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Research Papers

Pore-Scale Modeling of Non-Newtonian Shear-Thinning Fluids in Blood Oxygenator Design

[+] Author and Article Information
Kenny W. Q. Low

Advanced Sustainable Manufacturing
Technologies (ASTUTE) Project,
College of Engineering,
Swansea University,
Bay Campus, Fabian Way,
Swansea SA1 8EN, UK
e-mail: k.w.q.low@swansea.ac.uk

Raoul van Loon

College of Engineering,
Swansea University,
Bay Campus, Fabian Way,
Swansea SA1 8EN, UK
e-mail: r.vanloon@swansea.ac.uk

Samuel A. Rolland

Advanced Sustainable Manufacturing
Technologies (ASTUTE) Project,
College of Engineering,
Swansea University,
Bay Campus, Fabian Way,
Swansea SA1 8EN, UK
e-mail: s.rolland@swansea.ac.uk

Johann Sienz

Advanced Sustainable Manufacturing
Technologies (ASTUTE) Project,
College of Engineering,
Swansea University,
Bay Campus, Fabian Way,
Swansea SA1 8EN, UK
e-mail: j.sienz@swansea.ac.uk

1Corresponding author.

Manuscript received October 9, 2015; final manuscript received January 26, 2016; published online March 9, 2016. Assoc. Editor: Tim David.

J Biomech Eng 138(5), 051001 (Mar 09, 2016) (14 pages) Paper No: BIO-15-1501; doi: 10.1115/1.4032801 History: Received October 09, 2015; Revised January 26, 2016

This paper reviews and further develops pore-scale computational flow modeling techniques used for creeping flow through orthotropic fiber bundles used in blood oxygenators. Porous model significantly reduces geometrical complexity by taking a homogenization approach to model the fiber bundles. This significantly simplifies meshing and can avoid large time-consuming simulations. Analytical relationships between permeability and porosity exist for Newtonian flow through regular arrangements of fibers and are commonly used in macroscale porous models by introducing a Darcy viscous term in the flow momentum equations. To this extent, verification of analytical Newtonian permeability–porosity relationships has been conducted for parallel and transverse flow through square and staggered arrangements of fibers. Similar procedures are then used to determine the permeability–porosity relationship for non-Newtonian blood. The results demonstrate that modeling non-Newtonian shear-thinning fluids in porous media can be performed via a generalized Darcy equation with a porous medium viscosity decomposed into a constant term and a directional expression through least squares fitting. This concept is then investigated for various non-Newtonian blood viscosity models. The proposed methodology is conducted with two different porous model approaches, homogeneous and heterogeneous, and validated against a high-fidelity model. The results of the heterogeneous porous model approach yield improved pressure and velocity distribution which highlights the importance of wall effects.

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Figures

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Fig. 1

(a) A graphical description of the fiber bundles in the oxygenators which can be represented by a unit cell of ordered (b) square and (c) staggered arrays of cylinders

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Fig. 2

Computational grid for ordered (a) square (porosity 0.22, 0.50, and 0.90) and (b) staggered unit cell (porosity 0.10, 0.50, and 0.90)

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Fig. 3

Comparison of nondimensional permeability for square and staggered configurations due to (a) transverse and (b) parallel flows against analytical results based on Re = 0.1. Numerical results are indicated as (○) and () for square and staggered configurations, respectively. Analytical expression abbreviations are indicated as (Sq) and (St) for square and staggered configurations, respectively.

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Fig. 4

Normalized velocity magnitude contours of various ε for transverse and parallel flows past square and staggered configurations: (a) ε = 0.22—transverse flow, (b) ε = 0.50—transverse flow, (c) ε = 0.90—transverse flow, (d) ε = 0.10—transverse flow, (e) ε = 0.50—transverse flow, (f) ε = 0.90—transverse flow, (g) ε = 0.22—parallel flow, (h) ε = 0.50—parallel flow, (i) ε = 0.90—parallel flow, (j) ε = 0.10—parallel flow, (k) ε = 0.50—parallel flow, and (l) ε = 0.90—parallel flow

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Fig. 5

(a) and (b) Comparison of normalized velocity profile at the narrowing for Newtonian and non-Newtonian fluids of various ε for transverse flow past square and staggered configurations: (a) ε = 0.90—square and staggered and (b) normalized actual difference. (c) Normalized actual difference relative to non-Newtonian results of velocity profiles for the lowest porosity, 0.5 and 0.9 of square and staggered configurations.

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Fig. 6

(a) Equivalent viscosity computation for square and staggered configurations with transverse and parallel flows. (b) Normalized absolute difference based on their respective parallel flow equivalent viscosities.

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Fig. 7

The components of μ̂ according to their respective flow direction (transverse and parallel flows) for square, (○) & (), and staggered, (△) & (◇), configurations. The data are fitted via least squares and expressed in solid lines with the equation and R2 correlation.

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Fig. 8

(a) Comparison of dynamic viscosity (μ) as a function of shear rate (γ˙) for various non-Newtonian shear-thinning blood viscosity models ((a) non-Newtonian models (see Table 3)). The components in μ̂ are subjected to their respective flow direction (transverse and parallel flows) for square, (○) & (), and staggered, (△) & (◇), configurations of (b) Carreau–Yasuda model, (c) cross model, and (d) modified Casson model. The data are fitted via least squares and expressed in solid lines with the equation and R2 correlation.

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Fig. 9

Porous medium viscosity (μpm) data points for square and staggered configurations plotted against equivalent shear rate (γ˙eq) of various non-Newtonian shear-thinning blood viscosity models. Solid line represents non-Newtonian viscosity models from Table 3.

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Fig. 10

(a) Oxygenator domain of interest being modeled in two ways: (b) one with the inclusion of physical fibers (high-fidelity model) and (c) without the presence of fibers (porous model)

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Fig. 11

Pressure contours of (a) high-fidelity model, (b) porous model 1, and (c) porous model 2 normalized by the maximum pressure of the high-fidelity model

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Fig. 12

(a)–(c) Flow monitoring positions in the device. (d)–(e) High-fidelity model, (g)–(i) porous model 1, and (j)–(l) porous model 2 velocity magnitude contours, normalized by their respective maximum magnitude and modeling approaches, at these flow monitoring positions. Note that the fibers in the porous model are for illustration purposes.

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Fig. 13

Relative error normalized by the area-weighted mass flow of the high-fidelity model at three monitoring positions for (a)–(c) porous model 1 and (d)–(f) porous model 2. Note that the fibers in the porous model are for illustration purposes.

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