A Mathematical Model for the Laminar Entrance Region of a Newtonian Fluid in a Cylindrical Pipe

2020 ◽  
Vol 143 (3) ◽  
Author(s):  
Richard J. Gross ◽  
Nicholas G. Garafolo ◽  
Garrett R. McHugh
Keyword(s):  
Shear Stress ◽  
Pressure Drop ◽  
Laminar Flow ◽  
Wall Shear Stress ◽  
Velocity Profile ◽  
Newtonian Fluid ◽  
Axial Velocity ◽  
Entrance Region ◽  
Velocity Data ◽  
Cylindrical Pipe

Abstract This paper develops equations for velocity, pressure drop, and wall shear stress in the entrance or development region of a cylindrical pipe. The model quantifies the velocity and wall shear stress contributions to the entrance region pressure drop and illustrates how data are used to determine the numerical values of parameters needed to complete the model. It assumes a Newtonian fluid, laminar flow, steady-state, and a constant mass density fluid. The fluid axial velocity profile at the entrance region inlet is modeled by an equation that is close to a flat axial velocity and drops off to zero as the radius approaches the wall. The fluid velocity at the entrance region exit is modeled as the axial, fully developed, laminar flow parabolic velocity profile. The inlet velocity profile is multiplied by a decaying function F(x) that is unity at the entrance region inlet and decreases to zero at the entrance region exit. The exit velocity profile is multiplied by a growing function G(x) that is zero at the entrance region inlet and increases to unity at the entrance region exit. The pressure drop through the entrance region is expressed in terms of the wall viscous friction and the change in axial momentum of the fluid. Two mathematical models for F(x) and G(x) are presented. One is more advantageous when pressure drop data and a few centerline velocity data points are available, and the second is more advantageous when only velocity data are available.

Analysis of Steady Laminar Flow of an Incompressible Newtonian Fluid Through Curved Pipes of Small Curvature

1973 ◽  
Vol 95 (1) ◽  
pp. 75-80 ◽  
Author(s):  
M. Sankaraiah ◽  
Y. V. N. Rao
Keyword(s):  
Pressure Drop ◽  
Laminar Flow ◽  
Secondary Flow ◽  
Newtonian Fluid ◽  
Axial Velocity ◽  
Axial Pressure ◽  
Small Curvature ◽  
Curved Pipes ◽  
Incompressible Newtonian Fluid ◽  
Steady Laminar

Steady laminar flow of an incompressible Newtonian fluid through a curved pipe of small curvature is considered. The governing equations of flow are obtained in terms of secondary flow stream function and axial velocity component as suggested by Dean. A method of successive approximations is developed to solve these equations. The first five approximations are computed. The solution obtained is used to determine the axial velocity distribution, secondary flow pattern, axial pressure drop, and pressure distribution along the pipe wall. A semiempirical equation is obtained for axial pressure drop. The theoretical results obtained are compared with the available experimental data on axial pressure drop.

A Superposition Analysis of the Turbulent Boundary Layer in an Adverse Pressure Gradient

1951 ◽  
Vol 18 (1) ◽  
pp. 95-100
Author(s):  
Donald Ross ◽  
J. M. Robertson
Keyword(s):  
Boundary Layer ◽  
Shear Stress ◽  
Wall Shear Stress ◽  
Turbulent Boundary Layer ◽  
Velocity Profile ◽  
Pressure Gradient ◽  
Adverse Pressure Gradient ◽  
Pressure Recovery ◽  
Stress Coefficient ◽  
Wall Shear

Abstract As an interim solution to the problem of the turbulent boundary layer in an adverse pressure gradient, a super-position method of analysis has been developed. In this method, the velocity profile is considered to be the result of two effects: the wall shear stress and the pressure recovery. These are superimposed, yielding an expression for the velocity profiles which approximate measured distributions. The theory also leads to a more reasonable expression for the wall shear-stress coefficient.

