Blood flow in stenosed arteries with radially variable viscosity, peripheral plasma layer thickness and magnetic field

A novel approach of combined mathematical and computational models has been developed to investigate the oscillatory two-layered flow of blood through arterial stenosis in the presence of a transverse uniform magnetic field applied. Blood in the core region and plasma fluid in the peripheral layer region are assumed to obey the law of Newtonian fluid. An analytical solution is obtained for velocity profile and volumetric flow rate in the peripheral plasma region and also wall shear stress. Finite difference method is employed to solve the momentum equation for the core region. The numerical solutions for velocity, flow rate and flow resistance are computed. The effects of various parameters associated with the present flow problem such as radially variable viscosity, hematocrit, plasma layer thickness, magnetic field and pulsatile Reynolds number on the physiologically important flow characteristics namely velocity distribution, flow rate, wall shear stress and resistance to flow have been investigated. It is observed that the velocity increases with the increase of plasma layer thickness. An increase or a decrease in the velocity and wall shear stress against the increase in the value of magnetic parameter (Hartmann number) and hematocrit is dependent on the value of t. An increase in magnetic field leads to an increase in the flow resistance and it decreases with the increase in the plasma layer thickness and pulsatile Reynolds number. The information concerning the phase lag between the flow characteristics and how it is affected by the hematocrit, plasma layer thickness and Hartmann number has, for the first time, been added to the literature.     

Performance modeling and analysis of blood flow in elastic arteries

Nomenclature　　τ　　wallshearstressγshearrateτy yieldstressηc Cassonviscosityktheconsistencyindexnnon_Newtonianindexτp shearstressofthepthelementωangularvelocityRvessel’sradiusCwavespeedM　　magneticparameter (Hartmannnumber)u,w　velocitycomponentinther_andz_directions,respectivelyP　　pressureα　　unsteadinessparameter k , R meanparametersTp relaxationtimeofthepthelementρ densityIntroductionTheimportancetoatherogenesisofarterialflowphenomenasuchasflowseparation ,recirculationands…     

Two-phase non-linear model for blood flow in asymmetric and axisymmetric stenosed arteries

The pulsatile flow of a two-phase model for blood flow through axisymmetric and asymmetric stenosed narrow arteries is analyzed, treating blood as a two-phase model with the suspension of all the erythrocytes in the core region as the Herschel-Bulkley material and plasma in the peripheral layer as the Newtonian fluid. The perturbation method is applied to solve the resulting non-linear implicit system of partial differential equations. The expressions for various flow quantities are obtained. It is found that the pressure drop, plug core radius, wall shear stress increase as the yield stress or stenosis height increases. It is noted that the velocity increases, longitudinal impedance decreases as the amplitude increases. For asymmetric stenosis, the wall shear stress increases non-linearly with the increase of the axial distance. The estimates of the increase in longitudinal impedance to flow of the two-phase Herschel-Bulkley material are significantly lower than those of the single-phase Herschel-Bulkley material. The results show the advantages of two-phase flow over single-phase flow in small diameter arteries with stenosis.     

Asymptotic suction profiles for the Blasius and Sakiadis flow with constant and variable fluid properties

A theoretical study of the effect of variable fluid properties on the Blasius and Sakiadis flow with uniform suction at the asymptotic state is presented in this paper. The investigation concerns air and water taking into account the variation of their physical properties with temperature. Velocity and temperature profiles are presented as well as values of the displacement thickness, momentum thickness, shape factor, wall shear stress and Nusselt number for different temperatures of the plate and the ambient fluid. It is found that the nondimensional displacement thickness, momentum thickness, shape factor, absolute wall shear stress and Nusselt number are identical in both Blasius and Sakiadib flow at the asymptotic state for a fluid with constant properties. The same is valid for any fluid with variable properties if the temperature boundary conditions are the same in Blasius and Sakiadis flow.     

Numerical simulation of mass transfer to micropolar fluid flow past a stenosed artery

A mathematical model of unsteady non‐Newtonian blood flow together with the mass transfer through constricted arteries has been developed. The mass transport refers to the movement of atherogenic molecules, i.e. blood‐borne components, such as low‐density lipoproteins from flowing blood into the arterial walls or vice versa. The flowing blood is represented as the suspension of all erythrocytes assumed to be Eringen's micropolar fluid and the arterial wall is considered to be rigid having cosine‐shaped stenosis in its lumen. The mass transfer to blood is controlled by the convection–diffusion equation. The governing equations of motion accompanied by the appropriate choice of the boundary conditions are solved numerically by Marker and Cell method and the results obtained are checked for numerical stability with the desired degree of accuracy. The quantitative analysis carried out finally includes the respective profiles of the flow‐field and the mass concentration along with their distributions over the entire arterial segment as well. The key factors, such as the wall shear stress and Sherwood number, are also examined for further quantitative insight into the flow and the mass transport phenomena through arterial stenosis. The present results show consistency with several existing results in the literature which substantiate sufficiently to validate the applicability of the model under consideration. Copyright © 2010 John Wiley & Sons, Ltd.     

