  The mechanical properties of a thin, planar material, perfused by an embedded
flow network, can be changed locally and globally by the fluid transport and
storage, resulting in small or large-scale deformation, such as out-of-plane
buckling. Fluid absorption and storage eventually cause the material to locally
swell. Different parts can hydrate and swell unevenly, prompting a differential
expansion of the surface. In order to computationally study the hydraulically
induced differential swelling and buckling of such a membrane, we develop a
network model that describes both the membrane shape and fluid movement,
coupling mechanics with hydrodynamics. We simulate the time-dependent fluid
distribution in the flow network based on a spatially explicit resistor network
model with local fluid-storage capacitance. The shape of the surface is modeled
by a spring network produced by a tethered mesh discretization, in which local
bond rest lengths are adjusted instantaneously according to associated local
fluid content in the capacitors in a quasi-static way. We investigate the
effects of various designs of the flow network, including overall hydraulic
traits (resistance and capacitance) and hierarchical architecture (arrangement
of major and minor veins), on the specific dynamics of membrane shape
transformation. To quantify these effects, we explore the correlation between
local Gaussian curvature and relative stored fluid content in each hierarchy by
using linear regression, which reveals that stronger correlations could be
induced by less densely connected major veins. This flow-controlled mechanism
of shape transformation was inspired by the blooming of flowers through the
unfolding of petals. It can potentially offer insights for other reversible
motions observed in plants induced by differential turgor and water transport
through the xylem vessels, as well as engineering applications.
