Viscous flow in a soft valve

Keunhwan Park, A. Tixier, A.H. Christensen, S. F. Arnbjerg-Nielsen, M. A. Zwieniecki, K. H. Jensen*

*Corresponding author for this work

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Abstract

Fluid-structure interactions are ubiquitous in nature and technology. However, the systems are often so complex that numerical simulations or ad hoc assumptions must be used to gain insight into the details of the complex interactions between the fluid and solid mechanics. In this paper, we present experiments and theory on viscous flow in a simple bioinspired soft valve which illustrate essential features of interactions between hydrodynamic and elastic forces at low Reynolds numbers. The set-up comprises a sphere connected to a spring located inside a tapering cylindrical channel. The spring is aligned with the central axis of the channel and a pressure drop is applied across the sphere, thus forcing the liquid through the narrow gap between the sphere and the channel walls. The sphere's equilibrium position is determined by a balance between spring and hydrodynamic forces. Since the gap thickness changes with the sphere's position, the system has a pressure-dependent hydraulic resistance. This leads to a nonlinear relation between applied pressure and flow rate: flow initially increases with pressure, but decreases when the pressure exceeds a certain critical value as the gap closes. To rationalize these observations, we propose a mathematical model that reduced the complexity of the flow to a two-dimensional lubrication approximation. A closed-form expression for the pressure drop/flow rate is obtained which reveals that the flow rate Q depends on the pressure drop Δp, sphere radius a, gap thickness h0, and viscosity η as Q ∼η-1 a1/2h05/2 (1 - Δppc)5/2Δp, where the critical pressure Δpc scales with spring constant k as Δpckh0a-2. These predictions compared favourably to the results of our experiments with no free parameters.
Original languageEnglish
Article numberR3
JournalJournal of Fluid Mechanics
Volume836
Number of pages11
ISSN0022-1120
DOIs
Publication statusPublished - 2018

Keywords

  • Flow-structure interactions
  • Lubrication theory
  • Microfluidics

Cite this

Park, K., Tixier, A., Christensen, A. H., Arnbjerg-Nielsen, S. F., Zwieniecki, M. A., & Jensen, K. H. (2018). Viscous flow in a soft valve. Journal of Fluid Mechanics, 836, [R3]. https://doi.org/10.1017/jfm.2017.805
Park, Keunhwan ; Tixier, A. ; Christensen, A.H. ; Arnbjerg-Nielsen, S. F. ; Zwieniecki, M. A. ; Jensen, K. H. / Viscous flow in a soft valve. In: Journal of Fluid Mechanics. 2018 ; Vol. 836.
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keywords = "Flow-structure interactions, Lubrication theory, Microfluidics",
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Park, K, Tixier, A, Christensen, AH, Arnbjerg-Nielsen, SF, Zwieniecki, MA & Jensen, KH 2018, 'Viscous flow in a soft valve', Journal of Fluid Mechanics, vol. 836, R3. https://doi.org/10.1017/jfm.2017.805

Viscous flow in a soft valve. / Park, Keunhwan; Tixier, A.; Christensen, A.H.; Arnbjerg-Nielsen, S. F.; Zwieniecki, M. A.; Jensen, K. H.

In: Journal of Fluid Mechanics, Vol. 836, R3, 2018.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - Viscous flow in a soft valve

AU - Park, Keunhwan

AU - Tixier, A.

AU - Christensen, A.H.

AU - Arnbjerg-Nielsen, S. F.

AU - Zwieniecki, M. A.

AU - Jensen, K. H.

PY - 2018

Y1 - 2018

N2 - Fluid-structure interactions are ubiquitous in nature and technology. However, the systems are often so complex that numerical simulations or ad hoc assumptions must be used to gain insight into the details of the complex interactions between the fluid and solid mechanics. In this paper, we present experiments and theory on viscous flow in a simple bioinspired soft valve which illustrate essential features of interactions between hydrodynamic and elastic forces at low Reynolds numbers. The set-up comprises a sphere connected to a spring located inside a tapering cylindrical channel. The spring is aligned with the central axis of the channel and a pressure drop is applied across the sphere, thus forcing the liquid through the narrow gap between the sphere and the channel walls. The sphere's equilibrium position is determined by a balance between spring and hydrodynamic forces. Since the gap thickness changes with the sphere's position, the system has a pressure-dependent hydraulic resistance. This leads to a nonlinear relation between applied pressure and flow rate: flow initially increases with pressure, but decreases when the pressure exceeds a certain critical value as the gap closes. To rationalize these observations, we propose a mathematical model that reduced the complexity of the flow to a two-dimensional lubrication approximation. A closed-form expression for the pressure drop/flow rate is obtained which reveals that the flow rate Q depends on the pressure drop Δp, sphere radius a, gap thickness h0, and viscosity η as Q ∼η-1 a1/2h05/2 (1 - Δp/Δpc)5/2Δp, where the critical pressure Δpc scales with spring constant k as Δpc ∼ kh0a-2. These predictions compared favourably to the results of our experiments with no free parameters.

AB - Fluid-structure interactions are ubiquitous in nature and technology. However, the systems are often so complex that numerical simulations or ad hoc assumptions must be used to gain insight into the details of the complex interactions between the fluid and solid mechanics. In this paper, we present experiments and theory on viscous flow in a simple bioinspired soft valve which illustrate essential features of interactions between hydrodynamic and elastic forces at low Reynolds numbers. The set-up comprises a sphere connected to a spring located inside a tapering cylindrical channel. The spring is aligned with the central axis of the channel and a pressure drop is applied across the sphere, thus forcing the liquid through the narrow gap between the sphere and the channel walls. The sphere's equilibrium position is determined by a balance between spring and hydrodynamic forces. Since the gap thickness changes with the sphere's position, the system has a pressure-dependent hydraulic resistance. This leads to a nonlinear relation between applied pressure and flow rate: flow initially increases with pressure, but decreases when the pressure exceeds a certain critical value as the gap closes. To rationalize these observations, we propose a mathematical model that reduced the complexity of the flow to a two-dimensional lubrication approximation. A closed-form expression for the pressure drop/flow rate is obtained which reveals that the flow rate Q depends on the pressure drop Δp, sphere radius a, gap thickness h0, and viscosity η as Q ∼η-1 a1/2h05/2 (1 - Δp/Δpc)5/2Δp, where the critical pressure Δpc scales with spring constant k as Δpc ∼ kh0a-2. These predictions compared favourably to the results of our experiments with no free parameters.

KW - Flow-structure interactions

KW - Lubrication theory

KW - Microfluidics

U2 - 10.1017/jfm.2017.805

DO - 10.1017/jfm.2017.805

M3 - Journal article

VL - 836

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

SN - 0022-1120

M1 - R3

ER -

Park K, Tixier A, Christensen AH, Arnbjerg-Nielsen SF, Zwieniecki MA, Jensen KH. Viscous flow in a soft valve. Journal of Fluid Mechanics. 2018;836. R3. https://doi.org/10.1017/jfm.2017.805