Pulsatile flow of Herschel–Bulkley fluid through catheterized arteries – A mathematical model

The pulsatile flow of blood through catheterized artery has been studied in this paper by modeling blood as Herschel–Bulkley fluid and the catheter and artery as rigid coaxial circular cylinders. The Herschel–Bulkley fluid has two parameters, the yield stress θ and the power index n. Perturbation method is used to solve the resulting quasi-steady nonlinear coupled implicit system of differential equations. The effects of catheterization and non-Newtonian nature of blood on yield plane locations, velocity, flow rate, wall shear stress and longitudinal impedance of the artery are discussed. The existence of two yield plane locations is investigated and their dependence on yield stress θ, amplitude A, and time t are analyzed. The width of the plug core region increases with increasing value of yield stress at any time. The velocity and flow rate decrease, whereas wall shear stress and longitudinal impedance increase for increasing value of yield stress with other parameters held fixed. On the other hand, the velocity, flow rate and wall shear stress decrease but resistance to flow increases as the catheter radius ratio (ratio of catheter radius to vessel radius) increases with other parameters fixed. The results for power law fluid, Newtonian fluid and Bingham fluid are obtained as special cases from this model.     

Numerical computation of pulsatile flow through a locally constricted channel

This paper deals with the numerical solution of a pulsatile laminar flow through a locally constricted channel. A finite difference technique has been employed to solve the governing equations. The effects of the flow parameters such as Reynolds number, flow pulsation in terms of Strouhal number, constriction height and length on the flow behaviour have been studied. It is found that the peak value of the wall shear stress has significantly changed with the variation of Reynolds numbers and constriction heights. It is also noted that the Strouhal number and constriction length have little effect on the peak value of the wall shear stress. The flow computation reveals that the peak value of the wall shear stress at maximum flow rate time in pulsatile flow situation is much larger than that due to steady flow. The constriction and the flow pulsation produce flow disturbances at the vicinity of the constriction of the channel in the downstream direction.     

Laminar flow separation in an axi-symmetric sudden smooth expanded circular tube

This paper concerns with the investigation of laminar flow separation and its consequences in a tube over a smooth expansion under the axi-symmetric approximations. A co-ordinate stretching has been made to map the expanded tube into a straight tube. The two-dimensional unsteady Navier-Stokes equations are solved approximately by using primitive variables in staggered grid. A thorough quantitative analysis is performed through numerical simulations of the desired quantities such as wall shear stress, axial velocity, pressure distribution etc. These quantities are presented graphically and their consequences in the flow field are analysed in details. The dependence of the flow field on the physical parameter like expansion height d and on the Reynolds number has been investigated in details. It is interesting to note that the peak value of wall shear stress decreases with increasing height of expansion and also with the increasing Reynolds number.     

Numerical simulation of generalized newtonian blood flow past a couple of irregular arterial stenoses

This paper looked at the numerical investigations of the generalized Newtonian blood flow through a couple of irregular arterial stenoses. The flow is treated to be axisymmetric, with an outline of the stenoses obtained from a three dimensional casting of a mild stenosed artery, so that the flow effectively becomes two‐dimensional. The Marker and Cell (MAC) method is developed for the governing unsteady generalized Newtonian equations in staggered grid for viscous incompressible flow in the cylindrical polar co‐ordinates system. The derived pressure‐Poisson equation was solved using Successive‐Over‐Relaxation (S.O.R.) method and the pressure‐velocity correction formulae have been derived. Computations are performed for the pressure drop, the wall shear stress distribution and the separation region. The presented computations show that in comparison to the corresponding Newtonian model the generalized Newtonian fluid experiences higher pressure drop, lower peak wall shear stress and smaller separation region. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 960–981, 2011     

Effects of stenosis on arterial rheology through a mathematical model

The paper presents an analytical study of blood flow through a stenosed artery using a suitable mathematical model. The artery is modelled as an anisotropic viscoelastic cylindrical tube containing a non-Newtonian viscous incompressible fluid representing blood. The blood flow is assumed to be characterized by the Herschel–Bulkley model. The effect of the surrounding connective tissues on the motion of the arterial wall has been incorporated. Initially, the relevant solutions of the boundary value problem are obtained in the Laplace transform space, through the use of a suitable finite difference technique. Laplace inversion is carried out by employing suitable numerical techniques. Finally, the variations of the vascular wall displacements, the velocity distribution of the blood flow, the flux, the resistance to flow and the wall shear stress in the stenotic region are quantified through numerical computations and presented graphically.     