A Review On the Effect of Temporal Geometric Variations of the Coronary Arteries On the Wall Shear Stress and Pressure Drop

2021 ◽  
Author(s):  
Navid Freidoonimehr ◽  
Rey Chin ◽  
Anthony C. Zander ◽  
Maziar Arjomandi
Keyword(s):  
Shear Stress ◽  
Pressure Drop ◽  
Wall Shear Stress ◽  
Coronary Arteries ◽  
Cardiac Cycle ◽  
Temporal Variations ◽  
Hemodynamic Parameters ◽  
Clinical Environment ◽  
Cross Sectional ◽  
Wall Shear

Abstract Temporal variations of the coronary arteries during a cardiac cycle are defined as the superposition of the changes in the position, curvature, and torsion of the coronary artery axis markers and the variations in the lumen cross-sectional shape due to the distensible wall motion induced by the pulse pressure and contraction of the myocardium in a cardiac cycle. This review discusses whether the modelling the temporal variations of the coronary arteries is needed for the investigation of the hemodynamics specifically in time critical applications such as a clinical environment. The numerical modellings in the literature which model or disregard the temporal variations of the coronary arteries on the hemodynamic parameters are discussed. The results in the literature show that neglecting the effects of temporal geometric variations is expected to result in about 5\% deviation of the time-averaged pressure drop and wall shear stress values and also about 20\% deviation of the temporal variations of hemodynamic parameters, such as time-dependent wall shear stress and oscillatory shear index. This review study can be considered as a guide for the future studies to outline the conditions in which temporal variations of the coronary arteries can be neglected, while providing a reliable estimation of hemodynamic parameters.

scholarly journals Numerical Simulation of Dispersed Particle-Blood Flow in the Stenosed Coronary Arteries

2018 ◽  
Vol 2018 ◽  
pp. 1-16 ◽  
Author(s):  
Mongkol Kaewbumrung ◽  
Somsak Orankitjaroen ◽  
Pichit Boonkrong ◽  
Buraskorn Nuntadilok ◽  
Benchawan Wiwatanapataphee
Keyword(s):  
Numerical Simulation ◽  
Blood Flow ◽  
Shear Stress ◽  
Coronary Artery ◽  
Pressure Drop ◽  
Wall Shear Stress ◽  
Stokes Equations ◽  
Spherical Shape ◽  
Shear Stresses ◽  
Wall Shear

A mathematical model of dispersed bioparticle-blood flow through the stenosed coronary artery under the pulsatile boundary conditions is proposed. Blood is assumed to be an incompressible non-Newtonian fluid and its flow is considered as turbulence described by the Reynolds-averaged Navier-Stokes equations. Bioparticles are assumed to be spherical shape with the same density as blood, and their translation and rotational motions are governed by Newtonian equations. Impact of particle movement on the blood velocity, the pressure distribution, and the wall shear stress distribution in three different severity degrees of stenosis including 25%, 50%, and 75% are investigated through the numerical simulation using ANSYS 18.2. Increasing degree of stenosis severity results in higher values of the pressure drop and wall shear stresses. The higher level of bioparticle motion directly varies with the pressure drop and wall shear stress. The area of coronary artery with higher density of bioparticles also presents the higher wall shear stress.

scholarly journals Investigation of blood flow through an artery in the presence of overlapping stenosis

2017 ◽  
Vol 14 (1) ◽  
pp. 39-46 ◽  
Author(s):  
K. Maruthi Prasad ◽  
S. Thulluri ◽  
M. V. Phanikumari
Keyword(s):  
Blood Flow ◽  
Shear Stress ◽  
Pressure Drop ◽  
Wall Shear Stress ◽  
Couple Stress ◽  
Flow Characteristics ◽  
Couple Stress Fluid ◽  
Flow Equations ◽  
Wall Shear ◽  
Fluid Parameters

