A Modification of Murray's Law for Shear-Thinning Rheology

2015 ◽  
Vol 137 (5) ◽  
Author(s):  
Patrick M. McGah ◽  
Massimo Capobianchi
Keyword(s):  
Shear Stress ◽  
Wall Shear Stress ◽  
Volume Flow Rate ◽  
Shear Thinning ◽  
Arterial Tree ◽  
Branching Law ◽  
Typical Parameter ◽  
Wall Shear ◽  
Parameter Ranges ◽  
Minimum Work

This study reformulates Murray's well-known principle of minimum work as applied to the cardiovascular system to include the effects of the shear-thinning rheology of blood. The viscous behavior is described using the extended modified power law (EMPL), which is a time-independent, but shear-thinning rheological constitutive equation. The resulting minimization problem is solved numerically for typical parameter ranges. The non-Newtonian analysis still predicts the classical cubic diameter dependence of the volume flow rate and the cubic branching law. The current analysis also predicts a constant wall shear stress throughout the vascular tree, albeit with a numerical value about 15–25% higher than the Newtonian analysis. Thus, experimentally observed deviations from the cubic branching law or the predicted constant wall shear stress in the vasculature cannot likely be attributed to blood's shear-thinning behavior. Further differences between the predictions of the non-Newtonian and the Newtonian analyses are highlighted, and the limitations of the Newtonian analysis are discussed. Finally, the range and limits of applicability of the current results as applied to the human arterial tree are also discussed.

A New Hexahedral Mesher for a Numerically Efficient Patient Specific Analysis of Arterial Blood Flow and Wall Shear Stress

2010 ◽  
Author(s):  
G. De Santis ◽  
P. Mortier ◽  
M. De Beule ◽  
P. Segers ◽  
P. Verdonck ◽  
...  
Keyword(s):  
Shear Stress ◽  
Wall Shear Stress ◽  
Imaging Techniques ◽  
Arterial Blood Flow ◽  
Patient Specific ◽  
Arterial Tree ◽  
Arterial Blood ◽  
Current Measuring ◽  
Wall Shear

Atherosclerosis depends on systemic risk factors but manifests itself as geometrically focal plaques, which appear in regions of the arterial tree experiencing low and/or oscillating Wall Shear Stress (WSS) such as outer edges of vessels bifurcations and highly curved vessels. Because direct measurements of WSS (differential quantity) in vivo are difficult due to limited spatial resolution offered by current measuring technologies (ultrasound, phase contrast MRI), an indirect approach is often taken, integrating medical imaging techniques (biplane angiography, CT, MRI) with Computational Fluid Dynamics (CFD) for patient specific WSS profiling.

The Impact of Hemorheology on Wall Shear Stress in a Separated and Reattached Flow Region

2013 ◽  
Author(s):  
Khaled J. Hammad
Keyword(s):  
Shear Stress ◽  
Wall Shear Stress ◽  
Yield Stress ◽  
Flow Region ◽  
Shear Thinning ◽  
Progression Rate ◽  
Vortex Center ◽  
Wall Shear ◽  
Separated And Reattached Flow ◽  
The Impact

Wall-bounded separating and reattaching flows are encountered in biological applications dealing with blood flows through arteries and prosthetic devices. Separated and reattached flow regions have been associated in the past with the most common arterial disease, atherosclerosis. Previous studies suggest that local wall shear stress (WSS) patterns affect the location and progression rate of atherosclerotic lesions. A parametric study is performed to investigate the influence of hemorheology on the wall shear stress distribution in a separated and reattached flow region. Recent hemorheological studies quantified and emphasized the yield stress and shear-thinning non-Newtonian characteristics of unadulterated human blood. Numerical solutions to the governing equations that account for yield stress and shear-thinning rheological effects are obtained. A low WSS region is observed around the flow reattachment point while a peak WSS always exists close to the vortex center. The yield shear-thinning hemorheological model always results in the highest observed peak WSS. The yield stress impact on WSS distribution is most pronounced in the case of severe restrictions to the flow.

EFFECTS OF TRANSVERSE MAGNETIC FIELD ON THE PERISTALTIC TRANSPORT OF VISCOELASTIC FLUID WITH JEFFREY MODEL IN A FINITE LENGTH CHANNEL

2011 ◽  
Vol 25 (26) ◽  
pp. 3455-3471 ◽  
Author(s):  
D. TRIPATHI ◽  
T. HAYAT ◽  
N. ALI ◽  
S. K. PANDEY
Keyword(s):  
Magnetic Field ◽  
Shear Stress ◽  
Wall Shear Stress ◽  
Viscoelastic Fluid ◽  
Transverse Magnetic Field ◽  
Flow Rate ◽  
Volume Flow Rate ◽  
Transverse Magnetic ◽  
Volume Flow ◽  
Wall Shear

