Flexible fibers in shear flow approach attracting periodic solutions

The three-dimensional dynamics of a single non-Brownian flexible fiber in shear flow is evaluated numerically, in the absence of inertia. A wide range of ratios $A$ of bending to hydrodynamic forces and hundreds of initial configurations are considered. We demonstrate that flexible fibers in shear flow exhibit much more complicated evolution patterns than in the case of extensional flow, where transitions to higher-order modes of characteristic shapes are observed when $A$ exceeds consecutive threshold values. In shear flow, we identify the existence of an attracting steady configuration and different attracting periodic motions that are approached by long-lasting rolling, tumbling, and meandering dynamical modes, respectively. We demonstrate that the final stages of the rolling and tumbling modes are effective Jeffery orbits, with the constant parameter $C$ replaced by an exponential function that either decays or increases in time, respectively, corresponding to a systematic drift of the trajectories. In the limit of $C\to 0$, the fiber aligns with the vorticity direction and in the limit of $C\to \infty$, the fiber periodically tumbles within the shear plane. For moderate values of $A$, a three-dimensional meandering periodic motion exists, which corresponds to intermediate values of $C$. Transient, close to periodic oscillations are also detected in the early stages of the modes.

Figure 1

Evolution of a flexible fiber in shear flow: notation.

Figure 2

Different modes of the dynamics. Starting from different initial orientations of the initial end-to-end vector ${\mathbf{n}}_0$, (a) ${\mathrm{\Theta }}_0=5^{\circ }, {\mathrm{\Phi }}_0={10}^{\circ }$; (b) ${\mathrm{\Theta }}_0={45}^{\circ }, {\mathrm{\Phi }}_0={30}^{\circ }$; (c) ${\mathrm{\Theta }}_0={10}^{\circ }, {\mathrm{\Phi }}_0={10}^{\circ }$, fibers approach different attractors: (a) rolling (blue online), (b) tumbling (red online), (c) meandering (green online). Examples shown are for $A=10$. The orientation angle $\mathrm{\Theta }$ of the end-to-end vector $\mathbf{n}$ is plotted versus time. Shape evolution of the fibers is also shown in the $xy$ and $xz$ projections for the time window indicated. In (c) the transient squirming (close to periodic) sequence of shapes is shown, approximately repeating after $t\approx 305$.

Figure 3

Three evolving modes. Trajectories of the fiber end-to-end vector for the three modes with $A=10$, shown in Fig. 2: (a) rolling (blue online), (b) tumbling (red online), and (c) meandering (green online). A given initial condition is attracted to one of three characteristic orbits (rolling, tumbling and meandering), shown in Supplemental Material Movie 2 [61] and Sec. 4. Symbols: $•t=0$; $⧫t=5000$.

Figure 4

Sensitivity to a small change of the fiber initial orientation (indicated by the dot in each panel). The initial orientation in each column is the same, and there is only a small difference in initial orientations in the different columns, (a) ${\mathrm{\Theta }}_0$ = ${35}^{\circ },{\mathrm{\Phi }}_0$ = $0^{\circ }$, (b)  ${\mathrm{\Theta }}_0={35}^{\circ },{\mathrm{\Phi }}_0=5^{\circ }$, (c) ${\mathrm{\Theta }}_0$ = ${35}^{\circ },{\mathrm{\Phi }}_0$ = ${10}^{\circ }$. Each initial orientation is attracted to a different orbit: (a) rolling (blue online), (b) tumbling (red online), and (c) meandering (green online). The trajectories of the fiber end-to-end vector are shown for $A$ = 10 and $0\le t\le t_{\text{end}}$. Symbols: $• t=0$; $⧫t=t_{\text{end}}$, with $t_{\text{end}}=7740$ in (a), $t_{\text{end}}=25000$ in (b), and $t_{\text{end}}=20000$ in (c). The tumbling period $T_t$ = 181 and the meandering period $T_m$ = 735.

Figure 5

Transient, close to periodic squirming motion for $A$ = 10 and the same initial conditions as in Figs. 2, 3 and Supplemental Material Movie 1 [61]. The time range is $600\le t\le 1224$. Trajectories of the fiber end-to-end vector are close to periodic; shapes are a little different, but almost repeating after $t\approx 305$.

Figure 6

Three attractors of the fiber dynamics for $A$ = 10: rolling stationary configuration (blue diamonds), together with tumbling (red), and meandering (green) periodic orbits. The $xy, xz,$ and $yz$ projections of the end-to-end periodic trajectories are plotted for $t_{\text{end}}-800<t<t_{\text{end}}$, with $t_{\text{end}}={10}^5$ for meandering, and $t_{\text{end}}$ = 15 000 for tumbling trajectories.

Figure 7

Periodic shape sequences for the meandering (green online) and tumbling (red online) attracting periodic solutions with $A$ = 10, reached from the same initial orientations as in Figs. 2 and 3. Snapshots are taken at $t$ = $t_0+\tau$, with (a) $t_0$ = 99556, (b) $t_0$ = 4653, and $\tau$ as indicated.

