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TECHNICAL PAPERS: Fluids/Heat/Transport

Finite-Reynolds-Number Effects in Steady, Three-Dimensional Airway Reopening

[+] Author and Article Information
Andrew L. Hazel

School of Mathematics, University of Manchester, Manchester, UK

Matthias Heil

School of Mathematics, University of Manchester, Manchester, UKm.heil@maths.man.ac.uk

J Biomech Eng 128(4), 573-578 (Feb 02, 2006) (6 pages) doi:10.1115/1.2206203 History: Received September 12, 2005; Revised February 02, 2006

Motivated by the physiological problem of pulmonary airway reopening, we study the steady propagation of an air finger into a buckled elastic tube, initially filled with viscous fluid. The system is modeled using geometrically non-linear, Kirchhoff-Love shell theory, coupled to the free-surface Navier-Stokes equations. The resulting three-dimensional, fluid-structure-interaction problem is solved numerically by a fully coupled finite element method. Our study focuses on the effects of fluid inertia, which has been neglected in most previous studies. The importance of inertial forces is characterized by the ratio of the Reynolds and capillary numbers, ReCa, a material parameter. Fluid inertia has a significant effect on the system’s behavior, even at relatively small values of ReCa. In particular, compared to the case of zero Reynolds number, fluid inertia causes a significant increase in the pressure required to drive the air finger at a given speed.

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Copyright © 2006 by American Society of Mechanical Engineers
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Figures

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Figure 1

Bubble pressure versus capillary number in the absence of fluid inertia (zero Reynolds number) for generic system parameters (ν=0.49,h∕R=1∕20,σ=1,A∞=0.373; see Sec. 2 for parameter definitions). The dashed line is an asymptotic approximation for the behavior on the pushing branch. Inset figures illustrate tube and interface shapes on the two branches (adapted from (10)).

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Figure 2

(a) Bubble pressure versus capillary number for A∞=0.373, σ=1.0 and Re∕Ca=0,1,5,10. The markers show the results for Re∕Ca=10 on a refined mesh (83,000 degrees of freedom). They differ by less than 0.5% from the results at the standard resolution (43,000 degrees of freedom). The dashed line is the result from Hazel and Heil’s (10) asymptotic model for the pushing branch at zero Reynolds number. (b) The total viscous dissipation, Φ, versus capillary number for the same parameter values as in (a).

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Figure 3

Interface and wall shapes for Ca=4 and Re∕Ca=0 top, 10 (bottom). Contours of pressure in the planes x1=0 and x2=0 are also shown.

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Figure 5

Axial slices in the plane x1=0, showing the airway wall and the air-liquid interface for Re∕Ca=1 (solid lines) and Re∕Ca=0 (dashed lines). (a)A∞=0.373;(b)A∞=0.41; (c)A∞=0.45. In all three cases Ca=6 and σ=1.

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Figure 4

Bubble pressure versus dimensionless propagation speed for A∞=0.373, 0.41, 0.45 for σ=1 and (a)Re=0 and (b)Re∕Ca=1

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Figure 6

Transverse cross sections in the plane x3=4 for the same parameter values as in Fig. 5. Solid lines: Re∕Ca=1; dashed lines Re∕Ca=0. The outer lines represent the airway wall, the inner lines the air-liquid interface.

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