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．     

冠状动脉狭窄情况下的非牛顿血液流动和大分子传质

针对冠状动脉狭窄的情况,采用数值模拟方法求解了牛顿流体与非牛顿流体(幂次律流体和Casson流体)的定常与脉动的流场。在此基础上,求解了LDL(低密度脂肪蛋白)和Albumin(血清白蛋白)的浓度场。根据计算结果,详细讨论了壁面剪应力、非牛顿流效应、分子大小等因素对大分子传质的影响;并对牛顿流体与非牛顿流体、定常流动与脉动流动的大分子浓度场进行了比较,这些结果对于了解动脉硬化成因与流动特性和大分子传质的联系提供了较为丰富的信息。     

Rimming flow of a power-law fluid: Qualitative analysis of the mathematical model and analytical solutions

Rimming flow of a non-Newtonian fluid on the inner surface of a horizontal rotating cylinder is investigated. Simple lubrication theory is applied since the Reynolds number is small and liquid film is thin. For the steady-state flow of a power-law fluid the mathematical model reduces to a simple algebraic equation regarding the thickness of the liquid film. The qualitative analysis of this equation is carried out and the existence of two possible solutions is rigorously proved. Based on this qualitative analysis, different regimes of the rimming flow are defined and analyzed analytically. For the particular case, when the flow index in a power-law constitutive equation is equal to 1/2, the problem reduces to the fourth order algebraic equation which is solved analytically by Ferrari method.     

Analysis of fluid separation in microfluidic T-channels

The behaviour of a fluid, which may contain particle suspensions, flowing in micro-dimensional channels is governed by both viscous and surface tension forces as well as high shear rates and geometric effects such as bifurcations, constriction, and high surface-to-volume ratio. This paper discusses some of the key design factors affecting fluid behaviour in micro-engineered products containing a main channel, constriction and side channel bifurcations. Differences in fluid behaviour at the macro and micro-scales are discussed. The dynamic bulk fluid behaviour is characterised in terms of: (i) fluid properties, (ii) governing physics and (iii) microchannel geometric features.At this stage of the analysis the fluids are assumed to be Newtonian and single phase, where any particle suspension is represented through a bulk density and viscosity. Based on these assumptions Computational Fluid Dynamics (CFD) is used to investigate the effect of both product inlet and outlet boundary conditions on the bulk flow behaviour. Discussions are provided on how these boundary conditions can affect particle separation efficiency. In particular, the so called pull-design whereby the fluid is pulled out of the device at the outlet, is shown to offer better performance compared to the mode of operation where the fluid is pushed into the device at the inlet. It is also observed that increasing the pressure at the outlet of the main channel can achieve a balanced flow rate ratio which leads to a uniform separation performance among all bifurcations.     

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.     

Axisymmetric flow over a backward-facing step in an annular pipe

The incompressible flow of a Newtonian fluid over a backward-facing step is investigated numerically. The geometry is an annular pipe in which the radius of the inner cylinder decreases suddenly. Keeping the radial expansion ratio fixed axisymmetric flows are computed for outlet radius ratios from 0.1 to 1 (ratio of the inner to the outer outlet radius). The Reynolds number at which the flow separates from the outer cylinder decreases as the outlet radius ratio decreases for constant inlet geometry. The growth with Reynolds number of the recirculation zone on the inner outlet cylinder just behind the step is strongly reduced when the recirculation zone on the outer cylinder is established. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Flow Structure in a Technical Scale Reactor with an Internal Reboiler

In this contribution, results of experimental flow investigations in a technical scale cylindrical reactor with centrally located internal reboiler are presented. The fluid motion in the kettle is induced by natural circulation. Two different geometrical configurations are considered. First, the liquid level is set below the evaporator's outlet. Thus, only the liquid phase penetrates the liquid surface. The second configuration characterizes the a direct injection of the two phase mixture into bulk liquid. In both cases the liquid leaves the evaporator radially and horizontally towards the cylindrical wall. Gathered results reveal the presence of the big, toroidal vortex in the kettle annulus. Independently on the boundary conditions, self similarity of velocity profiles in the near wall region is observed. In contrast, investigations of the momentum transfer between the jet and the bulk liquid reveal strong dependency on the geometrical configuration of the setup. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

