3.1. Flow distribution

First, we justify the process of selecting the circular kirigami sheet described in § 2 as an experimental model. Although not included in this paper, preliminary experiments were conducted with a rectangular ribbon-type kirigami sheet with transverse cutting lines, the configuration adopted by Marzin et al. (Reference Marzin, Le Hay, de Langre and Ramananarivo2022). These experiments at high Reynolds number with water as the working fluid revealed that when the fluid force exceeds a certain threshold, the sheet undergoes complex three-dimensional deformation to resemble a saddle shape. With this model, however, it is difficult to quantify deformation and porosity. The porosity of the sheet is determined by transforming the cutting lines in the initial flat sheet into the holes of a complex, three-dimensional curved surface. Furthermore, due to the geometrical complexity of the structure, force balance analysis is also challenging. For these reasons, we adopted the circular kirigami sheet described by Carleton & Modarres-Sadeghi (Reference Carleton and Modarres-Sadeghi2024). Its initial two-dimensional planar shape satisfies both point and line symmetry, and its three-dimensional deformed shape retains axisymmetry as it undergoes steady translation in the quiescent fluid (figure 2). This axisymmetric geometry makes the quantification of kirigami sheet deformation easier and enables the estimation of porosity from the surface area and its variation.

Figure 2.

Raw images of the reconfigured kirigami sheet of diameter $D$ = 100 mm during steady translation at four different translation speeds $U$ : (a) $U$ = 5.0 cm s $^{-1}$ ; (b) $U$ = 10.0 cm s $^{-1}$ ; (c) $U$ = 15.0 cm s $^{-1}$ ; (d) $U$ = 20.0 cm s $^{-1}$ (sheet thickness $h$ = 0.4 mm, cutting ratio $\gamma$ = 5.0).

For a specific kirigami sheet with the fixed sheet thickness $h$ and cutting ratio $\gamma$ , the effect of the translation speed $U$ on quasi-static deformation is briefly reported for the cases of low Reynolds number (with glycerine as the working fluid). At the lowest translation speed tested in this study ( $U$ = 5.0 cm s $^{-1}$ ), a relatively small deformation is observed in the quasi-steady state (figure 2 a). Even though this deformation is the smallest among the tested cases, the cutting line at the edge of the circular sheet clearly transitions into a hole, with a three-dimensional, axisymmetric curved surface. The extent of this deformation becomes greater as the translation speed increases. The holes of the sheet continue to grow as the translation speed $U$ increases further (figure 2 b–d). This pronounced axisymmetric deformation in the quasi-steady state is clearly distinct from those reported in previous studies. Compared with the work of Carleton & Modarres-Sadeghi (Reference Carleton and Modarres-Sadeghi2024), the present circular sheet also exhibits axisymmetric deformation owing to the clamping of the outer edge by a rigid frame. However, since their working fluid was water, the Reynolds number range was significantly higher than in the present study, producing some oscillations that prevented the sheet from maintaining a truly quasi-static configuration. Lamoureux et al. (Reference Lamoureux, Fillion, Ramananarivo, Gosselin and Melancon2025a ) enforced a boundary condition that fixed the centre of the circular kirigami sheet, under which the kirigami sheet could exhibit non-axisymmetric deformation due to much higher Reynolds number.

To clarify the relationship between boundary conditions and the reconfiguration of the circular kirigami sheet, additional experiments were conducted using a rigid frame that fixed only the centre of the sheet with its outer edge free. According to the experiments, the boundary condition has a substantial influence on the reconfiguration behaviour. For relatively thin sheets (i.e. with low bending stiffness), large deformation leads to a breakdown of axisymmetry. In contrast, for thicker sheets (i.e. with high bending stiffness), cutting lines do not reconfigure into holes; instead, the structure deforms in a manner analogous to the bending of an elastic circular disk interacting with a high-Reynolds-number flow (Schouveiler & Eloy Reference Schouveiler and Eloy2013; Lamoureux et al. Reference Lamoureux, Fillion, Ramananarivo, Gosselin and Melancon2025a ). These findings indicate that the axisymmetric deformation observed in the present study is governed primarily by the edge-clamped boundary condition, rather than by the Reynolds-number regime.

