2.1. Bead model

The bead-spring model illustrated in figure 1(a) describes elastic and hydrodynamic interactions between $n$ numbered $i = 1,\ldots ,n$ spherical beads of identical radii $a$ ($i$th bead position is denoted as $\boldsymbol {R}_i$). In this work, we use three different bead models $\mathcal {M}_i,\ i=1,2,3$ (cf. table 1), which include combinations of two different descriptions of hydrodynamic interactions (HI), described below and in appendix A: the generalized Rotne–Prager–Yamakawa (GRPY) method (Wajnryb et al. Reference Wajnryb, Mizerski, Zuk and Szymczak2013; Zuk et al. Reference Zuk, Cichocki and Szymczak2017) and the multipole method with lubrication correction (Hydromultipole) (Cichocki, Ekiel-Jeżewska & Wajnryb Reference Cichocki, Ekiel-Jeżewska and Wajnryb1999; Ekiel-Jeżewska & Wajnryb Reference Ekiel-Jeżewska and Wajnryb2009) with two sets of constitutive laws specifying elastic interactions that are described next.

Table 1. Physical assumptions of the bead-spring models $\mathcal {M}_1$, $\mathcal {M}_2$ and $\mathcal {M}_3$.

The results presented in the following sections are based on the numerical simulations from the bead models $\mathcal {M}_1$ and $\mathcal {M}_2$. Both of them have the same long-distance asymptotics of the hydrodynamic interactions, and for close beads the Hydromultipole method is more precise than the GRPY. However, the computations based on the Hydromultipole algorithm require more time and memory than the GRPY approach. The GRPY model has been therefore used in simulations of very long fibres. For $n\le 100$ both $\mathcal {M}_1$ and $\mathcal {M}_2$ have been applied.

Qualitative results from the bead-spring models $\mathcal {M}_1$ and $\mathcal {M}_2$ are similar and in regimes of large $A$ and $n$ they are also the same quantitatively. A detailed comparison of the results obtained within both models is given in appendix B. The model $\mathcal {M}_3$ (see table 1) is applied there to explain that some differences between the models $\mathcal {M}_1$ and $\mathcal {M}_2$ are related to different expressions for the bending potential energy, in agreement with the predictions of Bukowicki & Ekiel-Jeżewska (Reference Bukowicki and Ekiel-Jeżewska2018).

2.1.1. Elastic interactions

An elastic interaction potential model (constitutive laws) specifies a sum $\mathcal {E}$ of all bead–bead interaction energies, which are used to calculate elastic forces $\boldsymbol {F}_i=-\boldsymbol {\nabla }_{\boldsymbol {R}_i} \mathcal {E}$ acting on each bead $i$. We assume that there are no elastic torques acting on the beads. For every pair of beads $i,j$ the connector vector $\boldsymbol {R}_{ij}=\boldsymbol {R}_j-\boldsymbol {R}_i$ points from the centre of the sphere $i$ towards the centre of sphere $j$.

For the first set of constitutive laws, set 1, between the centres of every two consecutive beads $i$ and $i+1$ we impose the FENE (finitely extensible nonlinear elastic) stretching potential energy (Warner Reference Warner1972)

(2.2)\begin{equation} E_{s,ij} = \frac{k'_s R_m^2}{2}\log \left[ 1 - \left(\frac{R_{ij}-2a}{R_m}\right)^2 \right],\end{equation}

where $j=i+1$, $k'_s$ is the spring stiffness, $R_m=0.2a$ is the maximum stretching from the equilibrium distance and $R_{ij}=|{\boldsymbol {R}}_{ij}|$. Between every triplet of beads $i-1,i,i+1$ we impose a harmonic bending potential energy,

(2.3)\begin{equation} E_{b,i} = \frac{A'}{2} (\theta_0 - \theta_i)^2,\end{equation}

where $\theta _i$ and $\theta _0$ are, respectively, the time-dependent and the equilibrium angles between connector vectors $\boldsymbol {R}_{i,i-1}$ and $\boldsymbol {R}_{i,i+1}$, and $A'=EI/L_0$ is the bending stiffness (per unit length), with the Young modulus $E$, the moment of inertia $I={\rm \pi} a^4/4$ and the distance $L_0$ between the centres of the consecutive beads. Because a fibre is straight, when in equilibrium, the angle $\theta _0={\rm \pi}$. In the set 1 of the constitutive laws we assume that $L_0=2a$.

