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Kinetic Theory Based Model for Blood Flow and its Viscosity

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Abstract

A kinetic theory based two phase flow model for plasma and red blood cells (RBCs) is shown to explain the Fahraeus–Lindqvist effect, the migration of red blood cells from the wall to the center in narrow tubes. The migration is caused by shear induced diffusion which in the kinetic theory based model is computed using a balance of granular temperature, the random kinetic energy for red blood cells per unit mass. The computed hematocrit distribution agrees with experimental measurements using a complete computational fluid dynamic model and an approximate fully developed flow solution. The model predicts the momentum and granular temperature boundary layers. The model computes the observed blood viscosity dependence on diameter and hematocrit.

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Abbreviations

C D :

Drag coefficient

D :

Optical density

d p :

Diameter of RBC

e :

Restitution coefficient

e w :

Wall restitution coefficient

\( \vec{g} \) :

Gravitational acceleration

g 0 :

Radial distribution function

Hct:

Hematocrit

\( {\vec{\vec{\rm I}}} \) :

Unit tensor

k :

Extinction coefficient of the absorbing medium

k s :

Granular conductivity

l :

Length of the light passing through

M s :

Molecular weight

n :

Normal component

P :

Pressure

P s :

Solid pressure

R :

Tube radius

Re p :

Reynolds number

q :

Flux of random kinetic energy

t :

Time

U ∞ :

Free stream velocity

\( \vec{\upsilon }_{\text{f}} \) :

Fluid phase velocity vector

\( \vec{\upsilon }_{\text{s}} \) :

Solid phase velocity vector

υ s,w :

RBC velocity in the direction parallel to the wall

υ sz :

Solid phase velocity in axial direction

υ z,max :

Maximum solid phase velocity

β :

Drag coefficient between particles

ρ f :

Fluid density

ρ s :

RBC density

ε f :

Volume fraction of fluid phase

ε s :

Volume fraction of solid phase

ε s,max :

Maximum solid volume fraction

\( {\vec{\vec{\tau}}}_{\text{f}} \) :

Stress tensor of fluid phase

τ rz :

The stress in the z direction acting on the surface perpendicular to the r direction

\( \vec{\vec{\tau }}_{\text{s}} \) :

Stress tensor of solid phase

θ :

Granular temperature

θ w :

Granular temperature at the wall

γ :

Energy dissipation due to inelastic collision of particles

γ w :

Energy dissipation due to inelastic collision between particle and the wall

μ f :

Fluid viscosity

μ s :

Solid viscosity

ξ s :

Bulk viscosity of solid phase

ϕ :

Specularity coefficient

ν :

Kinematic viscosity

δ :

Boundary layer thickness

δ T :

Granular temperature boundary layer thickness

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Acknowledgments

The award of an Illinois Institute of Technology Fieldhouse Fellowship to the second author was instrumental in enabling the research presented herein. We also thank Sofiane Benyahia of the US Department of Energy, NETL, for helping us to delete the fluid–particle interaction term in the granular temperature equation in FLUENT.

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  1. Department of Chemical and Biological Engineering, Illinois Institute of Technology, Chicago, IL, 60616, USA

    Dimitri Gidaspow & Jing Huang

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  1. Dimitri Gidaspow
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Correspondence to Dimitri Gidaspow.

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Gidaspow, D., Huang, J. Kinetic Theory Based Model for Blood Flow and its Viscosity. Ann Biomed Eng 37, 1534–1545 (2009). https://doi.org/10.1007/s10439-009-9720-3

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