The velocity field around the reconfigured kirigami sheet in the quasi-steady state provides important insights into the deformation and drag generation of the sheet at low Reynolds number. For flow visualisation on the $xy$ -plane passing through the centre of the sheet, two cases were considered as representatives: the small-deflection case ([ $U$ , $h$ , $\gamma$ ] = [15.0 cm s $^{-1}$ , 0.6 mm, 3.5]) and the large-deflection case ([ $U$ , $h$ , $\gamma$ ] = [15.0 cm s $^{-1}$ , 0.3 mm, 8.0]). In addition to the kirigami sheet, a solid shell structure with zero porosity was used as a reference model. Based on the quasi-static deformation profile of the kirigami sheet, a corresponding solid shell model was approximated to be an ideal truncated cone and fabricated using a three-dimensional (3-D) printer (ProJet MJP 2500, 3D SYSTEMS Inc.); the approximation of the deformed shape as the truncated cone is detailed in § 3.2. In the experiments, the fluid in the tank was initially stationary, while the rigid frame to which the kirigami sheet was attached was mechanically translated. For convenience of analysis, the coordinate system is fixed to the rigid frame hereafter, and in this coordinate system, the uniform free stream of velocity $U$ flows along the positive $x$ -axis.

We first examine the flow distribution for the small-deflection case with relatively small deformation (figure 3). In the low-Reynolds-number regime ( $ \textit{Re}_D$ = 12), because of strong viscous diffusion and the absence of flow separation at the edge of the solid shell model, the $x$ -directional velocity relative to the rigid frame, $u_{\textit{rel}}/U$ , is positive behind the solid shell, roughly ranging from 0.2 to 0.4 (figure 3 ai). When the wakes at low $ \textit{Re}_D$ are compared between the solid shell and the kirigami sheet, no significant difference is identified (figure 3 aii). This similarity can be attributed to relatively low porosity and viscosity-induced hydrodynamic blockage within the holes of the kirigami sheet. In contrast, in the high-Reynolds-number regime ( $ \textit{Re}_{D}$ = 15 000), a region where $u_{\textit{rel}}/U$ has negative values below $-0.2$ is widely distributed downstream of the solid shell model (white region in figure 3 bi). Even at high $ \textit{Re}_D$ , the $x$ -velocity distributions of the solid shell and the kirigami sheet appear to be similar (figure 3 bii). In the small-deflection case, the structural deformation of the kirigami sheet is relatively small, resulting in low porosity ( $\phi =0.12$ ); the definition of porosity is provided in § 3.2. When the degree of deformation and the porosity are sufficiently low, the $x$ -directional flow is effectively blocked by the kirigami sheet, detouring along the $y$ -direction. Thus, the distribution of $u_{\textit{rel}}/U$ at high $ \textit{Re}_D$ does not show a significant difference from that of the solid shell model, although the viscosity-induced hydrodynamic blockage is absent.

Figure 3.

Superimposed images of the model and the normalised relative $x$ -velocity field $u_{\textit{rel}}/U$ for the small-deflection case of [ $U$ , $h$ , $\gamma$ ] = [15.0 cm s $^{-1}$ , 0.6 mm, 3.5]: (a) $ \textit{Re}_{D}$ = 12 and (b) $ \textit{Re}_{D}$ = 15 000. Panels (i) and (ii) present the results of the corresponding solid shell and the kirigami sheet, respectively. A reference frame is fixed with the kirigami sheet.

Figure 4.

Superimposed images of the model and the normalised relative $x$ -velocity field $u_{\textit{rel}}/U$ for the large-deflection case of [ $U$ , $h$ , $\gamma$ ] = [15.0 cm s $^{-1}$ , 0.3 mm, 8.0]: (a) $ \textit{Re}_{D}$ = 12 and (b) $ \textit{Re}_{D}$ = 15 000. Panels (i) and (ii) present the results of the corresponding solid shell and the kirigami sheet, respectively. A reference frame is fixed with the kirigami sheet.

Figure 5.

Superimposed images of the model and the normalised relative $y$ -velocity field $v_{\textit{rel}}/U$ : (a) $ \textit{Re}_{D}$ = 12 and (b) $ \textit{Re}_{D}$ = 15 000. Panels (i) and (ii) are for the small-deflection case ([ $U$ , $h$ , $\gamma$ ] = [15.0 cm s $^{-1}$ , 0.6 mm, 3.5]) and the large-deflection case ([ $U$ , $h$ , $\gamma$ ] = [15.0 cm s $^{-1}$ , 0.3 mm, 8.0]), respectively.

