  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 $\Delta p$, sphere radius $a$, gap
thickness $h_0$, and viscosity $\eta$ as $Q\sim \eta^{-1}
a^{1/2}h_0^{5/2}\left(\Delta p_c-\Delta p\right)^{5/2}\Delta p$, where the
critical pressure $\Delta p_c$ scales with the spring constant $k$ and sphere
radius $a$ as $\Delta p_c\sim k a^{-2}$. These predictions compared favorably
to the results of our experiments with no free parameters.
