# Experimental and Numerical Study of Blood Flow in μ-vessels: Influence of the Fahraeus–Lindqvist Effect

^{1}

^{2}

^{*}

## Abstract

**:**

_{∞}) and the hematocrit (H

_{ct}). This correlation along with the lateral distribution of blood viscosity were used as input to a “two-regions” computational model. The reliability of the code was checked by comparing the experimentally obtained axial velocity profiles with those calculated by the computational fluid dynamics (CFD) simulations. We propose a methodology for calculating the friction loses during blood flow in μ-vessels, where the Fahraeus–Lindqvist effect plays a prominent role, and show that the pressure drop may be overestimated by 80% to 150% if the CFL is neglected.

## 1. Introduction

_{ct}, and flow conditions. The latter can be described by Re

_{∞}, which uses the asymptotic viscosity value, μ

_{∞}. As expected, the CFL exhibits a lower viscosity than the rest of the fluid and this facilitates blood flow through the microvessels. This is clearly pronounced by relevant studies that attempt to estimate wall shear stress [8,9] or pressure drop [10] values in such vessels. Although this phenomenon is significant for diameters less than 300 μm, most of the published work is focused on vessel diameters between 20 and 100 μm, e.g., [6], or on fairly different geometries [11].

- First, estimate the CFL extent. In the relevant published research, either the H
_{ct}value [12] or the vessel diameter [13] was considered when formulating correlations for the prediction of the CFL width. In this study, we investigated the effect of both Re_{∞}and H_{ct}on CFL characteristics in three microvessels with a 50, 100, and 170 μm hydraulic diameter. The experimental data were used for formulating a correlation for estimating the CFL width under the combined effect of the various flow conditions (i.e., Re_{∞}) and hematocrit values. - The second one is to investigate the blood velocity profile in such microvessels using the well-established micro-particle image velocimetry (μ-PIV) technique. We employed two different tracing methods, i.e., coloured RBCs or standard PIV tracers. Previous studies suggested that the two tracing methods give significant different results [14]. This issue is clarified in this study. The acquired experimental data was used for validating the computational fluid dynamics (CFD) code.
- The final one is to develop a simplified “two-regions” model using computational fluid dynamics (CFD) by utilizing the outcome of the experimental part of this study in order to set up reliable simulations, a common practice in the literature [15]. The scope of this final step was to calculate the blood flow characteristics as well as the overall pressure drop (ΔP) across the vessels. The CFL flow domain was solved by assuming the blood properties are that of a Newtonian fluid, whereas in the vessel core, the blood rheology is formulated using a non-Newtonian model. Simulations were executed for various blood velocities and vessel diameters.

## 2. Experimental Procedure

- Separation of RBCs from plasma by centrifugation (at 3200 rpm) of the whole blood sample.
- Purification of the separated RBCs by washing with saline water and centrifuging twice.

_{ct}. The composition of all fluids used in our experiments are presented in Table 1. The viscosity of the samples was measured using a cone plate rheometer (AR-G2, TA Instruments) at 24 °C. To verify the similarity with “real” blood, the measured viscosity of the samples was compared to the Quemada model (Equation (1)) [16]:

_{0}, k

_{∞}, and γ

_{c}are, in general, functions of hematocrit [17].

## 3. Cell Free Layer (CFL) and Velocity Measurements

- Spherical fluorescent particles (Invitrogen) with 1.1 μm mean diameter.
- RBCs coloured with fluorescent dye (Rhodamine B).

_{t}(Equation (4)), is a function of the density difference between the two phases, the particle size, and the viscosity of the fluid:

_{ct}10%, 20%, 30%, and 40% and Re

_{∞}0.3–6.0. The reference viscosity used for calculating the Re

_{∞}is the one obtained from the Quemada model for very high shear rates (γ > 1000 s

^{−1}). The effect of the channel diameter on the CFL was investigated by conducting experiments in the three μ-channels. The experimental setup used (Figure 3) employs the microscope of the μ-PIV set up with the same lens, but in this case, the flow was recorded by a high-speed camera.