For the second set of constitutive laws, set 2, between centres of every two consecutive beads $i$ and $i+1$ we impose a harmonic stretching potential energy

(2.4)\begin{equation} E_{s,ij} = \frac{k'_s}{2} (R_{ij}-L_0)^2,\end{equation}

with $j=i+1$ and the equilibrium distance $L_0$ between the bead centres usually close to $R_0=2a$ but a bit larger. Between every triplet of beads $i-1,i,i+1$ we impose a cosine (Kratky–Porod) bending potential energy

(2.5)\begin{equation} E_{b,i} = A' \left(1 + \frac{\boldsymbol{R}_{i,i-1}\boldsymbol{\cdot} \boldsymbol{R}_{i,i+i}}{|\boldsymbol{R}_{i,i-1}||\boldsymbol{R}_{i,i+i}|}\right) = A' (1 + \cos\theta_i ). \end{equation}

This potential energy is a widely used discrete approximation of the elastic bending stiffness, see e.g. Gauger & Stark (Reference Gauger and Stark2006).

Additionally when the GRPY model of hydrodynamic interactions is used we add the repulsive part of the Lennard-Jones potential energy

(2.6)\begin{equation} E_{R,ij} = 4 \epsilon'_{LJ} \left( \frac{\sigma_{LJ}}{R_{ij}} \right)^{12}\end{equation}

between the second nearest or further neighbour beads, where $\epsilon '_{LJ}$ determines the strength of the potential and $\sigma _{LJ}$ is the characteristic distance. We set $\sigma _{LJ}=2a$ and truncate the Lennard-Jones interaction range to $2.5 \sigma _{LJ}$. This potential acts to prevent large overlaps of the beads (comparable with $2a - R_{ij} \ll a$ for $R_{ij}<2a$). This is not necessary for the Hydromultipole model because the lubrication forces prevent the beads from overlapping.

2.1.2. Hydrodynamic interactions

In this work, we study translational motion of segments of a flexible fibre. In the framework of the bead-spring modelling, the translational motion of the fibre beads is determined by the theory of hydrodynamic interactions between spherical particles. We consider $n$ spherical particles in a fluid of viscosity $\mu _0$ subject to an incompressible external flow $\boldsymbol {V}_{\infty }(\boldsymbol{R})$. We investigate the case where the Reynolds number is much smaller than unity and the quasi-steady fluid velocity $\boldsymbol {V}(\boldsymbol {R})$ and pressure $p(\boldsymbol {R})$ are described by the Stokes equations (Oseen Reference Oseen1927; Kim & Karrila Reference Kim and Karrila1991).

The theory of hydrodynamic interactions is used to calculate the translational velocities $\boldsymbol {U}_i$ of the particles, which are in turn necessary to integrate the particle trajectories. In our case the external flows are linear, and there are no torques applied to the particles. Therefore the translational velocities $\boldsymbol {U}_{i}$ satisfy the relations,

(2.7)\begin{equation} \boldsymbol{U}_{i} = \boldsymbol{V}_{\infty}(\boldsymbol{R}_i) + \sum_{j=1}^n \left(\boldsymbol{\mu}^{tt}_{ij} \boldsymbol{\cdot} \boldsymbol{F}_j + \boldsymbol{\mu}^{td}_{ij} : \boldsymbol{E}_{\infty}\right), \end{equation}

where $\boldsymbol {F}_j$ is the total external force exerted on the particle $j$ and $\boldsymbol {E}_{\infty }=(\boldsymbol {\nabla } \boldsymbol {V}_{\infty }+ (\boldsymbol {\nabla } \boldsymbol {V}_{\infty })^{\textrm {T}})/2$ denotes the rate-of-strain tensor of the external fluid flow $\boldsymbol {V}_{\infty }$. For the shear flow given by (2.1),

(2.8)\begin{equation} \boldsymbol{E}_{\infty} = \frac{\dot{\gamma}}{2}\left(\begin{array}{@{}ccc@{}} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{array}\right). \end{equation}