The effects of an overlapping stenosis on blood flow characteristics in an artery have been studied. Blood has been represented by a couple stress fluid. The flow equations have been linearised and the expressions for pressure drop, resistance to the flow and wall shear stress have been derived. The results are shown graphically. It is observed that the resistance to the flow, pressure drop and wall shear stress increases with height and length of the stenosis. And it is noticed that the resistance to the flow and pressure drop decreases with couple stress fluid parameters. But wall shear stress increases with couple stress fluid parameters.

scholarly journals Distribution of Wall Shear Stress in Right-Angle Branches during Laminar Flow

2005 ◽  
Vol 48 (4) ◽  
pp. 468-476
Author(s):  
Hisashi FUJII ◽  
Ryuhei YAMAGUCHI ◽  
Kazuo TANISHITA
Keyword(s):  
Shear Stress ◽  
Laminar Flow ◽  
Wall Shear Stress ◽  
Wall Shear

Temporal and Spatial Variations of Wall Shear Stress in the Entrance Region of Microvessels

2015 ◽  
Vol 137 (6) ◽  
Author(s):  
Othmane Oulaid ◽  
Junfeng Zhang
Keyword(s):  
Shear Stress ◽  
Wall Shear Stress ◽  
Stable State ◽  
Entrance Region ◽  
Channel Inlet ◽  
Long Distance ◽  
The Mean ◽  
Temporal And Spatial ◽  
Wall Shear ◽  
Cell Free Layer

Using a simplified two-dimensional divider-channel setup, we simulate the development process of red blood cell (RBC) flows in the entrance region of microvessels to study the wall shear stress (WSS) behaviors. Significant temporal and spatial variation in WSS is noticed. The maximum WSS magnitude and the strongest variation are observed at the channel inlet due to the close cell-wall contact. From the channel inlet, both the mean WSS and variation magnitude decrease, with a abrupt drop in the close vicinity near the inlet and then a slow relaxation over a relatively long distance; and a relative stable state with approximately constant mean and variation is established when the flow is well developed. The correlations between the WSS variation features and the cell free layer (CFL) structure are explored, and the effects of several hemodynamic parameters on the WSS variation are examined. In spite of the model limitations, the qualitative information revealed in this study could be useful for better understanding relevant processes and phenomena in the microcirculation.

Analysis of Flow Disturbance in a Stenosed Carotid Artery Bifurcation Using Two-Equation Transitional and Turbulence Models

2008 ◽  
Vol 130 (6) ◽  
Author(s):  
F. P. P. Tan ◽  
G. Soloperto ◽  
S. Bashford ◽  
N. B. Wood ◽  
S. Thom ◽  
...  
Keyword(s):  
Shear Stress ◽  
Carotid Artery ◽  
Laminar Flow ◽  
Wall Shear Stress ◽  
Turbulence Models ◽  
Carotid Bifurcation ◽  
Patient Specific ◽  
Shear Stresses ◽  
Zone Length ◽  
Wall Shear

In this study, newly developed two-equation turbulence models and transitional variants are employed for the prediction of blood flow patterns in a diseased carotid artery where the growth, progression, and structure of the plaque at rupture are closely linked to low and oscillating wall shear stresses. Moreover, the laminar-turbulent transition in the poststenotic zone can alter the separation zone length, wall shear stress, and pressure distribution over the plaque, with potential implications for stresses within the plaque. Following the validation with well established experimental measurements and numerical studies, a magnetic-resonance (MR) image-based model of the carotid bifurcation with 70% stenosis was reconstructed and simulated using realistic patient-specific conditions. Laminar flow, a correlation-based transitional version of Menter’s hybrid k‐ϵ∕k‐ω shear stress transport (SST) model and its “scale adaptive simulation” (SAS) variant were implemented in pulsatile simulations from which analyses of velocity profiles, wall shear stress, and turbulence intensity were conducted. In general, the transitional version of SST and its SAS variant are shown to give a better overall agreement than their standard counterparts with experimental data for pulsatile flow in an axisymmetric stenosed tube. For the patient-specific case reported, the wall shear stress analysis showed discernable differences between the laminar flow and SST transitional models but virtually no difference between the SST transitional model and its SAS variant.

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