Evolution of instabilities in filament buckling processes

In this work we study the dynamical buckling process of a thin filament immersed in a highly viscous medium. We perform an experimental study to track the shape evolution of the filament during a constant velocity compression. Numerical simulations reproduce the dynamical features observed from the experimental data and allow quantifying the filament's load. We observe that both the filament's load and the wave number evolve in a stepwise manner. In order to achieve a physical insight of the process, we apply a theoretical model to describe the buckling of a filament in a viscous medium. We solve a hydrodynamic equation in terms of normal modes for clamped-clamped boundary conditions and constant applied load. We find a good agreement between experimental data and simulations, suggesting that the proposed mechanistic model captures the essential features underlying the dynamical buckling process.

Figure 1

Wave numbers ${\kappa }_{-}$ and ${\kappa }_{+}$ (dash and continued thin lines) as a function of compressing force for $n=1$ symmetric ($\sigma =+1$) mode. Thick line: Effective wave number. Thick dash line corresponds to mean wave number $\kappa =\sqrt{p/2}=\sqrt{2{\pi }^{2}P/P_c}$. Inset: Sketch of a normal mode and inflection points in circular dots.

Figure 2

Schematic view of the experimental set-up: 1, horizontal cell; 2, filament; 3, syringe pump; 4, acrylic rod; 5, piston; 6, camera.

Figure 3

Snapshots of two experiments. [(a) and (b)] $v_0=0.062$ cm/s, $t=10.0$ s, and 33.3 s, respectively. [(c) and (d)] $v_0=5$ cm/s, $t=0.0625$ s, and 0.25 s. The experimental tracking [left panels (green)] and the numerical simulation [right panels (yellow)] are plotted as thin lines 1 cm below the filament to facilitate visual inspection. Horizontal and vertical scale bars represent 3 cm and 1 cm, respectively.

Figure 4

Forces computed at the moving (continuous line) and fixed (dash line) ends of the filament, and effective wave number (dots) as a function of time, for $v_0=0.062$ cm/s.

Figure 5

Wave number as a function of average force for the two simulations of the experiments at $v_0=0.062$ cm/s (circles) and $v_0=5.0$ cm/s (triangles). The continous curve corresponds to the theoretical value ${\kappa }_{+}$ for the symmetric mode $n=1$.