There are different methods to evaluate the translational–translational $\boldsymbol {\mu }^{tt}_{ij}$ and translational–dipolar $\boldsymbol {\mu }^{td}_{ij}$ mobility matrices. The most precise is the multipole expansion, corrected for lubrication, in order to speed up the convergence (Durlofsky, Brady & Bossis Reference Durlofsky, Brady and Bossis1987; Cichocki et al. Reference Cichocki, Felderhof, Hinsen, Wajnryb and Bławzdziewicz1994; Sangani & Mo Reference Sangani and Mo1994; Cichocki et al. Reference Cichocki, Ekiel-Jeżewska and Wajnryb1999; Ekiel-Jeżewska & Wajnryb Reference Ekiel-Jeżewska and Wajnryb2009) through the inverse-power expansion in the inter-particle distance (Kim & Karrila Reference Kim and Karrila1991). The analytical Rotne–Prager–Yamakawa approximation is also often used (Rotne & Prager Reference Rotne and Prager1969).

In this work we evaluate the mobility matrices as functions of the positions of all the beads using the two methods outlined in appendix A. First, we apply the analytical Rotne–Prager–Yamakawa approximation of the translational–translational mobility $\boldsymbol {\mu }^{tt}_{ij}$ (Rotne & Prager Reference Rotne and Prager1969), generalized also for the translational–dipolar mobility matrix $\boldsymbol {\mu }^{td}_{ij}$ (Kim & Karrila Reference Kim and Karrila1991) and implemented in the GRPY numerical program. Second, we use the precise multipole method corrected for lubrication, implemented in the numerical code Hydromultipole. The GRPY procedure is less precise, when particle surfaces are closer than the radius of the smaller particle, but computationally much faster than the Hydromultipole algorithm. Both methods will be briefly outlined in appendix A.

The equations of motion for the positions $\boldsymbol {R}_i$ of the beads are

(2.9)\begin{equation} \dot{\boldsymbol{R}}_{i} = \boldsymbol{U}_{i}, \end{equation}

with $\boldsymbol {U}_i$ dependent on the positions $\boldsymbol {R}_{j}$ of all the bead centres $j=1,\ldots ,n$, and given by (2.7).

The equations of motion (2.9) are solved numerically with the use of dimensionless variables. We choose as a characteristic length the bead diameter $2a$. The total length of the fibre at equilibrium is $L$, which in the case of the $\mathcal {M}_1$ model is fixed to $L=2na$ so that the fibre aspect ratio is $n$. We choose as a time scale the inverse of the shear rate $\dot {\gamma }^{-1}$ and the forces are normalized with ${\rm \pi} \mu _0 \dot {\gamma } (2a)^2$. The above introduces the dimensionless stretching stiffness $k_s=k'_s/({\rm \pi} \mu _0 \dot {\gamma } (2a))$, $\epsilon _{LJ}=\epsilon '_{LJ}/({\rm \pi} \mu _0 \dot {\gamma } (2a)^3)$ and the bending stiffness

(2.10)\begin{equation} A=A'/({\rm \pi} \mu_0 \dot{\gamma} (2a)^3)=EI/({\rm \pi} \mu_0 \dot{\gamma} L_0 (2a)^3). \end{equation}

For the GRPY approach, $L_0=2a$. Note that for the Hydromultipole treatment of the hydrodynamic interactions, the dimensionless bending stiffness ratio $A$ used here is slightly different from the corresponding parameter $EI/({\rm \pi} \mu _0 \dot {\gamma } (2a)^4)$ used by Słowicka et al. (Reference Słowicka, Wajnryb and Ekiel-Jeżewska2013, Reference Słowicka, Wajnryb and Ekiel-Jeżewska2015, Reference Słowicka, Stone and Ekiel-Jeżewska2020) and denoted there by the same letter. To adjust for this difference, all the numerical values of the bending stiffness based on the Hydromultipole codes taken from earlier works were in this paper divided by $L_0/(2a)$ (typically equal to 1.02 or 1.01, see appendix B).