The effect of hydrodynamic blockage on the velocity field is more pronounced in the large-deflection case, with relatively large deformation and an increased hole size. Despite a notable hole size within the kirigami sheet and high porosity ( $\phi =0.52$ ), the overall flow distribution is similar to that of the solid shell model at low $ \textit{Re}_D$ (figure 4 a). This result is consistent with the hydrodynamic blockage effect reported in previous studies using highly porous models (Lee & Kim Reference Lee and Kim2017; Lee et al. Reference Lee, Lee and Kim2020a ,Reference Lee, Lee and Kim b , Reference Lee, Mahravan and Kim2024). The notion of hydrodynamic blockage does not refer to the flow being physically obstructed by a solid bluff structure. Rather, for a porous structure, the overlap of thick shear layers in the low-Reynolds-number regime can give rise to a virtual fluid barrier even when holes or gaps are present, which makes the structure hydrodynamically opaque. Through this hydrodynamic blockage, the structure regulates its overall flow permeability, thereby preventing the fluid from penetrating the gaps and, instead, forcing it to bypass them. However, at high $ \textit{Re}_D$ , the incoming flow penetrates the holes of the kirigami sheet, producing localised regions of strong $x$ -velocity $u_{\textit{rel}}/U$ near the holes. Thus, the velocity distribution differs significantly from that of the solid shell model (figure 4 b).

The dependence of the $y$ -velocity distribution on the Reynolds number is examined in addition to the $x$ -velocity. At $ \textit{Re}_D$ = 12, the normalised $y$ -velocity $v_{\textit{rel}}/U$ near the holes ranges roughly from $-0.2$ to 0.2 for both the small-deflection case and the large-deflection case, demonstrating the effective hydrodynamic blockage (figure 5 a). By contrast, at $ \textit{Re}_D$  = 15 000, regions of high $y$ -velocity emerge in the vicinity of the holes, having upward velocity in the $y\gt 0$ domain and downward velocity in the $y\lt 0$ domain (figure 5 b). The magnitude of the $y$ -velocity varies with the extent of sheet deformation. For the case of smaller deformation, the flow velocity through the holes is higher due to the reduced hole size, considering that the total flow rate through the entire sheet remains similar at high $ \textit{Re}_D$ regardless of the extent of deformation. Accordingly, the magnitude of $v_{\textit{rel}}/U$ around the holes in the large-deflection case (figure 5 bii) is generally smaller than that in the small-deflection case (figure 5 bi). Even when the kirigami sheet reaches quasi-static deformation, the corresponding wake flow can be unsteady at high $ \textit{Re}_D$ (Carleton & Modarres-Sadeghi Reference Carleton and Modarres-Sadeghi2024). Nevertheless, the instantaneous $x$ - and $y$ -velocity fields presented herein are sufficient to highlight a distinct difference in flow patterns between low- and high-Reynolds-number regimes for the kirigami sheet.

Figure 6.

Schematic illustrating the definition of inlet flow rate $Q_{\textit{in}}$ (calculated based on the blue arrows) and outlet flow rate $Q_{\textit{out}}$ (calculated based on the red arrows).

In addition to the qualitative description of the velocity distribution, the hydrodynamic blockage must be quantified. Recently, Lee & Kim (Reference Lee and Kim2024) proposed a quantitative metric to characterise hydrodynamic blockage by appropriately defining the flow rate passing through a simple cylinder-array model. Following this approach, we obtain the flow rate penetrating the deformed kirigami sheet from the velocity field data. On the $xy$ -plane, the cross-section of the deformed sheet can be approximated as an isosceles trapezoid (figure 6). To quantify the extent to which the fluid leaks through the holes of the deformed kirigami sheet, we define the outlet flow rate $Q_{\textit{out}}$ as

(3.1) \begin{align} Q_{\textit{out}}=\int \boldsymbol{n} \boldsymbol{\cdot }\boldsymbol{u}_{\textit{rel}}\,{\rm d}l, \end{align}

where $\boldsymbol{u}_{\textit{rel}}$ is the relative flow velocity vector, $l$ is the line for the integral and $\boldsymbol{n}$ is the outward unit vector perpendicular to $l$ (figure 6). The offset between the isosceles-trapezoid approximation of the sheet (solid line in the figure) and the integral line $l$ (dashed line) is set to $0.1D$ . Actually, it is more reasonable to estimate a three-dimensional volumetric flow rate by revolving the two-dimensional planar PIV data and assuming flow axisymmetry. However, this assumption becomes invalid in the high-Reynolds-number regime due to wake unsteadiness (figure 4). Therefore, we simply evaluate the outflow $Q_{\textit{out}}$ using a two-dimensional line integral on the $xy$ -plane. As the primary purpose of this quantification is not to acquire the exact volumetric flow rate, but rather to evaluate the hydrodynamic blockage effect, this approach provides a consistent metric for relative comparison across all Reynolds-number conditions.