## 4. Experimental Results

_{t}= 6.2 × 10

^{−4}m/s) is comparable to their horizontal velocity, resulting in the settling of the RBCs before exiting the test section. Thus, instead of a dispersed flow, a separated flow is maintained, where the majority of the RBCs migrate near the bottom, leaving the upper part of the vessel filled with saline. However, in the second method, i.e., the method employing the tracing particles, the settling velocity is much lower (due to the smaller diameter compared to RBCs) and thus they can follow the flow even for very low flow rates.

_{ct}are shown in Figure 7, where for the same volumetric flow rate, the maximum velocity decreases and the velocity profile becomes blunter as the H

_{ct}is increased. A similar behaviour was also observed in the numerical work of Fedosov et al. [20] and Sriram et al. [21], who both found a great reduction of the maximum axial velocity as H

_{ct}increased.

_{mean}= Q/A. The detected velocity fluctuations can be attributed to the fact that due to the depth of the field (i.e., approximately 3 μm), particles located on different planes may contribute to the measurement. As expected, the fluctuations are more pronounced in the vicinity of the walls, due to higher axial velocity gradients. The present results clearly show that the RMS value increases with the hematocrit, a fact that is in accordance with the findings of Lima et al. [22] and Lauri et al. [11].

_{ct}, which is in accordance with Namgung et al. [6]. This can be attributed to the fact that as the concentration of the RBCs increases, the viscous forces increase and hinder the movement of the cells in the middle of the channel. Additionally, the normalized CFL decreases as the diameter of the microchannel increases, a fact that is also confirmed by Namgung et al. [6] The results also showed that in the flow rate range where migration started, there is still a low cell concentration near the wall and not a total absence of cells.

_{∞}> 4. This can be explained as follows. As the RBCs are migrating towards the centre of the vessel, their concentration increases up to the point where viscous forces hinder additional migration. Using response surface methodology (RSM), a second order polynomial equation, also known as the quadratic model (Equation (5)) [23], is formulated:

_{0}, a

_{j}, a

_{jj}and a

_{ij}are the unknown coefficient of the polynomial equation, whereas x

_{i},x

_{j}are the selected design variables. In our case, we calculated the quantity CFL/D as a function of Re

_{∞}and H

_{ct}(Equation (6)). The values of the coefficients of the polynomial were determined using the experimental data for two (k = 2) variables and are given in Table 2:

_{∞}0.3 to 6 and H

_{ct}10% to 40% (Figure 11 and Figure 12). The accuracy of the equation was calculated by comparing the correlation with the experimental results. For Re

_{∞}> 4, the CFL/D quantity approaches a constant value, which depends on H

_{ct}. The novelty in our approach was the inclusion of two parameters, namely Re

_{∞}and H

_{ct}, in the proposed correlation, a combination that is often omitted in existing research [12,13] but adds a significant amount of credibility in the proposed model.

## 5. CFD Simulations

_{b}= 1050 kg/m

^{3}). In the CFL zone, blood was modelled as a Newtonian fluid having the viscosity of human blood plasma (μ

_{p}= 1.2 cP), whereas, in the core area blood, it was modelled as non-Newtonian shear thinning fluid. This was achieved through a modified Quemada model, where local μ is expressed by Equation (1), K is expressed by Equation (2), and its coefficients (${k}_{0},{k}_{\infty},{\gamma}_{c}$) are expressions of ${H}_{ct}$ as formulated by Das et al. [17], described in Equations (9)–(11):

_{∞}values are low, the flow is laminar. Thus, the direct numerical simulation (DNS) model was used, while the high-resolution advection scheme was employed for the discretization of the momentum equations. The simulations were run in steady state, the vessel walls were considered smooth, and a no-slip boundary condition was imposed. At the inlet, constant velocity, i.e., a Dirichlet boundary condition, was imposed. The outlet pressure was set constant and equal to atmospheric. As for the meshing procedure, an optimum grid density was chosen by performing a grid-dependency study. The inlet pressure was used as a parameter for the grid-dependency study. Typical results are presented in Table 4. Furthermore, the mesh was refined along the CFL border, where viscosity changes are more pronounced. Consequently, cells with a maximum face size of 2.5 μm were used, while mesh refinement was applied in areas where viscosity changes were pronounced, e.g., near the wall.