2.1.3. Why a fibre aligned with the flow moves out of this position

To answer this question, we will use (2.7) to analyse the velocities of the beads for a fibre aligned with the flow and at the elastic equilibrium. We will use the standard pairwise-additive Rotne–Prager–Yamakawa (RPY) approximation for the distinct mobility matrices $\boldsymbol {\mu }^{tt}_{ij}$ and $\boldsymbol {\mu }^{td}_{ij}$ with $i \ne j$ (Kim & Karrila Reference Kim and Karrila1991). From the geometric symmetry we can write down the tensorial form of the mobility matrices for a pair of particles $i$ and $j$ (Kim & Karrila Reference Kim and Karrila1991),

(2.11a)\begin{gather} \boldsymbol{\mu}^{tt}_{ij} = A (R_{ij}) \boldsymbol{d}_{ij} \boldsymbol{d}_{ij} + B (R_{ij}) (\boldsymbol{I} - \boldsymbol{d}_{ij}\boldsymbol{d}_{ij}), \end{gather}

(2.11b)\begin{gather}\boldsymbol{\mu}^{td}_{ij} = C (R_{ij}) \big( \boldsymbol{d}_{ij} \boldsymbol{d}_{ij} - \tfrac{1}{3}\boldsymbol{I} \big) \boldsymbol{d}_{ij} + D (R_{ij}) \overline{\boldsymbol{d}_{ij}(\boldsymbol{I}-\boldsymbol{d}_{ij}\boldsymbol{d}_{ij})}, \end{gather}

where $\boldsymbol {I}$ is the unit tensor, $\boldsymbol {d}_{ij} = \boldsymbol {R}_{ij}/|\boldsymbol {R}_{ij}|$, and $\overline {\boldsymbol {d}_{ij}(\boldsymbol {I}-\boldsymbol {d}_{ij}\boldsymbol {d}_{ij})}$ is a third rank tensor symmetric and traceless in the first and second Cartesian components, i.e. $\overline {\boldsymbol {d}_{ij} (\boldsymbol {I}-\boldsymbol {d}_{ij}\boldsymbol {d}_{ij})} _{\alpha \beta \gamma } = \frac {1}{2}( d_\alpha \delta _{\beta \gamma } + d_\beta \delta _{\alpha \gamma })-d_\alpha d_\beta d_\gamma$, where the Cartesian components are labelled with the Greek letters. Within the RPY approximation, the translational– translational self-mobility matrix

(2.12)\begin{equation} \boldsymbol{\mu}^{tt}_{ii}=\frac{1}{6 {\rm \pi}\mu_0 a} {\boldsymbol I}\end{equation}

and the translational–dipolar self-mobility matrix vanishes, $\boldsymbol {\mu }^{td}_{ii}=\boldsymbol {0}$.

Our goal now is to investigate the initial configuration, when the fibre is parallel to the flow. In this case $\boldsymbol {d}_{ij} = \pm \hat {\boldsymbol {e}}_x$, with the minus sign for the beads with labels $i>j$. Since the fibre is at the elastic equilibrium, the external forces vanish, $\boldsymbol {F}_j=\boldsymbol {0}$, and the only contribution to velocity in the direction perpendicular to the flow comes from the translational–dipolar mobility. From (2.11b) it follows that the contribution to the velocity $\boldsymbol {U}_{i}$ of particle $i$ from the translational–dipolar mobility $\boldsymbol {\mu }^{td}_{ij}$ acting on the strain tensor $\boldsymbol {E}_{\infty }$ (where $[\boldsymbol {A}:\boldsymbol {B}]_{ij} = A_{ik} B_{kj}$) consists of two terms proportional to

(2.13a,b)\begin{align} \left(\boldsymbol{d}_{ij} \boldsymbol{d}_{ij}-\frac{1}{3}\boldsymbol{I}\right) \boldsymbol{d}_{ij}: \boldsymbol{E}_{\infty} = \frac{\dot{\gamma}}{2}\left(\begin{array}{c} 0 \\ 0 \\ 1/3 \end{array}\right)\quad \mbox{and}\quad \overline{\boldsymbol{d}_{ij} (\boldsymbol{I} -\boldsymbol{d}_{ij} \boldsymbol{d}_{ij})}: \boldsymbol{E}_{\infty}= \frac{\dot{\gamma}}{2} \left(\begin{array}{c} 0 \\ 0 \\ -1 \end{array}\right), \end{align}

respectively. Therefore, there exist contributions to the bead velocities perpendicular to the flow, and this is why the fibre moves out of the position aligned with the flow. In the next section, we will show that the largest are perpendicular velocities of the first and last beads, at the initial configuration aligned with the flow, and also later when the fibre is slightly deflected. We will also demonstrate that this effect can be considered as the result of a hydrodynamic force exerted by the shear flow on the fibre.

