Viscous flow in a soft valve

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 setup 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 non-linear 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 ∆p, sphere radius a, gap thickness h 0 , and viscosity η as Q ∼ η −1 a 1/2 h 5/2 0 (∆p c − ∆p) 5/2 ∆p, where the critical pressure ∆p c scales with the spring constant k and sphere radius a as ∆p c ∼ ka −2 . These predictions compared favorably to the results of our experiments with no free parameters.

Solid mechanics of the torus–margo in conifer intertracheid bordered pits

“…Park et al . (2018) estimated that the pressure difference required to close the pit, for a pit based upon measurements from Capron et al . (2014), would be about 0.5 MPa.…”

“…Domec et al (2006) estimated that earlywood pits in Douglas-fir required pressure differences across the pit membrane from 0.02 to 0.47 MPa, although they suggested that pits in the roots required less pressure to seal. Park et al (2018) estimated that the pressure difference required to close the pit, for a pit based upon measurements from Capron et al (2014), would be about 0.5 MPa.…”

Viscous flow in a slit between two elastic plates

“…Another example is Ledesma-Alonso et al [14], who conducted experiments on valves consisting of two flexible plates, to study under which conditions they block backflow. Finally, Park et al [15] found a nonlinear pressure-drop-flow-rate relation across a valve comprising a sphere connected to a spring in a tapering channel.…”

Tunable flow asymmetry and flow rectification with bio-inspired soft leaflets

“…This flow asymmetry, coupled with self-regulated contractions of muscles surrounding the lymphatic channel [5], induces distributed pumping along the lymphatic channel. Although flow-response asymmetry has been described as a consequence of a nonlinear interaction between a laminar flow and deformable asymmetric leaflets [3,[9][10][11][12][13][14][15][16], our understanding of the relationship between the flow unidirectional response and the leaflets' geometry and mechanics remains limited to empirical models [3,17].…”

“…Here, inspired by the lymphatic system [3,5,17], and building on the field of fluid-structure interaction of flexible structures [3,[9][10][11][12][13][14][15][16], we establish how flow asymmetry (i) emerges from an interplay between geometric asymmetry and nonlinearity induced by a fluid-structure interaction, and (ii) can be used to induce flow rectification, thus passively converting an oscillatory flow into a net fluid transport. The paper is organized as follows.…”