A two-dimensional model of pulsating flow in the basilar artery

The flow in the basilar artery is modelled by a pulsating flow of a viscous fluid in a plane straight semi-infinite channel with rigid walls. To model the merging flow from the two vertebral arteries, the prescribed initial velocity profile exhibits two separate maximum values. Numerical results are presented for the downstream velocity, the wall shear and the time-dependent inlet length. Finally, the biomechanical implications of the results are discussed.     

Mathematical modeling of pulsatile flow of non-Newtonian fluid in stenosed arteries

The pulsatile flow of blood through mild stenosed artery is studied. The effects of pulsatility, stenosis and non-Newtonian behavior of blood, treating the blood as Herschel–Bulkley fluid, are simultaneously considered. A perturbation method is used to analyze the flow. The expressions for the shear stress, velocity, flow rate, wall shear stress, longitudinal impedance and the plug core radius have been obtained. The variations of these flow quantities with different parameters of the fluid have been analyzed. It is found that, the plug core radius, pressure drop and wall shear stress increase with the increase of yield stress or the stenosis height. The velocity and the wall shear stress increase considerably with the increase in the amplitude of the pressure drop. It is clear that for a given value of stenosis height and for the increasing values of the stenosis shape parameter from 3 to 6, there is a sharp increase in the impedance of the flow and also the plots are skewed to the right-hand side. It is observed that the estimates of the increase in the longitudinal impedance increase with the increase of the axial distance or with the increase of the stenosis height. The present study also brings out the effects of asymmetric of the stenosis on the flow quantities.     

Investigation of physiological pulsatile flow in a model arterial stenosis using large-eddy and direct numerical simulations

Physiological pulsatile flow in a 3D model of arterial stenosis is investigated by using large eddy simulation (LES) technique. The computational domain chosen is a simple channel with a biological type stenosis formed eccentrically on the top wall. The physiological pulsation is generated at the inlet using the first harmonic of the Fourier series of pressure pulse. In LES, the large scale flows are resolved fully while the unresolved subgrid scale (SGS) motions are modelled using a localized dynamic model. Due to the narrowing of artery the pulsatile flow becomes transition-to-turbulent in the downstream region of the stenosis, where a high level of turbulent fluctuations is achieved, and some detailed information about the nature of these fluctuations are revealed through the investigation of the turbulent energy spectra. Transition-to-turbulent of the pulsatile flow in the post stenosis is examined through the various numerical results such as velocity, streamlines, velocity vectors, vortices, wall pressure and shear stresses, turbulent kinetic energy, and pressure gradient. A comparison of the LES results with the coarse DNS are given for the Reynolds number of 2000 in terms of the mean pressure, wall shear stress as well as the turbulent characteristics. The results show that the shear stress at the upper wall is low just prior to the centre of the stenosis, while it is maximum in the throat of the stenosis. But, at the immediate post stenotic region, the wall shear stress takes the oscillating form which is quite harmful to the blood cells and vessels. In addition, the pressure drops at the throat of the stenosis where the re-circulated flow region is created due to the adverse pressure gradient. The maximum turbulent kinetic energy is located at the post stenosis with the presence of the inertial sub-range region of slope −5/3.     