2.1.4. Hydrodynamic force acting on the tip of the fibre initially aligned with the flow

We now move on to the discussion of the hydrodynamic force exerted by the shear flow on the tip of a fibre aligned with the flow or already slightly deflected from the alignment. In the following, we are going to provide the theoretical explanation for the initial stage of the bending process in terms of the elastica, based on the assumption that a constant hydrodynamic force is exerted on the fibre end by the shear flow. In the standard use of the elastica equations the existence of such a force has not been yet taken into account. Here, we use the general framework of the theory of hydrodynamic interactions presented in the previous sections to explain the physical origin of this force, and to estimate its value numerically (with the bead model $\mathcal {M}_1$).

In the bead models, the tip force can be found rewriting (2.7)

(2.14)\begin{equation} \dot{\boldsymbol{R}}_i - \boldsymbol{V}_{\infty}(\boldsymbol{R}_i) = \boldsymbol{\mu}^{tt}_{ii} \boldsymbol{\cdot} \left( \boldsymbol{F}_i + (\boldsymbol{\mu}^{tt}_{ii})^{-1} \sum_{j} \boldsymbol{\mu}^{td}_{ij} : \boldsymbol{E}_{\infty}\right)+\sum_{j \neq i} \left(\boldsymbol{\mu}^{tt}_{ij} \boldsymbol{\cdot} \boldsymbol{F}_j\right), \end{equation}

where $\boldsymbol {\mu }^{tt}_{ii}$ is the translational self-mobility matrix, and defining the hydrodynamic force acting on bead $i$ as

(2.15)\begin{equation} \boldsymbol{F}_{Hi} = (\boldsymbol{\mu}^{tt}_{ii})^{-1} \sum_{j} \boldsymbol{\mu}^{td}_{ij} : \boldsymbol{E}_{\infty}. \end{equation}

The dimensionless form is $\boldsymbol {f}_{Hi}=\boldsymbol {F}_{Hi}/( {\rm \pi}\mu _0 \dot {\gamma } (2a)^2 )$.

Our goal is to investigate ${\boldsymbol F}_{Hi}$ at the early stage of the evolution, when the fibre, initially aligned with the flow, slowly moves out of this configuration, but still remains almost parallel to the flow. We will now show that, for the fibre almost aligned with the flow, the hydrodynamic forces $\boldsymbol {F}_{Hi}$, defined by (2.15), are almost perpendicular to the flow direction $\hat {\boldsymbol {e}}_x$. We will also provide some theoretical arguments why the value of ${\boldsymbol F}_{Hi}$ is the largest at the ends of the fibre.

The hydrodynamic forces ${\boldsymbol F}_{Hi}$ given by (2.15) are proportional to the shear rate $\dot {\gamma }$. Moreover, the force $\boldsymbol {F}_{Hi}$ is perpendicular to the flow and parallel to the $z$ direction of the flow gradient, ${\boldsymbol F}_{Hi} \approx \hat {\boldsymbol e}_z \boldsymbol{\cdot} {\boldsymbol F}_{Hi} \hat {\boldsymbol e}_z$. Therefore, they displace the fibre beads away from the position aligned with the flow. From the explicit expressions for the functions $C$ and $D$ in (2.11b), given e.g. by Kim & Karrila (Reference Kim and Karrila1991), it follows that for the first bead $\hat {\boldsymbol e}_z \boldsymbol{\cdot} {\boldsymbol F}_{H1} > 0$ and for the last bead $\hat {\boldsymbol e}_z \boldsymbol{\cdot} {\boldsymbol F}_{Hn} < 0$. This means that, owing to the hydrodynamic forces (2.15), the fibre follows the rotational component of the shear flow.

It is also known that $C \propto R_{ij}^{-2}$ and $D\propto R_{ij}^{-4}$, see e.g. Kim & Karrila (Reference Kim and Karrila1991). Therefore, the major contribution to $F_{Hi}$ comes from relatively close beads $j$. Additionally, $\boldsymbol {\mu }^{td}_{ij}$ is antisymmetric in $\boldsymbol {d}_{ij}$, which means that the terms in (2.15) corresponding to equally distant left and right neighbours will cancel. Therefore, the total force $F_{Hi}$ is close to zero for $i$ in the middle part of the fibre, and it increases when $i$ is closer to the fibre ends. For longer fibres, the force $F_{Hi}$ is non-negligible only for $i$ close to one of the fibre ends, and it only weakly depends on the total fibre length because it comes from unbalanced local interactions between the bead $i$ and close beads $j$.