A flow blockage ratio is defined as $\eta = 1 - Q_{\textit{out}}/Q_{\textit{in}}$ , where the inlet flow rate, $Q_{\textit{in}} = \textit{UD}$ is the flow rate of the free stream velocity on the vertical line of length $D$ (figure 6). Here, $\eta = 1$ indicates complete blockage of the flow, whereas $\eta = 0$ corresponds to perfect leakage. Because the wake behind the kirigami sheet is not completely steady, $\eta$ is also expected to vary over time. Therefore, $\eta$ is sampled from five consecutive flow-field datasets at a time interval of $\Delta t = 0.5$ s. For the large-deflection case with high porosity, the mean of $\eta$ is 0.29 for $ \textit{Re}_D = 15\,000$ and 0.83 for $ \textit{Re}_D = 12$ . Despite the substantial deformation and significant hole size, $\eta$ remains notably high at low Reynolds number, highlighting the dominance of hydrodynamic blockage.

The dominance of hydrodynamic blockage can also be explained by comparing the thickness of the local shear layer around the sheet segment (segment between adjacent cutting lines) with the characteristic hole size of the reconfigured kirigami sheet. To scale the local shear layer thickness (viscous diffusion length), first the Reynolds number $ \textit{Re}_d$ based on the distance between adjacent cutting lines, $d$ , should be considered. In the low-Reynolds-number regime of our study, $ \textit{Re}_d$ is between 0.20 and 0.75. Because the shear layer thickness around the sheet segment scales as $\delta _{sl} \sim d/Re_d$ in this $ \textit{Re}_d$ range (Proudman & Pearson Reference Proudman and Pearson1957; Lee et al. Reference Lee, Lee and Kim2020b ), $\delta _{sl}$ is estimated to be $O(d)$ , which is comparable to the hole size, indicating effective flow blockage within the holes by strong viscous diffusion. Conversely, in the high-Reynolds-number regime ( $ \textit{Re}_d = 250$ –1000) where the shear layer thickness scales as $\delta _{sl} \sim d/Re_d^{1/2}$ , it is $O(10^{-2}d)$ , which is much smaller than the hole size. Thus, the flow can easily pass through the holes.

3.2. Deflection and porosity

To characterise the deformation of the kirigami sheet in viscosity-dominant conditions, a new dimensionless parameter is proposed based on the force balance. In studies of fluid–structure interactions, a governing dimensionless parameter is typically obtained by balancing the fluid force imposed on the structure with the elastic force within the structure. At high Reynolds number, the fluid force scales with the dynamic pressure of the surrounding flow, $\rho _{\kern-1pt f}U^2$ . Accordingly, the interactions of a flow and an elastic structure have been investigated in terms of the Cauchy number $C_y = \rho _{\kern-1pt f} U^{2}/E$ , where $E$ is the Young’s modulus of the structure (i.e. Gosselin et al. Reference Gosselin, de Langre and Machado-Almeida2010; Gosselin & de Langre Reference Gosselin and de Langre2011; Lamoureux et al. Reference Lamoureux, Fillion, Ramananarivo, Gosselin and Melancon2025a , Reference Lamoureux, Ramananarivo, Melancon and Gosselinb ) or its variations, such as the dimensionless velocity that represents the ratio of the fluid inertia force and the bending force (i.e. Alben, Shelley & Zhang Reference Alben, Shelley and Zhang2002; Kim et al. Reference Kim, Cossé, Cerdeira and Gharib2013; Jung, Song & Kim Reference Jung, Song and Kim2021; Lee et al. Reference Lee, Kim and Kim2022, Reference Lee, Jung, Lee and Kim2023).

However, at low Reynolds number, the viscous force dominates over the inertial force, and the scaling of the fluid force should be appropriately modified. For our kirigami sheet in the low-Reynolds-number regime, the holes within the sheet are hydrodynamically blocked and the flow distribution is similar to that of the solid shell model with the same diameter $D$ ; see figures 3(a) and 4(a). Thus, rather than the internal hole size or the cutting line spacing $d$ , the diameter of the sheet, $D$ , is a more appropriate length when scaling the viscous stress acting on it at low Reynolds number; $\tau \sim \mu U / D$ . To scale the viscous fluid force acting on the kirigami sheet, it is necessary to determine the effective area. Although the deformed kirigami sheet is not flat, the projected frontal area can be used to represent the effective area, without considering the porosity of the sheet, because the holes within the sheet are blocked hydrodynamically. Although the holes are not completely blocked, porosity is neglected in calculating the effective area, regardless of the degree of structural deformation. The effective area $A_{\textit{eff}}$ then scales as $D^{2}$ without considering porosity, and the viscous fluid force scales as

