Numerical investigation of the non-Newtonian blood flow in a bifurcation model with a non-planar branch

The non-Newtonian fluid flow in a bifurcation model with a non-planar daughter branch is investigated by using finite element method to solve the three-dimensional Navier–Stokes equations coupled with a non-Newtonian constitutive model, in which the shear thinning behavior of the blood fluid is incorporated by the Carreau–Yasuda model. The objective of this study is to investigate the influence of the non-Newtonian property of fluid as well as of curvature and out-of-plane geometry in the non-planar daughter vessel on wall shear stress (WSS) and flow phenomena. In the non-planar daughter vessel, the flows are typified by the skewing of the velocity profile towards the outer wall, creating a relatively low WSS at the inner wall. In the downstream of the bifurcation, the velocity profiles are shifted towards the flow divider. The low WSS is found at the inner walls of the curvature and the lateral walls of the bifurcation. Secondary flow patterns that swirl fluid from the inner wall of curvature to the outer wall in the middle of the vessel are also well documented for the curved and bifurcating vessels. The numerical results for the non-Newtonian fluid and the Newtonian fluid with original Reynolds number and the corresponding rescaled Reynolds number are presented. Significant difference between the non-Newtonian flow and the Newtonian flow is revealed; however, reasonable agreement between the non-Newtonian flow and the rescaled Newtonian flow is found. Results of this study support the view that the non-planarity of blood vessels and the non-Newtonian properties of blood are an important factor in hemodynamics and may play a significant role in vascular biology and pathophysiology.     

Pulsatile non-Newtonian flow characteristics in a three-dimensional human carotid bifurcation model.

Numerical analysis of flow phenomena and wall shear stresses in the human carotid artery bifurcation has been carried out using a three-dimensional geometrical model. The primary aim of this study is the detailed discussion of non-Newtonian flow velocity and wall shear stress during the pulse cycle. A comparison of non-Newtonian and Newtonian results is also presented. The applied non-Newtonian behavior of blood is based on measured dynamic viscosity. In the foreground of discussion are the flow characteristics in the carotid sinus. The investigation shows complex flow patterns especially in the carotid sinus where flow separation occurs at the outer wall throughout the systolic deceleration phase. The changing sign of the velocity near the outer sinus wall results in oscillating shear stress during the pulse cycle. At the outer wall of the sinus at maximum diameter level the shear stress ranges from -1.92 N/m2 to 1.22 N/m2 with a time-averaged value of 0.04 N/m2. At the inner wall of the sinus at maximum diameter level the shear stress range is from 1.16 N/m2 to 4.18 N/m2 with a mean of 1.97 N/m2. The comparison of non-Newtonian and Newtonian results indicates unchanged flow phenomena and rather minor differences in the basic flow characteristics.     

The influence of the non-Newtonian properties of blood on the flow in large arteries: steady flow in a carotid bifurcation model.

Laser Doppler anemometry experiments and finite element simulations of steady flow in a three dimensional model of the carotid bifurcation were performed to investigate the influence of non-Newtonian properties of blood on the velocity distribution. The axial velocity distribution was measured for two fluids: a non-Newtonian blood analog fluid and a Newtonian reference fluid. Striking differences between the measured flow fields were found. The axial velocity field of the non-Newtonian fluid was flattened, had lower velocity gradients at the divider wall, and higher velocity gradients at the non-divider wall. The flow separation, as found with the Newtonian fluid, was absent. In the computations, the shear thinning behavior of the analog blood fluid was incorporated through the Carreau-Yasuda model. The viscoelastic properties of the fluid were not included. A comparison between the experimental and numerical results showed good agreement, both for the Newtonian and the non-Newtonian fluid. Since only shear thinning was included, this seems to be the dominant non-Newtonian property of the blood analog fluid under steady flow conditions.     

The influence of the non-Newtonian properties of blood on the flow in large arteries: unsteady flow in a 90 degrees curved tube.