Figure 8

Modes of flexible fiber dynamics for different initial orientations $({\mathrm{\Phi }}_0,{\mathrm{\Theta }}_0)$ and different values of the bending stiffness ratio $A$; (a) $A$ = 4, (b) $A$ = 10, (c) $A$ = 40, (d) ${\mathrm{\Phi }}_0$ = ${10}^{\circ }$. The color indicates convergence to a specific attractor: rolling (blue), tumbling (red), periodic meandering (green dots), or periodic squirming (green diamonds for $A$ = 7.3).

Figure 9

Oscillations of $tan\mathrm{\Theta }\left(t\right)$ are approximately periodic with an amplitude that changes exponentially with time. Main panel, lower curve (blue online): rolling mode with $({\mathrm{\Theta }}_0,{\mathrm{\Phi }}_0)$ = $(5^{\circ },{10}^{\circ })$; upper curve (red online): tumbling mode with $({\mathrm{\Theta }}_0,{\mathrm{\Phi }}_0)$ = $({45}^{\circ },{30}^{\circ })$. The slopes of the solid straight lines are $-t/1253$ and $t/170$, respectively. Inset: Characteristic time $T$ of the oscillations is almost time independent. $T\left(t\right)$ is the time difference between the $i\mathrm{th}$ and $(i+2)\mathrm{th}$ maxima, and the $i\mathrm{th}$ maximum is at time $t$. $T_r$: triangles (blue online); $T_t$: dots (red online).

Figure 10

Rolling (a) and tumbling (b) modes interpreted by Eq. (16) as effective Jeffery solutions. (a) ${\mathrm{\Theta }}_0$ = $5^{\circ }, {\mathrm{\Phi }}_0$ = ${10}^{\circ }, C_0$ = 0.0101. (b) ${\mathrm{\Theta }}_0$ = ${45}^{\circ }, {\mathrm{\Phi }}_0$ = ${30}^{\circ }, C_0$ = 0.463. Solid lines: numerical results. Dashed lines: Jeffery approximation given by Eqs. (11) and (13) with $r=r_r=73.0$ and $r=r_t=28.8$.

Figure 11

Rolling (blue) and tumbling (red) modes evolving toward $\mathrm{\Theta }=0^{\circ }$ and $\mathrm{\Theta }={90}^{\circ }$, respectively, from two different initial conditions. Red: ${\mathrm{\Theta }}_0={15}^{\circ }, {\mathrm{\Phi }}_0={90}^{\circ }, C_0=0.268$. Blue: ${\mathrm{\Theta }}_0={50}^{\circ }, {\mathrm{\Phi }}_0=0^{\circ }, C_0=0.0163$. Black: Jeffery solutions with $C_0=0.0195, r=73.0$ (bottom), and $C_0=0.25, r=28.8$ (top).

Figure 12

Periodic meandering solution (green dots) compared to Jeffery orbits (black solid lines) of a rigid effective spheroid with $C=0.25, r=28.8$ (top) and $C=0.0195, r=73.0$ (bottom). Here $A=10, {\mathrm{\Theta }}_0={10}^{\circ }, {\mathrm{\Phi }}_0={10}^{\circ }$, and $19164\le t\le 19531$.

Figure 13

Shapes of fibers which perform periodic tumbling motion are shown at the flipping time, defined as the time when the end-to-end vector is perpendicular to the flow. The values of the bending stiffness $A$ are indicated. Initially, $({\mathrm{\Theta }}_0,{\mathrm{\Phi }}_0)=({90}^{\circ },0^{\circ })$. A detailed study of the quantitative features of the fiber shapes as a function of $A$ will be published in the future.

Figure 14

Fiber center-of-mass periodic trajectory for $A=10$.

Figure 15

Example of close to periodic meandering motion: trajectories of the fiber end-to-end vector. Here $A$ = 11, ${\mathrm{\Theta }}_0={55}^{\circ }, {\mathrm{\Phi }}_0=0^{\circ }$, and the time range is $10565\le t\le 12565$.

Figure 16

Long-time effects for periodic meandering mode with $A$ = 10: trajectories of the fiber end-to-end vector. Dots (brown online) and diamonds (cyan online) mark positions, respectively, at the beginning and at the end of the indicated time range.

Figure 17

The long-time trajectories of the fiber end-to-end vector. Comparison between fibers with different values of $l_0$ and $A$ (as indicated), for the same initial orientation $({\mathrm{\Theta }}_0,{\mathrm{\Phi }}_0)=({10}^{\circ },{10}^{\circ })$. The plots are made for $5000\le t\le 10000$ (upper row) and for $t_{\text{end}}-5000\le t\le t_{\text{end}}$ (lower row), where $t_{\text{end}}=5·{10}^4$ for $A=9$ and 11, and $t_{\text{end}}={10}^5$ for $A=10$. See also Table 1.