Chaos and irreversibility of a flexible filament in periodically driven Stokes flow

The flow of Newtonian fluid at low Reynolds number is, in general, regular and time-reversible due to absence of nonlinear effects. For example, if the fluid is sheared by its boundary motion that is subsequently reversed, then all the fluid elements return to their initial positions. Consequently, mixing in microchannels happens solely due to molecular diffusion and is very slow. Here, we show, numerically, that the introduction of a single, freely floating, flexible filament in a time-periodic linear shear flow can break reversibility and give rise to chaos due to elastic nonlinearities, if the bending rigidity of the filament is within a carefully chosen range. Within this range, not only the shape of the filament is spatiotemporally chaotic, but also the flow is an efficient mixer. Overall, we find five dynamical phases: the shape of a stiff filament is time-invariant—either straight or buckled; it undergoes a period-two bifurcation as the filament is made softer; becomes spatiotemporally chaotic for even softer filaments but, surprisingly, the chaos is suppressed if bending rigidity is decreased further.

Figure 1

Sketch of numerical experiment: Initially the filament is straight. It rotates and translates while advected by $\mathit{U}=\dot{\gamma }y\widehat{\mathrm{x}}$ with $\dot{\gamma }=Ssin\left(\omega t\right)$ over the first-half of the cycle. In the second half the filament rotates and translates back but in addition buckles if its elastoviscous number is large enough. The flow $\mathit{U}$ at $t=T/4$ (top panel) and $t=3T/4$ (bottom panel) are shown as red arrows.

Figure 2

Kaleidoscope of dynamical behavior: (A) Periodic buckling ($\overline{\mu }=1.46\times {10}^6, \sigma =1.5$): The filament is buckled after 8 cycles and repeats itself after every cycle afterwards. Till cycle 8 the filament is not straight but slightly deformed from its initial straight shape. This small deformation is barely noticeable in this figure. (B) Two-period ($\overline{\mu }=0.67\times {10}^6, \sigma =0.75$): The shape of the filament is either of two shapes, which are mirror images of each other, at odd and even cycles. (C, D) The two types of Complex phases: (C) For $\overline{\mu }=3.35\times {10}^6, \sigma =1.5$: The filament never repeats itself. (D) For $\overline{\mu }=6.7\times {10}^6, \sigma =0.75$): The filament shows spatiotemporally complex behavior but for late cycles, the filament (almost) repeats at the end of every cycle ($nT$) ($t=55T,65T,75T$) but not at any other time. To illustrate, we show the snapshots at $t=55.7T,65.7T,75.7T$—the shape of the filament is different from one another. Note, the filament shows maximum buckling, not at the end of a cycle, but at times in-between the cycles, i.e., at $t=(n+p)T$, where $n$ is an integer and $0<p<1$. (E) Complex transients ($\overline{\mu }=16.75\times {10}^6, \sigma =1.5$): Filament shows complex behavior for early periods but repeats itself at late times.

Figure 3

Phase diagram from time-dependent numerical simulations in the $\overline{\mu }-\sigma$ parameter space; $(\sigma =\frac{\omega }{S},\overline{\mu }=\frac{8\pi \eta S{L}^4}{B})$. Here, $\omega$ is rate of change of strain, $S$ is strain rate, $\eta$ is the viscosity, $L$ is length of the filament, and $B$ is the bending modulus. Initially, the filament is freely suspended in the shear flow; see Fig. 1. We show five different dynamical phases in the system represented by five symbols. Straight ($•$): The filament comes back to the initial position in the straight configuration after every period. Periodic buckling ($▾$): The filament comes back in the buckled configuration after every period. Two-period ($\blacksquare$): The filament repeats its configuration not after every but after two-period. Complex ($★$): The filament buckles into complex shape with very high mode of buckling instability. Complex–transients ($⧫$): Filament shows long transients with complex shape but at late times, the shape of the filament repeats itself. The boundary between the complex and complex-transient phase is difficult to clearly demarcate.

Figure 4

Evolution in the complex phase: ($\overline{\mu }=3.35\times {10}^6, \sigma =1.5$) The filament shows complex behavior and does not repeat in real space or configuration space ($t<45T$) 9. (A) Curvature ($\kappa$) as a function of material coordinate, $s$, of the filament at early cycles, $t=T,10T,19T,$ and $28T$. (B) Image of the filament at the same times. (C, D) Same as (A, B) but for late times. (E, F) Phase portrait of tracer velocity at a fixed Eulerian point for late cycles $t=45T$–$75T$.

