Page 211 - Computational Modeling in Biomedical Engineering and Medical Physics

200 Computational Modeling in Biomedical Engineering and Medical Physics

behavior and its cellular components (red blood cells mainly) indicate that different
biorheological models are recommendable [Shaw (2017)]: Casson models (Aroesty and
Gross, 1972) are relevant for the hemodynamic od microvessel of diameter
130 1300 μm, and the Herschel Bulkley model (Priyadharshini and Ponalagusamy,
2015) is adequate for the microvessel of radius 20 1000 μm(Misra et al., 1992; Pries
et al., 2000). However, it is appealing to approximate blood as a homogeneous,
Newtonian fluid, and its flow incompressible. The percentage of hematocrit (cells) is
B40% 50%, which gives a relative viscosity of 3 4, an equivalent kinematic viscosity
3
2
ν 5 0.032 cm /s, and density ρ 5 1.05 g/cm are, for instance, used by Hunter (1972).
For oscillatory flows with typically blunt velocity profiles the recommended the
profile
α 22α 1 ð R 1 ð R
r
2
u 5 U 1 2 α21 ; U 5 2rudr; α 5 2ru dr;
2 2 α R R 2 0 RU 2 0 ð6:18Þ
replaces the Hagen Poiseuille (parabolic) profile, which is characteristic for swirl-free
and axisymmetric Newtonian fluids. In Eq. (6.18) R is the radius of the circular vessel
(the characteristic length recommended by Barnard et al., 1966), u 5 u(r) the axial
velocity, and U the mean axial velocity is a constant for the particular profile assumed.
The value α 5 1.1 provides a velocity profile that matches satisfactorily empirical data.
A similar profile was proposed by Smith et al. (2002)

r
ξ
u 5 U 1 2 ; ð6:19Þ
R
to overcome the difficulty of actually solving the transport equations for a complex,
non-Newtonian fluid, for example, ξ 5 9(Nacev et al., 2011; Köstler, 2016).
However, this model is about the streamwise flow, essentially one dimensional. In
MDT, though, the action of the magnetic field may be orthogonal to it, and this con-
tributes to enhance (accelerate) or oppose (decelerate) it. Indirectly this streamwise
effect may result in recirculation cells hence some flow orthogonal to it. If the MD is
to cross the epithelial membrane, then the magnetic field force orthogonal to the
streamwise flow is of interest.
In our study we use a Carreau-type rheological model (Morega et al., 2013a,b,c;
Akbar and Nadeem, 2014)

h i ð n21Þ=2
η 5 η 1 η 2 η N 11 λγ Þ ; ð6:20Þ
0 2

N
0
ð
that provides for the blunt velocity profile specific to microcirculation. Here
η N 5 0.0032 Pa s is the infinite shear rate viscosity, η 0 5 45.6 Pa s is the zero shear
rate viscosity, γ is the shear rate tensor, λ 5 10 s, n 5 0.344 are model parameters.
0