This paper investigates the peristaltic flow of viscoelastic fluid represented by Jeffrey model in presence of transverse magnetic field under long wavelength and low Reynolds number assumptions. The expressions of pressure gradient, volume flow rate, average volume flow rate and local wall shear stress are obtained. The effects of transverse magnetic field, electrical conductivity (i.e., Hartman number M), relaxation time and retardation time on pressure difference, local wall shear stress, and mechanical efficiency of peristaltic pump are discussed. Reflux limit for viscoelastic fluid is also found and the effects of all parameters on reflux phenomena are discussed. Comparative study of integral and nonintegral number of waves propagate in a train is presented.

scholarly journals Effects of Disturbed Flow on Vascular Endothelium: Pathophysiological Basis and Clinical Perspectives

2011 ◽  
Vol 91 (1) ◽  
pp. 327-387 ◽  
Author(s):  
Jeng-Jiann Chiu ◽  
Shu Chien
Keyword(s):  
Blood Flow ◽  
Shear Stress ◽  
Wall Shear Stress ◽  
Flow Patterns ◽  
Arterial Tree ◽  
Disturbed Flow ◽  
Hemodynamic Forces ◽  
Venous Diseases ◽  
Wall Shear

Vascular endothelial cells (ECs) are exposed to hemodynamic forces, which modulate EC functions and vascular biology/pathobiology in health and disease. The flow patterns and hemodynamic forces are not uniform in the vascular system. In straight parts of the arterial tree, blood flow is generally laminar and wall shear stress is high and directed; in branches and curvatures, blood flow is disturbed with nonuniform and irregular distribution of low wall shear stress. Sustained laminar flow with high shear stress upregulates expressions of EC genes and proteins that are protective against atherosclerosis, whereas disturbed flow with associated reciprocating, low shear stress generally upregulates the EC genes and proteins that promote atherogenesis. These findings have led to the concept that the disturbed flow pattern in branch points and curvatures causes the preferential localization of atherosclerotic lesions. Disturbed flow also results in postsurgical neointimal hyperplasia and contributes to pathophysiology of clinical conditions such as in-stent restenosis, vein bypass graft failure, and transplant vasculopathy, as well as aortic valve calcification. In the venous system, disturbed flow resulting from reflux, outflow obstruction, and/or stasis leads to venous inflammation and thrombosis, and hence the development of chronic venous diseases. Understanding of the effects of disturbed flow on ECs can provide mechanistic insights into the role of complex flow patterns in pathogenesis of vascular diseases and can help to elucidate the phenotypic and functional differences between quiescent (nonatherogenic/nonthrombogenic) and activated (atherogenic/thrombogenic) ECs. This review summarizes the current knowledge on the role of disturbed flow in EC physiology and pathophysiology, as well as its clinical implications. Such information can contribute to our understanding of the etiology of lesion development in vascular niches with disturbed flow and help to generate new approaches for therapeutic interventions.

scholarly journals Pulsatile flow of shear-dependent fluid in a stenosed artery

2012 ◽  
Vol 39 (3) ◽  
pp. 209-231 ◽  
Author(s):  
Shankar Mandal ◽  
Swati Mukhopadhyayy ◽  
G.C.Z. Layek
Keyword(s):  
Blood Flow ◽  
Shear Stress ◽  
Wall Shear Stress ◽  
Blood Viscosity ◽  
Shear Thinning ◽  
Fluid Viscosity ◽  
Stenosed Artery ◽  
Flowing Blood ◽  
Wall Shear ◽  
Shear Thinning Fluid

This paper aims to investigate the blood flow in a bell-shaped constricted rigid tube, modeled as stenosed artery. The flow is assumed to be axi-symmetric, laminar and of oscillatory type. A mathematical model of shear-thinning fluid corresponding to the shear-dependent blood viscosity (mainly due to the behavior of the red blood cells in suspension of the flowing blood) is considered. The governing equations of motion are presented with the help of stream function-vorticity and are solved numerically by finite-difference technique. The shear-thinning fluid model for the flowing blood has significant contribution in the dynamics of oscillatory blood flow. The results reveal that the arterial wall shear stress reduced significantly and the peak value of the wall shear stress at the maximum area reduction is comparatively low for Newtonian fluid viscosity. The lengths of recirculating regions formed after the constriction are reduced for the shear-thinning blood viscosity model and also for its different material parameters.

scholarly journals Numerical Study of Non-Newtonian Blood Behavior in the Abdominal Aortic Bifurcation of a Patient-Specific at Rest

