Numerical simulation of the non-Newtonian blood flow through a mechanical aortic valve

This work focuses on the comparison between Newtonian and non-Newtonian blood flows through a bileaflet mechanical heart valve in the aortic root. The blood, in fact, is a concentrated suspension of cells, mainly red blood cells, in a Newtonian matrix, the plasma, and consequently its overall behavior is that of a non-Newtonian fluid owing to the action of the cells’ membrane on the fluid part. The common practice, however, assumes the blood in large vessels as a Newtonian fluid since the shear rate is generally high and the effective viscosity becomes independent of the former. In this paper, we show that this is not always the case even in the aorta, the largest artery of the systemic circulation, owing to the pulsatile and transitional nature of the flow. Unexpectedly, for most of the pulsating cycle and in a large part of the fluid volume, the shear rate is smaller than the threshold level for the blood to display a constant effective viscosity and its shear thinning character might affect the system dynamics. A direct inspection of the various flow features has shown that the valve dynamics, the transvalvular pressure drop and the large-scale features of the flow are very similar for the Newtonian and non-Newtonian fluid models. On the other hand, the mechanical damage of the red blood cells (hemolysis), induced by the altered stress values in the flow, is larger for the non-Newtonian fluid model than for the Newtonian one.     

A transient natural convection of micropolar fluids over a vertical cylinder

A transient free convective boundary layer flow of micropolar fluids past a semi-infinite cylinder is analysed in the present study. The transformed dimensionless governing equations for the flow, microrotation and heat transfer are solved by using the implicit scheme. For the validation of the current numerical method heat transfer results for a Newtonian fluid case where the vortex viscosity is zero are compared with those available in the existing literature, and an excellent agreement is obtained. The obtained results concerning velocity, microrotation and temperature across the boundary layer are illustrated graphically for different values of various parameters and the dependence of the flow and temperature fields on these parameters is discussed. An increase in the vortex viscosity tends to increase the magnitude of microrotation and thus decreases the peak velocity of fluid flow. An increase in the vortex viscosity in micropolar fluids is shown to decrease the heat transfer rate.     

Sensitivity of shear rate in artificial grafts using automatic differentiation

An accurate numerical simulation of blood requires the solution of incompressible Navier–Stokes equations coupled with specific constitutive models. We consider a generalized Newtonian fluid model in which viscosity depends on shear rate, accounting for the shear‐thinning behavior of blood. Previous work on the design of an artificial graft indicated that there is an influence of the fluid model on the solution of the partial differential equation‐constrained shape optimization problem. Therefore, we carry out a sensitivity analysis of the actual implementation of the flow solver using automatic differentiation (AD). We compare the sensitivities of shear rate with respect to viscosity for different configurations and validate the truncation‐error‐free sensitivities obtained from AD with those based on divided differencing and, if available, with analytic derivatives. Copyright © 2009 John Wiley & Sons, Ltd.     

A Two-Layer Mathematical Model of Blood Flow in Porous Constricted Blood Vessels

In this paper, we discussed a mathematical model for two-layered non-Newtonian blood flow through porous constricted blood vessels. The core region of blood flow contains the suspension of erythrocytes as non-Newtonian Casson fluid and the peripheral region contains the plasma flow as Newtonian fluid. The wall of porous constricted blood vessel configured as thin transition Brinkman layer over layered by Darcy region. The boundary of fluid layer is defined as stress jump condition of Ocha-Tapiya and Beavers–Joseph. In this paper, we obtained an analytic expression for velocity, flow rate, wall shear stress. The effect of permeability, plasma layer thickness, yield stress and shape of the constriction on velocity in core & peripheral region, wall shear stress and flow rate is discussed graphically. This is found throughout the discussion that permeability and plasma layer thickness have accountable effect on various flow parameters which gives an important observation for diseased blood vessels.     

Dispersion in a Bingham plastic fluid: Effects of peripheral layer

The dispersion of a soluble matter in a plastic fluid flowing through a tube and a channel has been analysed by taking into account the variations of viscosity, diffusivity and yield stress. It has been shown that in the special case of a Bingham fluid, surrounded by a peripheral layer of a Newtonian fluid, the effective dispersion coefficient with which the solute disperses across a plane moving with the mean speed of the flow decreases with the viscosity of the peripheral layer fluid but increases as the molecular diffusion coefficient of this layer decreases. Further, the effective dispersion coefficient also decreases as the yield stress of the Bingham fluid increases.     