_{ct}between 10% and 40%, and Re

_{∞}between 0.3 and 5. In Figure 14b, the blood viscosity distribution is illustrated for one of the cases tested (D = 100 μm, Re

_{∞}= 0.8, H

_{ct}= 30%). In the CFL zone, depicted in blue colour, the viscosity is that of blood plasma, whereas in the non-Newtonian core, the viscosity gradually decreases as the radial distance increases.

_{∞}= 0.3, H

_{ct}= 37%). There seems to be a relatively good agreement between the experimental and computational results, bound with deviations less than 5%.

_{CFL}) in various vessels was calculated across the whole length of the vessel and was compared with three values, namely:

- The ΔP
_{HP}, calculated analytically using the Hagen–Poiseuille equation for straight circular tubes and considering the blood as a Newtonian fluid (e.g., μ_{∞}= 3.5 cP for H_{ct}= 40%). - The ΔP
_{nCFL}calculated numerically considering that there is no CFL and that the blood behaves as a non-Newtonian fluid (in the same way the non-Newtonian core was modelled) [29]. - The ΔP
_{[10]}calculated iteratively using the equations proposed in [10], considering blood as a Casson fluid.

## 6. Conclusions

_{∞}) are known, someone can utilize the proposed correlation (Equation (6)) to predict the CFL width. In the CFL zone, blood is modelled as a Newtonian fluid having the viscosity of blood plasma, whereas in the core area, blood is modelled as a non-Newtonian shear thinning fluid. Consequently, a complicated problem, the Fahraeus–Lindqvist effect, which occurs during blood flow, is now simplified to a single-phase flow simulation that can converge relatively fast without the need of high computational power. The results of the CFD simulation give valuable information, like the pressure drop, in blood μ-vessels and wall shear stress estimation.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Q | volumetric flow rate, mL/h |

U | blood axial velocity, m/s |

U_{t} | settling velocity, m/s |

g | acceleration of gravity, m/s^{2} |

CFL | cell free layer width, m |

D | hydraulic diameter, m |

H_{ct} | hematocrit, dimensionless |

Re_{∞} | Reynolds number based on μ_{∞}, dimensionless |

Greek letters | |

γ | shear rate, s^{−1} |

γ* | pseudo shear rate, s^{−1} |

ΔP_{CFL} | numerically calculated pressure drop, Pa |

ΔP_{nCFL} | numerically calculated pressure drop without CFL modelling, Pa |

ΔP_{HP} | pressure drop calculated using the Hagen-Poiseuille correlation, Pa |

ΔP_{[10]} | pressure drop calculated using the correlation proposed in [10], Pa |