A numerical and experimental investigation of unsteady entry flow in a 90 degrees curved tube is presented to study the impact of the non-Newtonian properties of blood on the velocity distribution. The time-dependent flow rate for the Newtonian and the non-Newtonian blood analog fluid were identical. For the numerical computation, a Carreau-Yasuda model was employed to accommodate the shear thinning behavior of the Xanthan gum solution. The viscoelastic properties were not taken into account. The experimental results indicate that significant differences between the Newtonian and non-Newtonian fluid are present. The numerical results for both the Newtonian and the non-Newtonian fluid agree well with the experimental results. Since viscoelasticity was not included in the numerical code, shear thinning behavior of the blood analog fluid seems to be the dominant non-Newtonian property, even under unsteady flow conditions. Finally, a comparison between the non-Newtonian fluid model and a Newtonian fluid at a rescaled Reynolds number is presented. The rescaled Reynolds number, based on a characteristic rather than the high-shear rate viscosity of the Xanthan gum solution, was about three times as low as the original Reynolds number. Comparison reveals that the character of flow of the non-Newtonian fluid is simulated quite well by using the appropriate Reynolds number.     

Numerical investigation of the non-Newtonian pulsatile blood flow in a bifurcation model with a non-planar branch

The pulsatile flow of non-Newtonian fluid in a bifurcation model with a non-planar daughter branch is investigated numerically by using the Carreau-Yasuda model to take into account the shear thinning behavior of the analog blood fluid. The objective of this study is to deal with the influence of the non-Newtonian property of fluid and of out-of-plane curvature in the non-planar daughter vessel on wall shear stress (WSS), oscillatory shear index (OSI), and flow phenomena during the pulse cycle. The non-Newtonian property in the daughter vessels induces a flattened axial velocity profile due to its shear thinning behavior. The non-planarity deflects flow from the inner wall of the vessel to the outer wall and changes the distribution of WSS along the vessel, in particular in systole phase. Downstream of the bifurcation, the velocity profiles are shifted toward the flow divider, and low WSS and high shear stress temporal oscillations characterized by OSI occur on the outer wall region of the daughter vessels close to the bifurcation. Secondary motions become stronger with the addition of the out-of-plane curvature induced by the bending of the vessel, and the secondary flow patterns swirl along the non-planar daughter vessel. A significant difference between the non-Newtonian and the Newtonian pulsatile flow is revealed during the pulse cycle; however, reasonable agreement between the non-Newtonian and the rescaled Newtonian flow is found. Calculated results for the pulsatile flow support the view that the non-planarity of blood vessels and the non-Newtonian properties of blood are an important factor in hemodynamics and may play a significant role in vascular biology and pathophysiology.     

Numerical simulation of non-Newtonian blood flow dynamics in human thoracic aorta

Three non-Newtonian blood viscosity models plus the Newtonian one are analysed for a patient-specific thoracic aorta anatomical model under steady-state flow conditions via wall shear stress (WSS) distribution, non-Newtonian importance factors, blood viscosity and shear rate. All blood viscosity models yield a consistent WSS distribution pattern. The WSS magnitude, however, is influenced by the model used. WSS is found to be the lowest in the vicinity of the three arch branches and along the distal walls of the branches themselves. In this region, the local non-Newtonian importance factor and the blood viscosity are elevated, and the shear rate is low. The present study revealed that the Newtonian assumption is a good approximation at mid-and-high flow velocities, as the greater the blood flow, the higher the shear rate near the arterial wall. Furthermore, the capabilities of the applied non-Newtonian models appeared at low-flow velocities. It is concluded that, while the non-Newtonian power-law model approximates the blood viscosity and WSS calculations in a more satisfactory way than the other non-Newtonian models at low shear rates, a cautious approach is given in the use of this blood viscosity model. Finally, some preliminary transient results are presented.     

Flow investigations in a model of a three-dimensional human artery with Newtonian and non-Newtonian fluids. Part I

Together with biochemical factors, fluid mechanical factors play a role in atherogenesis and the deposition of blood platelets at bends and bifurcations in human arteries. Hence, flow patterns were investigated in a simplified 3-dimensional model of a human renal artery bifurcation using Newtonian (aqueous glycerol) and non-Newtonian (aqueous solution of polyacrylamide) fluids. Studies were carried out in steady as well as pulsatile flow at inflow Reynolds numbers of 498 and 951 with flow rate ratios main tube V1: right branch V4: left branch V3 of 1: 0.25: 0.25 and 1: 0.18: 0.18 respectively. The velocity distribution proximal and distal to the bifurcations was measured using a laser-Doppler anemometer. In steady flow, zones of flow separation and reverse flow were observed distal to the bifurcations. In pulsatile flow using non-Newtonian fluids, there was a significant enlargement of these zones. Differences between the Newtonian and non-Newtonian fluids occurred especially distal to the bifurcations. Shear stresses along all measuring positions were computed from the velocity gradients.     

