3.1. Classification of dynamics over a half-period

In an attempt to illustrate the distinct features of an oscillatory shear flow compared to its steady counterpart, let us first consider the more straightforward case of a rigid non-Brownian rod in the limit of vanishing aspect ratio $\epsilon \rightarrow 0$. A rigid rod will rotate and perform a periodic tumbling motion about its centre in a steady, planar shear flow, while translating with the fluid. The tumbling motion is described by Jeffery's model (Jeffery Reference Jeffery1922), which predicts that the tumbling frequency is proportional to the shear rate, $\nu _t \sim \dot {\gamma }$ (Jeffery Reference Jeffery1922; Tornberg & Shelley Reference Tornberg and Shelley2004). When the rotational diffusion of the rod is taken into account, the tumbling dynamics is also affected by the additional time scale $D_r^{-1}$. As reported in previous works (Puliafito & Turitsyn Reference Puliafito and Turitsyn2005; Kobayashi & Yamamoto Reference Kobayashi and Yamamoto2010; Harasim et al. Reference Harasim, Wunderlich, Peleg, Kröger and Bausch2013), as long as $\dot \gamma \gg D_r$, thermal fluctuations dominate over the effects of shear only in a restricted angular region around the flow axis, where the effect of the noise is to drive the rod across the line $\theta =0$ in a finite time and thus initiate a new tumbling cycle. In a steady shear flow, therefore, a Brownian filament alternates between fast deterministic phases, in which it undergoes both compressional and extensional viscous stresses, and relatively long diffusive phases dominated by thermal fluctuations around the flow-aligned state. During each tumble, any deformation in the compressive quadrant of the flow is followed by relaxation in the extensional part before a new rotational cycle begins. Therefore, each tumbling phase starts with a thermally fluctuating, flow-aligned straight conformation with $\theta _0 \approx 0$.

The situation is altered fundamentally in an oscillatory flow. The mean filament orientation now oscillates with a frequency set by the flow's characteristic time period $T$. For a rigid non-Brownian rod in the plane of the flow, Jeffery's equation reads

(3.1)\begin{equation} \frac{{\rm d} \theta}{{\rm d} t} ={-} \sin^2 \theta(t) \sin (2 {\rm \pi}t/ \rho). \end{equation}

The orientational amplitude and frequency are thus governed by the single dimensionless number $\rho =\dot {\gamma }_m T$, which, unlike in steady shear, can be varied independently by changing either the maximum shear rate $\dot {\gamma }_m$ or the time period $T$. In addition, the periodic flow reversal allows the filaments to explore different orientations at the beginning of each oscillation. For a fixed time period and shear rate, this initial orientation determines the relative fraction of time that the filament spends in the compressive or extensile quadrants before the flow reverses at $t = T/2$. As long as the initial filament orientation, as well as its orientation at reversal, remains sufficiently far from the flow direction, the influence of rotational diffusion should remain negligible. We thus expect the deformation dynamics to be sensitive to both $\rho$ and the initial orientation $\theta _0 \equiv \theta (t=0)$ at the start of the cycle and to be well described by Jeffery dynamics.

Also, the orientational dynamics for the flexible filaments is affected by $\bar {\mu }_{m}$ with the possibility of large deformations. These deformations lead to a non-reciprocal conformational evolution. As a result, the flow reversal at $t=T/2$ does not generally lead to retracing of the morphological dynamics of the first half-period, which implies that $\theta (T) \neq \theta _0$ in general. Thus it is helpful to first restrict our analysis to the half-period $0< t< T/2$, where we have precise control over all the dynamical parameters of the problem. Note that we consider only filaments that are not deformed at $t=0$. We reserve the discussion of the filament evolution over an entire time period for § 3.4.