Multiphase non‐Newtonian effects on pulsatile hemodynamics in a coronary artery

Hemodynamic stresses are involved in the development and progression of vascular diseases. This study investigates the influence of mechanical factors on the hemodynamics of the curved coronary artery in an attempt to identify critical factors of non‐Newtonian models. Multiphase non‐Newtonian fluid simulations of pulsatile flow were performed and compared with the standard Newtonian fluid models. Different inlet hematocrit levels were used with the simulations to analyze the relationship that hematocrit levels have with red blood cell (RBC) viscosity, shear stress, velocity, and secondary flow. Our results demonstrated that high hematocrit levels induce secondary flow on the inside curvature of the vessel. In addition, RBC viscosity and wall shear stress (WSS) vary as a function of hematocrit level. Low WSS was found to be associated with areas of high hematocrit. These results describe how RBCs interact with the curvature of artery walls. It is concluded that although all models have a good approximation in blood behavior, the multiphase non‐Newtonian viscosity model is optimal to demonstrate effects of changes in hematocrit. They provide a better stimulation of realistic blood flow analysis. Copyright © 2008 John Wiley & Sons, Ltd.     

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 generalized equation for rheology of emulsions and suspensions of deformable particles subjected to simple shear at low Reynolds number

We present analyses to provide a generalized rheological equation for suspensions and emulsions of non-Brownian particles. These multiparticle systems are subjected to a steady straining flow at low Reynolds number. We first consider the effect of a single deformable fluid particle on the ambient velocity and stress fields to constrain the rheological behavior of dilute mixtures. In the homogenization process, we introduce a first volume correction by considering a finite domain for the incompressible matrix. We then extend the solution for the rheology of concentrated system using an incremental differential method operating in a fixed and finite volume, where we account for the effective volume of particles through a crowding factor. This approach provides a self-consistent method to approximate hydrodynamic interactions between bubbles, droplets, or solid particles in concentrated systems. The resultant non-linear model predicts the relative viscosity over particle volume fractions ranging from dilute to the the random close packing in the limit of small deformation (capillary or Weissenberg numbers) for any viscosity ratio between the dispersed and continuous phases. The predictions from our model are tested against published datasets and other constitutive equations over different ranges of viscosity ratio, volume fraction, and shear rate. These comparisons show that our model, is in excellent agreement with published datasets. Moreover, comparisons with experimental data show that the model performs very well when extrapolated to high capillary numbers (C a?1). We also predict the existence of two dimensionless numbers; a critical viscosity ratio and critical capillary numbers that characterize transitions in the macroscopic rheological behavior of emulsions. Finally, we present a regime diagram in terms of the viscosity ratio and capillary number that constrains conditions where emulsions behave like Newtonian or Non-Newtonian fluids.     

Numerical simulations of linear viscoelasticity of monodisperse emulsions of Newtonian drops in a Newtonian fluid from dilute to concentrated regime

The bulk viscoelastic properties of monodisperse emulsions of Newtonian drops in a Newtonian matrix subjected to small amplitude oscillatory shear (SAOS) flow are investigated by means of arbitrary Lagrangian Eulerian finite element method 3D numerical simulations. Volume fractions of the suspended phase from the dilute to the concentrated regime (up to 30 %), and a range of several orders of magnitude of the drops-to-matrix viscosity ratio and of the frequency of the oscillatory flow are examined; the eventual presence of slip between the two fluids is also considered. The computational results are compared with theory, yielding a quantitative agreement with Oldroyd (Proc R Soc Lond A 218:122–132, 1953) predictions in a wide range of values of the considered parameters, even well beyond the dilute regime, and also in the cases with slip.     

Time-dependent plane Poiseuille flow of a Johnson–Segalman fluid

We numerically solve the time-dependent planar Poiseuille flow of a Johnson–Segalman fluid with added Newtonian viscosity. We consider the case where the shear stress/shear rate curve exhibits a maximum and a minimum at steady state. Beyond a critical volumetric flow rate, there exist infinite piecewise smooth solutions, in addition to the standard smooth one for the velocity. The corresponding stress components are characterized by jump discontinuities, the number of which may be more than one. Beyond a second critical volumetric flow rate, no smooth solutions exist. In agreement with linear stability analysis, the numerical calculations show that the steady-state solutions are unstable only if a part of the velocity profile corresponds to the negative-slope regime of the standard steady-state shear stress/shear rate curve. The time-dependent solutions are always bounded and converge to different stable steady states, depending on the initial perturbation. The asymptotic steady-state velocity solution obtained in start-up flow is smooth for volumetric flow rates less than the second critical value and piecewise smooth with only one kink otherwise. No selection mechanism was observed either for the final shear stress at the wall or for the location of the kink. No periodic solutions have been found for values of the dimensionless solvent viscosity as low as 0.01.     