A non-Newtonian fluid flow model for blood flow through a catheterized artery—Steady flow

In this paper the effects catheterization and non-Newtonian nature of blood in small arteries of diameter less than 100 μm, on velocity, flow resistance and wall shear stress are analyzed mathematically by modeling blood as a Herschel–Bulkley fluid with parameters n and θ and the artery and catheter by coaxial rigid circular cylinders. The influence of the catheter radius and the yield stress of the fluid on the yield plane locations, velocity distributions, flow rate, wall shear stress and frictional resistance are investigated assuming the flow to be steady. It is shown that the velocity decreases as the yield stress increases for given values of other parameters. The frictional resistance as well as the wall shear stress increases with increasing yield stress, whereas the frictional resistance increases and the wall shear stress decreases with increasing catheter radius ratio k (catheter radius to vessel radius). For the range of catheter radius ratio 0.3–0.6, in smaller arteries where blood is modeled by Herschel–Bulkley fluid with yield stress θ = 0.1, the resistance increases by a factor 3.98–21.12 for n = 0.95 and by a factor 4.35–25.09 for n = 1.05. When θ = 0.3, these factors are 7.47–124.6 when n = 0.95 and 8.97–247.76 when n = 1.05.     

Power law fluid model for blood flow through a tapered artery with a stenosis

In the present paper, blood flow through a tapered artery with a stenosis is analyzed, assuming the flow is steady and blood is treated as non-Newtonian power law fluid model. Exact solution has been evaluated for velocity, resistance impedance, wall shear stress and shearing stress at the stenosis throat. The graphical results of different types of tapered arteries (i.e. converging tapering, diverging tapering, non-tapered artery) have been examined for different parameters of interest. Some special cases of the problem are also presented.     

弹性动脉血管中血液流动特性的模拟和分析

研究了两个不同的非牛顿血液流动模型:低粘性剪切简单幂律模型和低粘性剪切及粘弹性振荡流的广义Maxwell模型．同时利用这两个非牛顿模型和牛顿模型,研究了磁场中刚性和弹性直血管中血液的正弦型脉动．在生理学条件下,大动脉中血液的弹性对其流动性态似乎并不产生影响,单纯低粘性剪切模型可以逼真地模拟这种血液流动．利用高剪切幂律模型模拟弹性血管中的正弦型脉动流,发现在同一压力梯度下,与牛顿流体相比较,幂律流体的平均流率和流率变化幅度都更小．控制方程用Crank-Niclson方法求解．弹性动脉中血液受磁场作用是产生此结果的直观原因．在主动脉生物流的模拟中,与牛顿流体模型比较,发现在匹配流率曲线上,幂律模型的平均壁面剪切应力增大,峰值壁面剪切应力减小．讨论了弹性血管横切磁场时的血液流动,评估了血管形状和表面不规则等因素的影响．     

Effects of pulsatile flow and straight length on low-density lipoprotein transport in a curved arterial wall

Abnormal accumulation of macromolecules such as low-density lipoproteins (LDLs) in the arterial wall causes narrowing and blockage of vessels, which leads to atherosclerosis. Effects of pulsatile nature of blood flows as well as the initial length on transport of the LDL species in the arterial boundary layer region are analyzed numerically in the present work. The set of governing equations consisting of continuity, Navier-Stokes, and species transport is solved using a projection method based on the second-order central difference discretization. The obtained results are in excellent agreement with the pertinent data. The computational results imply that the flow field and concentration distribution are time dependent but the variation of the filtration velocity can be ignored. The LDL concentration boundary layer thickness decreases in the outer part and increases in the inner part for both with or without straight length. Presence of initial straight length generates about 26% growth in the boundary layer thickness, although its effect on the LDL surface concentration (LSC) is negligible. The maximum LSC is related to the regions with minimum wall shear stress in the inner part of the curved artery, which have more potential for formation of atherosclerosis. A new numerical correlation between the LSC and boundary layer thickness is proposed and examined.     