Figure 2 illustrates the role of the initial orientation $\theta _0$ on the emergent morphological dynamics over the half-period. The mean orientation angle $\theta$ in the $x$–$y$ shear plane is measured clockwise from the negative $x$-axis; see figure 1(d). For $0< t< T/2$, the shear rate is positive, which results in compressive viscous stresses for $0^{\circ }< \theta <90^{\circ }$, and extensile stresses for $90^{\circ }<\theta <180^{\circ }$. In all these examples, we fixed $\ell _p/L = 1.3$ and $\rho = \dot {\gamma }_m T = 13$. The elastoviscous number was fixed at $\bar {\mu }_{m} = 10^4$, which is significantly higher than the buckling threshold of $\bar {\mu }_{m}^c = 306.8$ for steady shear. In figure 2(a), we show snapshots of filament conformations over the half time period from both experiments and simulations, which show very good qualitative agreement. Quantitative agreement is observed in figures 2(b,c), where for each horizontal pair, we respectively plot the time evolution of $\omega$ and $\theta$ from experiments (coloured symbols) and simulations (solid lines in corresponding colours). We also report in figure 2(c) the evolution of the angle $\theta (t)$ expected for a non-Brownian rigid rod obeying Jeffery's equation (3.1) (black dashed line). The good agreement between experiments, simulations and the rigid-rod model indicates that Jeffery's model for orientational dynamics works well even in cases where the polymers are deformed. This also confirms that orientational fluctuations play only a secondary role, in agreement with the fact that we focus on relatively small values of $\rho$ (in the range 1–20), for which the filaments do not have enough time to get aligned with the flow axis where Brownian effects are expected to dominate.

Figure 2.

The role of initial orientation $\theta _0$ on the emerging dynamics of filaments over a half time period. We classify four distinct dynamical behaviours (a) by the characteristic evolution of the anisotropy parameter $\omega$ (b) and orientation $\theta (t)$ defined in terms of the end-to-end vector (c). The evolution of $\theta (t)$ is further compared to a rigid non-Brownian rod model (RR (3.1)) as shown by the black dashed line. The symbols correspond to experiments (E) and compare well with the direct simulations (S) indicated by solid lines. Yellow/grey regions in (c) correspond to compression and extension, respectively. In (b), $\omega _c=0.95$ is the value of $\omega$ below which a deformation is classified as a buckling event. Parameter values: $\bar {\mu }_{m} = 10^4$, $\rho = 13$ and $\ell _p/L = 1.3$.

Depending on the initial orientation $\theta _0$ of the filament, we identify four types of dynamical behaviours, each characterized by distinct morphological dynamics over the course of the half-period. Their characteristics can be summarized as follows.

1. (i) Continuous buckling (CB). For a small initial angle close to the flow axis, $\theta _0 \sim 14^{\circ }$ (blue parts of figure 2), the filament buckles in the presence of compressive viscous loading. Here, the filament is contained entirely in the compressional quadrant over the half-period, and never enters the extensional quadrant. After initiation of buckling, it keeps getting compressed, and $\omega$ reaches a minimum when $t=T/2$.

2. (ii) Buckling then partial stretching (BTPS). Upon slightly increasing $\theta _0 \sim 16^{\circ }$ (magenta parts of figure 2), we transition to a regime where the filament rotates into the extensional quadrant of the flow, as illustrated by the angle $\theta$ in figure 2(c). At this point, the filament stops deforming and gets partially stretched out. This can be seen from the final conformation as well as from the parameter $\omega$.

3. (iii) Buckling then stretching (BTS). Upon further increase of $\theta _0 \sim 25^{\circ }$ (orange parts of figure 2), we observe a dynamics similar to BTPS with a key difference: since the filament spends less time in the compressional quadrant, its deformation is smaller when entering the extensional quadrant (at around $t=T/4$), so that it has enough time to get stretched entirely before the end of the half-period. This is indicated by a distinct minimum in the evolution of $\omega$ followed by an evolution towards a straight conformation with $\omega \approx 1$.

4. (iv) Tumbling (T). For a large $\theta _0 \sim 46^{\circ }$ (yellow parts of figure 2), the filament spends little time in the compressive quadrant of the flow. As a result, it is found to tumble without any observable deformation, in a way akin to a rigid Brownian rod. This is quantified further by the evolution of the anisotropy parameter $\omega$ that remains close to unity.

We emphasize that we need to distinguish buckling-induced large deformations from fluctuation-induced conformational changes in both our experiments and simulations. This becomes a challenging task for $L \sim \ell _p$. In all of the examples shown here, we have chosen a threshold of $\omega _c = 0.95$ (dash-dotted lines in figure 2b) below which we classify a filament shape as buckled. As we will see in the following subsections, this choice does not significantly alter the major conclusions of our analysis.