恒磁场对刚性圆直管中脉动流的影响*

本文研究了恒磁场对于刚性圆直管中脉动流的影响,并根据现有的实验资料考虑了磁场对于血液粘度的影响,给出了恒磁场作用下刚性圆直管脉动流的分析解以及恒磁场对刚性圆直管中的流速分布、流量以及阻抗的影响的计算结果.这些结果对于深入研究磁场对于血液动力学的影响具有一定参考价值.     

Maxwell流体在震荡的矩形输送管道中的流动

分析了不可压缩Maxwell流体在震荡矩形截面管道中的非稳定流动问题.利用Fourier变换和Laplace变换作为数学工具,提出了问题的解,该解可以看成稳态解和暂态解之和.大倍数时,暂态消失,解可以表示为稳态解.在极限情况的案例中给出了Newton流体的解.当震荡频率不存在时,得到了Maxwell流体在震荡矩形截面管道中流动问题的解.最后,以图形形式给出不同参数时,矩形管道正弦震荡达到稳态所需要的时间.同时,分别描绘了x和y变化时的速度曲线.     

On Symmetric Boundary Conditions in the Context of Viscoelastic Fluid Flows

In order to reduce the numerical cost of three dimensional flow problems with geometrical symmetry, the use of symmetric boundary conditions is standard. For Newtonian fluid flow problems this approximation is usually appropriate, particularly when the Reynolds number is small. In the case of viscoelastic fluid flow simulations with stabilization techniques, such as the so-called DEVSS and/or Log-Conformation tensor methods, at high Deborah number flows this implementation is not straightforward, as in the Newtonian case. It is well known that viscoelastic models (e.g. Maxwellian models), show (purely) elastic flow instabilities when the Deborah number is increased above a critical value, even under creeping flow conditions. In this work we present numerical simulations with different stabilization techniques and different differential viscoelastic models at high Deborah number flows. As a test-case, we compare the flow in a full two-dimensional cross-slot geometry to show the asymmetrical behavior of the viscoelastic fluid flow. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the short-wave nature of Richtmyer–Meshkov instability

In the case of a variable period (wavelength) of a perturbed interface, the instability and stability of Richtmyer–Meshkov vortices in perfect gas and incompressible perfect fluid, respectively, are investigated numerically and analytically. Taking into account available experiments, the instability of the interface between the argon and xenon in the case of a relatively small period is modeled. An estimate of the magnitude of the critical period is given. The nonlinear (for arbitrary initial conditions) stability of the corresponding steady-state vortex flow of perfect fluid in a strip (vertical periodic channel) in the case of a fairly large period is shown.     

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.     

Non-reflecting boundary conditions for the compressible Euler and Navier-Stokes equations in the finite volume method

The paper is concerned with the numerical implementation of the inlet and outlet boundary conditions in the finite volume method for the solution of the 3D Euler and Navier-Stokes equations. The explicit time marching procedure is described. The classical Riemann problem is modified for physically relevant boundary conditions with the aim to keep conservation laws. This technique was used in [2]. The initial condition in the Riemann problem is replaced by the suitable one-sided boundary condition. This results in the acceleration of the numerical method itself. On the inlet the pressure and the density and the angle of attack or velocity vector and the entropy are prescribed. On the outlet the pressure or normal component of the velocity or temperature or mass flow are investigated in such a way to obtain the unique solution of the modified Riemann problem. Various combinations of inlet and outlet boundary conditions are investigated. This results in the sufficiently precise approximation of real flow boundary conditions. Numerical examples illustrating the usefulness of the proposed approach for cascade flow are presented. Another numerical example is shown in [3]. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Bifurcating periodic solutions of wind-driven circulation equations

The existence of bifurcating periodic flows in a quasi-geostrophic mathematical model of wind-driven circulation is investigated. In the model, the Ekman number r and Reynolds number R control the stability of the motion of the fluid. Through rigorous analysis it is proved that when the basic steady-state solution is independent of the Ekman number, then a spectral simplicity condition is sufficient to ensure the existence of periodic solutions branching off the basic steady-state solution as the Ekman number varies across its critical value for constant Reynolds number. When the basic solution is a function of Ekman number, an additional condition is required to ensure periodic solutions.     