Shear rheometry and visualization of glass fiber suspensions

The nonlinear rheological behavior of short glass fiber suspensions has been investigated in this work by rotational rheometry and flow visualization. A Newtonian and a Boger fluid (BF) were used as suspending media. The suspensions exhibited shear thinning in the semidilute regime and weaker shear thinning in the transition to the concentrated one. Normal stresses and relative viscosity were higher for the BF suspensions than for the Newtonian ones presumably due to enhanced hydrodynamic interactions resulting from BF elasticity. In addition, relative viscosity of the suspensions increased rapidly with fiber content, suggesting that the rheological behavior in the concentrated regime is dominated by mechanical contacts between fibers. Visualization of individual fibers and their interactions under flow allowed the detection of aggregates, which arise from adhesive contacts. The orientation states of the fibers were quantified by a second order tensor and fast Fourier transforms of the flow field images. Fully oriented states occurred for shear rates around 20 s^− 1. Finally, the energy required to orient the fibers was higher in step forward than in reversal flow experiments due to a change in the spatial distribution of fibers, from isotropic to planar oriented, during the forward experiments.     

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 fluid flow in a porous tube: a model for blood flow in capillaries

A theoretical study of blood flow, under the influence of a body force, in a capillary is presented. Blood is modeled as a two-phase fluid consisting of a core region of suspension of all erythrocytes, represented by a micropolar fluid and a plasma layer free from cells modeled as a Newtonian fluid. The capillary is modeled as a porous tube consisting of a thin transition Brinkman layer overlying a porous Darcy region. Analytical expressions for the pressure, microrotation, and velocities for the different regions are given. Plots of pressure, microrotation, and velocities for varying micropolar parameters, hydraulic resistivity, and Newtonian fluid layer thickness are presented. The overall system was found to be sensitive to variations in micropolar coupling number. It was also discovered that high values of hydraulic resistivity result in an overall slower velocity of the micropolar and Newtonian fluid.     

基于舌/上颚微间隙下流体流动行为研究

为探讨口腔环境下流体的流动行为,采用数值方法与流变试验深入研究舌/上颚微间隙下流体流量的影响因素. 建立舌/上颚微间隙的简化模型及Reynolds方程,通过数值方法获取微间隙下流量变化;在DHR-2流变仪上研究非牛顿流体的黏度与剪切率的变化,探讨牛顿流体和非牛顿流体的流量影响. 结果表明:牛顿流体流量平方的倒数同载荷和黏度比值和时间均呈线性函数关系;所制备的非牛顿流体近似为幂律流体,其黏度随脂肪含量的增加而增大,而非牛顿流体流量率先高于后低于等效牛顿流体,其研究结果将为特定人群功能产品的研发提供技术支持.      

Squeezing flow of a highly viscous incompressible liquid pressed between slightly inclined lubricated wide plates

The theoretical force-height relationships of Newtonian and pseudo plastic liquids compressed between slightly tilted frictionless plates are compared with those produced when the plates are perfectly parallel. It is shown that a very small inclination angle can distort the flow curve to such an extent that a Newtonian liquid will appear as a pseudo plastic fluid, and a pseudo plastic liquid as having a flow index considerably smaller than its true one. The shape of the biaxial elongational viscosity vs biaxial strain rate relationship is also highly sensitive to the plates' inclination angle. Thus, if an experimental force-height relationship is used to determine a material's biaxial elongational viscosity, an unsuspected slight tilt will result in a considerable underestimate of the viscosity. A slight tilt will also produce an apparent strain rate dependency in a Newtonian liquid, which obviously does not exist. The mathematical model developed to reach these conclusions was tested with commercial mayonnaise, a self-lubricating fluid. A reasonable agreement was found between the predicted force-height relationships and those experimentally determined at tilts of 1°, 3°, and 5°. Received: 4 August 2000 Accepted: 21 August 2000     