Figure 3 indicates the same dynamical transitions as a function of the time period $\rho$, for a fixed $\theta _0$ and $\bar {\mu }_{m}$. As $\rho$ is increased, the orientation amplitude increases (figure 3c). This results in a transition from tumbling for the smallest $\rho$ (yellow) to the occurrence of CB (blue) and eventually BTS (orange) dynamics for the intermediate and largest values of $\rho$, respectively.

Figure 3.

The role of the dimensionless time period $\rho$ on the emerging dynamics over a half time period. The initial orientation in all these examples is fixed at $\theta _0 \approx 25^{\circ }$. Parameters: $\ell _p/L = 1.3$ and $\bar {\mu }_{m} = 5 \times 10^3$. (a) Snapshots from experiments (E) and simulations (S) for increasing values of $\rho$. (b) Evolution of the anisotropy parameter $\omega (t)$ and (c) of the orientation $\theta (t)$. Colour panels and legends are the same as in figure 2.

As highlighted in figures 2 and 3, the two critical aspects of an oscillatory flow are the possibilities of (i) exploring many different initial orientations of the filament, and (ii) controlling independently the angular amplitude of the filament motion. In very strong flows, we can show that these two aspects beget higher-order buckling modes absent in steady shear. Figure 4 illustrates morphological dynamics as a function of the elastoviscous number $\bar {\mu }_{m}$ for a fixed orientation and time period. Identical to steady shear, we find that filaments can have a characteristic C-shaped Euler buckling as $\bar {\mu }_{m}$ is increased. However, unique to the oscillatory forcing is the emergence of S- and W-shaped configurations at large $\bar {\mu }_{m}$ (figures 4b,c). These higher-order buckling modes along with global tumbling of the filament backbone are not observed in steady shear where the filaments are always aligned with the flow direction at the beginning of any time period, and as a result undergo a tank-treading motion with hairpin configurations for sufficiently large $\bar {\mu }_{m}$ (Liu et al. Reference Liu, Chakrabarti, Saintillan, Lindner and du Roure2018).

Figure 4.

Various buckling modes and conformational dynamics observed with increasing values of $\bar {\mu }_{m}$, for a fixed initial orientation $\theta _0 \approx 20^{\circ }$ and dimensionless time period $\rho \approx 13$.

3.2. Suppression of buckling instabilities in strong flows

A remarkable feature in figure 3 is the absence of buckling instabilities at small $\rho$ (yellow), even though the filament is contained entirely in the compressional quadrant of the flow and the elastoviscous number exceeds the theoretical buckling threshold, $\bar {\mu }_{m} \gg \bar {\mu }_{m}^c$. A linear stability analysis in the spirit of Becker & Shelley (Reference Becker and Shelley2001) would suggest the emergence of deformed conformations. However, in the presence of a high-frequency periodic forcing, we observe suppression of finite-size deformations. Interestingly, as illustrated in the yellow parts of figure 2, filament buckling is absent even at large $\rho$ for initial orientations close to $45^{\circ }$, for which the compression rate is maximum.

These two examples highlight the subtle interplay of two nonlinear effects beyond the predictions of a linear stability theory. First, filament orientational dynamics dictate the duration and strength of the effective compressive viscous loading acting along the filament backbone, depending on both filament orientation and instantaneous flow strength. This is sensitive to the initial orientation $\theta _0$ and the time period $\rho$. Second is the coupling between the duration of filament exposure to compressive forces and the characteristic time scale over which deformations grow to be detectable over thermal fluctuations. In § 3.3, we will develop a weakly nonlinear Landau theory that accounts for all of these effects, but first, we focus on their quantitative characterization from experiments and simulations.