Significance of plaque morphology in modifying flow characteristics in a diseased coronary artery: Numerical simulation using plaque measurements from intravascular ultrasound imaging

The paper deals with numerical investigation of the effect of plaque morphology on the flow characteristics in a diseased coronary artery using realistic plaque morphology. The morphological information of the lumen and the plaque is obtained from intravascular ultrasound imaging measurements of 42 patients performed at Cleveland Clinic Foundation, Ohio. For this data, study of Bhaganagar et al. (2010) [1] has revealed the stenosis for 42 patients can be categorized into four types – type I (peak-valley), type II (ascending), type III (descending), and type IV (diffuse). The aim of the present study is to isolate the effect of shape of the stenosis on the flow characteristics for a given degree of the stenosis. In this study, we conduct fluid dynamic simulations for the four stenosis types (type I–IV) and analyze the differences in the flow characteristics between these types. Finely refined tetrahedral mesh for the 3-D solid model of the artery with plaques has been generated. The 3-D steady flow simulations were performed using the turbulence (k–ε) model in a finite volume based computational fluid dynamics solver. The axial velocity, the radial velocity, turbulence kinetic energy and wall shear stress profiles of the plaque have been analyzed. From the axial and radial velocity profiles results the differences in the velocity patterns are significantly visible at proximal as well as distal to the throat, region of maximum stenosis. Turbulent kinetic energy and wall shear stress profiles have revealed significant differences in the vicinity of the plaque. Additional unsteady flow simulations have been performed to validate the hypothesis of the significance of plaque morphology in flow alterations in diseased coronary artery. The results revealed the importance of accounting for plaque morphology in addition to plaque height to accurately characterize the turbulent flow in a diseased coronary artery.     

Nonlinear mathematical analysis for blood flow in a constricted artery under periodic body acceleration

This study analyses the pulsatile flow of blood through mild stenosed narrow arteries, treating the blood in the core region as a Casson fluid and the plasma in the peripheral layer as a Newtonian fluid. Perturbation method is employed to solve the resulting coupled implicit system of non-linear partial differential equations. The expressions for shear stress, velocity, wall shear stress, plug core radius, flow rate and longitudinal impedance to flow are obtained. The effects of pulsatility, stenosis depth, peripheral layer thickness, body acceleration and non-Newtonian behavior of blood on these flow quantities are discussed. It is noted that the plug core radius, wall shear stress and longitudinal impedance to flow increase as the yield stress and stenosis depth increase and they decrease with the increase of the body acceleration, pressure gradient, width of the peripheral layer thickness. It is observed that the plug flow velocity and flow rate increase with the increase of the pulsatile Reynolds number, body acceleration, pressure gradient and the width of the peripheral layer thickness and the reverse behavior is found when the yield stress, stenosis depth and lead angle increase. It is also recorded that the wall shear stress and longitudinal impedance to flow are considerably lower for the two-fluid Casson model than that of the single-fluid Casson model. It is found that the presence of body acceleration and peripheral layer influences the mean flow rate and mean velocity by increasing their magnitude significantly in the arteries.     

流固耦合作用下狭窄颈动脉内非Newton血流分析

采用计算流体力学方法分别对6种狭窄率的颈动脉内非Newton瞬态血流进行流固耦合数值分析.研究了狭窄率对颈动脉内血流动力学分布的影响,以探索狭窄率与颈动脉内粥样斑块形成的关系.结果表明,狭窄率不同的颈动脉内血流动力学分布特性明显不同,与0.05,0.1,0.2,0.3和04这5种狭窄率的颈动脉内血流动力学分布特性相比,狭窄率为0.5的颈动脉内血流动力学分布独特,狭窄部位附近区域存在面积较大的低速涡流区;复杂血流作用下,该区域分布低壁面压力,异常壁面切应力,较大管壁形变量和von Mises应力;血流速度低使血液中脂质、纤维蛋白等大分子易沉积,低壁面压力引起的明显“负压”效应引发脑部供血障碍,异常壁面切应力作用下粥样斑块易破裂与脱落,并堵塞脑血管,较大的von Mises应力易引起应力集中,导致血管破裂,为脑卒中发生提供有利条件.因此,狭窄率越大对颈动脉内血流动力学分布的影响越显著,促进颈动脉粥样斑块形成与发展,并引发缺血性脑卒中.     

