# A Simplified Model for Predicting Friction Factors of Laminar Blood Flow in Small-Caliber Vessels

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## Abstract

**:**

## 1. Introduction

^{−1}, it is not valid for blood flow in small vessels [5], where the Reynolds numbers are low, and the flow is laminar.

## 2. Experimental Setup and Procedure

^{®}, Sarasota, FL, USA), a three-way valve and a digital pressure transducer (68035, Cole Palmer, Vernon Hills, IL, USA).

#### Blood Analogue

_{y}is the yield stress and n

_{N}is the viscosity corresponding to high shear rates (asymptotic value). The yield stress is a measure of the amount of energy required to break down the aggregates of red blood cells formed at very low shear rates. Merrill [8] extensively investigated blood rheology and confirmed the strong relationship between viscosity and hematocrit (H

_{t}) and suggested that the terms n

_{N}and τ

_{y}of Equation (1) could be expressed as functions of hematocrit, i.e.,

_{p}is the viscosity of the plasma,

_{tc}is the critical hematocrit below which the yield stress (τ

_{y}) can be considered negligible. For normal blood, H

_{tc}ranges between 4 and 8 and A is a constant, ranging between 0.6 × 10

^{−7}and 1.2 × 10

^{−7}Pa. These expressions are employed hereafter for predicting blood viscosity as a function of hematocrit [8]. In this study, the values selected for A and H

_{tc}were 0.9 × 10

^{−7}Pa, and 6, respectively, i.e., the middle values of the corresponding range.

^{3}.

_{t}of ~55% can be simulated by a 30% v/v aqueous glycerol solution that contains 0.035% w/v xanthan gum, i.e., a polysaccharide that acts as a rheology modifier and renders the fluid non-Newtonian. The viscosity curve of the fluid was measured in our laboratory via a magnetic rheometer (AR-G2, TA Instruments, Sussex, UK), for shear rates between 1–1000 s

^{−1}(Figure 3) resulting in an excellent fit shown by a Casson-type curve (Equation (1)).

## 3. Numerical Simulations

^{3}), while the diameter of the conduit was a parametric variable and was set to be between 0.3–1.8 mm. With the aim of reducing memory consumption and CPU time and as the geometry consisted of two symmetry planes, only one fourth of the domain was used (Figure 4).

#### 3.1. Code Validation

#### 3.2. Numerical Procedure

^{®}package. The design variables selected along with the imposed upper and lower bounds are presented in Table 1. The upper bound of the vessel inside diameter corresponds to the larger arterioles and venules of an adult male [12], while the corresponding lower bound was chosen to be 500 μm to avoid the consequences of the Fahraeus-Lindqvist effect [12]. Due to this effect in small vessels (smaller than 300 μm), red blood cells tend to drift towards the central axis of the vessel, forming a cell-free layer called a plasma layer along the vascular wall. This effect results in an apparent blood viscosity, which declines substantially with decreasing diameter [13]. The hematocrit range that was chosen is typical for healthy adult humans [8] and the blood flow rate bounds imposed are typical for such μ-vessels.

## 4. Results

_{∞}is a Reynolds number that uses the asymptotic value (μ

_{∞}) of blood viscosity. From Figure 8 it is obvious that by no means can the correlation for Newtonian fluids be applied to non-Newtonian fluids because it underestimates ΔP by 30%.

^{−0.266}Re*

^{−1.064}

_{calc}), calculated by Equation (10) with the values that resulted from the CFD simulations (f

_{CFD}). It was proven that Equation (10) could predict the Fanning friction factor with 10% uncertainty.

## 5. Concluding Remarks

- For a given volumetric flow rate (Q) and vessel inside diameter (D), the pseudo-shear rate (γ*) is calculated using Equation (5).
- For a given hematocrit value (H
_{t}), an effective viscosity (μ*), that corresponds to the pseudo-shear rate is estimated using Equations (1)–(3). - The corresponding Re* and Bm numbers are calculated by Equations (4) and (9).
- The Fanning friction factor (f) is then calculated by the proposed correlation (Equation (10)).
- Finally, the pressure drop (ΔP/L) is calculated using Equation (6).