To this end, we performed simulations by systematically varying $\rho \in [2,18]$ and $\theta _0 \in [0,{\rm \pi} /2)$ while keeping $\bar {\mu }_{m} = 2\times 10^4 \gg \bar {\mu }_{m}^c$ and $\ell _p/L = 1.3$ fixed. For each combination of $(\rho,\theta _0)$, we performed 50 numerical simulations with different random seeds over one half-period for statistical averaging. This then allows us to define and estimate a probability of buckling as

(3.2)\begin{equation} \mathcal{P}_B(\rho,\theta_0) = \left.\frac{\mathcal{N}(\omega(t) \le \omega_c)}{\mathcal{N}_{{tot}}}\right|_{(\bar{\mu}_{m},\ell_p/L)}, \end{equation}

where $\mathcal {N}(\omega (t) \le \omega _c)$ is the number of cases for which the filament buckled at some point during the half-period, and $\mathcal {N}_{{tot}} = 50$. It is important to point out that the buckled filament conformations will arise due to the various morphological dynamics characterized in the previous subsection. In contrast to the previous characterization, here we focus on understanding the role of flow frequency and filament orientation in the initiation of the buckling instability, independent of the precise nature of the underlying morphological dynamics. The probability of observing specific types of morphological dynamics for a particular set of parameters and initial condition is discussed in Appendix B.

Figure 5(a) displays the buckling probability as a function of the initial filament orientation $\theta _0$ for three different values of the dimensionless time period $\rho$. For $\theta _0 \gtrsim 70^{\circ }$, instability growth is not sufficient to lead to detectable deformations, and irrespective of $\rho$, the buckling probability $\mathcal {P}_B$ is small. For intermediate initial angles, buckling probabilities are significant, and we note that for a given $\theta _0$, the probability of buckling $\mathcal {P}_B$ increases with increasing $\rho$. When initial angles get close to $\theta _0 \sim 0^{\circ }$, a decrease in the buckling probability is again observed.

Figure 5.

(a) Buckling probability $\mathcal {P}_B$ as a function of $\theta _0$ as obtained from simulations (50 independent runs) for three different dimensionless time periods $\rho$. (b) The ensemble-averaged minimum value $\omega _m$ of the deformation parameter, corresponding to the maximum deformation, from the same simulations. (c,d) Phase charts of $\mathcal {P}_B$ and $\omega _{m}$ versus $(\rho,\theta _0)$ for a larger set of dimensionless periods $\rho \in [2,18]$. In (d), the maximum deformation from simulations (isolines) is also compared to experimental data (symbols) under identical conditions. The vertical coloured lines in (c,d) indicate the constant $\rho$ slices shown in (a,b). Parameter values: $\bar {\mu }_{m} = 2\times 10^4$, $\ell _p/L = 1.3$.

Figure 5(b) shows the ensemble-averaged minimum of $\omega (t)$ reached during the evolution over the half-period and serves as a measure of the deformation. The existence of strongly deformed filament conformations with a small value of $\langle \omega _m \rangle$ correlates with a probability of buckling close to unity. These behaviours are further corroborated in the phase plots shown in figures 5(c,d), where we varied systematically the dimensionless period in the range $\rho =2\unicode{x2013}18$. In figure 5(d), superimposed on the simulations, we also indicate experimental results under identical flow conditions and find good quantitative agreement. Note that here, due to intrinsic difficulties in gathering experimental data with identical values of the elastoviscous number, dimensionless period and initial orientation, the ensemble-averaged simulations (isolines) are compared with only single experiments (symbols).

The observations in figure 5 are in qualitative agreement with the nonlinear effects discussed above. For initial orientations larger than $\theta _0 \gtrsim 70^{\circ }$, the time spent by the filament in the compressional quadrant is not sufficient for the instability to grow, regardless of the value of $\rho$. For small initial angles, the growth of the instability is limited by the combined effect of the reduced rotational excursion for these initial angles, which impedes the filament from reaching maximum compression, and the time-varying flow.

3.3. Weakly nonlinear Landau theory

In the previous subsection, we highlighted that even in strong flows, filaments may not undergo detectable buckling instabilities, in contrast to linear stability theory. To explain this finding quantitatively, we propose here a weakly nonlinear Landau theory for filament deformations.

