Mathematical modelling of blood flow through an overlapping arterial stenosis

Of concern in the paper is a theoretical study of blood flow in an arterial segment in the presence of a time-dependent overlapping stenosis using an appropriate mathematical model. A remarkably new shape of the stenosis in the realm of the formation of the arterial narrowing caused by atheroma is constructed mathematically. The artery is simulated as an elastic (moving wall) cylindrical tube containing a viscoelastic fluid representing blood. The unsteady flow mechanism of the present investigation is subjected to a pulsatile pressure gradient arising from the normal functioning of the heart. The equations governing the motion of the system are sought in the Laplace transform space and their relevant solutions supplemented by the suitable boundary conditions are obtained numerically in the transformed domain through the use of an appropriate finite difference technique. Laplace inversion is also carried out by employing numerical techniques. A thorough quantitative analysis is performed at the end of the paper for the flow velocity, the flux, the resistive impedances, and the wall shear stresses together with their variations with the time, the pressure gradient, and the severity of the stenosis in order to illustrate the applicability of the present mathematical model under consideration.     

Analysis of blood flow through a modelled artery with an aneurysm

The intention of the present work is to carry out a systematic analysis of flow features in a tube, modelled as artery, having a local aneurysm in presence of haematocrit. The arterial model is treated to be axi-symmetric and rigid. The blood, flowing through the modelled artery, is treated to be Newtonian and non-homogeneous. For a thorough quantitative analysis of the flow characteristics such as wall pressure, flow velocity, wall shear stress, the unsteady incompressible Navier-Stokes equations in cylindrical polar co-ordinates under the laminar flow conditions are solved by using the finite-difference method. Finally, the numerical illustrations presented in this paper provide an effective measure to estimate the combined influence of haematocrit and aneurysm on flow characteristics. It is found that the magnitude of wall shear stress and also the length of separation increase with increasing values of the haematocrit parameter. The length of flow separation increases but the peak value of wall shear stress decreases with the increasing length of aneurysm. The peak value of wall shear stress as well as the length of separation increases with the increasing height of the aneurysm.     

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.     

Numerical simulation of Casson fluid flow through differently shaped arterial stenoses

The present investigation deals with a mathematical model of blood flow through an asymmetric (about its narrowest point) arterial constriction obtained from casting of a mildly stenosed artery. The flowing blood is represented as the suspension of all red cells (erythrocytes) in plasma assumed to be Casson fluid, while the arterial wall is considered to be rigid having differently shaped stenoses in its lumen arising from various types of abnormal growth or plaque formation. The governing equations of motion accompanied by the appropriate choice of the boundary conditions are solved numerically by Marker and Cell method in order to compute the physiologically significant quantities with desired degree of accuracy. The necessary checking for numerical stability has been incorporated in the algorithm for better precision of the results computed. The quantitative analyses have been carried out finally with the inclusion of the respective profiles of the flow field over the entire arterial segment as well. The key factors such as the wall shear stress, the pressure drop and the velocity profiles are exhibited graphically and examined thoroughly for qualitative insight into blood flow phenomena through arterial stenosis.     

Application of Adomian's approximation to blood flow through arteries in the presence of a magnetic field

The present investigation deals with the application of Adomian's decomposition method to blood flow through a constricted artery in the presence of an external transverse magnetic field which is applied uniformly. The blood flowing through the tube is assumed to be Newtonian in character. The expressions for the two-term approximation to the solution of stream function, axial velocity component and wall shear stress are obtained in this analysis. The numerical solutions of the wall shear stress for different values of Reynold number and Hartmann number are shown graphically. The solution of this theoretical result for a particular Hartmann number is compared with the integral method solution of Morgan and Young [17].     

A one dimensional model of blood flow through a curvilinear artery

We present a one-dimensional model describing the blood flow through a moderately curved and elastic blood vessel. We use an existing two dimensional model of the vessel wall along with Navier−Stokes equations to model the flow through the channel while taking factors, namely, surrounding muscle tissue and presence of external forces other than gravity into account. Our model is obtained via a dimension reduction procedure based on the assumption of thinness of the vessel relative to its length. Results of numerical simulations are presented to highlight the influence of different factors on the blood flow.     

