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TECHNICAL PAPERS: Fluids/Heat/Transport

# Effect of Gravity on Liquid Plug Transport Through an Airway Bifurcation Model

[+] Author and Article Information
Y. Zheng, J. C. Anderson, V. Suresh

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

J. B. Grotberg

Department of Biomedical Engineering,  University of Michigan, Ann Arbor, MI 48109grotberg@umich.edu

J Biomech Eng 127(5), 798-806 (Apr 20, 2005) (9 pages) doi:10.1115/1.1992529 History: Received December 17, 2004; Revised April 20, 2005

## Abstract

Many medical therapies require liquid plugs to be instilled into and delivered throughout the pulmonary airways. Improving these treatments requires a better understanding of how liquid distributes throughout these airways. In this study, gravitational and surface mechanisms determining the distribution of instilled liquids are examined experimentally using a bench-top model of a symmetrically bifurcating airway. A liquid plug was instilled into the parent tube and driven through the bifurcation by a syringe pump. The effect of gravity was adjusted by changing the roll angle $(ϕ)$ and pitch angle $(γ)$ of the bifurcation ($ϕ=γ=0deg$ was isogravitational). $ϕ$ determines the relative gravitational orientation of the two daughter tubes: when $ϕ≠0deg$, one daughter tube was lower (gravitationally favored) compared to the other. $γ$ determines the component of gravity acting along the axial direction of the parent tube: when $γ≠0deg$, a nonzero component of gravity acts along the axial direction of the parent tube. A splitting ratio $Rs$, is defined as the ratio of the liquid volume in the upper daughter to the lower just after plug splitting. We measured the splitting ratio, $Rs$, as a function of: the parent-tube capillary number $(Cap)$; the Bond number (Bo); $ϕ$; $γ$; and the presence of pre-existing plugs initially blocking either daughter tube. A critical capillary number $(Cac)$ was found to exist below which no liquid entered the upper daughter $(Rs=0)$, and above which $Rs$ increased and leveled off with $Cap$. $Cac$ increased while $Rs$ decreased with increasing $ϕ$, $γ$, and Bo for blocked and unblocked cases at a given $Cap>Cac$. Compared to the nonblockage cases, $Rs$ decreased (increased) at a given $Cap$ while $Cac$ increased (decreased) with an upper (lower) liquid blockage. More liquid entered the unblocked daughter with a blockage in one daughter tube, and this effect was larger with larger gravity effect. A simple theoretical model that predicts $Rs$ and $Cac$ is in qualitative agreement with the experiments over a wide range of parameters.

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## Figures

Figure 1

Schematic of the experimental setup. The roll angle, ϕ, and pitch angle, γ describe the orientation of the bifurcation plates with respect to gravity. The branch angle of the daughter tube with respect to the parent tube is indicated by θ.

Figure 2

(a) An image of the plug immediately after it has entered two daughter branches and split. (b) Image of an experiment with blockage in daughter A.

Figure 3

Rs vs Cap for γ=0deg and different ϕ using LB-400-X oil. ϕ=15deg: ◆ (experiments), — theory); ϕ=30deg: ∎¯ (experiments), – – – (theory); ϕ=60deg: ∣▴¯ (experiments), ⋯ (theory).

Figure 4

Rs vs Cap for for ϕ=30deg and different γ using LB-400-X oil. γ=−15deg: ◇ (experiments), — (theory); γ=0deg: ∣◆ (experiments), – – – (theory); γ=15deg: ∎ (experiments), ⋯ (theory); γ=30deg: ▴ (experiments), –--– (theory).

Figure 5

Rs vs Cap for γ=0deg and different blockage conditions using LB-400-X oil with a pitch angle γ=0deg and a roll angle of ϕ=15deg (panel a), ϕ=30deg (panel b), and ϕ=60deg (panel c). Blockage in lower daughter: ◆ (experiment), — (theory); No blockage: ∎ (experiment), – – – (theory); Blockage in upper daughter: ▴∣ (experiment), ⋯ (theory).

Figure 6

Rs vs Cap at ϕ=60deg, γ=0deg for different Bo. Glycerin Bo=0.78: ∎̱ (experiments), — (theory); LB-400-X Bo=1.26: ◆¯ (experiments), – – – (theory).

Figure 7

Schematic of plug flow in the bifurcation plates. a1, a2, and a3 are the radius of the parent, upper and lower daughter tube respectively. P1 is the pressure of the pumped air. P2 and P3 are atmospheric pressure. L1, L2, and L3 are the plug lengths in the parent, upper, and lower daughter tube, respectively. π1, π0, π2, and π3 are the pressures in the liquid plug at the rear interface, bifurcation zone, front meniscus of upper daughter, and front meniscus of lower daughter, respectively. Up, U2, and U3 are the plug velocities of rear meniscus in parent tube and front menisci in upper daughter and lower daughter tubes.

Figure 8

Experimental results of effect of parent plug volume on splitting ratio, Rs, and capillary number, Cap for LB-400-X oil at ϕ=30deg, γ=0deg ∎: L0=3cm(L0∕a1=15), ◆: L0=1.5cm(L0∕a1=7.5).

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