Our theory neglects Brownian fluctuations, and we scale time with the inverse shear rate $\dot {\gamma }_m^{-1}$. In this case, the non-Brownian SBT equation becomes

(3.3)\begin{equation} \bar{\mu}_{m}\,\partial_{t} \boldsymbol{x}(s, t)=\bar{\mu}_{m}\, \boldsymbol{u}_{\infty}(t)-\boldsymbol{\varLambda} \boldsymbol{\cdot}\left[\boldsymbol{x}_{s s s s}-\left(T \boldsymbol{x}_{s}\right)_{s}\right], \end{equation}

where ${\boldsymbol u}_\infty (t) = [\sin (2 {\rm \pi}t/\rho ) y, 0, 0 ]$. We start our analysis by recapitulating the linear stability analysis (Becker & Shelley Reference Becker and Shelley2001). The base state of the filament is an undeformed straight rod. Suppose that at any instant, the rod makes an angle $\theta (t)$ with the negative $x$-axis. We describe the instantaneous orientation of the rod by the director ${\boldsymbol p}(t) = -\cos \theta \,\hat {\boldsymbol {x}} + \sin \theta \,\hat {\boldsymbol {y}}$. We also define the vector orthogonal to the director ${\boldsymbol p}^\perp (t) = \sin \theta \,\hat {\boldsymbol {x}} + \cos \theta \,\hat {\boldsymbol {y}}$. The orientation of the rod is assumed to evolve according to Jeffery's equation (3.1). Associated with this is a time-varying parabolic tension profile given by

(3.4)\begin{equation} T_0(s,t) = \frac{\bar{\mu}_{{inst}}(t)}{8}\left(s^2 - \frac{1}{4}\right), \end{equation}

where

(3.5)\begin{equation} \bar{\mu}_{{inst}}(t) = \bar{\mu}_{m} \sin (2 {\rm \pi}t/ \rho) \sin (2 \theta) \end{equation}

is the instantaneous elastoviscous number experienced by the filament as it rotates in the unsteady flow. In order to first understand the linear stability, we perturb the filament in the transverse direction. The perturbed state is given by ${\boldsymbol x} = s {\boldsymbol p} + h_\perp (s,t)\,{\boldsymbol p}^\perp$, where $h_\perp (s,t) \sim {O}(\varepsilon )$ with $\varepsilon \ll 1$. Linearizing around the base state results in a linear equation for $h_\perp (s,t)$ given by $\bar {\mu }_{m}\,\partial _t h_\perp = \bar {\mu }_{{inst}}(t)\,\mathcal {L}[h_\perp ]/2 - \partial _s^4 h_\perp$, where $\mathcal {L}$ is the differential operator

(3.6)\begin{equation} \mathcal{L}[f] = \left[1 + s \partial_s + \tfrac{1}{4}\left(s^2-\tfrac{1}{4}\right) \partial_s^2 \right]f(s). \end{equation}

Following Becker & Shelley (Reference Becker and Shelley2001), we use the ansatz of normal modes and write $h(s,t) = \phi (s) \exp (\sigma t)$, where $\sigma$ is the growth rate. This leads to an eigenvalue problem for the growth rate $\sigma$ and the eigenfunction $\phi (s)$, from which we find that there is an Euler buckling instability when the elastoviscous number exceeds $\bar {\mu }_{m}^c \approx 306.8$. At the onset of this instability, we have $\sigma = 0$ and $\phi = \phi ^c(s)$. This means that the eigenfunction at the critical threshold satisfies

(3.7)\begin{equation} \mathcal{L}[ \phi^c] = \frac{2}{\bar{\mu}_{m}^c}\,\partial_s^4 \phi^c. \end{equation}

We now proceed to describe the nonlinear dynamics of the filament away from the instability threshold. For that purpose, we develop a weakly nonlinear theory in the vicinity of the first bifurcation. We expand the transverse perturbation on the basis of the first mode of deformation $\phi ^c(s)$ as $h_\perp (s,t) = A(t)\,\phi ^c(s)$, where $A(t) \sim {O}(\varepsilon )$. Transverse perturbations induce changes in length along the axis of the filament. As we will see, these length changes are higher order in $A(t)$. We account for this by introducing axial perturbations $h_{\parallel }(s,t)$. The perturbed filament conformation can then be written as

(3.8)\begin{equation} {\boldsymbol x}(s,t) = \left(s + h_\parallel\right) {\boldsymbol p} + h_\perp {\boldsymbol p}^\perp. \end{equation}

Similarly, we perturb the tension around its base state $T_0(s,t)$ as

(3.9)\begin{equation} T(s,t) = T_0(s,t) + \tilde{T}(s,t). \end{equation}

We choose to describe the filament backbone and its mean orientation with the rigid-rod model, which is supported by the observations of figure 2. So at any instant, we assume that the orientation $\theta (t)$ follows (3.1). We next outline the set of steps to obtain the evolution equation for the amplitude $A(t)$.