(3.2) \begin{align} F_{\kern-1pt f} \sim \mu \textit{UD}. \end{align}

The viscous fluid force should be balanced by the bending force of the kirigami sheet. For kirigami structures, the presence of cutting lines introduces geometric discontinuity that reduces the effective bending stiffness $B_{\textit{eff}}$ . Hence, the scaling of the bending force must account for the cutting ratio. Although the detailed geometry of the kirigami sheet may vary, the previous study on the interaction of fluids and kirigami structures introduced geometric parameters to capture variations in the effective bending stiffness arising from the cutting pattern (Marzin et al. Reference Marzin, Le Hay, de Langre and Ramananarivo2022). As described in § 2, the kirigami sheet was designed using a single design parameter, cutting ratio $\gamma$ . However, in general, multiple parameters may be employed in the design of a circular kirigami sheet (Lamoureux et al. Reference Lamoureux, Fillion, Ramananarivo, Gosselin and Melancon2025a ). In addition to the cutting ratio $\gamma$ , one may consider the number of sectors $N$ , by which the circular sheet is divided into unit segments, and the distance between adjacent cutting lines $d$ , which is normalised by the sheet diameter $D$ (distance ratio $d^{*} = d/D$ ).

In the present study, we assume that the effective bending stiffness $B_{\textit{eff}}=B_{\textit{eff}}(E, h, \gamma , N, d^{*})$ can be expressed as the product of the bending stiffness of a flat circular sheet (uncut sheet) and an empirical function $f(\gamma , N, d^{*})$ , where $\gamma$ , $N$ and $d^{*}$ denote the cutting ratio, number of sectors and distance ratio, respectively. The bending force then scales as $F_{b} \sim f(\gamma , N, d^{*}) E h^{3}/D$ . Based on mechanical loading tests of circular kirigami sheets reported by Tani et al. (Reference Tani, Hong, Tomizawa, Lepoivre, Bico and Roman2024), the relationship between point loading and the resulting deflection was examined for our model, and an empirical function of the form $f(\gamma , N, d^{*}) = C N^4 d^{*2} / \gamma$ was derived, where $C$ is a proportionality constant; see Appendix A for the details of the empirical function. Substituting this relation into the scaling of the bending force yields

(3.3) \begin{align} F_{b} \sim Eh^3 N^4 d^{*2} /\gamma D. \end{align}

From the ratio of the viscous fluid force (3.2) to the bending force (3.3), the dimensionless velocity $U^*$ is given as

(3.4) \begin{align} U^{*} =\mu \textit{UD}^{2} \gamma / E h ^{3} N^{4} d^{*2}. \end{align}

This dimensionless velocity has a form similar to that of the dimensionless parameters proposed in prior studies on the flow-induced deformation of an elastic structure in a highly viscous flow (Pezzulla et al. Reference Pezzulla, Strong, Gallaire and Reis2020; Luna Lin et al. Reference Luna Lin, Pezzulla and Reis2023). The key distinction here is that the design parameters of the circular kirigami (cutting ratio, number of sectors, distance ratio) are explicitly embedded in the formulation.

We quantitatively analyse the quasi-static deformation of the kirigami sheet in the low-Reynolds-number regime. Although the actual deformation of the sheet is geometrically complex, the deformed surface can be simply approximated as the surface of a truncated cone owing to the axisymmetry of the deformed surface. The surface of a truncated cone is represented by the variables shown in figure 7. The displacement of the sheet in the direction opposite to the translation motion, relative to the rigid frame, is defined as $\delta$ , which corresponds to the height of the truncated cone. For a given kirigami sheet design, the magnitude of the streamwise deflection $\delta$ increases almost linearly with the translation speed $U$ (figure 8). This result matches the scaling of the viscous fluid force, $F_{\kern-1pt f} \sim \mu \textit{UD}$ (3.2), where the imposed fluid force is proportional to the translation speed. For a given cutting ratio $\gamma$ , $\delta$ decreases as the sheet thickness $h$ increases because of the higher bending stiffness (figure 8 a). For a given $h$ , the increase in $\gamma$ leads to a greater $\delta$ due to the reduction in the effective bending stiffness caused by the cutting lines (figure 8 b); note that the bending force scales inversely with the cutting ratio, $F_{b} \sim Eh^3 N^4 d^{*2}/\gamma D$ (3.3).

Figure 7.