The viscous squeeze-film behaviour of shear-thinning fluids

A finite element technique is proposed to predict the purely viscous squeeze-film behaviour of an arbitrary shear-thinning fluid confined between parallel discs and subjected to a constant load. The technique requires establishment of the distribution of viscosity in the gap. The variable viscosity is modelled by a discrete number of Newtonian fluids, with each fluid lying in a region bounded by lines of constant shear rate. Each of these Newtonian regions is further divided into regions which appear as “finite element” rectangles in the r-z plane. The equations governing squeeze-film flow are applied to this finite element network and an ordinary differential equation is ultimately derived which governs the gap decrease with time. Solving this equation is not simple because the coefficients of two terms change as the gap decreases. When the number of Newtonian fluids is sufficient, the technique predicts the squeeze-film time of a power-law fluid to within a fraction of a percent. Application of the technique to synovial fluid viscosity prevents the cartilage surfaces from touching for only a fraction of a second.     

Classical Solvability of the Coupled System Modelling a Heat-Convergent Poiseuille-Type Flow

We consider the coupled system of two nonlinear scalar parabolic equations modelling a simple uni-directional Poiseuille-type flow of a homogeneous incompressible Newtonian fluid whose viscosity is a temperature-dependent function. The energy balance equation of this system takes into account the phenomena of the viscous energy dissipation. We prove existence of a classical solution to this system on an arbitrary interval of time. The smooth solution turns out to be unique in a wider class of weak solutions.     

Effective Rheology of Two-Phase Flow in Three-Dimensional Porous Media: Experiment and Simulation

We present an experimental and numerical study of immiscible two-phase flow of Newtonian fluids in three-dimensional (3D) porous media to find the relationship between the volumetric flow rate (Q) and the total pressure difference (\(\Delta P\)) in the steady state. We show that in the regime where capillary forces compete with the viscous forces, the distribution of capillary barriers at the interfaces effectively creates a yield threshold (\(P_t\)), making the fluids reminiscent of a Bingham viscoplastic fluid in the porous medium. In this regime, Q depends quadratically on an excess pressure drop (\(\Delta P-P_t\)). While increasing the flow rate, there is a transition, beyond which the overall flow is Newtonian and the relationship is linear. In our experiments, we build a model porous medium using a column of glass beads transporting two fluids, deionized water and air. For the numerical study, reconstructed 3D pore networks from real core samples are considered and the transport of wetting and non-wetting fluids through the network is modeled by tracking the fluid interfaces with time. We find agreement between our numerical and experimental results. Our results match with the mean-field results reported earlier.     

Direct numerical simulation of the turbulent energy balance and the shear stresses in power-law fluid flows in pipes

The results of direct numerical simulation of turbulent flows of non-Newtonian pseudoplastic fluids in a straight pipe are presented. The data on the distributions of the turbulent stress tensor components and the shear stress and turbulent kinetic energy balances are obtained for steady turbulent flows at the Reynolds numbers of 10⁴ and 2×10⁴. As distinct from Newtonian fluid flows, the viscous shear stresses turn out to be significant even far from the wall. In power-law fluid flows the mechanism of the energy transport from axial to transverse component fluctuations is suppressed. It is shown that with decrease in the fluid index the turbulent transfer of the momentum and the velocity fluctuations between the wall layer and the flow core reduces, while the turbulent energy flux toward the wall increases. The earlier-proposed models for the average viscosity and the non-Newtonian one-point correlations are in good agreement with the data of direct numerical simulation.     

Viscosity of a Newtonian fluid calculated from the deformation of droplets covered with a surfactant under a linear shear flow

The viscosity of small fluid droplets covered with a surfactant is determined using drop deformation techniques. This method, proposed by Hu and Lips, is here extended to the case of the presence of a surface-active adsorpted at the liquid–liquid interface, to consider more general scenarios. In these experiments, a droplet is sheared by another immiscible fluid of known viscosity, both Newtonian liquids. From the steady-state deformation and retraction mechanisms, the droplet viscosity is calculated using an equation derived from the theories of Taylor and Rallison. Although these theories were expressed for surfactant-free interfaces, they can be applied when a surfactant is present in the system if the sheared droplet reaches reliable steady-state deformations and the surfactant attains its equilibrium adsorption concentration. These determinations are compared to bulk viscosities measured in a rheometer for systems with different viscosity ratios and surfactant concentrations. Very good agreement between both determinations is found for drops more viscous than the continuous phase.