1. (i) Inextensibility. Filament inextensibility implies that ${\boldsymbol x}_s \boldsymbol {\cdot } {\boldsymbol x}_s = 1$, which yields

   (3.10)\begin{equation} {\boldsymbol x}_s \boldsymbol{\cdot} {\boldsymbol x}_s = 1 + \left[2 \partial_s h_{{\parallel}} + (\partial_s h_\perp)^2 \right] + (\partial_s h_{{\parallel}})^2. \end{equation}
   We first note that in the absence of any axial perturbations, the inextensibility condition was satisfied up to ${O}(\varepsilon ^2)$. This points to the fact that $h_{\parallel } \sim {O}(\varepsilon ^2)$. We can now satisfy inextensibility up to ${O}(\varepsilon ^4)$ and obtain $h_{\parallel }(s,t)$ by setting the second term in (3.10) to zero. Thus we obtain
   (3.11)\begin{equation} h_\parallel(s,t) ={-}\frac{1}{2}\int_{{-}1/2}^s \left[\partial_{s'} h_\perp(s',t) \right]^2 \mathrm{d} s'= {A(t)^2}\,v(s), \end{equation}
   where we have defined $v(s) = -\frac {1}{2}\int _{-1/2}^{s} \phi ^c_{s'}(s')^2\,{\mathrm {d}} s'$.

2. (ii) Tension perturbation. In order to solve for the tension, we first derive an auxiliary equation from the SBT using the fact that $\partial _t {\boldsymbol x}_s \boldsymbol {\cdot } {\boldsymbol x}_s = 0$. This provides a second-order ordinary differential equation for the tension $T(s,t)$ as

   (3.12)\begin{equation} 2 T_{s s}- ({\boldsymbol x}_{ss}\boldsymbol{\cdot} {\boldsymbol x}_{ss}) T={-}\bar{\mu}_{m} {\boldsymbol x}_s \boldsymbol{\cdot} \partial_s {\boldsymbol u} -6 {\boldsymbol x}_{s s s} \boldsymbol{\cdot} {\boldsymbol x}_{s s s}-7 {\boldsymbol x}_{s s} \boldsymbol{\cdot} {\boldsymbol x}_{s s s s} \equiv \mathcal{R}. \end{equation}
   Retaining terms up to $A(t)^3$, the equation for the tension perturbation becomes
   (3.13)\begin{align} 2 \tilde{T}_{ss} &= \left(\bar{\mu}_{m}\,A(t) \cos(2 \theta)\,\phi^c_s + A(t)^2 \left[-\bar{\mu}_{m} \sin(2 \theta)\,(\phi^c_s)^2 - 6 (\phi^c_{sss})^2 - 7 \phi^c_{ss} \phi^c_{ssss} \right] \right) \nonumber\\ &\quad \times\sin(2 {\rm \pi}t/ \rho)+ A(t)^2\,(\phi^c_{ss})^2\,T_0(s,t). \end{align}
   The above linear system can be solved numerically along with the boundary condition $\tilde {T}(s=\pm 1/2) = 0$.

3. (iii) Amplitude equation at the linear order. Before deriving the nonlinear amplitude equation, we first consider the nature of the linearized amplitude equation. At the linear order, we have

   (3.14)\begin{equation} \bar{\mu}_{m}\,\frac{{\mathrm{d}} A}{{\mathrm{d}} t}\,\phi^c(s) = \left(\bar{\mu}_{{inst}}(t)\,\mathcal{L}[\phi^c]/2 - \partial^4_s \phi^c \right)A(t). \end{equation}
   On using (3.7), we can simplify the above equation and obtain
   (3.15)\begin{equation} \frac{{\mathrm{d}} A}{{\mathrm{d}} t} = \left\langle \phi^c,\partial^4_s \phi^c \right\rangle \left[\frac{\bar{\mu}_{{inst}}(t) - \bar{\mu}_{m}^c}{\bar{\mu}_{m}\bar{\mu}_{m}^c} \right] A(t), \end{equation}
   where $\langle f,g \rangle$ is the inner product of two functions. The above relation amounts to recasting the results of the linear stability analysis, and predicts exponential growth of the amplitude $A(t)$ whenever $\bar {\mu }_{{inst}} > \bar {\mu }_{m}^c$.