Mass transfer to blood flowing through arterial stenosis

The present investigation deals with a mathematical model representing the mass transfer to blood streaming through the arteries under stenotic condition. The mass transport refers to the movement of atherogenic molecules, that is, blood-borne components, such as oxygen and low-density lipoproteins from flowing blood into the arterial walls or vice versa. The blood flowing through the artery is treated to be Newtonian and the arterial wall is considered to be rigid having differently shaped stenoses in its lumen arising from various types of abnormal growth or plaque formation. The nonlinear unsteady pulsatile flow phenomenon unaffected by concentration-field of the macromolecules is governed by the Navier–Stokes equations together with the equation of continuity while that of mass transfer is controlled by the convection-diffusion equation. The governing equations of motion accompanied by appropriate choice of the boundary conditions are solved numerically by MAC(Marker and Cell) method and checked numerical stability with desired degree of accuracy. The quantitative analysis carried out finally includes the respective profiles of the flow-field and concentration along with their distributions over the entire arterial segment as well. The key factors like the wall shear stress and Sherwood number are also examined for further qualitative insight into the flow and mass transport phenomena through arterial stenosis. The present results show quite consistency with several existing results in the literature which substantiate sufficiently to validate the applicability of the model under consideration.      

Mass transfer to blood flowing through arterial stenosis

The present investigation deals with a mathematical model representing the mass transfer to blood streaming through the arteries under stenotic condition. The mass transport refers to the movement of atherogenic molecules, that is, blood-borne components, such as oxygen and low-density lipoproteins from flowing blood into the arterial walls or vice versa. The blood flowing through the artery is treated to be Newtonian and the arterial wall is considered to be rigid having differently shaped stenoses in its lumen arising from various types of abnormal growth or plaque formation. The nonlinear unsteady pulsatile flow phenomenon unaffected by concentration-field of the macromolecules is governed by the Navier–Stokes equations together with the equation of continuity while that of mass transfer is controlled by the convection-diffusion equation. The governing equations of motion accompanied by appropriate choice of the boundary conditions are solved numerically by MAC(Marker and Cell) method and checked numerical stability with desired degree of accuracy. The quantitative analysis carried out finally includes the respective profiles of the flow-field and concentration along with their distributions over the entire arterial segment as well. The key factors like the wall shear stress and Sherwood number are also examined for further qualitative insight into the flow and mass transport phenomena through arterial stenosis. The present results show quite consistency with several existing results in the literature which substantiate sufficiently to validate the applicability of the model under consideration.     

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     

Factors Influencing the Disturbed Flow Patterns Downstream of Curved Atherosclerotic Arteries

Pulsatile blood flows in curved atherosclerotic arteries are studied by com- puter simulations.Computations are carried out with various values of physiological parameters to examine the effects of flow parameters on the disturbed flow patterns downstream of a curved artery with a stenosis at the inner wall.The numerical re- suits indicate a strong dependence of flow pattern on the blood viscosity and inlet flow rate,while the influence of the inlet flow profile to the flow pattem in downstream is negligible.     

磁场、多孔性和各向异性动脉壁有多处狭窄段时对血液流动的影响

建立一个血液流动的数学模型:多孔介质在磁场作用下,血液流过一段有多处狭窄段的弹性动脉;用一个各向异性的弹性圆柱形管道模拟动脉,用粘性不可压缩的导电流体表示血液,动脉有轻微的局部性狭窄,形成一段内腔局部变窄的动脉,并完成该模型的数学分析．详细阐述了血管壁参数对血液流动的影响,参数包括纵向和圆周向的粘弹性应力分量Tt和Tθ、血管壁的各向异性度γ、血管及其周边结缔组织的总质量M、完全栓管中粘性约束的贡献C和弹性约束的贡献K,并用图形表示壁面剪切应力的分布、径向和轴向的速度等．还研究了狭窄形状参数m、渗透率常数κ、Hartmann数Ha和血管狭窄区的最大高度δ,对血液流动特征的影响．研究表明,流动受到周边结缔组织（动脉壁运动）的影响式微,血管壁的各向异性度,是确定动脉材料的一个重要指标．进一步发现壁剪切力分布,随着Tt和γ的增加而增加,随着Tθ,M,C和K的增加而减少．壁面剪切应力分布的传播,以及壁面处阻力阻抗的传播,栓管与自由管相比要低得多;狭窄段咽喉处的剪切应力分布特性,完全栓管和自由管正相反．靠近中心线的俘获区大小,随着渗透率κ的增加而增大;随着Hartmann数Ha的增加而减小．最后,狭窄段非对称时,逐渐形成俘获区;狭窄段对称时,不出现俘获区,各向同性自由管（无初始应力）中俘获区的大小,比完全栓管中的小得多．     