μ | dynamic viscosity, Pa s |

μ_{∞} | dynamic viscosity for high shear rates (asymptotic), Pa s |

μ_{p} | viscosity of plasma, Pa s |

ρ_{p} | particle density, kg/m^{3} |

ρ_{l} | blood density, kg/m^{3} |

## References

- Tuma, R.F.; Duran, W.N.; Ley, K. Microcirculation; Academic Press: Cambridge, MA, USA, 2011; ISBN 978-0-08-056993-2. [Google Scholar]
- Anastasiou, A.D.; Spyrogianni, A.S.; Koskinas, K.C.; Giannoglou, G.D.; Paras, S.V. Experimental investigation of the flow of a blood analogue fluid in a replica of a bifurcated small artery. Med. Eng. Phys.
**2012**, 34, 211–218. [Google Scholar] [CrossRef] [PubMed] - Fåhræus, R.; Lindqvist, T. The viscosity of the blood in narrow capillary tubes. Am. J. Physiol.-Leg. Content
**1931**, 96, 562–568. [Google Scholar] [CrossRef] - Albrecht, K.H.; Gaehtgens, P.; Pries, A.; Heuser, M. The Fahraeus effect in narrow capillaries (i.d. 3.3 to 11.0 μm). Microvasc. Res.
**1979**, 18, 33–47. [Google Scholar] [CrossRef] - Ong, P.K.; Namgung, B.; Johnson, P.C.; Kim, S. Effect of erythrocyte aggregation and flow rate on cell-free layer formation in arterioles. Am. J. Physiol. Heart Circ. Physiol.
**2010**, 298, H1870–H1878. [Google Scholar] [CrossRef] [PubMed] - Namgung, B.; Ju, M.; Cabrales, P.; Kim, S. Two-phase model for prediction of cell-free layer width in blood flow. Microvasc. Res.
**2013**, 85, 68–76. [Google Scholar] [CrossRef] - Pries, A.R.; Schönfeld, D.; Gaehtgens, P.; Kiani, M.F.; Cokelet, G.R. Diameter variability and microvascular flow resistance. Am. J. Physiol. Heart Circ. Physiol.
**1997**, 272, H2716–H2725. [Google Scholar] [CrossRef] - Sriram, K.; Intaglietta, M.; Tartakovsky, D.M. Non-Newtonian Flow of Blood in Arterioles: Consequences for Wall Shear Stress Measurements. Microcirculation
**2014**, 21, 628–639. [Google Scholar] [CrossRef] - Balogh, P.; Bagchi, P. Three-dimensional distribution of wall shear stress and its gradient in red cell-resolved computational modeling of blood flow in in vivo-like microvascular networks. Physiol. Rep.
**2019**, 7, e14067. [Google Scholar] [CrossRef] - Chandran, K.B.; Rittgers, S.E.; Yoganathan, A.P.; Rittgers, S.E.; Yoganathan, A.P. Biofluid Mechanics: The Human Circulation, 2nd ed.; CRC Press: Boca Raton, MA, USA, 2012; ISBN 978-0-429-10607-1. [Google Scholar]
- Lauri, J.; Bykov, A.; Fabritius, T. Quantification of cell-free layer thickness and cell distribution of blood by optical coherence tomography. J. Biomed. Opt.
**2016**, 21, 040501. [Google Scholar] [CrossRef] - Sriram, K.; Vázquez, B.Y.S.; Yalcin, O.; Johnson, P.C.; Intaglietta, M.; Tartakovsky, D.M. The effect of small changes in hematocrit on nitric oxide transport in arterioles. Antioxid. Redox Signal.
**2011**, 14, 175–185. [Google Scholar] [CrossRef] - Al-Khazraji, B.K.; Jackson, D.N.; Goldman, D. A Microvascular Wall Shear Rate Function Derived From In Vivo Hemodynamic and Geometric Parameters in Continuously Branching Arterioles. Microcirculation
**2016**, 23, 311–319. [Google Scholar] [CrossRef] [PubMed] - Pitts, K.L.; Fenech, M. High speed versus pulsed images for micro-particle image velocimetry: A direct comparison of red blood cells versus fluorescing tracers as tracking particles. Physiol. Meas.
**2013**, 34, 1363–1374. [Google Scholar] [CrossRef] [PubMed] - Sharan, M.; Popel, A.S. A two-phase model for flow of blood in narrow tubes with increased effective viscosity near the wall. Biorheology
**2001**, 38, 415–428. [Google Scholar] [PubMed] - Kim, S.; Kong, R.L.; Popel, A.S.; Intaglietta, M.; Johnson, P.C. Temporal and spatial variations of cell-free layer width in arterioles. Am. J. Physiol. Heart Circ. Physiol.
**2007**, 293, H1526–H1535. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Das, B.; Johnson, P.C.; Popel, A.S. Effect of nonaxisymmetric hematocrit distribution on non-Newtonian blood flow in small tubes. Biorheology
**1998**, 35, 69–87. [Google Scholar] [CrossRef] - Batchelor, G.K. An Introduction to Fluid Dynamics; Cambridge University Press: Cambridge, UK, 2000; ISBN 978-0-521-66396-0. [Google Scholar]
- Wereley, S.T.; Meinhart, C.D. Micron-Resolution Particle Image Velocimetry. In Microscale Diagnostic Techniques; Breuer, K.S., Ed.; Springer: Berlin/Heidelberg, Germany, 2005; pp. 51–112. ISBN 978-3-540-26449-1. [Google Scholar]
- Fedosov, D.; Caswell, B.; Popel, A.S.; Karniadakis, G. Blood Flow and Cell-Free Layer in Microvessels. Microcirculation
**2010**, 17, 615–628. [Google Scholar] [CrossRef] [Green Version] - Sriram, K.; Tsai, A.G.; Cabrales, P.; Meng, F.; Acharya, S.A.; Tartakovsky, D.M.; Intaglietta, M. PEG-albumin supraplasma expansion is due to increased vessel wall shear stress induced by blood viscosity shear thinning. Am. J. Physiol. Heart Circ. Physiol.
**2012**, 302, H2489–H2497. [Google Scholar] [CrossRef] [Green Version] - Lima, R.; Wada, S.; Takeda, M.; Tsubota, K.; Yamaguchi, T. In vitro confocal micro-PIV measurements of blood flow in a square microchannel: The effect of the haematocrit on instantaneous velocity profiles. J. Biomech.
**2007**, 40, 2752–2757. [Google Scholar] [CrossRef] [Green Version] - Box, G.E.P.; Wilson, K.B. On the Experimental Attainment of Optimum Conditions. J. R. Stat. Soc. Ser. B Methodol.
**1951**, 13, 1–45. [Google Scholar] [CrossRef] - Reinke, W.; Gaehtgens, P.; Johnson, P.C. Blood viscosity in small tubes: Effect of shear rate, aggregation, and sedimentation. Am. J. Physiol.
**1987**, 253, H540–H547. [Google Scholar] [CrossRef] - Fåhraeus, R. The suspension stability of the blood. Physiol. Rev.
**1929**, 9, 241–274. [Google Scholar] [CrossRef] - Pries, A.R.; Secomb, T.W.; Gaehtgens, P.; Gross, J.F. Blood flow in microvascular networks. Experiments and simulation. Circ. Res.
**1990**, 67, 826–834. [Google Scholar] [CrossRef] - Lih, M.M. A mathematical model for the axial migration of suspended particles in tube flow. Bull. Math. Biophys.
**1969**, 31, 143–157. [Google Scholar] [CrossRef] - Gaehtgens, P.; Meiselman, H.J.; Wayland, H. Velocity profiles of human blood at normal and reduced hematocrit in glass tubes up to 130 μ diameter. Microvasc. Res.
**1970**, 2, 13–23. [Google Scholar] [CrossRef] - Mouza, A.A.; Skordia, O.D.; Tzouganatos, I.D.; Paras, S.V. A Simplified Model for Predicting Friction Factors of Laminar Blood Flow in Small-Caliber Vessels. Fluids
**2018**, 3, 75. [Google Scholar] [CrossRef]