4. (iv) Landau equation for the amplitude. In order to derive a nonlinear amplitude equation, we make use once again of the SBT equation. Using our ansatz for the perturbations, we project the equations onto the transverse direction. Retaining all the terms up to order $A(t)^3$ and using the solution from the perturbed tension equation, we can obtain a nonlinear amplitude equation of the form

   (3.16)\begin{equation} \frac{{\mathrm{d}} A}{{\mathrm{d}} t} = \left\langle \phi^c,\partial^4_s \phi^c \right\rangle \left[\frac{\bar{\mu}_{{inst}}(t) - \bar{\mu}_{m}^c}{\bar{\mu}_{m}\bar{\mu}_{m}^c} \right] A(t) + \frac{1}{\bar{\mu}_{m}}\left\langle \phi^c, {\varOmega} \right\rangle. \end{equation}
   In this equation, $\varOmega$ captures the effect of nonlinearities and is given by
   (3.17)\begin{align} {\varOmega} = A(t)^2\,v(s) \sin^2 \theta + A(t) \left[2 \phi^c_s \tilde{T}_s + \tilde{T} \phi^c_{ss}\right] - A(t)^3 \left[\phi^c_{s} v_{ssss} + (\phi^c_s)^2 \phi^c_{ssss}\right]. \end{align}
   It is worth pointing out that the operator in (3.7) used for the stability analysis is not self-adjoint, and as a result the eigenfunctions of (3.7) are not orthonormal. However, this subtlety does not show up in the present analysis since we are interested only in the first mode of deformation. It would be interesting to formulate a theory that expands upon this analysis and incorporates the dynamics of higher-order buckling modes. This would require the use of the adjoint eigenmodes as done in our past work (Chakrabarti et al. Reference Chakrabarti, Liu, LaGrone, Cortez, Fauci, du Roure, Saintillan and Lindner2020).

5. (v) Numerical solution. We integrate (3.16) numerically using a fourth-order Runge–Kutta scheme. The amplitude equation is complemented with (3.1), which provides the instantaneous orientation of the filament. At every time step, we solve the linear system for the tension perturbation $\tilde {T}(s)$ to compute the nonlinear terms ${\varOmega }$.

In our numerical explorations, we varied $\rho$ and $\theta _0$, and studied the evolution of filament deformation predicted from the nonlinear theory. We initiated the simulations by perturbing the filament backbone as ${\boldsymbol x}(s,t=0) = s {\boldsymbol p} + A_0\,\phi ^c(s)\,{\boldsymbol p}^\perp$, where $A_0$ is the initial perturbation amplitude. For all our simulations, we chose $A_0 = 0.02$ for the initial amplitude, which is the typical magnitude of transverse fluctuations of a freely suspended actin filament with $\ell _p/L \sim {O}(1)$ as considered in our experiments. We characterize an event as buckling in our theory whenever the amplitude exceeds $A \approx 0.04$, which corresponds to an anisotropy parameter $\omega _c \approx 0.95$. We show the phase chart of minimum $\omega _m$ computed from the theory in figure 6(a). It predicts that buckling is favoured only for certain ranges of $\theta _0$, and that its probability increases with increasing $\rho$, an observation consistent with our experiments and Brownian simulations shown in § 3.2.

Figure 6.

(a) Phase plot of the maximum deformation for $\bar {\mu }_{m} = 3.5 \times 10^3$ as computed from the nonlinear Landau theory outlined in § 3.3. The two vertical coloured lines indicate slices of $\rho = 6$ (b) and $\rho = 16$ (c), respectively, for which we show the evolution of the growth rate $\sigma (t)$ and the perturbation amplitude $A(t)$ as functions of the instantaneous orientation of the filament, for several values of the initial angle $\theta _0$.