(a) Raw image of the reconfigured kirigami sheet with diameter $D$ = 100 mm and (b) approximation of the kirigami sheet as a truncated cone.

Figure 8.

(a) Streamwise deflection $\delta$ versus translation speed $U$ for (a) different sheet thicknesses $h$ and a fixed cutting ratio $\gamma$ = 8.0, and for (b) different $\gamma$ and a fixed $h$ = 0.4 mm.

The undeformed reference area of the sheet is $S_0 = \pi D^2/4$ . After flow-induced reconfiguration, the lateral surface area of the truncated cone is expressed as $A_1 = \pi (D+a)[(D-a)^2/4 + \delta ^2]^{1/2}/2$ and its right surface area (truncated area) is given as $A_2 = \pi a^2/4$ (figure 7). The total area of the deformed surface is $S = A_1 + A_2$ . The difference between the deformed and initial surface areas can be approximated as the total area of the holes formed on the deformed sheet: $G = S - S_0$ . The porosity $\phi$ of the sheet is then defined by normalising the hole area $G$ using the deformed surface area $S$ : $\phi = G/S = 1 - S_0/S$ . Consequently, the porosity is obtained from the initial diameter $D$ and the variables $\delta$ and $a$ describing the deformed shape:

(3.5) \begin{align} \phi = 1-\frac {D^2}{a^2 + (D+a)[(D-a)^2+4\delta ^2]^{1/2}}. \end{align}

In previous studies on fluid–structure interactions involving elastic porous structures, the porosity of the structure was defined using the geometric parameters of the initial undeformed configuration, and thus it was independent of flow conditions and the mechanical properties of the structure (Guttag et al. Reference Guttag, Karimi, Falcón and Reis2018; Jin et al. Reference Jin, Kim, Cheng, Barry and Chamorro2020; Lee et al. Reference Lee, Mahravan and Kim2024). Because the porous state did not change dramatically during the reconfiguration process, the porosity based on the initial geometry was sufficient. In contrast, for the kirigami sheet, the porosity of the initial geometry is zero and the porosity is dependent on the reconfiguration.

To examine the combined effects of the flow conditions and effective bending stiffness on the streamwise deflection and porosity, the dimensionless velocity $U^*$ introduced in (3.4) is employed. This dimensionless parameter incorporates all key variables ( $\mu , U, D, \gamma , N, d, E, h$ ) that determine the reconfiguration. When the normalised streamwise deflection $\delta /D$ (figure 9 a) and the reconfiguration-dependent porosity $\phi$ (figure 9 b), which is calculated based on (3.5), are plotted with respect to $U^{*}$ , the data obtained from all the combinations of $U$ , $h$ and $\gamma$ considered in this study tend to collapse onto a single curve. When $U^*$ is low ( $U^*\lt 110$ ), $\delta /D$ and $\phi$ increase almost linearly with $U^*$ , but the slopes of $\delta /D$ and $\phi$ begin to decline near $U^{*} = 110$ , having a saturated value of roughly $\phi = 0.45$ beyond $U^{*} = 310$ .

Figure 9.

(a) Normalised streamwise deflection $\delta /D$ and (b) reconfiguration-dependent porosity $\phi$ versus dimensionless velocity $U^*$ .

The data in figure 9 do not perfectly collapse onto a single curve. This result can be explained from two perspectives. First, there is a limitation of the function $f(\gamma ,N,d^*)$ . This function is empirically determined from deflection measurements under a point load applied at the centre of the kirigami sheet. However, the kirigami sheet experiences fluid force over its entire surface, including the central region, so the function may not fully account for distributed hydrodynamic loading. Second, the simple force scaling described earlier is fundamentally based on the assumption that the structural deformation is sufficiently small; i.e. the system has a linear response. However, the deformation of the kirigami sheet can be large and highly nonlinear. Once $U^{*}$ exceeds a certain threshold, the data exhibit deviations due to this nonlinearity. Here, linear and nonlinear responses are simply from the kinematic standpoint based on deflection and porosity, rather than a fluid-dynamic standpoint. For all cases considered in this study, the flow is not in the linear Stokes regime and fluid force is not proportional to translation speed $U$ .

In figure 9, the number of sectors, $N$ , and the distance ratio, $d^{*}$ , are fixed at $N=4$ and $d^{*}=0.05$ , respectively. Therefore, it cannot be verified from the figure that the dimensionless velocity $U^*$ successfully characterises the changes in the sheet kinematics when $N$ and $d^*$ vary. To resolve this issue, we fabricated additional kirigami sheets and performed supplementary experiments to obtain the data of $\delta /D$ and $\phi$ for different values of $N$ and $d^*$ . These additional data match well with the original data of figure 9 in the small-deflection regime, although some deviations exist in the large-deflection regime; see figure 13 and Appendix B for details.

