Influence of Blood Rheology and Outflow Boundary Conditions in Numerical Simulations of Cerebral Aneurysms - Mathematical Methods and Models in Biomedicine - page 148

Tabl e 1 Some generalized Newtonian models for blood viscosity and correspond-
ing constants
Model

Viscosity model Model constants for blood

μ 0 =

. , μ ∞ = .

032
λ = 10 . 03 s , n = 0 . 344

0 456 0

˙ ˙

2

) ( n − 1 ) / 2

Carreau F ( γ )=( 1 +( λ γ )

μ 0 = 0 . 618 , μ ∞ = 0 . 034

m

) − 1

F ( γ )=(

˙ +( λ

γ )

˙

Cross 1

λ = 7 . 683 s , m = 0 . 810

+ ( + λ

γ )

˙

1 log 1 μ = 1 . 10 , μ ∞ = 0 . 035 0

Yeleswarapu F ( γ )= ˙

1 + λ γ ˙ λ = 45 . 23 s

2

λ − 2
2

1 +( λ 1 ˙ γ )

1

μ = 0 . 426 , μ ∞ = μ λ 0 0

Oldroyd μ ( γ )= μ ˙ 0

2

1 +( λ 2 ˙ γ ) λ = 1 . 09 s , λ = 3 . 349 s 1 2

⎧
⎨

ρ ∂

u

˙

t + ρ (

u · ∇ ) u + ∇ p − div ( 2 μ ( γ ) D ( u )) = 0 ,

∂

(8)

in Ω ,∀ t > 0 ,

⎩

div u = 0 .

A variety of non-Newtonian viscosity functions μ ( γ ) ˙ can be used, only differing

on the functional dependence of the viscosity μ on the shear rate ˙ γ . To model blood

flow, the focus is put on bounded viscosity functions of the form

μ ( γ )= μ ∞ +( μ 0 − μ ∞ ) ˙ F ( γ ) , ˙ (9)

where the constants μ 0 and μ ∞ are the asymptotic viscosities at zero, μ 0 =

lim γ → 0 μ ( γ ) ˙ , and infinity, μ ∞ = lim γ → ∞ μ ( γ ) ˙ , shear rate. F ( γ ) ˙ is a continuous and

monotonic function such that

lim
˙ 0 F

( γ )= ˙ 0 , lim

˙

F ( γ )= ˙ 1 . (10)

γ → γ → ∞

The definition of function F characterizes the generalized Newtonian model.

Tab le 1 was taken from [ 9 ] and shows several possible viscosity functions.
The values of the parameters there displayed, corresponding to an hematocrit
H t =

( γ ) ˙

37 ◦ C, were obtained from in vitro blood experimen-
tal data, as described in [ 9 ]. To set the parameters values, a nonlinear least squares
fitting was applied [ 9 , 13 ]. Notice that, with such parameters, all the viscosity
functions in Table 1 correspond to shear-thinning models.
Other generalized Newtonian models for blood viscosity, like the power-law
and the Carreau-Yasuda model, have been frequently used to describe blood flow
(for further details on these models, see [ 22 ]). In this work, following [ 9 , 13 ], the
Carreau viscosity function is used, with the parameters provided in Table 1 .

40 % and temperature T =