of the microvasculature in vivo in terms of intravascular fluid flow velocity. However, wall shear stress is lower compared to in vivo references. The morphological analysis also highlights the model's physiological similarities (vessel length and tortuosity) and shortcomings (vessel radius and surface‐over‐volume ratio). The addition of the third dimension in our analysis produced significant differences in the metrics assessed compared to 2D estimations. It enabled the computation of new indices, such as vessel eccentricity. These μVES capabilities can find application in analyses of different in vitro vascular models, as well as in vivo and ex vivo microvasculature.

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An efficient semi‐implicit method for three‐dimensional non‐hydrostatic flows in compliant arterial vessels

Fambri

Dumbser

Casulli

2014

Numer Methods Biomed Eng

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Blood flow in arterial systems can be described by the three-dimensional Navier-Stokes equations within a time-dependent spatial domain that accounts for the elasticity of the arterial walls. In this article, blood is treated as an incompressible Newtonian fluid that flows through compliant vessels of general cross section. A three-dimensional semi-implicit finite difference and finite volume model is derived so that numerical stability is obtained at a low computational cost on a staggered grid. The key idea of the method consists in a splitting of the pressure into a hydrostatic and a non-hydrostatic part, where first a small quasi-one-dimensional nonlinear system is solved for the hydrostatic pressure and only in a second step the fully three-dimensional non-hydrostatic pressure is computed from a three-dimensional nonlinear system as a correction to the hydrostatic one. The resulting algorithm is robust, efficient, locally and globally mass conservative, and applies to hydrostatic and non-hydrostatic flows in one, two and three space dimensions. These features are illustrated on nontrivial test cases for flows in tubes with circular or elliptical cross section where the exact analytical solution is known. Test cases of steady and pulsatile flows in uniformly curved rigid and elastic tubes are presented. Wherever possible, axial velocity development and secondary flows are shown and compared with previously published results.

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Pulsatile Flow in Tubes of Elliptic Cross Sections

Cited by 45 publications

References 6 publications

Flow in Elliptical Vessels Calculated for a Physiological Waveform

Flow in Elliptical Vessels Calculated for a Physiological Waveform

A three‐dimensional method for morphological analysis and flow velocity estimation in microvasculature on‐a‐chip

An efficient semi‐implicit method for three‐dimensional non‐hydrostatic flows in compliant arterial vessels

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