Deformation of a flexible fiber settling in a quiescent viscous fluid

The equilibrium state of a flexible fiber settling in a viscous fluid is examined using a combination of macroscopic experiments, numerical simulations, and scaling arguments. We identify three regimes having different signatures on this equilibrium configuration of the elastic filament: weak and large deformation regimes wherein the drag is proportional to the settling velocity as expected in Stokes flow and an intermediate elastic reconfiguration regime where the filament deforms to adopt a shape with a smaller drag which is no longer linearly proportional to the velocity.

Figure 1

Experimental chronophotographies of an elastic filament (length $2\ell =8.3\mathrm{cm}$) settling in a viscous fluid for different initial conditions. The time step between two consecutive images is taken as $\mathrm{\Delta }t\simeq \ell /U$, where $U$ is the final steady settling speed. Note that the fiber is not perfectly homogeneous, resulting in a systematic slight asymmetry of the shape.

Figure 2

Experimental chronophotographies in the stationary state, where the fiber shape and velocity remain constant. All filaments have similar material properties ($a, {\rho }_s, EI$) such that the settling velocity of an equivalent rigid filament $U_{\perp }$ is the same. From panel (a) to (f), the length of the fiber increases, i.e., $\mathcal{B}$ increases ($\mathcal{B}=57,111,207,222,329,439,549$). The time between successive photos is $10a/U_{\perp }$; a difference in traveled distance thus indicates a difference in velocity.

Figure 3

Experimental setup. The filament is initially maintained at the center of the tank and aligned with the plane defined by the dotted line. The shaded area corresponds to the plane of view of the camera.

Figure 4

Image analysis (typical profiles obtained experimentally): the instantaneous shape $y\left(x\right)$ of the filament is extracted from the images (a), the deformation $d\left(s\right)$ is measured along the arc length (b), and the maximum deflection $d_{\max }$ as well as the velocity $u$ are followed with time (c). The final stationary shape [inset in (c)] is symmetric and has a maximum deflection $\delta$ and an end-to-end distance $\lambda$.

Figure 5

Sketch of the bead-spring model wherein the filament is modeled as a chain of spherical beads connected by spring.

Figure 6

Final shapes of the filaments $\frac{y(x/\ell )}{\ell }$ (left column) and same but rescaled by the maximum amplitude $\frac{y(x/\ell )}{\delta }$ (right column) for experiments (top), graphs (a) and (b), bead-spring modeling noted (BSM) (middle), graphs (c) and (d), and selected comparison between experiments and numerical predictions (bottom), graphs (e) and (f).

Figure 7

Scaled maximum amplitude, $\delta /\ell$, versus $\mathcal{B}$. Comparison of the slender-body and bead-spring models presented in Sec. 4 with numerical results from previous models [9, 12, 16] at similar ${\kappa }^{-1}=30$ and 100.

Figure 8

Scaled maximum amplitude $\delta /\ell$ (top) and end-to-end distance $\lambda /\ell$ (bottom) as functions of $\mathcal{B}$ (left), graphs (a) and (c), and $\mathcal{V}$ (right), graphs (b) and (d), for the experimental, analytical (slender-body model), and numerical (BSM) results.

Figure 9

Scaled velocity, $U/U_{\perp }$, versus $\mathcal{B}$ (left), graph (a), and $\mathcal{V}$ (right), graph (b), for the experimental and numerical (BSM) data.

Figure 10

Dimensionless drag, $\mathcal{B}$, versus dimensionless velocity, $\mathcal{V}$, for (a) the numerical (BSM) and experimental data of various aspect ratios and (b) for ${\kappa }^{-1}=151$ with the fiber profiles (the color of the profile matches the corresponding dot on the curve).