In order to explain this behaviour, we now plot the evolution of the growth rate $\sigma (t)$ and the amplitude of the perturbation $A(t)$ as functions of the orientation angle of the filament $\theta (t)$ from the nonlinear model. Typical data for two values of $\rho$ and several initial orientations $\theta _0$ are reported respectively in figures 6(b) ($\rho =6$) and 6(c) ($\rho =16$).

A number of observations can be made from these figures. First, the range of orientation angles traversed by the filament varies strongly as a function of the initial orientation, even for a fixed value of $\rho$ (compare, for example, the blue and red curves in figure 6b). This is linked to the fact that the Jeffery dynamics is slow close to alignment with the negative flow direction, but faster for increasing initial angles. Filaments starting aligned close to the negative flow direction thus do not rotate very far compared to filaments with initial angles closer to 45$^\circ$. Second, filaments that start close to 90$^\circ$ will spend little time in the compressive quadrant, where positive growth rates can exist. In a time-independent shear flow, all filaments will experience maximum compression and thus a maximum growth rate at 45$^\circ$. In the case of an oscillatory flow, the flow forcing itself is time-dependent and the maximum growth rate results from a combination of the orientation angle and the time-dependent flow intensity, as is clear from the figures where maximum growth rates are not necessarily observed at 45$^\circ$, even for filaments sweeping by this orientation. Comparing figures 6(b) ($\rho =6$) and 6(c) ($\rho =16$), it is obvious that larger values of $\rho$ lead to larger growth rates over more extended ranges of angles. It is also interesting to note that negative growth rates are observed even in the compressive quadrant, for situations where $\bar {\mu }_{inst}$ remains below the buckling threshold.

As pointed out above, to observe a buckling instability, it is not sufficient for the filament to experience a positive growth rate; the perturbation also needs sufficient time to grow so as to lead to a measurable signature in the anisotropy parameter $\omega$. This can be seen from the bottom plots of figures 6(b,c), which show the instantaneous amplitude resulting from the cumulated growth of the perturbation during rotation of the filament through the compressive quadrant. These examples thus underscore a key aspect of our experimental measurements: linear instability does not always correspond to detectable deformations of the filament.

This critical distinction between linear instability and measurable deformations also applies to steady shear flows, even if to a lower extent, but seems to have been unreported. We elaborate further on this point in Appendix C.

3.4. Dynamics over one full period

So far, we have focused on the deformation dynamics in the first half of the forcing period, considering only the particular case of a straight filament at $t=0$. This has helped us to understand the role of the time period $\rho$ and the initial orientation $\theta _0$ that can be both controlled independently and varied systematically. We now discuss briefly the filament dynamics over one entire period of the oscillatory flow. The key aspect here is that the orientation and the deformation of the filament at the beginning of the second half of the period is governed by its deformation and orientational dynamics during the first half, giving rise to new, interesting dynamics. In particular, one also has to consider cases in which the filament is already deformed at the beginning of the second half-cycle. The interplay between oscillation period, flow strength, initial orientation and deformation of the filament leads to a very rich spectrum of dynamical modes. Typical examples are reported in figure 7. In figure 7(a), the filament undergoes a continuous buckling (CB) event during the first half-cycle. The filament remains oriented in the extensional quadrant when the flow is reversed. As a result, it is stretched by the flow during the second half-cycle, and the anisotropy parameter increases continuously. As $\rho$ is increased (figure 7b), the dynamics in the first half of the period transitions to a BTPS event. Consequently, when the shear is reversed, the filament first experiences compression, deforming from an already buckled state. As the filament enters the extensional quadrant, it is then stretched out by the flow.

Figure 7.

Dynamical features over a complete period of tumbling. During the second half-period, the filament can be either continuously stretched by the flow (a) or compressed once again (b). Parameter values: $\ell _p/L = 1.3$ and $\bar {\mu } = 1 \times 10^4$.

These examples give a glimpse of the complex evolution of the filament deformation over multiple cycles. Since the deformation in general brings about irreversible orientational dynamics ($\theta (T) \neq \theta _0$), the asymptotic filament dynamics is expected to be characterized by the emergence of a chaotic behaviour, even in the athermal limit and at small Reynolds number.