Hydrodynamic interactions change the buckling threshold of parallel flexible sheets in shear flow

Buckling induced by viscous flow changes the shape of sheetlike nanomaterial particles suspended in liquids. This instability at the particle scale affects collective behavior of suspension flows and has many technological and biological implications. Here, we investigated the effect of viscous hydrodynamic interactions on the morphology of flexible sheets. By analyzing a model experiment using thin sheets suspended in a shear cell, we found that a pair of sheets can bend for a shear rate ten times lower than the buckling threshold defined for a single sheet. This effect is caused by a lateral hydrodynamic force that arises from the disturbance flow field induced by the neighboring sheet. The lateral hydrodynamic force removes the buckling instability but massively enhances the bending deformation. For small separations between sheets, lubrication forces prevail and prevent deformation. Those two opposing effects result in a nonmonotonic relation between distances and shear rate for bending. Our study suggests that the morphology of sheetlike particles in suspensions is not purely a material property but also depends on particle concentration and microstructure.

Figure 1

(a) Schematic of the shear cell. (b) Schematic of a buckled sheet viewed along the vorticity direction of the shear flow and definition of the midpoint orientation angle $\theta$ and midpoint curvature $\overline{\kappa }$. The compressional quadrant and the extensional quadrant are shown. In this schematic, the sheet is oriented in the compressional quadrant ($-\pi /2<\theta <0$).

Figure 2

(a) Normalized midpoint curvature $\overline{\kappa }L$ (crosses) and midpoint orientation angle $\theta$ (triangular markers) versus rescaled time $\dot{\gamma }t$ for $E_v\simeq 21$. Time has been shifted so that $\dot{\gamma }t=0$ corresponds to the orientation $\theta =0$. The black line is Jeffery's prediction $\theta \left(t\right)=arctan\left(\dot{\gamma }t\right)$ [32]. (b) Maximum normalized curvature versus elastoviscous number for a single sheet. The dark and light grey regions delimit the rigid limit and the buckling region, respectively. The measured critical elastoviscous number from this diagram is $E_v^c\simeq 11$. (c) Normalized curvature versus normalized time from dynamic simulations of a single sheet for different elastoviscous numbers.

Figure 3

Comparison between single sheet dynamics and dynamics of a pair of parallel sheets below the single-sheet buckling threshold. (a) Experimental normalized curvature $\overline{\kappa }L$ versus normalized time $\dot{\gamma }t$ for a single sheet (in black) and a sheet pair separated by $d/L\simeq 0.2$ (in red) for $E_v\simeq 3.6$. (b) Images of a pair of parallel sheets at two selected times. (c) Normalized curvature $\overline{\kappa }L$ versus normalized time $\dot{\gamma }t$ for simulation of a single sheet (in black) and a pair of sheets separated by $d/L=0.1$ (in red) for $E_v=7$. (d) Simulated shapes of a pair of parallel sheets corresponding to the two selected times of panel (b).

Figure 4

Hydrodynamic interactions from simulation. (a) Lateral force induced by the first sheet on the second sheet in the case of two parallel sheets in a shear flow, oriented in the compressional quadrant (at ${\theta }_0=-\pi /2+\pi /10$), for $d/L=1,0.7,\mathrm{and}0.5$. (b) Magnitude of the lateral force at the center of the sheet versus the separation distance when the sheets are oriented in the compressional quadrant (at ${\theta }_0=-\pi /2+\pi /10$). The line is the best fit $y=A{x}^{\alpha }$ with $A\simeq 0.02$ and $\alpha \simeq -1.1$. (c) Vector plot of the 2D disturbance flow. The rectangle represents the sheet. (d) Magnitude of the disturbance flow velocity $u_{\text{dist}}$ induced by a sheet in a 2D compressional flow, versus the distance $r$ measured orthogonally to the sheet.

Figure 5

Left panel: sketch of the hydrodynamic forces distribution in the single and two sheet cases. The black line segments represent the sheets. The green arrows represent the compressional or extensional tangential forces. The red arrows represent the dipolar lateral forces. For $E_v<E_v^c$ the single sheet remains straight while the pair of sheets adopts concave and convex shapes. Right panel: morphology diagram. Maximum normalized curvature for different normalized separation distances $d/L$ and elastoviscous numbers $E_v=\eta \dot{\gamma }{L}^3/B$. The black squares correspond to deformation below the experimental resolution ${\kappa }_r=0.02/L$, the color circles to a finite curvature. The shadow regions are guides for the eye. The experimental data for single sheets of Fig. 2 are reported in correspondence to $d/L=\infty$. The equation of the dashed line is $d/L=K{E}_v/\left[{\kappa }_{r}L(E_v^c-E_v)\right]$, with $E_v^c=11, {\kappa }_{r}L=0.02$, and $K=0.02$. The dashed line is plotted for $d/L>0.05$.