**Figure 1.**Experimental data of “blood” viscosity vs. shear rate compared with the Quemada correlation (H

_{ct}= 40%).

**Figure 2.**Micro-particle image velocimetry (μ-PIV) experimental setup used in this study (

**a**): high-speed camera, (

**b**): microscope, (

**c**): syringe pump.

**Figure 3.**Depiction of the: (

**a**) cell free layer (CFL) measurement setup used in our study, (

**b**) detail of the test section.

**Figure 4.**Typical images for: (

**a**) Q = 0.5 mL/h; (

**b**) Q = 1.2 mL/h. (D = 100 μm, H

_{ct}= 20%) and (

**c**) identification of the CFL boundaries and measurement of its width (Δ).

**Figure 5.**Typical velocity profiles for H

_{ct}= 10% for both tracing methods (microparticles vs. RBCs) employed (D = 170 μm, Q = 5 mL/h). Comparison shows that the RBC slip velocity is practically zero.

**Figure 6.**Typical images: (

**a**) at the bottom of the micro-channel, (

**b**) in the middle, and (

**c**) at the top of the channel. (D = 170 μm, H

_{ct}= 10%, Q = 0.01 mL/h).

**Figure 7.**Typical velocity profiles at D = 170 μm for three different H

_{ct}values (Q = 10 mL/h) using the micro particle method.

**Figure 8.**RMS/U

_{mean}variation for H

_{ct}= 10% and 20% with respect to the radial position (Q = 5 mL/h, D = 170 μm).

**Figure 10.**Typical variation of CFL standard deviation (SD) vs. pseudo shear rate (H

_{ct}= 10%, D = 100 μm).

**Figure 11.**Comparison of the current correlation (Equation (6)) with CFL experimental data of this study for: (

**a**) H

_{ct}= 10% and (

**b**) H

_{ct}= 20% (7% error bars).

**Figure 12.**CFL thickness dependence on H

_{ct}and Re

_{∞}based on the proposed correlation (Equation (6)).