Jeffrey流体流过有限长圆柱管时磁流体动力学的蠕动流

研究Jeffrey流体流过有限长管道时的蠕动流.在外加磁场作用时,流体呈导电性.分析是在长波长和低Reynolds数近似假设下完成.得到了压力梯度、体积流量、平均体积流量和局部壁面剪应力的表达式.研究了松弛时间、延迟时间和Hartman数,对压力、局部壁面剪应力以及蠕动泵机械效率的影响.还研究了回流现象,调查了沿管道壁波数非整倍数时的传播情况,研究有限长管道传播的内在特性.     

Buoyant Poiseuille–Couette flow with viscous dissipation in a vertical channel

Steady combined forced and free convection is investigated in a vertical channel having a wall at rest and a moving wall subjected to a prescribed shear stress. The moving wall is thermally insulated, while the wall at rest is kept at a uniform temperature. The analysis deals with the fully–developed parallel flow regime. The governing equations yield a boundary value problem, that is solved analytically by employing a power series expansion of the velocity field with respect to the transverse coordinate. It is shown that the nonlinear interplay between buoyancy and viscous dissipation may determine the existence of dual solutions of the boundary value problem corresponding to fixed values of the applied shear stress on the moving wall and of the hydrodynamic pressure gradient. It is shown that a nontrivial fully separated flow may occur such that the hydrodynamic pressure gradient is zero and the shear stress vanishes on both walls. E. Magyari: On leave from Institute of Building Technology, ETH – Zürich     

A micropolar fluid model of blood flow through a tapered artery with a stenosis

A nonlinear two‐dimensional micropolar fluid model for blood flow in a tapered artery with a single stenosis is considered. This model takes into account blood rheology in which blood consists of microelements suspended in plasma. The classical Navier–Stokes theory is inadequate to describe the microrotations or particles' spin of such suspension in a viscous medium. The governing equations involving unsteady nonlinear partial differential equations are solved using a finite difference scheme. A quantitative analysis performed through numerical computation shows that the axial velocity profile and the flow rate decrease and the wall shear stress increases once the artery is narrower in the presence of the polar effect. Furthermore, the taper angle certainly bears the potential to influence the velocity and the flow characteristics to considerable extent. Copyright © 2010 John Wiley & Sons, Ltd.     

Study of magnetohydrodynamic pulsatile flow in a constricted channel

The present work reports the study of steady and pulsatile flows of an electrically conducting fluid in a differently shaped locally constricted channel in presence of an external transverse uniform magnetic field. The governing nonlinear magnetohydrodynamic equations simplified for low conducting fluids are solved numerically by finite difference method using stream function-vorticity formulation. The analysis reveals that the flow separation region is diminished with increasing values of magnetic parameter. It is noticed that the increase in the magnetic field strength results in the progressive flattening of axial velocity. The variations of wall shear stress with increasing values of the magnetic parameter are shown for both steady and pulsatile flow conditions. The streamline and vorticity distributions in magnetohydrodynamic flow are also shown graphically and discussed.     

Effect of constriction height on flow separation in a two-dimensional channel

We studied numerically the effect of the constriction height on viscous flow separation past a two-dimensional channel with locally symmetric constrictions. A numerically stable scheme in primitive variables (velocity and pressure) for the solution of two-dimensional incompressible time-dependent Navier–Stokes equations is employed using finite-difference approximation in staggered grid. The wall shear stresses at different heights of the constriction are computed and presented graphically. It is noticed that the maximum stress and the length of the recirculating region associated with two shear layers of the constriction increase with the increase of the area reduction of the constriction. The critical Reynolds number for symmetry breaking bifurcation for the 50%, 60% and 70% area reduction are obtained numerically. The flow field separates after the symmetry breaking bifurcation and the symmetry of the flow depends on the Reynolds number and the height of the constriction.