Gas absorption with first order chemical reaction in a laminar falling film over a reacting solid wall

In the present work, a general case of gas absorption with first order irreversible chemical reaction in a liquid film, for laminar flow over a solid wall, has been analyzed theoretically. First order chemical reaction between the diffused solute and the wall is also considered. Laplace transform followed by power series method has been applied to solve the governing equations. Thereafter, the obtained analytical solution of the developed general model has been successfully verified by an explicit numerical scheme. The general model has also been reduced to six simplified cases, tackled by previous workers and an excellent agreement in the solutions is observed. Moreover, the results are validated by the experimental data available in the literature. The obtained concentration profiles in both the phases have been used to find the absorption rates and enhancement factor.     

Effects of variable viscosity on pulsatile flow of blood in a tapered stenotic flexible artery

The objective of the present study is to investigate the effects of variable viscosity on incompressible laminar pulsatile flow of blood through an overlapping doubly constricted tapered artery. To mimic the realistic situation, wall of the artery is taken to be flexible, and physiologically relevant pulsatile flow is introduced. The governing equations of blood flow are made dimensionless. A coordinate transformation is used to make the overlapping doubly constricted wall geometry of tube to a straight tube. Taking advantage of the Stream function–Vorticity formulation, the system of partial differential equations is then solved numerically by finite difference approximations. Effects of Reynolds number, Strouhal number, degree of contraction, tapering angle, and viscosity parameters are presented graphically and analyzed. The results show that formation of stenosis and tapering disturb the flow field significantly, and degree of stenosis is more important in influencing blood flow compared with tapering.     

肾动脉变窄和流-固结构相互作用对非Newton脉动流的影响——一个真实的腹部主动脉和肾动脉模型

研究肾动脉狭窄(RAS)对血液流动和血管壁的影响.根据CT扫描图像,重建腹部主动脉和肾动脉的解剖模型,通过模型的脉动流进行了仿真计算,计算中考虑了流体-固体结构的相互作用(FSI).研究RAS对血管壁剪切应力和位移的影响,RAS使得肾动脉中流量减少,肾素-血管紧缩素系统可能被激活,从而导致严重的高血压.     

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.     

A numerical simulation of unsteady blood flow through multi-irregular arterial stenoses

An unsteady mathematical model to study the characteristics of blood flowing through an arterial segment in the presence of a couple of stenoses with surface irregularities is developed. The flow is treated to be axisymmetric, with an outline of the stenoses obtained from a three dimensional casting of a mildly stenosed artery [1], so that the flow effectively becomes two-dimensional. The governing equations of motion accompanied by appropriate choice of boundary and initial conditions are solved numerically by MAC (Marker and Cell) method in cylindrical polar coordinate system in staggered grids and checked numerical stability with desired degree of accuracy. The pressure-Poisson equation has been solved by successive-over-relaxation (SOR) method and the pressure–velocity correction formulae have been derived. The flexibility of the arterial wall has also been accounted for in the present investigation. Further, in-depth study in the flow pattern reveals that the separation Reynolds number for the multi-irregular stenoses is lower than those for cosine-shaped stenoses and a long single irregular stenosis. The present results predict the excess pressure drop across the cosine stenoses than the irregular ones and show quite consistency with several existing results in the literature which substantiate sufficiently to validate the applicability of the model under consideration.     

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 theoretical study of filter cake washing

A problem on filter cake washing assuming a geometrical model of the pore spaces, in which axial dispersion occurs in the flow channel and mass transfer through a resistance is taking place from the cross channels to the flow channel, is solved by deriving the differential equations and the auxiliary conditions corresponding to the geometrical model and solving the mathematical problem utilizing the Laplace transform and a method for its numerical inversion that utilizes positive values of the transform variable. The numerical results are checked by comparison with asymptotic expansions and by recalculating the results utilizing other values of the transform variable. The results are presented both numerically with four significant figures and graphically.     

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．     

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.