The 7th Asian Symposium on Computational Heat Transfer and Fluid Flow, Tokyo, Sep. 3-7, 2019

ASCHT2019-199

Numerical simulation of blood modeled as a fluid-particulate mixtureaKrishna Chandrana, Indranil Saha Dalalb*, Kazuya Tatsumic and Krishnamurthy Muralidhara
Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, Indiab
Department of Chemical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, Indiac
Department of Mechanical Engineering and Science, Kyoto University, Kyoto 606-8501, Japan* Corresponding author, E-mail: indrasd@iitk.ac.in
Abstract

Migration of red blood cells (RBC) towards the center of the capillaries from the walls due to shear-induced diffusion(SID) is a recently studied phenomenon. It results in a cell-free layer near the walls of the capillaries. While numerous
works are available in literature for the SID of particulate laden flow, a blood rheology model which predicts accurately
the RBC concentration distribution along with a mass conserving velocity field has not been addressed. In the present
study, a mathematical model for the transport of red blood cells (RBC) represented as a particulate is presented. Itproposes a coupled species-momentum transport model to predict the RBC distribution inside a tubular geometry and inpatient-specific human arteries.
Keywords : Shear-induced diffusion, red blood cells, particulate, hematocrit, cell free layer

1. Introduction
Blood is a complex fluid primarily constituting of erythrocytes (RBC), leukocytes (WBC) and platelets. Blood flow
inside arteries and capillaries can cause WBCs and platelets to separate from RBCs and tend to move towards the wallby a process called margination. On the other hand, RBCs tend to migrate towards the center of the arteries which cause
RBC depletion near the wall leading to what is known as the cell free layer at the wall [1]. Under normal physiologicalconditions blood behaves as a suspension and hence the interaction between the blood particulates especially RBCs
become more important to understand its rheology. RBC being a non-spherical and deformable particle, it tends to havean irreversible effect due to mutual interactions resulting in anisotropic non-linear shear induced diffusion (SID) [2].
Only a few experimental measurements [3] of SID coefficient are available in the literature since the diffusivity at thatscale is small. Rusconi and Stone [4] conducted experiments to show strong cross-stream shear induced diffusion forplatelet-like particles. Their experiments provided quantitative results for a wide range of shear rates. Grandchamp et al.
[5] developed a mathematical model for the transport of RBCs from experiments. Their results showed that thediffusion of RBCs are primarily in the shear and vorticity directions.
Understanding rheology is important in deriving appropriate constitutive models for blood. Apostolidis and Berris [6]proposed a hematocrit dependent stress-strain constitutive model to characterize the non-Newtonian behaviour of blood.
The study showed that the Casson constitutive model fits the best where the apparent viscosity is dependent on thehematocrit concentration. In the present work, a coupled RBC-momentum transport model for blood is proposed wherethe velocity and pressure fields are obtained from the Navier-Stokes equations and the RBC concentration is obtained
from the model developed by Grandchamp et al. [5]. The momentum equation is coupled to the RBC transport equationthrough a hematocrit dependent fluid viscosity model [6].

2. Mathematical model
Following [5], the governing equations for velocity, pressure and RBC concentration are given as:   u      u  u       pI          u 

 t


(1)

   u  = 0(2)




 u  y, z   Jy  Jz
t
x
yz

(3)

Equations (1) and (2) are the momentum and the continuity equations whereas Eq. (3) gives the RBC transport equation
where the fluxes Jy and Jz are given as [5]:  J y   R21  f 3uz2  f 2u 2y    f 2  f 3  u yuz  ; J z   R 2 1  f 3u 2y  f 2uz2    f 2  f 3  u yuz yz
z
y 



The constants f2 and f3 are dimensionless diffusivities in the shear and vorticity directions. Further, R is the radius of
RBC and Г is the magnitude of the strain rate tensor [5]. The hematocrit-dependent viscosity model is given as:

The 7th Asian Symposium on Computational Heat Transfer and Fluid Flow, Tokyo, Sep. 3-7, 20192  T eff   y  C  ; C  n p 1  2.0703    3.722  2   exp  7.0276 1  0   T  (4)

The constants T0=296.16 K and np=1.67×10-3 Pa-s.

3. Results and Discussion
Figure 1 shows the outcome of validation studies carried out in the present work. As a first step, the RBC concentration
profile given in [7] for the same geometry was extracted and was curve fitted to obtain a close match as shown in Fig.
1a. Next, this RBC concentration distribution was given as the initial condition and Eqs. (1) and (2) were solved
simultaneously with the RBC dependent viscosity model Eq. (4). Figure 1b shows that the computed velocity profile isin good agreement with the result obtained in [7] for blood flow inside a pipe of diameter 40 μm. The mass conserving
velocity field is shown to match well with [7].

Figure 1: Comparison of (a) RBC concentration profile and (b) velocity profile for a flow inside a tube of diameter
40 μm with the literature [7].

4. Conclusions
A mathematical model is proposed to solve a coupled nonlinear species-momentum transport model. Velocity andpressure fields are solved using the Navier-Stokes equations while the RBC transport equation is solved using themathematical model proposed by Grandchamp et al. [5]. Casson hematocrit dependent stress-strain constitutive modelis used to couple the momentum and RBC transport equations via the effective viscosity. For a prescribed hematocrit
distribution, the velocity profile is shown to match well with the literature for a flow through a circular tube. Further
simulations will consider pulsatile flow and non-circular geometries.

Acknowledgements
All simulations were carried out at the High Performance Computing facility of IIT Kanpur, India.

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