Instead of $U^*$ based on the viscous fluid force, we may consider an alternative dimensionless parameter based on the inertial fluid force that scales with dynamic pressure, $F_{\kern-1pt f} \sim \rho _{\kern-1pt f}U^2D^2$ , commonly adopted for the fluid force scaling in the high-Reynolds-number regime (Kim et al. Reference Kim, Cossé, Cerdeira and Gharib2013; Jung et al. Reference Jung, Song and Kim2021; Lee et al. Reference Lee, Jung, Lee and Kim2023). However, compared with $U^*$ , this alternative dimensionless parameter is not suitable for characterising $\delta /D$ and $\phi$ ; see Appendix C for details.

3.3. Drag force

The hydrodynamic blockage discussed in § 3.1 dramatically influences the drag force imposed on the reconfigured kirigami sheet in the quasi-steady state. The dimensional drag of the kirigami sheet, $F_{\textit{kirigami}}$ , is plotted with respect to the translation speed $U$ for four sheet thicknesses $h$ and two cutting ratios $\gamma$ in figure 10. The drag force acting on a flat rigid disk representing the undeformed configuration is also included (black dashed lines in figure 10); the diameter of the flat rigid disk is identical to that of the kirigami sheet. For each case, the drag averaged over three trials is presented. In the low-Reynolds-number regime $ \textit{Re}_{D}$ = 4–15 (using glycerine), the drag force of the reconfigured kirigami sheet is comparable to that of the undeformed flat disk, regardless of $h$ and $\gamma$ (figure 10 a). Interestingly, a considerable portion of the data points lie above the black dashed line, indicating that, despite the formation of holes, the drag force increases as a result of flow-induced reconfiguration. In contrast, in the high-Reynolds-number regime $ \textit{Re}_{D}$ = 5000–20 000 (using water), the drag force of the reconfigured kirigami sheet is clearly less than that of the flat rigid disk (figure 10 b). This reduction is obviously attributed to hole formation in the kirigami sheet and significant flow permeation through the holes.

Figure 10.

Drag force acting on the kirigami sheet, $F_{\textit{kirigami}}$ , with respect to translation speed $U$ : (a) low-Reynolds-number regime ( $ \textit{Re}_{D}$ = 4–15) and (b) high-Reynolds-number regime ( $ \textit{Re}_{D}$ = 5000–20 000). Panels (i) and (ii) present the results of a low cutting ratio $\gamma = 2.6$ and a high cutting ratio $\gamma = 8.0$ , respectively. The black dashed lines denote the drag force acting on the flat rigid disk. The schematic inside the panel shows the unit structure corresponding to the cutting ratio $\gamma$ of each model.

The drag of the kirigami sheet in the low-Reynolds-number regime exhibits non-intuitive behaviour with respect to the cutting ratio $\gamma$ and sheet thickness $h$ . For instance, for a sheet with a thickness of $h = 0.6$ mm, the drag is larger than that of a flat rigid disk at $\gamma = 2.6$ (figure 10 ai), but becomes smaller at $\gamma = 8.0$ (figure 10 aii). Interestingly, this trend is reversed for a sheet with a thickness of $h = 0.4$ mm. These opposing results do not appear in the high-Reynolds-number regime, where the lower $h$ generally results in the smaller drag at each $\gamma$ (figure 10 bi, bii). At high Reynolds number, a thinner sheet undergoes greater deformation, which consequently forms larger holes. Because the fluid readily passes through these holes, the relationship between the drag and the sheet thickness is straightforward. However, at low Reynolds number, the drag is governed by a strong coupling between the structural reconfiguration and the hydrodynamic blockage. Because both $h$ and $\gamma$ are key design parameters determining the reconfiguration and hydrodynamic blockage, their effects on the drag are intertwined, making it difficult to isolate the influence of a single parameter.

Figure 11.

Drag ratio $F_{{kirigami}} / F_{\textit{disk}}$ with respect to dimensionless velocity $U^{*}$ in the low-Reynolds-number regime ( $ \textit{Re}_{D}$ = 4–15).