Analysis of non-Newtonian effects within an aorta-iliac bifurcation region

The geometry of the arteries at or near arterial bifurcation influences the blood flow field, which is an important factor affecting arteriogenesis. The blood can act sometimes as a non-Newtonian fluid. However, many studies have argued that for large and medium arteries, the blood flow can be considered to be Newtonian. In this work a comprehensive investigation of non-Newtonian effects on the blood fluid dynamic behavior in an aorta-iliac bifurcation is presented. The aorta-iliac geometry is reconstructed with references to the values reported in Shah et al. (1978); the 3D geometrical model consists of three filleted cylinders of different diameters. Governing equations with the appropriate boundary conditions are solved with a finite-element code. Different rheological models are used for the blood flow through the lumen and detailed comparisons are presented for the aorta-iliac bifurcation. Results are presented in terms of the velocity profiles in the bifurcation zone and Wall Shear Stress (WSS) for different sides of the bifurcation both for male and female geometries, showing that the Newtonian fluid assumption can be made without any particular loss in terms of accuracy with respect to the other more complex rheological models.     

Predicted wall shear rate gradients in T-type arteriolar bifurcations

The purpose of this study was to examine the theoretical impact of the local bifurcation geometry on the shear rate gradient in a divergent arteriolar-type bifurcation. Newtonian flow through an arteriolar bifurcation was modeled using 3-dimensional computational fluid dynamics (CFD). Branching angles of 30 degrees, 50 degrees, 70 degrees, 90 degrees, 110 degrees, 130 degrees, and 150 degrees were studied at a Reynolds number (Re) of 0.01 in seven separate models. Both the flow split (30%) and the branch to main vessel diameter ratio (4/5) were held constant. Velocity profiles were predicted to deviate significantly from a parabolic form, both immediately before and after the branch. This deviation was shown to be a function of the local bifurcation geometry of each model, which consisted of a branching angle and associated feed-branch intersection shape. Immediately before and after the branch, the shear rate along the lateral branching wall was predicted to exceed (5-fold) that calculated for fully developed flow in the feed. In vivo data were from the anesthetized (pentobarbital, 70 mg/kg) hamster cremaster muscle preparation. Red blood cells were used as flow markers in arteriolar branch points (n = 74) show that a significant gradient in shear rate occurs at the locations and branch shapes predicted by the computational model. Thus, for low Re divergent flow, the gradient in shear rate measured for non-Newtonian conditions, is approximated by a finite element fluid dynamics model of Newtonian flow.     

Vascular wall flow-induced forces in a progressively enlarged aneurysm model

The current study is focused on the numerical investigation of the flow field induced by the unsteady flow in the vicinity of an abdominal aortic aneurysm model. The computational fluid dynamics code used is based on the finite volume method, and it has already been used in various bioflow studies. For modelling the rheological behaviour of blood, the Quemada non-Newtonian model is employed, which is suitable for simulating the two-phase character of blood namely a suspension of blood cells in plasma. For examining its non-Newtonian effects a comparison with a corresponding Newtonian flow is carried out. Furthermore, the investigation is focused on the distribution of the flow-induced forces on the interior wall of the aneurysm and in order to study the development of the distribution with the gradual enlargement of the aneurysm, three different degrees of aneurysm-growth have been assumed. Finally and for examining the effect of the distribution on the aneurysm growth, a comparison is made between the pressure and wall shear-stress distributions at the wall for each growth-degree.     

Vascular wall flow-induced forces in a progressively enlarged aneurysm model

The current study is focused on the numerical investigation of the flow field induced by the unsteady flow in the vicinity of an abdominal aortic aneurysm model. The computational fluid dynamics code used is based on the finite volume method, and it has already been used in various bioflow studies. For modelling the rheological behaviour of blood, the Quemada non-Newtonian model is employed, which is suitable for simulating the two-phase character of blood namely a suspension of blood cells in plasma. For examining its non-Newtonian effects a comparison with a corresponding Newtonian flow is carried out. Furthermore, the investigation is focused on the distribution of the flow-induced forces on the interior wall of the aneurysm and in order to study the development of the distribution with the gradual enlargement of the aneurysm, three different degrees of aneurysm-growth have been assumed. Finally and for examining the effect of the distribution on the aneurysm growth, a comparison is made between the pressure and wall shear-stress distributions at the wall for each growth-degree.     