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Bm | Bingham number, - |

D | Inside vessel diameter, m |

f | Fanning friction factor, - |

f_{CFD} | Fanning friction factor from CFD simulations, - |

f_{calc} | Fanning friction factor from Equation (7), - |

H_{t} | Hematocrit, % |

H_{tc} | Critical hematocrit, % |

L | Length, m |

n_{p} | Plasma viscosity, Pa∙s |

P | Pressure, Pa |

Q | Volumetric flow rate, mm^{3}/s |

Re_{∞} | Reynolds number corresponding to μ_{∞} |

Re* | Effective Reynolds number (Equation (4)), - |

U | Mean velocity, m/s |

x | Axial coordinate, m |

Greek letters | |

γ* | Pseudo shear rate, s^{−1} |

ΔP | Pressure drop, Pa |

μ | Blood viscosity, Pa∙s |

μ* | Effective viscosity, Pa∙s |

μ_{∞} | Asymptotic viscosity value, Pa∙s |

ρ | Blood density, kg/m^{3} |

τ | Shear stress, Pa |

τ_{y} | Yield stress, Pa |

## References

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**Figure 7.**Typical simulation results of the pressure distribution across a small vessel (L = 50 mm, D = 0.4 mm and Q = 1 mm

^{3}/s).

**Figure 8.**Comparison of experimental results for the blood analogue with the theoretical prediction of ΔP using the correlation for Newtonian fluids (f = 16/Re

_{∞}).

Parameter | Lower Bound | Upper Bound |
---|---|---|

Vessel inside diameter (mm) | 0.50 | 1.80 |

Hematocrit (%) | 35 | 50 |

Blood flow rate (mm^{3}/s) | 7.0 | 88.0 |

Design Points | Verification Points | |||||||
---|---|---|---|---|---|---|---|---|

Box-Behnken | Additional Points | |||||||

Q | D | H_{t} | Q | D | H_{t} | Q | D | H_{t} |

mm^{3}/s | mm | % | mm^{3}/s | mm | % | mm^{3}/s | mm | % |

72.0 | 1.15 | 35 | 6.0 | 0.40 | 35 | 6.7 | 0.40 | 55 |

72.0 | 1.15 | 50 | 8.5 | 0.40 | 35 | 7.0 | 0.50 | 40 |

72.0 | 1.80 | 43 | 11.0 | 0.40 | 35 | 10.0 | 0.40 | 55 |

72.0 | 0.50 | 43 | 13.0 | 0.40 | 35 | 11.7 | 0.40 | 55 |

80.0 | 0.50 | 35 | 16.0 | 0.40 | 35 | 15.0 | 0.40 | 55 |

80.0 | 0.50 | 50 | 18.0 | 0.40 | 35 | 20.0 | 0.30 | 45 |

80.0 | 1.15 | 43 | 21.0 | 0.40 | 35 | 33.4 | 0.40 | 55 |

80.0 | 1.80 | 35 | 23.0 | 0.40 | 35 | 40.0 | 0.90 | 50 |

80.0 | 1.80 | 50 | 25.0 | 0.40 | 35 | 66.8 | 0.40 | 55 |

88.0 | 1.15 | 35 | 45.0 | 0.40 | 35 | 68.0 | 1.20 | 35 |

88.0 | 1.15 | 50 | 50.0 | 0.60 | 37 | 83.0 | 0.60 | 37 |

88.0 | 1.80 | 43 | 7.0 | 0.30 | 43 | 83.5 | 0.40 | 55 |

88.0 | 0.50 | 43 | - | - | - | - | - | - |

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**MDPI and ACS Style**

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.
https://doi.org/10.3390/fluids3040075

**AMA Style**

Mouza AA, Skordia OD, Tzouganatos ID, Paras SV.
A Simplified Model for Predicting Friction Factors of Laminar Blood Flow in Small-Caliber Vessels. *Fluids*. 2018; 3(4):75.
https://doi.org/10.3390/fluids3040075

**Chicago/Turabian Style**

Mouza, Aikaterini A., Olga D. Skordia, Ioannis D. Tzouganatos, and Spiros V. Paras.
2018. "A Simplified Model for Predicting Friction Factors of Laminar Blood Flow in Small-Caliber Vessels" *Fluids* 3, no. 4: 75.
https://doi.org/10.3390/fluids3040075