Mathematical modeling and parametric investigation of blood flow through a stenosis artery

In this study, a mathematical model is formulated to examine the blood flow through a cylindrical stenosed blood vessel. The stenosis disease is caused because of the abnormal narrowing of flow in the body. This narrowing causes serious health issues like heart attack and may decrease blood flow in the blood vessel. Mathematical modeling helps us analyze such issues. A mathematical model is considered in this study to explore the blood flow in a stenosis artery and is solved numerically with the finite difference method. The artery is an elastic cylindrical tube containing blood defined as a viscoelastic fluid. A complete parametric analysis has been done for the flow velocity to clarify the applicability of the defined problem. Moreover, the flow characteristics such as the impedance, the wall shear stress in the stenotic region, the shear stresses in the throat of the stenosis and at the critical stenosis height are discussed. The obtained results show that the intensity of the stenosis occurs mostly at the highest narrowing areas compared with all other areas of the vessel, which has a direct impact on the wall shear stress. It is also observed that the resistive impedance and wall shear pressure get the maximum values at the critical height of the stenosis.

     

Magnetohydrodynamic approach of non-Newtonian blood flow with magnetic particles in stenosed artery

The non-Newtonian blood flow, together with magnetic particles in a stenosed artery, is studied using a magneto-hydrodynamic approach. The wall slip condition is also considered. Approximate solutions are obtained in series forms under the assumption that the Womersley frequency parameter has small values. Using an integral transform method, analytical solutions for any values of the Womersley parameter are obtained.Numerical simulations are performed using MATHCAD to study the influence of stenosis and magnetic field on the flow parameters. When entering the stenosed area, blood velocity increases slightly, but increases considerably and reaches its maximum value in the stenosis throat. It is concluded that the magnitude of axial velocity varies considerably when the applied magnetic field is strong. The magnitude of maximum fluid velocity is high in the case of weak magnetic fields. This is due to the Lorentz's force that opposes motion of an electrically conducting fluid. The effect of externally transverse magnetic field is to decelerate the flow of blood. The shear stress consistently decreases in the presence of a magnetic field with increasing intensity.     

磁场和Hall电流对狭窄动脉中血液流动的影响

A micropolar model for blood simulating magnetohydrodynamic flow through a horizontally nonsymmetric but vertically symmetric artery with a mild stenosis is presented. To estimate the effect of the stenosis shape, a suitable geometry has been considered such that the horizontal shape of the stenosis can easily be changed just by varying a parameter referred to as the shape parameter. Flow parameters, such as velocity, the resistance to flow （the resistance impedance）, the wall shear stress distribution in the stenotic region, and its magnitude at the maximum height of the stenosis （stenosis throat）, have been computed for different shape parameters, the Hartmann number and the Hall parameter. This shows that the resistance to flow decreases with the increasing values of the parameter determining the stenosis shape and the Hail parameter, while it increases with the increasing Hartmann number. The wall shear stress and the shearing stress on the wall at the maximum height of the stenosis possess an inverse characteristic to the resistance to flow with respect to any given value of the Hartmann number and the Hall parameter. Finally, the effect of the Hartmann number and the Hall parameter on the horizontal velocity is examined.     

Influence of magnetic field and thermal radiation by natural convection past vertical cone subjected to variable surface heat flux

A numerical study is performed to examine the heat transfer characteristics of natural convection past a vertical cone under the combined effects of magnetic field and thermal radiation.The surface of the cone is subjected to a variable surface heat flux.The fluid considered is a gray,absorbing-emitting radiation but a non-scattering medium.With approximate transformations,the boundary layer equations governing the flow are reduced to non-dimensional equations valid in the free convection regime.The dimensionless governing equations are solved by an implicit finite difference method of Crank-Nicolson type which is fast convergent,accurate,and unconditionally stable.Numerical results are obtained and presented for velocity,temperature,local and average wall shear stress,and local and average Nusselt number in air and water.The present results are compared with the previous published work and are found to be in excellent agreement.     

Peristaltic flow of MHD Jeffrey fluid through finite length cylindrical tube

The present investigation studies the peristaltic flow of the Jeffrey fluid through a tube of finite length. The fluid is electrically conducting in the presence of an applied magnetic field. Analysis is carried out under the assumption of long wavelength and low Reynolds number approximations. Expressions of the pressure gradient, volume flow rate, average volume flow rate, and local wall shear stress are obtained. The effects of relaxation time, retardation time, Hartman number on pressure, local wall shear stress, and mechanical efficiency of peristaltic pump are studied. The reflux phenomenon is also investigated. The case of propagation of a non-integral number of waves along the tube walls, which are inherent characteristics of finite length vessels, is also examined.     