Boundary conditions at the cartilage-synovial fluid interface for joint lubrication and theoretical verifications

The objective of this study is to establish and verify the set of boundary conditions at the interface between a biphasic mixture (articular cartilage) and a Newtonian or non-Newtonian fluid (synovial fluid) such that a set of well-posed mathematical problems may be formulated to investigate joint lubrication problems. A "pseudo-no-slip" kinematic boundary condition is proposed based upon the principle that the conditions at the interface between mixtures or mixtures and fluids must reduce to those boundary conditions in single phase continuum mechanics. From this proposed kinematic boundary condition, and balances of mass, momentum and energy, the boundary conditions at the interface between a biphasic mixture and a Newtonian or non-Newtonian fluid are mathematically derived. Based upon these general results, the appropriate boundary conditions needed in modeling the cartilage-synovial fluid-cartilage lubrication problem are deduced. For two simple cases where a Newtonian viscous fluid is forced to flow (with imposed Couette or Poiseuille flow conditions) over a porous-permeable biphasic material of relatively low permeability, the well known empirical Taylor slip condition may be derived using matched asymptotic analysis of the boundary layer at the interface.     

Numerical study of the impact of non-Newtonian blood behavior on flow over a two-dimensional backward facing step

Endothelial cell (EC) responsiveness to shear stress is essential for vasoregulation and plays a role in atherogenesis. Although blood is a non-Newtonian fluid, EC flow studies in vitro are typically performed using Newtonian fluids. The goal of the present study was to determine the impact of non-Newtonian behavior on the flow field within a model flow chamber capable of producing flow disturbance and whose dimensions permit Reynolds and Womersley numbers comparable to those present in vivo. We performed two-dimensional computational fluid dynamic simulations of steady and pulsatile laminar flow of Newtonian and non-Newtonian fluids over a backward facing step. In the non-Newtonian simulations, the fluid was modeled as a shear-thinning Carreau fluid. Steady flow results demonstrate that for Re in the range 50-400, the flow recirculation zone downstream of the step is 22-63% larger for the Newtonian fluid than for the non-Newtonian fluid, while spatial gradients of shear stress are larger for the non-Newtonian fluid. In pulsatile flow, the temporal gradients of shear stress within the flow recirculation zone are significantly larger for the Newtonian fluid than for the non-Newtonian fluid. These findings raise the possibility that in regions of flow disturbance, EC mechanotransduction pathways stimulated by Newtonian and non-Newtonian fluids may be different.     

Non-Newtonian Blood Flow in Left Coronary Arteries with Varying Stenosis: A Comparative Study

This paper presents Computational fluid dynamic (CFD) analysis of blood flow in three different 3-D models of left coronary artery (LCA). A comparative study of flow parameters (pressure distribution, velocity distribution and wall shear stress) in each of the models is done for a non-Newtonian (Carreau) as well as the Newtonian nature of blood viscosity over a complete cardiac cycle. The difference between these two types of behavior of blood is studied for both transient and steady states of flow. Additionally, flow parameters are compared for steady and transient boundary conditions considering blood as non-Newtonian fluid. The study shows that the highest wall shear stress (WSS), velocity and pressure are found in artery having stenosis in all the three branches of LCA. The use of Newtonian blood model is a good approximation for steady as well as transient blood flow boundary conditions if shear rate is above 100 s-1. However, the assumption of steady blood flow results in underestimating the values of flow parameters such as wall shear stress, pressure and velocity.     

Flush mounted hot film anemometer measurement of wall shear stress distal to a tri-leaflet valve for Newtonian and non-Newtonian blood analog fluids

Wall shear stress has been measured by flush-mounted hot film anemometry distal to an Ionescu-Shiley tri-leaflet valve under pulsatile flow conditions. Both Newtonian (aqueous glycerol) and non-Newtonian (aqueous polyacrylamide) blood analog fluids were investigated. Significant differences in the axial distribution of wall shear stress between the two fluids are apparent in flows having nearly identical Reynolds numbers. The Newtonian fluid exhibits a (peak) wall shear rate which is maximized near the valve seat (30 mm) and then decays to a fully developed flow value (by 106 mm). In contrast, the shear rate of the non-Newtonian fluid at 30 mm is less than half that of the Newtonian fluid and at 106 mm is more than twice that of the Newtonian fluid. It is suggested that non-Newtonian rheology influences valve flow patterns either through alterations in valve opening associated with low shear separation zones behind valve leaflets, or because of variations in the rate of jet spreading. More detailed studies are required to clarify the mechanisms. The Newtonian wall shear stresses for this valve are low. The highest value observed anywhere in the aortic chamber was 2.85 N/m2 at a peak Reynolds number of 3694.     