In the low-Reynolds-number regime, the majority of drag ratio data $F_{\textit{kirigami}} / F_{\textit{disk}}$ cluster around unity in the broad range of $U^{*}$ considered in this study (figure 11); $F_{\textit{disk}}$ is the drag acting on the flat rigid disk. The drag ratio is actually similar to the reconfiguration number, commonly used to quantify how reconfiguration affects drag force in general elastic structures (Gosselin et al. Reference Gosselin, de Langre and Machado-Almeida2010; Gosselin & de Langre Reference Gosselin and de Langre2011; Satheesh & Huera-Huarte Reference Satheesh and Huera-Huarte2019; Song et al. Reference Song, Yoo, Ham and Kim2022; Lamoureux et al. Reference Lamoureux, Ramananarivo, Melancon and Gosselin2025b ). The drag force of the kirigami sheet does not significantly deviate from that of the flat rigid disk when significant structural deformation and hole formation occur, and the drag ratio can be greater than unity even under high $U^*$ . In contrast, the drag ratio is near 0.5 in most cases in the high-Reynolds-number regime, as can be seen in figure 10(b).

When elastic structures such as sheets and fibres undergo reconfiguration under flow, the drag generally declines from that of their initial configuration. However, a few studies have reported the opposite effect. Under specific flow conditions and structure designs, the frontal area of the deforming structure can instead become greater after reconfiguration, enhancing the drag; for example, a model of multiple fibres attached to a central structure at high Reynolds number (Gosselin & de Langre Reference Gosselin and de Langre2011) and an array of elastically mounted cylinders under strong hydrodynamic blockage (Lee et al. Reference Lee, Mahravan and Kim2024). In the present study, the radial expansion of the outer edge of the kirigami sheet is constrained by the rigid frame, and the physical frontal area decreases due to hole formation. Nevertheless, the drag of the reconfigured kirigami sheet can exceed that of the flat rigid disk with the same diameter in many experimental cases of low Reynolds number, by virtue of strong hydrodynamic blockage.

Furthermore, to more clearly evaluate the blockage effect of the kirigami sheet, drag measurements were performed for the specific truncated-cone-shaped solid shell model described in figure 4, which corresponds to the large-deflection case ([ $U$ , $h$ , $\gamma$ ] = [15 cm s $^{-1}$ , 0.3 mm, 8.0]). The kirigami sheet of the large-deflection case exhibits a drag ratio of $F_{\textit{kirigami}} / F_{\textit{disk}} = 1.15$ despite the formation of large holes and the high porosity ( $\phi = 0.52$ ). Although the kirigami sheet in the large-deflection case has a thickness of $h = 0.3$ mm, the solid shell model could not be fabricated with the same thickness because the printed structure could not maintain its shape and failed due to the mechanical properties of the printer filament. Therefore, the shell was fabricated with a larger thickness of $h = 1.0$ mm to ensure structural integrity. The drag force acting on the solid shell model is 0.21 N. For reference, the drag force of the reconfigured kirigami sheet of the large-deflection case is 0.18 N. Although the hydrodynamic blockage is strong, it does not completely prevent flow through the holes, so the kirigami sheet produces slightly lower drag than the solid shell model. Nevertheless, it is notable that in the low-Reynolds-number regime, the reconfigured kirigami sheet can retain approximately 90 % of the drag generated by the corresponding solid shell model. Because of its many holes, the reconfigured kirigami sheet has a considerably lower mass than the corresponding solid shell model and can modulate its shape passively in response to external conditions. Along with these inherent advantages, the generation of comparable drag suggests that the kirigami sheet holds potential for future application to the design of small-scale drag-based transport and propulsion systems in viscosity-dominant environments.

If a drag-based propulsion system is designed using a kirigami sheet, the sheet should move freely in response to the external flow. From this perspective, it is necessary to examine the stability of the kirigami sheet in the absence of any constraining parts such as the rigid frame. We conducted simple free-fall experiments under low-Reynolds-number conditions. A weight of mass was attached at the centre of the kirigami sheet, which was similar to the model of Lamoureux et al. (Reference Lamoureux, Fillion, Ramananarivo, Gosselin and Melancon2025a ), and the sheet was released in a vertical glass tank filled with glycerine. Depending on the bending stiffness of the sheet, the reconfigured shape differed markedly; however, the falling trajectory remained nearly vertical, indicating highly stable descent. These observations contrast with those from free-fall experiments performed in air (Lamoureux et al. Reference Lamoureux, Fillion, Ramananarivo, Gosselin and Melancon2025a ), in which a pronounced horizontal drift occurred during the descent of the kirigami sheet. Although the experimental model considered here is simple and idealised, these results suggest that systems designed with kirigami sheets may exhibit a high level of stability in viscosity-dominated environments.