Flapping instability of elastic disks in Stokes flows

Through a combination of experiments and simulations, this study reveals that a freely suspended elastic disk in a low-Reynolds-number shear flow undergoes a subcritical flapping instability driven by finite extensibility, exhibiting rich oscillatory dynamics with implications for understanding the behavior of sheet-like particles such as 2D polymers.

The Gist

Imagine you are watching a tiny, flexible dinner plate floating in a thick, slow-moving river of honey. If the river flows gently, the plate acts like a rigid coin: it spins smoothly, just like a coin rolling on a table. This is what scientists have known for over a century: in slow, thick fluids, small objects usually just spin in predictable loops.

But this paper reveals a surprising secret: if the honey flows just a little faster, the plate doesn't just spin—it starts to flap.

Here is the story of what happens, broken down into simple ideas:

1. The Setup: A Flexible Plate in Thick Honey

The researchers took very thin, flexible disks (made of a soft, rubbery material) and placed them in a thick fluid (glycerol). They set the disks up so they were lying flat, parallel to the direction the fluid was moving.

They asked a simple question: What happens when we speed up the flow?

2. The Surprise: The "Flapping" Dance

When the flow was slow, the disk spun flat and steady. But once the flow crossed a certain "tipping point," the disk suddenly started to bend and flap.

Instead of staying flat, the disk would curve up like a smile, then curve down like a frown, over and over again, while it spun. The researchers call this the "flapping regime."

Think of it like a flag in the wind, but instead of being attached to a pole, the flag is floating freely and bending itself into a "C" shape, then flipping that shape upside down, all while spinning.

3. Why Does It Happen? The "Squeeze and Stretch" Game

The paper explains that this happens because of a tug-of-war between two forces:

• The Fluid: As the disk spins, different parts of it get squeezed (compressed) and pulled (stretched) by the flowing honey.
• The Disk: The disk tries to stay flat because it's stiff, but it's also flexible enough to bend.

When the flow is strong enough, the "squeezing" force wins. The disk buckles (like a soda can being crushed) in the parts being squeezed. But because the disk is slightly stretchy (it has "finite extensibility"), it can't just stay in a perfect flat circle; it has to twist into a saddle shape (like a Pringles chip) to accommodate the bending. This creates a rhythmic, flapping motion.

4. The Computer Simulations: Finding Hidden Moves

The researchers used powerful computers to simulate this process. They found that the behavior is even more complex than what they saw in the lab:

• The "Wiggling" Mode: Before the disk starts flapping, there is a hidden, unstable state where the disk just wiggles slightly in an "S" shape. In the real world, this wiggling is so hard to trigger that they didn't see it, but the computer found it.
• The "Flapping" Mode: This is the main event they observed. It requires a specific "push" to start. Once it starts, it keeps going for a long time.
• The "Tipping Point": If the flow gets too strong, the disk stops flapping and reorients itself to face the flow directly, like a leaf settling on a stream.

5. Why This Matters

This discovery changes how we understand how thin, sheet-like things behave in fluids.

• The Analogy: Imagine you thought a piece of paper in a stream would just spin. This paper shows that under the right conditions, that paper might actually start doing a rhythmic dance, bending up and down.
• The Real-World Connection: This helps scientists understand how new, ultra-thin materials (like graphene or 2D polymers) behave when they are processed in liquids. It also helps explain how certain biological sheets might move in fluids.

In short: The paper shows that a flexible disk in a slow, thick fluid doesn't just spin; if the flow is strong enough, it starts a rhythmic, self-sustaining dance of bending up and down, a behavior that only happens because the disk is flexible enough to bend but stretchy enough to twist.

Technical Summary

Problem Statement
The prediction of dynamics for small anisotropic objects suspended in shear flow is a fundamental problem in hydrodynamics. While Jeffery's theory accurately describes the periodic orbits of rigid spheroids in simple shear flows, the behavior of deformable objects introduces complex fluid-structure interactions that can break time-reversibility and symmetry. Previous research has extensively covered flexible fibers and low-aspect-ratio vesicles, but there remains a critical gap in understanding how flexible, sheet-like particles with large aspect ratios orient and deform in shear flow. Specifically, the dynamics of a thin sheet initially oriented with its plane parallel to the flow plane are not well understood; while symmetry arguments suggest trivial rolling motion, the potential for elasto-viscous instabilities to induce non-trivial dynamical states remains an open question. This study addresses the dynamics of a freely suspended, thin elastic disk in a shear flow, focusing on the transition from rigid rotation to deformable, oscillatory states.

