LXXII.<i>The stream-line motion of fluid in a curved pipe</i>(Second paper)

Dean, W. R.

doi:10.1080/14786440408564513

Cited by 1,044 publications

(436 citation statements)

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“…[12,29,42]) and analytical studies (e.g. [11,40]) have shown that vessel curvature can have a strong influence on the shape of the velocity profile. The analytical models proposed by Waters et al are interesting but, however, limited to steady flow for low Dean numbers, which does not correspond to physiological flow [40].…”

Section: Discussionmentioning

confidence: 99%

Model-based assessment of dynamic arterial blood volume flow from ultrasound measurements

Leguy

Bosboom

Hoeks

et al. 2009

Med Biol Eng Comput

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To assess in clinical practice arterial blood volume flow (BVF) from ultrasound measurements, the assumption is commonly made that the velocity profile can be approximated by a quasi-static Poiseuille model. However, pulsatile flow behaviour is more accurately described by a Womersley model. No clinical studies have addressed the consequences on the estimated dynamics of the BVF when Poiseuille rather than Womersley models are used. The aim of this study is to determine the influence of assumed Poiseuille profile instead of Womersley profile on the estimation and intrasubject variability of dynamical parameters of the BVF. For this purpose, a low number of volunteers sufficed. Brachial artery centerline velocity waveform and vessel diameter were measured with ultrasound within a small group of six volunteers. Within subjects, the intra-and inter-registration variability of BVF parameters estimates did not significantly differ. Poiseuille profiles compared to Womersley underestimates the maximum BVF by 19%, the maximum retrograde volume flow by 32% and the rise time by 18%. It can be concluded that when estimating in a straight vessel the dynamic properties of the BVF, Womersley profiles should preferably be chosen.

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Section: Discussionmentioning

confidence: 99%

Model-based assessment of dynamic arterial blood volume flow from ultrasound measurements

Leguy

Bosboom

Hoeks

et al. 2009

Med Biol Eng Comput

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“…In some cases, it is possible to reduce the set of parameters to a small number, a classic example of this being Dean flow. 10,11,15 However, the relevance of these solutions to flow in real blood vessels is not yet fully known, and hence the mechanisms responsible for some of the complex flow phenomena that are observed in the human vasculature are not well understood. 18,[24][25][26] Usually, 3D simulations produce such large quantities of data that they are unlikely to be of clinical use unless methods are available to simplify our understanding of the flow dynamics.…”

Section: Introductionmentioning

confidence: 99%

“…Experimental, theoretical, and numerical studies have been carried out to study laminar flow in idealised geometries, such as flow around bends of constant curvature (e.g., Refs. [8][9][10][11][12][13][14][15][16][17][18][19] or helical tubes of constant curvature and torsion (e.g., Refs. 15 and 20-23).…”

Section: Introductionmentioning

confidence: 99%

Reducing the data: Analysis of the role of vascular geometry on blood flow patterns in curved vessels

et al. 2012

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“…2͒. Dean 11 used a slightly different definition, but the results as presented in this article are adapted to the coordinate system definition of Topakoglu 12 and Siggers and Waters. 13 The most relevant results to this study obtained by the authors with respect to the axial velocity profiles are briefly shown, together with the equations, which relate the flow in a curved tube to the flow in a straight tube.…”

Section: B Introduction To the Theoretical Backgroundmentioning

confidence: 99%

“…The solutions obtained by Dean, 11 Topakoglu,12 and Siggers and Waters 13 will be evaluated more extensively and compared with each other. These authors derived analytical solutions for small Dean numbers ͑DnӶ 1͒ and assumed that the analytical solution for a curved tube is just a small disturbance on the Poiseuille flow of a straight tube, with the flow being driven by the pressure gradient.…”

Section: B Introduction To the Theoretical Backgroundmentioning

confidence: 99%

Estimation of volume flow in curved tubes based on analytical and computational analysis of axial velocity profiles

et al. 2009

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To monitor biomechanical parameters related to cardiovascular disease, it is necessary to perform correct volume flow estimations of blood flow in arteries based on local blood velocity measurements. In clinical practice, estimates of flow are currently made using a straight-tube assumption, which may lead to inaccuracies since most arteries are curved. Therefore, this study will focus on the effect of curvature on the axial velocity profile for flow in a curved tube in order to find a new volume flow estimation method. The study is restricted to steady flow, enabling the use of analytical methods. First, analytical approximation methods for steady flow in curved tubes at low Dean numbers (Dn) and low curvature ratios (δ) are investigated. From the results a novel volume flow estimation method, the cos θ-method, is derived. Simulations for curved tube flow in the physiological range (1≤Dn≤1000 and 0.01≤δ≤0.16) are performed with a computational fluid dynamics (CFD) model. The asymmetric axial velocity profiles of the analytical approximation methods are compared with the velocity profiles of the CFD model. Next, the cos θ-method is validated and compared with the currently used Poiseuille method by using the CFD results as input. Comparison of the axial velocity profiles of the CFD model with the approximations derived by Topakoglu [J. Math. Mech. 16, 1321 (1967)] and Siggers and Waters [Phys. Fluids 17, 077102 (2005)] shows that the derived velocity profiles agree very well for Dn≤50 and are fair for 50<Dn≤100, and this result applies for 0.01≤δ≤0.16, while Dean’s [Philos. Mag. 5, 673 (1928)] approximation only coincides for δ=0.01. For higher Dean numbers (Dn>100), no analytical approximation method exists. In the position of the maximum axial velocity, a shift toward the inside of the curve is observed for low Dean numbers, while for high Dean numbers, the position of the maximum velocity is located at the outer curve. When the position of the maximum velocity of the axial velocity profile is given as a function of the Reynolds number, a “zero-shift point” is found at Re=21.3. At this point the shift in the maximum axial velocity to the outside of the curve, caused by the difference in axial pressure gradient, balances the shift to the inside of the curve, caused by the centrifugal forces (radial pressure gradient). Comparison of the volume flow estimation of the cos θ-method with the Poiseuille method shows that for Dn≤100 the Poiseuille method is sufficient, but for Dn≥100 the cos θ-method estimates the volume flow nearly three times better. For δ=0.01 the maximum deviation from the exact flow is 4% for the cos θ-method, while this is 12.7% for the Poiseuille method in the plane of symmetry. The axial velocity profile measured at a certain angle from the symmetry plane results in a maximum estimation error of 6.2% for Dn=1000 and δ=0.16. The results indicate that the estimation of the volume flow through a curved tube from a given asymmetrical axial velocity profile is more precise with the cos θ-method than the Poiseuille method, which is currently used in clinical practice.

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LXXII.The stream-line motion of fluid in a curved pipe(Second paper)

Cited by 1,044 publications

References 1 publication

Model-based assessment of dynamic arterial blood volume flow from ultrasound measurements

Model-based assessment of dynamic arterial blood volume flow from ultrasound measurements

Reducing the data: Analysis of the role of vascular geometry on blood flow patterns in curved vessels

Estimation of volume flow in curved tubes based on analytical and computational analysis of axial velocity profiles

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