局部狭窄血管中血液振荡流的速度和切应力

本文建立一种分析局部缓慢狭窄血管中血液振荡流的数学模型，给出了血液的轴向流速，径向流速和切应力的包含压力梯度项的解析表达式，并讨论了血管内由局部狭窄引起的压力梯度沿轴向变化的规律。文章以局部余弦狭窄为例进行数值计算，详细讨论上游均匀管段压力梯度的定常部分和不同次谐波对狭窄管段内流速和切应力的影响。数值结果表明，与均匀管情况相比，在狭窄段内，血液振荡流轴向流速无论平均值还是脉动幅值均明显增大，且径向流速不再为零。但径向流速仍远小于轴向流速。同时，切应力也不再仅由轴向流速梯度提供，径向流速梯度也将产生切应力，但是在计算管壁切向上的切应力时，径向流速梯度的贡献仍相当大。与均匀管管壁切应力沿流运方向保持恒定不同。狭窄管管壁切应力（平均值和脉动值）将随着狭窄高度的增大而增大，在狭窄最大高度处达到最大，因而沿流动方向产生了较大的切应力梯度。     

Effects of renal artery stenosis on realistic model of abdominal aorta and renal arteries incorporating fluid-structure interaction and pulsatile non-Newtonian blood flow

The effects of the renal artery stenosis(RAS) on the blood flow and vesselwalls are investigated.The pulsatile blood flow through an anatomically realistic model ofthe abdominal aorta and renal arteries reconstructed from CT-scan images is simulated,which incorporates the fluid-structure interaction(FSI).In addition to the investigationof the RAS effects on the wall shear stress and the displacement of the vessel wall,it isdetermined that the RAS leads to decrease in the renal mass flow.This may cause theactivation of the renin-angiotension system and results in severe hypertension.     

狭窄血管内偶应力流体流动分析

本文考察了血管狭窄对血液流动的影响,血液以偶应力流体表示,并在求解过程中采用了在管壁上流体质点无相对涡量的边界条件,结果表明,和Young的经典工作相比流动阻抗和壁切应力大于同样程度狭窄下牛顿流体的相应值,偶应力流体对狭窄的敏感性大于牛顿流体;在狭窄发展过程中,偶应力流体的流量要小于牛顿流体的流量,和牛顿流体相比,这些结果更符合生理实际。     

A comparative assessment of fluid flow and heat transfer parameters in laminar flow around a circular cylinder and a sphere

Theoretical studies have been carried out for a comparative assessment of hydrodynamic boundary layer thickness, displacement thickness and shear stress at the wall for laminar flow around a circular cylinder and a sphere with the help of the approximate method due to Karman and Pohlhausen for two dimensional flow and the method as applied to bodies of revolution based on the work of F. W. Scholkemeier, respectively. Thermal boundary layer thickness and Nusselt number have been evaluated around the surface of the solids. Comparison is made with available solutions. The graphical presentation of the results depicts a concise and relative assessment of fluid flow and heat-transfer parameters for flow around cylinder and sphere.     

窦部对称狭窄对颈动脉内流场影响的数值研究

以颈动脉分岔血管为例，采用数值方法研究了窦部环缩狭窄之后的流场分布情况，并和正 常血管情况下的流场分布进行了比较. 结果表明，采用环缩方式给颈动脉分岔血管施加对称 的狭窄改变了颈动脉窦内流场，特别是壁面剪应力的分布规律. 低剪应力区出现在狭窄段之 后的窦内，并且沿整个周向均匀分布. 根据低剪应力和动脉粥样硬化的关系，指出： 若人为地给颈动脉窦内施加对称狭窄，则脂质沉积将在狭窄下游的窦内沿周向轴对称 发展. 为了更真实地反映颈动脉窦内的狭窄，建议根据动脉血管中的实际狭窄情况，采用非 对称的狭窄分布模式.     

Effects of magnetic field,porosity, and wall properties for anisotropically elastic multi-stenosis arteries on blood flow characteristics