Methodology
The authors employed a combined approach of experiments and numerical simulations to investigate the system:

• Experimental Setup: Circular disks were fabricated from thin films of silicon elastomer (radius $a \approx 1.14$ cm, thickness $h \approx 180$ $\mu$m) and suspended in glycerol within a belt-driven shear cell. The disks were density-matched to the fluid to prevent settling. The shear rate ($\dot{\gamma}$) was controlled between $0.4$ and $10$s$^{-1}$. Deformation was measured by projecting a laser sheet at a fixed angle ($\approx 45^\circ$) onto the disk and analyzing the distortion of the intersection line via high-speed imaging. The Reynolds number was kept low (order of 1), ensuring Stokes flow conditions.
• Numerical Simulations: A regularized Stokeslet method was used to solve the steady Stokes equation coupled with the elastic deformation of a zero-thickness sheet. The model accounted for hydrodynamic interactions between sheet portions and included in-plane stretching (modeled as a neo-Hookean solid) and out-of-plane bending. The simulations tracked a mesh of nodes representing material points on the sheet surface.
• Parameters: The dynamics were characterized by two dimensionless numbers: the capillary number ($Ca = \eta \dot{\gamma} a / G$), representing the ratio of viscous to in-plane stretching stresses, and the elasto-viscous number ($EV = \eta \dot{\gamma} a^3 / K_b$), representing the ratio of viscous to bending stresses. The study focused on nearly inextensible disks ($Ca = 0.01$) and varied $EV$ to observe transitions in behavior. Linear stability analysis using Floquet theory was also performed to identify instability thresholds.

Key Results
The study reveals a rich phenomenology of fluid-structure interactions beyond the predictions of rigid-body dynamics:

1. Flapping Instability: Beyond a critical elasto-viscous number ($EV \approx 24$ in experiments, $EV \approx 14$ in simulations), the disk ceases to rotate rigidly and enters a "flapping" regime. In this state, the disk curves up and down periodically relative to the horizontal shear plane while rotating. This motion is characterized by a saddle-shaped deformation with negative Gaussian curvature.
2. Mechanism: The flapping is driven by the competition between hydrodynamic compression and extension in the shear flow. Material points in the compressional quadrants buckle, while those in the extensional quadrants relax. The finite extensibility of the disk is identified as a key ingredient, allowing for the non-zero Gaussian curvature required for this saddle shape; ideal inextensible disks would theoretically forbid such shapes.
3. Bifurcation Diagram: Numerical simulations revealed a complex bifurcation diagram as a function of $EV$:
   • $EV < 22$: The disk exhibits trivial in-plane rolling.
   • $EV \approx 22$: A supercritical linear buckling instability occurs, leading to a "wiggling" motion (an S-shaped deformation where the curvature axis oscillates). This mode was predicted by linear stability analysis but was not observed in the experiments, likely due to the nature of initial perturbations.
   • $EV \approx 14 - 64$: A subcritical instability leads to the stable "flapping" regime (C-shaped deformation). The simulations identified a saddle-node bifurcation at $EV \approx 14$, where a stable flapping branch and an unstable branch coexist.
   • $EV > 38$ (Simulations) / $EV > 59$ (Experiments): At higher $EV$, the periodic deformations destabilize, and the disk reorients, eventually aligning with the flow direction (normal in the flow plane).
4. Asymmetry: Experiments showed a pronounced downward flapping (asymmetry), whereas free-space simulations produced symmetric flapping. The authors attribute this asymmetry to the presence of a free surface in the experimental setup, which was confirmed by additional simulations incorporating a nearby free surface boundary.

Significance and Claims
The paper claims to demonstrate that fluid-structure interactions at low Reynolds numbers can lead to significantly richer dynamics than previously expected for sheet-like particles. Specifically, it establishes that:

• A thin elastic disk initially parallel to the shear plane can undergo a subcritical instability leading to sustained periodic flapping, rather than simple rolling or immediate reorientation.
• The finite extensibility of the sheet is a crucial factor enabling these dynamics by permitting saddle-shaped deformations.
• The system exhibits a complex bifurcation structure involving both supercritical (wiggling) and subcritical (flapping) instabilities, which are sensitive to initial conditions and perturbations.

The authors position these findings as a contribution to the understanding of the flow dynamics of emerging sheet-like materials, such as 2D polymers and 2D crystalline nanomaterials (e.g., graphene, InSe, WSe$_2$), particularly in the context of their processing in viscous fluids. The work serves as a foundational step for investigating more complex systems, including suspensions of flexible 2D materials and biological sheet-like macromolecules.