Computational fluid dynamics in abdominal aorta bifurcation: non-Newtonian versus Newtonian blood flow in a real case study

Hemodynamic in abdominal aorta bifurcation was investigated in a real case using computational fluid dynamics. A Newtonian and non-Newtonian (Walburn-Schneck) viscosity models were compared. The geometrical model was obtained by 3D reconstruction from CT-scan and hemodynamic parameters obtained by laser-Doppler. Blood was assumed incompressible fluid, laminar flow in transient regime and rigid vessel wall. Finite volume-based was used to study the velocity, pressure, wall shear stress (WSS) and viscosity throughout cardiac cycle. Results obtained with Walburn-Schneck’s model, during systole, present lower viscosity due to shear thinning behavior. Furthermore, there is a significant difference between the results obtained by the two models for a specific patient. During the systole, differences are more pronounced and are preferably located in the tortuous regions of the artery. Throughout the cardiac cycle, the WSS amplitude between the systole and diastole is greater for the Walburn-Schneck’s model than for the Newtonian model. However, the average viscosity along the artery is always greater for the non-Newtonian model, except in the systolic peak. The hemodynamic model is crucial to validate results obtained with CFD and to explore clinical potential.     

Non-Newtonian blood flow in human right coronary arteries: transient simulations

This study looks at pulsatile blood flow through four different right coronary arteries, which have been reconstructed from biplane angiograms. A non-Newtonian blood model (the Generalised Power Law), as well as the usual Newtonian model of blood viscosity, is used to study the wall shear stress in each of these arteries over the entire cardiac cycle. The difference between Newtonian and non-Newtonian blood models is also studied over the whole cardiac cycle using the recently generalised global non-Newtonian importance factor. In addition, the flow is studied by considering paths of massless particles introduced into the flow field. The study shows that, when studying the wall shear stress distribution for transient blood flow in arteries, the use of a Newtonian blood model is a reasonably good approximation. However, to study the flow within the artery in greater detail, a non-Newtonian model is more appropriate.     

Wall shear stress in backward-facing step flow of a red blood cell suspension.

An experimental investigation of the wall shear stress distribution downstream of a backward-facing step is carried out. The wall shear stress distribution was determined by measuring the deformation of a gel layer, attached to the wall downstream of the step. Speckle pattern interferometry was applied to measure the deformation of the gel layer. The measured deformation, combined with the properties of the gel layer, served as an input for a finite element solid mechanics computation to determine the stress distribution in the gel layer. The wall shear stress, required to generate the measured deformation of the gel layer, was determined from these computations. A Newtonian buffer solution and a non-Newtonian red blood cell suspension were used as measuring fluids. The deformation of the gel layer was determined for a Newtonian buffer solution to evaluate the method and to obtain the properties of the gel layer. Subsequently, the wall shear stress distribution for the non-Newtonian red blood cell suspension was determined for three different flow rates. The inelastic non-Newtonian Carreau-Yasuda model served as constitutive model for the red blood cell suspension. Using this model, the velocity and wall shear stress distribution were computed by means of a finite element fluid mechanics computation. From the comparison between the numerical and the experimental results, it can be concluded that wall shear stresses, induced by the red blood cell suspension, can be modeled accurately by employing a Carreau-Yasuda model.     

The effects of non-Newtonian viscoelasticity and wall elasticity on flow at a 90 degrees bifurcation

To study the fundamentals of hemodynamics in arteries, the flow parameters: pulsatility, elasticity and non-Newtonian viscoelasticity were considered in detail in a 90 degrees-T-bifurcation of a rigid and elastic model. The velocity distribution 2.5 mm behind the bifurcation in the straight tube was measured with a laser-Doppler-anemometer. The fluid used was an aqueous glycerine solution and a viscoelastic Separan mixture. Flow visualization studies were done with a sheet of laser light in the plane of the bifurcation. The velocity distribution was measured for both steady and pulsatile flows with a laser-Doppler-anemometer in a backward scattered way. From the velocity measurements the shear gradients were calculated. Substantial differences were found in the flow behavior of Newtonian and non-Newtonian fluids, especially behind the bifurcation in the main tube, where secondary flows and flow separation started. Also, differences due to the elastic and rigid wall could be seen. Very high shear gradients were found in the flow between main flow and the separation zone which can lead to a damage of the blood cells.