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TECHNICAL PAPERS: Fluids/Heat/Transport

# Pulsatile Blood Flow and Oxygen Transport Past a Circular Cylinder

[+] Author and Article Information
Jennifer R. Zierenberg, Hideki Fujioka

Department of Biomedical Engineering, University of Michigan, Ann Arbor, MI 48109-2099

Ronald B. Hirschl, Robert H. Bartlett

Department of Surgery, University of Michigan Medical Center, Ann Arbor, MI 48109

James B. Grotberg1

Department of Biomedical Engineering, 2200 Bonisteel Boulevard, 1107 Gerstacker Bldg., University of Michigan, Ann Arbor, MI 48109-2099grotberg@umich.edu

1

Corresponding author.

J Biomech Eng 129(2), 202-215 (Apr 03, 2007) (14 pages) doi:10.1115/1.2485961 History: Received May 01, 2006; Revised September 05, 2006; Online April 03, 2007

## Abstract

The fundamental study of blood flow past a circular cylinder filled with an oxygen source is investigated as a building block for an artificial lung. The Casson constitutive equation is used to describe the shear-thinning and yield stress properties of blood. The presence of hemoglobin is also considered. Far from the cylinder, a pulsatile blood flow in the $x$ direction is prescribed, represented by a time periodic (sinusoidal) component superimposed on a steady velocity. The dimensionless parameters of interest for the characterization of the flow and transport are the steady Reynolds number (Re), Womersley parameter $(α)$, pulsation amplitude $(A)$, and the Schmidt number (Sc). The Hill equation is used to describe the saturation curve of hemoglobin with oxygen. Two different feed-gas mixtures were considered: pure $O2$ and air. The flow and concentration fields were computed for Re=5, 10, and 40, $0≤A≤0.75$, $α=0.25$, 0.4, and Schmidt number, Sc=1000. The Casson fluid properties result in reduced recirculations (when present) downstream of the cylinder as compared to a Newtonian fluid. These vortices oscillate in size and strength as $A$ and $α$ are varied. Hemoglobin enhances mass transport and is especially important for an air feed which is dominated by oxyhemoglobin dispersion near the cylinder. For a pure $O2$ feed, oxygen transport in the plasma dominates near the cylinder. Maximum oxygen transport is achieved by operating near steady flow (small $A$) for both feed-gas mixtures. The time averaged Sherwood number, $Sh̿$, is found to be largely influenced by the steady Reynolds number, increasing as Re increases and decreasing with $A$. Little change is observed with varying $α$ for the ranges investigated. The effect of pulsatility on $Sh̿$ is greater at larger Re. Increasing Re aids transport, but yields a higher cylinder drag force and shear stresses on the cylinder surface which are potentially undesirable.

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## Figures

Figure 1

Schematic of unsteady flow over and transport from a single cylinder

Figure 2

Schematic of computational grid with three distinct regions

Figure 3

Streamlines for steady flow over a cylinder: (a)Re=10, Newtonian; (b)Re=40, Newtonian; (c)Re=10, Casson; and (d)Re=40, Casson

Figure 4

Shear stress fields for steady flow over a cylinder: (a)Re=10, Casson; and (b)Re=40, Casson

Figure 5

Concentration fields in mL O2∕mL blood for a pure oxygen feed gas, PO2cyl′=713mmHg, (Sc=1000, PO2∞′=40mmHg) for steady flow over a cylinder: (a)Re=10, Newtonian blood plasma without hemoglobin; (b)Re=40, Newtonian blood plasma without hemoglobin; (c)Re=10, Casson blood with hemoglobin; (d)Re=40, Casson blood with hemoglobin

Figure 6

PO2/concentration boundary layer near cylinder at θ=π∕2 for steady flow over a cylinder (Sc=1000, PO2∞′=40mmHg): (a) for a pure oxygen feed gas, PO2cyl′=713mmHg and (b) for an air feed gas, PO2cyl′=156mmHg

Figure 7

Saturation boundary layer near cylinder at θ=π∕2 for steady flow over a cylinder (Sc=1000, PO2∞′=40mmHg): (a) for a pure oxygen feed gas, PO2cyl′=713mmHg and (b) for an air feed gas, PO2cyl′=156mmHg

Figure 8

Streamline (left column) and oxygen concentration (right column) fields in mL O2∕mL blood for a Casson fluid for Re=10, α=0.25, A=0.75, Sc=1000, PO2∞′=40mmHg, and pure oxygen feed gas, PO2cyl′=713mmHg, which are plotted every one-quarter cycle: (a)t=0; (b)t=π∕2; (c)t=π; and (d)t=3π∕2

Figure 9

Shear stress fields for a Casson fluid for Re=10, α=0.25, A=0.75, which are plotted every one-quarter cycle: (a)t=0; (b)t=π∕2; (c)t=π; and (d)t=3π∕2

Figure 10

The effect of pulsation amplitude on the local Sherwood number, Sh, for a Casson fluid for Re=10 and 40; α=0.25; Sc=1000; PO2∞′=40mmHg; and PO2cyl′=713mmHg: (a)Re=10, t=π∕2; (b)Re=10, t=3π∕2; (c)Re=40, t=π∕2; (d)Re=40, t=3π∕2. Note: θ=0 defines the rear stagnation point and θ=π defines the front stagnation point.

Figure 11

The effect of pulsation amplitude on the surface averaged Sherwood number, Sh¯, for a Casson fluid for α=0.25; Sc=1000; PO2∞′=40mmHg; and PO2cyl′=713mmHg: (a)Re=10; (b)Re=40

Figure 12

The effect of pulsation amplitude on the local Sherwood number, Sh, for a Casson fluid for Re=10 and 40; α=0.25; Sc=1000; PO2∞′=40mmHg; and PO2cyl′=156mmHg: (a)Re=10, t=π∕2; (b)Re=10, t=3π∕2; (c)Re=40, t=π∕2; (d)Re=40, t=3π∕2. Note: θ=0 defines the rear stagnation point and θ=π defines the front stagnation point.

Figure 13

The effect of pulsation amplitude on the surface averaged Sherwood number, Sh̿, for a Casson fluid for α=0.25; Sc=1000; PO2∞′=40mmHg; and PO2cyl′=156mmHg: (a)Re=10; and (b)Re=40

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