To evaluate $\boldsymbol {f}_{Hi}$ numerically, we use the pairwise-additive GRPY approximation for the mobility matrices. As argued above, in the stage when fibre is only slightly deflected from the straight line, at leading order, $\boldsymbol {f}_{Hi}$ is directed along $\hat {\boldsymbol {e}}_z$. In figure 2(a) we plot the dimensionless hydrodynamic force $\hat {\boldsymbol {e}}_{z} \boldsymbol{\cdot} \boldsymbol {f}_{Hi}$ as a function of the bead label $i$ for three different fibre lengths $n$. It is clear that the force is well localized close to the fibre ends.

Figure 2. Hydrodynamic forces normal to the fibre, acting on bead $i$, calculated from (2.15) with the GRPY model of the hydrodynamic interactions. (a) Spatial distribution of the forces on the beads along the fibre. (b) The force on the first bead as a function of the fibre aspect ratio $n$.

The orientation of $\boldsymbol {f}_{Hi}$ follows the rotational component of the shear flow. As the fibre gets longer, the force is more localized. Regardless of the fibre length, the end beads support the largest forces, an order of magnitude larger than the forces acting on the other beads. The magnitude of the force acting on the first bead, $\boldsymbol {f}_{H1} \boldsymbol{\cdot} \hat {\boldsymbol {e}}_z$, initially changes non-monotonically as a function of $n$ (see figure 2b), until it reaches a limiting value $\,f_H \approx 0.16$. Indeed, we observe a localized, length-independent tip force perpendicular to the flow. We will use this observation later to construct a modified elastica model, applicable for a fibre initially aligned with the flow. Now it is time to present the standard Euler–Bernoulli beam, based on the local slender-body theory.

2.2. The elastica and local slender-body theory

To rationalize the results of numerical simulations from the bead-spring simulations the inextensible elastica model (Duprat & Stone Reference Duprat and Stone2015; Lindner & Shelley Reference Lindner and Shelley2015) is used with the local slender-body theory (SBT) (Cox Reference Cox1970; Keller & Rubinow Reference Keller and Rubinow1976; Johnson Reference Johnson1980) to account for the drag forces acting along the fibre. Within the local SBT, in contrast to the bead models, the full long-ranged hydrodynamic interactions are not incorporated, nor is the finite but small thickness of the filament. The last feature is especially important for fibres that are aligned with the flow, as will be discussed in detail later. Similarly as in the bead model, for the elastica we also neglect Brownian motion and buoyancy forces. The fibre has a circular cross-section of radius $a$ and length $L=2na$ where $\epsilon = {a}/{L} = {1}/{2n} \ll 1$. We denote as $\boldsymbol {R}(S,T)$ the dimensional position of a fibre segment at the arc length $S$ at time $T$. The equation of motion of each filament segment as a result of the applied elastic force density $\boldsymbol {F}(S,T)$ per unit length, under the steady undisturbed flow $\boldsymbol {V}_{\infty }$ can be expressed using the SBT (Cox Reference Cox1970; Duprat & Stone Reference Duprat and Stone2015; Lindner & Shelley Reference Lindner and Shelley2015)

(2.16)\begin{equation} \dot{\boldsymbol{R}} - \boldsymbol{V}_{\infty}(\boldsymbol{R}) = \frac{\ln(\epsilon^{-1})}{4 {\rm \pi}\mu_0} \left( \boldsymbol{I} + \boldsymbol{R}_S \boldsymbol{R}_S \right) \boldsymbol{\cdot} \boldsymbol{F}(S,T), \end{equation}

or alternatively

(2.17)\begin{equation} \frac{2 {\rm \pi}\mu_0}{\ln(\epsilon^{-1})} ( 2 \boldsymbol{I} - \boldsymbol{R}_S \boldsymbol{R}_S ) \boldsymbol{\cdot} ( \dot{\boldsymbol{R}} - \boldsymbol{V}_{\infty}(\boldsymbol{R}) ) = \boldsymbol{F}(S,T), \end{equation}

