Bending of elastic fibres in viscous flows: the influence of confinement

• Published in: Journal of fluid mechanics 2013-04, Vol.720, p.517-544
• Main Authors: Wexler, Jason S., Trinh, Philippe H., Berthet, Helene, Quennouz, Nawal, du Roure, Olivia, Huppert, Herbert E., Lindner, Anke, Stone, Howard A.
• Format: Article
• Language: English
• Subjects: Applied fluid mechanics; Boundary conditions; Exact sciences and technology; Fibers; Flow rates; Fluid dynamics; Fluidics; Fundamental areas of phenomenology (including applications); Low flow; Mathematical models; Physics; Viscosity
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/jfm.2013.49

We present a mathematical model and corresponding series of microfluidic experiments examining the flow of a viscous fluid past an elastic fibre in a three-dimensional channel. The fibre’s axis lies perpendicular to the direction of flow and its base is clamped to one wall of the channel; the sidewalls of the channel are close to the fibre, confining the flow. Experiments show that there is a linear relationship between deflection and flow rate for highly confined fibres at low flow rates, which inspires an asymptotic treatment of the problem in this regime. The three-dimensional problem is reduced to a two-dimensional model, consisting of Hele-Shaw flow past a barrier, with boundary conditions at the barrier that allow for the effects of flexibility and three-dimensional leakage. The analysis yields insight into the competing effects of flexion and leakage, and an analytical solution is derived for the leading-order pressure field corresponding to a slit that partially blocks a two-dimensional channel. The predictions of our model show favourable agreement with experimental results, allowing measurement of the fibre’s elasticity and the flow rate in the channel.