**Figure 13.**Comparison of Equation (6) with Reinke et al.’s [24] experimental data for CFL thickness (10% error bars).

**Figure 14.**(

**a**) Radial viscosity distribution (H

_{ct}= 10%, Re

_{∞}= 4.4); (

**b**) graphical representation of the viscosity distribution on a cross section (Re

_{∞}= 0.8, H

_{ct}= 40%).

**Figure 15.**Comparison of computational and experimental velocity results for three cases with D = 170 μm: (

**a**) H

_{ct}= 20%, Re

_{∞}= 2.1; (

**b**) H

_{ct}= 20%, Re

_{∞}= 4.2; (

**c**) H

_{ct}= 10%, Re

_{∞}= 2.1.

**Figure 16.**Comparison of numerically calculated axial velocity with published experimental data [28] (D = 80 μm, Re

_{∞}= 0.3, H

_{ct}= 37%).

Fluid | Red Blood Cell (RBC) | Saline | EDTA |
---|---|---|---|

H_{10} | 10 | 89.5 | 0.5 |

H_{20} | 20 | 79.5 | 0.5 |

H_{30} | 30 | 69.5 | 0.5 |

H_{40} | 40 | 59.5 | 0.5 |

Coefficient | Value | Coefficient | Value |
---|---|---|---|

a_{0} | 0.066384 | a_{11} | 0.000002 |

a_{1} | −0.000887 | a_{22} | −0.001414 |

a_{2} | 0.017796 | a_{12} | −0.000005 |

Coefficient | Value | Coefficient | Value |
---|---|---|---|

${m}_{0}$ | 0.275363 | ${m}_{1}$ | 0.100158 |

${b}_{0}$ | 1.3435 | ${b}_{1}$ | −2.803 |

${b}_{2}$ | 2.711 | ${b}_{3}$ | −0.6479 |

${c}_{0}$ | −6.1508 | ${c}_{1}$ | 27.923 |

${c}_{2}$ | −25.60 | ${c}_{3}$ | 3.697 |

Maximum Cell Face Size (μm) | Inlet Pressure (Pa) |
---|---|

5.0 | 2610 |

4.0 | 2578 |

3.0 | 2555 |

2.5 | 2553 |

Re_{∞} | D (μm) | H_{ct} (%) | ΔP_{CFL} (Pa) | ΔP_{nCFL} (Pa) | ΔP_{HP} (Pa) | ΔP_{[10]} (Pa) | ΔP_{nCFL}/ΔP_{CFL} | ΔP_{HP}/ΔP_{CFL} |
---|---|---|---|---|---|---|---|---|

0.8 | 100 | 30 | 132 | 238 | 302 | 184 | 1.8 | 2.29 |

1.4 | 170 | 40 | 42 | 103 | 105 | 67 | 2.5 | 2.50 |

2.5 | 170 | 40 | 105 | 187 | 186 | 117 | 1.8 | 1.77 |

4.9 | 170 | 40 | 204 | 377 | 372 | 229 | 1.8 | 1.82 |

1.2 | 50 | 20 | 1463 | 2753 | 2984 | 1629 | 1.9 | 2.04 |

1.1 | 50 | 40 | 1466 | 2772 | 3256 | 1978 | 1.9 | 2.22 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Stergiou, Y.G.; Keramydas, A.T.; Anastasiou, A.D.; Mouza, A.A.; Paras, S.V.
Experimental and Numerical Study of Blood Flow in *μ*-vessels: Influence of the Fahraeus–Lindqvist Effect. *Fluids* **2019**, *4*, 143.
https://doi.org/10.3390/fluids4030143

**AMA Style**

Stergiou YG, Keramydas AT, Anastasiou AD, Mouza AA, Paras SV.
Experimental and Numerical Study of Blood Flow in *μ*-vessels: Influence of the Fahraeus–Lindqvist Effect. *Fluids*. 2019; 4(3):143.
https://doi.org/10.3390/fluids4030143

**Chicago/Turabian Style**

Stergiou, Yorgos G., Aggelos T. Keramydas, Antonios D. Anastasiou, Aikaterini A. Mouza, and Spiros V. Paras.
2019. "Experimental and Numerical Study of Blood Flow in *μ*-vessels: Influence of the Fahraeus–Lindqvist Effect" *Fluids* 4, no. 3: 143.
https://doi.org/10.3390/fluids4030143