2017 ◽  
Vol 10 (1) ◽  
pp. 279-285 ◽  
Author(s):  
Carlos Oliveira ◽  
Armando A. Soares ◽  
André Simões ◽  
Sílvia Gonzaga ◽  
Abel Rouboa
Keyword(s):  
Shear Stress ◽  
Wall Shear Stress ◽  
Cardiovascular Health ◽  
Numerical Study ◽  
Shear Thinning ◽  
Patient Specific ◽  
Aortic Bifurcation ◽  
Low Velocity ◽  
Abdominal Aortic ◽  
Wall Shear

Background:The interaction of blood flow with walls of blood vessels is central for the development and maintenance of cardiovascular health. The analysis of wall shear stress is, therefore, fundamental in hemodynamic studies.Objective:The aim of this work is to study numerically the influence of the shear thinning blood properties on the hemodynamics in the abdominal aortic bifurcation for a patient-specific at rest.Methods:Were tested two models for the blood dynamic viscosity, one Newtonian and other non-Newtonian, with dependence on hematocrit and total protein minus albumin.Results and Conclusion:The results show the shear thinning behavior influence on the velocity distribution and wall shear stress. Furthermore, wall shear stress values are globally lower for non-Newtonian blood model at high velocity values than those for the Newtonian blood model. However, for low velocity values this behavior is inverted.

Wall shear stress gradient topography in the normal left coronary arterial tree: possible implications for atherogenesis

2004 ◽  
Vol 20 (5) ◽  
pp. 587-596 ◽  
Author(s):  
Thomas M. Farmakis ◽  
Johannes V. Soulis ◽  
George D. Giannoglou ◽  
George J. Zioupos ◽  
George E. Louridas
Keyword(s):  
Shear Stress ◽  
Wall Shear Stress ◽  
Stress Gradient ◽  
Arterial Tree ◽  
Wall Shear Stress Gradient ◽  
Shear Stress Gradient ◽  
Wall Shear

scholarly journals Effect of compliance and hematocrit on wall shear stress in a model of the entire coronary arterial tree

2009 ◽  
Vol 107 (2) ◽  
pp. 500-505 ◽  
Author(s):  
Yunlong Huo ◽  
Ghassan S. Kassab
Keyword(s):  
Shear Stress ◽  
Phase Separation ◽  
Wall Shear Stress ◽  
Coronary Arteries ◽  
Flow Rate ◽  
Large Scale ◽  
Polynomial Function ◽  
Arterial Tree ◽  
Order Polynomial ◽  
Wall Shear

A hemodynamic analysis is implemented in the entire coronary arterial tree of diastolically arrested, vasodilated pig heart that takes into account vessel compliance and blood viscosity in each vessel of a large-scale simulation involving millions of vessels. The feed hematocrit (Hct) is varied at the inlet of the coronary arterial tree, and the Fahraeus-Lindqvist effect and phase separation are considered throughout the vasculature. The major findings are as follows: 1) vessel compliance is the major determinant of nonlinearity of the pressure-flow relation, and 2) changes in Hct influence wall shear stress (WSS) in epicardial coronary arteries more significantly than in transmural and perfusion arterioles because of the Fahraeus-Lindqvist effect. The present study predicts the flow rate as a second-order polynomial function of inlet pressure due to vessel compliance. WSS in epicardial coronary arteries increases >15% with an increase of feed Hct from 45% to 60% and decreases >15% with a decrease of feed Hct from 45% to 30%, whereas WSS in small arterioles is not affected as feed Hct changes in this range. These findings have important implications for acute Hct changes under vasodilated conditions.

Inflow Conditions and the Wall Shear Stress Characteristics of a Biofluid in Separated and Reattached Flow Regions

2014 ◽  
Author(s):  
Khaled J. Hammad
Keyword(s):  
Shear Stress ◽  
Wall Shear Stress ◽  
Yield Stress ◽  
Human Blood ◽  
Flow Separation ◽  
Shear Thinning ◽  
Vortex Center ◽  
Wall Shear ◽  
Separated And Reattached Flow ◽  
Inflow Conditions

The influence of inflow conditions and human blood rheology on the wall shear stress distribution in a confined separated and reattached flow region is investigated. The governing mass and momentum conservation equations along with the Herschel-Bulkley rheological model are solved numerically using a finite-difference scheme. A parametric study is performed to reveal the influence of uniform and fully-developed inflow velocity profiles on the wall shear stress (WSS) characteristics using hemorheological models that account for the yield stress and shear-thinning non-Newtonian characteristics of human blood. The highest WSS or WSSmax, is always observed inside the flow separation region at a location corresponding to that of the corner vortex center. Uniform inflow results in higher WSSmax values in comparison with fully-developed inflow for moderate upstream flow restrictions. The opposite trend is observed for severe flow restrictions. Uniform inflow always results in smaller flow separation regions and WSSmax values at locations closer to the flow restriction plane. The yield shear-thinning hemorheological model always results in the highest observed peak WSS. The yield stress impact on WSS distribution is most pronounced in the case of severe restrictions to the flow.

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