A mathematical model for blood flow through an elastic artery with multistenosis under the effect of a magnetic field in a porous medium is presented. The considered arterial segment is simulated by an anisotropically elastic cylindrical tube filled with a viscous incompressible electrically conducting fluid representing blood. An artery with mild local narrowing in its lumen forming a stenosis is analyzed. The effects of arterial wall parameters represent viscoelastic stresses along the longitudinal and circumferential directions T _t and T _θ , respectively. The degree of anisotropy of the vessel wall γ, total mass of the vessel, and surrounding tissues M and contributions of the viscous and elastic constraints to the total tethering C and K respectively on resistance impedance, wall shear stress distribution, and radial and axial velocities are illustrated. Also, the effects of the stenosis shape m, the constant of permeability X, the Hartmann number H _α and the maximum height of the stenosis size δ on the fluid flow characteristics are investigated. The results show that the flow is appreciably influenced by surrounding connective tissues of the arterial wall motion, and the degree of anisotropy of the vessel wall plays an important role in determining the material of the artery. Further, the wall shear stress distribution increases with increasing T _t and γ while decreases with increasing T _θ , M, C, and K. Transmission of the wall shear stress distribution and resistance impedance at the wall surface through a tethered tube are substantially lower than those through a free tube, while the shearing stress distribution at the stenosis throat has inverse characteristic through totally tethered and free tubes. The trapping bolus increases in size toward the line center of the tube as the permeability constant X increases and decreases with the Hartmann number Ha increased. Finally, the trapping bolus appears, gradually in the case of non-symmetric stenosis, and disappears in the case of symmetric stenosis. The size of trapped bolus for the stream lines in a free isotropic tube (i.e., a tube initially unstressed) is smaller than those in a tethered tube.     

Jeffrey-Hamel flow of Leslie-Ericksen nematic liquids

A similarity solution of the Leslie-Ericksen equations for nematic liquid crystals is obtained for flow between converging and diverging planar walls (Jeffrey-Hamel flow). There are three regions in the flow: extensional or compressional flow near the centerline, shear near the wall, and a wall boundary layer in which elastic stresses control the transition from the wall-induced orientation to the bulk behavior. The boundary layer thickness is obtained in closed form; the scaling with the Ericksen number depends on whether or not the boundary layer extends into the region of extensional flow. Imposition of a magnetic field with an azimuthal component in a converging flow can result in a Freedericksz-like transition from radial to transverse orientation at the center line at a critical field strength. This transition provides a new means to measure the irrotational viscosity λ₂.     

Transient MHD stagnation flow of a non-Newtonian fluid due to impulsive motion from rest

The transient boundary layer flow and heat transfer of a viscous incompressible electrically conducting non-Newtonian power-law fluid in a stagnation region of a two-dimensional body in the presence of an applied magnetic field have been studied when the motion is induced impulsively from rest. The non-linear partial differential equations governing the flow and heat transfer have been solved by the homotopy analysis method and by an implicit finite-difference scheme. For some cases, analytical or approximate solutions have also been obtained. The special interest are the effects of the power-law index, magnetic parameter and the generalized Prandtl number on the surface shear stress and heat transfer rate. In all cases, there is a smooth transition from the transient state to steady state. The shear stress and heat transfer rate at the surface are found to be significantly influenced by the power-law index N except for large time and they show opposite behaviour for steady and unsteady flows. The magnetic field strongly affects the surface shear stress, but its effect on the surface heat transfer rate is comparatively weak except for large time. On the other hand, the generalized Prandtl number exerts strong influence on the surface heat transfer. The skin friction coefficient and the Nusselt number decrease rapidly in a small interval 0<t^*<1 and reach the steady-state values for t^*≥4.     

Finite element computations of two-dimensional arterial flow in the presence of a transverse magnetic field

A finite element solution of the Navier-Stokes equations for steady flow under the magnetic effect through a double-branched two-dimensional section of a three-dimensional model of the canine aorta is discussed. The numerical scheme involves transforming the physical co-ordinates to a curvilinear boundary-fitted co-ordinate system. The shear stress at the wall is calculated for a Reynolds number of 1000 with the branch-to-main aortic flow rate ratio as a parameter. The results are compared with earlier works involving experimental data and found to be in reasonable qualitative agreement. The steady flow, shear stress and branch flow under the effect of a magnetic field have been discussed in detail.     

Steady laminar mixed convection stagnation-point flow of a nanofluid over a vertical permeable surface in the presence of a magnetic field

A similarity solution for a steady laminar mixed convection boundary layer flow of a nanofluid near the stagnation point on a vertical permeable plate with a magnetic field and a buoyancy force is obtained by solving a system of nonlinear ordinary differential equations. These equations are solved analytically by using a new kind of a powerful analytic technique for nonlinear problems, namely, the homotopy analysis method (HAM). Three different types of nanoparticles, namely, copper (Cu), alumina (Al₂O₃), and titanium oxide (TiO₂), with water as the base fluid are considered. The influence of the volume fraction of nanoparticles, permeability parameter, magnetic parameter, and mixed convection parameter on the surface shear stress and surface heat transfer, as well as on the velocity and temperature profiles, is considered. It is observed that the skin friction coefficient and the local Nusselt number increase with the nanoparticle volume fraction for all types of nanoparticles considered in this study. The greatest values of the skin friction coefficient and the local Nusselt number are obtained for Cu nanoparticles.