Flow-network-controlled shape transformation of a thin membrane through differential fluid storage and surface expansion

The mechanical properties of a thin, planar material, perfused by an embedded flow network, have been suggested to be potentially changeable locally and globally by fluid transport and storage, which can result in both small- and large-scale deformations such as out-of-plane buckling. In these processes, 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 quasistatic 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.

Figure 1

Biological inspiration of flow-controlled shape transformation. In each panel, the changes proceed from left to right. (a) Flower blooming of Easter lily (Lilium longiflorum). Snapshots taken by a Canon EOS 1200D camera. (b) The flow of fluorescence dye in the xylem venation of plantain lily (Hosta sp.) petals in a flower that has already bloomed, opened for visualization. Images captured by a Nikon D3300 camera. See the Appendix for the experimental approach to create these fluorescent images. (c) Cartoon illustration of the effect of a flow network on petal deformation. As incoming liquid flux (yellow) fills up empty vessels (black), the originally flat surface (1) expands differentially and buckles out of the plane (2), and eventually resumes a flatter shape as a uniform saturation of the surface is being approached (3).

Figure 2

The hydrodynamic-mechanical coupling model, consisting of (a) a fluid-storing flow network and (b) an overlapping mechanical network, which models the shape changes. In (a), the same hierarchy composed of major (thicker) and minor veins is used to induce the shapes in Fig. 4. The specific hydraulic connectivity for the resistors and capacitors is shown between a node $i$ and one of its neighbors $n\left(i\right)$. Note that the vein segment highlighted with a red box in the network of (a) is in fact the wire linking pressures $P_i$ and $P_{n\left(i\right)}$ through a resistor $R_{n\left(i\right),i}$, while the underlying circuit for baseline pressure $P_s$ is not included in the diagram of the large network. At node $i, C_i$ is the local capacitor with voltage $V_i$ and associated resistor $R_i^{\left(c\right)}$. In (b), examples of nodes ($j$ and $k$), bonds, and faces ($\alpha$ and $\beta$), as well as their normals (${\mathit{n}}_{\alpha }$ and ${\mathit{n}}_{\beta }$) are shown. The positions of $j$ and $k$ in the mechanical network are the same as $n\left(i\right)$ and $i$ in the flow network, respectively. (c) Specific coupling methods and simulation procedures.

Figure 3

Numerical simulation results of time-dependent shape transformations induced by fluid expansion in a uniform flow network. Different resistance parameters $R$ (but identical capacitance $C$ with associated resistance $R_c$) are used for (a)–(c) to generate the 3D shapes at different time $t$, whose side views are shown below front views to illustrate possible buckling. With larger resistance, the fluid expansion becomes slower (indicated by larger time constant $\tau$) and the buckling lingers for a longer time.

Figure 4

Dynamic simulation results of shape transformations induced by flow networks with “forklike” hierarchies, whose hydraulic resistances of major ($R_{\text{maj}}$) and minor veins ($R_{\text{min}}$) are chosen (in addition to $R_c=1$ and $C=2$) so that the total cost of network is conserved. The 3D visualizations at different times $t$ illustrate the instant fluid content distribution (whose average spreading speed is characterized by time constant $\tau$) according to the same color bar of Fig. 3. Note that the 3D renderings, which are symmetric about their midlines, are at an angle with the paper to show the subsequent buckling from their original planes.

Figure 5

Dynamic simulation results (with time $t$) of shape transformations induced by flow networks with “leaflike” hierarchies. See the caption of Fig. 4 for detailed information of the parameters.

Figure 6

Dynamic simulation results (with time $t$) of shape transformations induced by flow networks with loopy hierarchies. See the caption of Fig. 4 for detailed information of the parameters.

Figure 7

Examples of correlation between local relative fluid content and Gaussian curvature over the surface obtained by linear regression at time $t=25$, for the two loopy hierarchical designs from Figs. 6 and 6, respectively. Both the linear regression equation and correlation coefficient $r$ are shown with each plot. (a) A small slope and low degree of correlation (small $r$ value), and (b) a large slope and high degree of correlation. On the right-hand side of each correlation plot, the 2D plots illustrate both the instant distribution of relative fluid content and that of the estimated Gaussian curvature of the obtained shape.

Figure 8

The correlation between local relative fluid content and Gaussian curvature over the surface, as represented by the linear regression results (slope, $y$ intercept, and correlation coefficient $r$) for each “forklike” hierarchy at each integer time point from $t=1$ to 50. On the right side, 2D plots of fluid content (RFC) and Gaussian curvature (GC) distributions are shown for several hierarchies at certain time points (corresponding to some shapes in Fig. 4). The symbols 4(a)–4(e) represent vein patterns with the same labels as in Fig. 4.

Figure 9

Linear regression results for each “leaflike” hierarchy, with 2D plots corresponding to some shapes in Fig. 5. The symbols 5(a)–5(d) represent vein patterns with the same labels as in Fig. 5. See the caption of Fig. 8 for more information on the parameters.

Figure 10

Linear regression results for each loopy hierarchy, with 2D plots corresponding to some shapes in Fig. 6. The symbols 6(a)–6(d) represent vein patterns with the same labels as in Fig. 6. See the caption of Fig. 8 for more information on the parameters.