Figure 5

Solutions of stroboscopic map in real space for $\sigma =0.75$ for different values of $\overline{\mu }$. We find multiple coexisting solutions as we increase $\overline{\mu }$ (black symbols from left to right) indicating the complexity of the system. We compare this solutions with the solutions obtained at late times from the evolution code at the same points in the phase diagram.

Figure 6

Mixing of passive tracer: (A–C) Positions of tracer particles at different times for the filament in complex phase ($\overline{\mu }=3.3\times {10}^6, \sigma =1.5$). Initially, the tracers are placed on concentric circles, color coded by their distance from the center of the circles. The mixing of the colors show the mixing of the scalars. (D) Mean-square displacement (MSD), $\langle {\rho }^2\rangle \left(qT\right)$, defined in Eq. (10), as a function of $q$ in log-log scale. We also plot two lines with slopes 1 and 2. (E) Cumulative probability distribution function, $1-\mathcal{Q}\left(\mathrm{\Delta }s\right)$ as a function of $\mathrm{\Delta }s=\sqrt{X_1^2+X_2^2}$ ($X_1$ and $X_2$ are in-plane coordinate of the tracers).

Figure 7

Schematic of a freely jointed bead-rod chain. We show $a>d$ for illustration, but we use $a=d$ for our simulation.

Figure 8

Comparison with experimental results: We reproduce the experimental results of Ref. [20]. The filament lies along the $x$ axis. It is advected by a flow $\mathit{U}=(\dot{\gamma }y,0)$, where $\dot{\gamma }$ is the shear rate. The flow is constant in time. We observe different dynamical behavior for different $\overline{\mu }$. The gray background are figures from Ref. [20], and the white background are results of our simulation. Initially, we add small perturbation to the filament but ignore thermal fluctuations. The ${\overline{\mu }}_{\mathrm{E}}$ and ${\overline{\mu }}_{\mathrm{S}}$ are the values of $\overline{\mu }$ from experiments and our simulations, respectively.

Figure 9

Evolution of the filament in complex phase ($\overline{\mu }=3.35\times {10}^6, \sigma =1.5$). The filament shows complex behavior for cycles at intermediate times: (A) real space; (B) configurational space. (C, D) For late cycles also, the filament either does not repeat itself or comes close to repeating itself with very high time-period. The filament shows maximum compression at the end of cycle. (E, F) Phase portrait of tracer velocity late cycles $t=45T$–$75T$. For comparison we also show the phase portrait for a case in the straight phase (orange).

Figure 10

Discrete sine transform for two-period phase (A), and complex phase (B).

Figure 11

Evolution of the filament in the complex phase ($\overline{\mu }=6.7\times {10}^6, \sigma =0.75$) The filament shows complex behavior at early cycles respectively in real space and configuration space. However, for late times ($t=65T,70T,75T$), the filament almost repeats itself at $nT$, where $n$ is an integer (A, B). Also note that, highest bending energy of the filament is at $(n+p)T, p\ne 0$ instead of $nT$ (C, D). The filament is shown stroboscopically for $p=0.7$ respectively in real space and configuration space. We observe that the filament does not repeat itself. (E, F) Phase portrait in $(x,y)$ and $(y,z)$ of Eulerian velocity at one point.

Figure 12

Evolution in the complex-transient phase: ($\overline{\mu }=3.35\times {10}^7, \sigma =1.5$). The shape of the filament in panels (A, C, E) and the corresponding curvature is shown in panels (B, D, F). For early cycles ($t \le$ 60T), the filament shows complex behavior and does not repeat itself (A, B), similar to the complex phase (see Fig. 9). For late cycles ($t=70T\cdots 100T$), although the filament does not repeat itself, it comes very close after every cycle (C–F). The filament shape, from the end of one cycle to next, changes very slowly, e.g., panel (E) shows the shape at $t=80.8T,90.8T,100.8T$. In panel (G), we show the shapes at $t=80T,130T\cdots$. Over such a long timescale, the shape does change. (I, J) Phase portrait of velocity at an Eulerian point for the late cycles.

Figure 13

Flowchart for Newton-Krylov iteration. $\mathbb{K}$ is coordinate transformation from real to curvature space using Eq. (A6). Similarly, ${\mathbb{K}}^{-1}$ is the inverse coordinate transformation from curvature to real space. We use the notation: $\mathcal{J}$ is jacobian of the operator $\mathcal{N}$, described in Eq. (D2). We use $\mathrm{tol}=0.01.$