Viscous two‐fluid flows in perturbed unbounded domains

Two stationary plane free boundary value problems for the Navier‐Stokes equations are studied. The first problem models the viscous two‐fluid flow down a perturbed or slightly distorted inclined plane. The second one describes the viscous two‐fluid flow in a perturbed or slightly distorted channel. For sufficiently small data and under certain conditions on parameters the solvability and uniqueness results are proved for both problems. The asymptotic behaviour of the solutions is investigated. For the second problem an example of nonuniqueness is constructed. Computational results of flow problems that are very close to the above problems are presented. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Electrical simulation of the circulatory system

An electrical model of the arterial part of the human vascular system is proposed. This model is used to investigate the impedance, the shape of the pressure and flow waves, and the static characteristics of the vascular system under normal conditions and in the case of artificial circulation. In order to simulate the ACA pump, the problem of the quasi-one-dimensional flow of a viscous fluid in a tube with wave-type variation of the radius is considered. A comparison of the results with the results of direct measurements in man shows that they are in qualitative and quantitative agreement. Simulation also reveals certain characteristics of pulse-wave propagation in artificial circulation. An explanation of the effects observed is proposed, and their possible influence on the activity of the organism is discussed.     

The effects of symmetric/asymmetric boundary conditions on the flow of an internally heated fluid

The two-dimensional flow of an internally heated fluid in a circular vessel has been investigated by using a finite-control-volume numerical model. It has been found that, when symmetry constraints are imposed along the vertical midplane of the vessel, two distinct flow patterns can he predicted for the same operating conditions, a jet-momentum-dominated pattern and buoyancy-dominated pattern. These patterns occur in an operating region where momentum and buoyancy forces are of comparable magnitude. However, when the symmetry constraints are removed and the full vessel cross section is modeled, only the buoyancy-dominated pattern is observed. Results for the two cases are described and possible reasons for the differences in behavior are discussed.     

Optimal Boundary Control of Steady-State Flow of a Viscous Inhomogeneous Incompressible Fluid

We study the problem of optimal boundary control of two-dimensional steady-state flow of a viscous inhomogeneous incompressible fluid. The role of control is played by the values of the velocity on a part of the boundary of the domain considered. On the remaining part of the boundary, the vector of flow velocity and the fluid density are given. We seek the fluid density as a scalar function (determined by the initial data) of the stream function, study the solvability of the problem, and obtain necessary optimality conditions.     

Discrete storage processes and their Poisson flow and fluid flow approximations

Consider discrete storage processes that are modulated by environmental processes. Environmental processes cause interruptions in the input and/or output processes of the discrete storage processes. Due to the difficulties encountered in the exact analysis of such discrete storage systems, often Poisson flow and/or fluid flow models with the same modulating environmental processes are proposed as approximations for these systems. The analysis of Poisson flow and fluid flow models is much easier than that of the discrete storage processes. In this paper we give sufficient conditions under which the content of the discrete storage processes can be bounded by the Poisson flow and the fluid flow models. For example, we show that Poisson flow models and the fluid flow models developed by Kosten (and by Anick, Mitra and Sondhi) can be used to bound the performance of infinite (finite) source packetized voice/data communication systems. We also show that a Poisson flow model and the fluid flow model developed by Mitra can be used to bound the buffer content of a two stage automatic transfer line. The potential use of the bounding techniques presented in this paper, of course, transcends well beyond these examples.Supported in part by NSF grant DMS-9308149.