where $\dot {\boldsymbol {R}}={\partial \boldsymbol {R}}/{\partial T}$, $\boldsymbol {R}_S = {\partial \boldsymbol {R}}/{\partial S}$ and the relative motion of the filament is obtained by applying the mobility tensor, proportional to the anisotropic tensor $( \boldsymbol {I} + \boldsymbol {R}_S \boldsymbol {R}_S )$, to the elastic force $\boldsymbol {F}(S,T)$ on the fibre. Here, we consider shear flow $\boldsymbol {V}_{\infty }(\boldsymbol {R}) = \dot {\gamma } \text {Z} \hat {\boldsymbol {e}}_x$, where $\text {Z}=\hat {\boldsymbol {e}}_z \boldsymbol{\cdot} \boldsymbol {R}$. For the elastic fibre we use the notation illustrated in figure 1(b), i.e. $\hat {\boldsymbol {e}}_n$ denotes a unit vector normal to the fibre in the shear plane and $\hat {\boldsymbol {e}}_s$ denotes a unit vector tangent to the fibre. The inextensibility condition $|\boldsymbol {R}_S|=1$ results in $\hat {\boldsymbol {e}}_s = \boldsymbol {R}_S$ and implies the Frenet formulas ${\partial \hat {\boldsymbol {e}}_{s}}/{\partial S} = K \hat {\boldsymbol {e}}_n,\ {\partial \hat {\boldsymbol {e}}_{n}}/{\partial S} = - K \hat {\boldsymbol {e}}_s$, where $K$ is the local curvature and we have assumed that the fibre shape is planar.

In the elastica model the elastic forces acting on the unit segment of the fibre are (Audoly & Pomeau Reference Audoly and Pomeau2000; Audoly Reference Audoly2015)

(2.18)\begin{equation} \boldsymbol{F}(S,T) =( - E I K_S \hat{\boldsymbol{e}}_n+ \varSigma \hat{\boldsymbol{e}}_s)_S, \end{equation}

where $\varSigma (S,T)$ is the tension in the filament (satisfying inextensibility), $E$ is the Young modulus and $I$ is the moment of inertia ($I={\rm \pi} a^4 /4$), as earlier. Alternatively, the force density per unit length can be expressed as $\boldsymbol {F}(S,T) = - EI {\boldsymbol R}_{SSSS} + (\mathcal {T} \hat {\boldsymbol {e}}_s)_S$, see e.g. Tornberg & Shelley (Reference Tornberg and Shelley2004) and Lindner & Shelley (Reference Lindner and Shelley2015). It is easy to check that $\varSigma = \mathcal {T}+EIK^2$.

With the use of the Frenet formulas it is convenient to write separately the equations of motion in the normal and tangential directions, respectively,

(2.19a)\begin{gather} \frac{4 {\rm \pi}\mu_0}{\ln(\epsilon^{-1})} \hat{\boldsymbol{e}}_n \boldsymbol{\cdot} ( \dot{\boldsymbol{R}} -\dot{\gamma} \hat{\boldsymbol{e}}_x (\hat{\boldsymbol{e}}_z \boldsymbol{\cdot} \boldsymbol{R}) ) = - E I K_{SS} + \varSigma K, \end{gather}

(2.19b)\begin{gather}\frac{2 {\rm \pi}\mu_0}{\ln(\epsilon^{-1})} \hat{\boldsymbol{e}}_s \boldsymbol{\cdot} ( \dot{\boldsymbol{R}} - \dot{\gamma} \hat{\boldsymbol{e}}_x (\hat{\boldsymbol{e}}_z \boldsymbol{\cdot} \boldsymbol{R}) ) = \left( \varSigma + \frac{E I}{2} K^2 \right)_{S}. \end{gather}

We write the dimensionless form (lowercase symbols) of (2.19) by expressing length in the units of $2a$ and time in the units $\dot {\gamma }^{-1}$, as in § 2.1, to find

(2.20a)\begin{gather} \eta \hat{\boldsymbol{e}}_n \boldsymbol{\cdot} ( \dot{\boldsymbol{r}} - \hat{\boldsymbol{e}}_x (\hat{\boldsymbol{e}}_z \boldsymbol{\cdot} \boldsymbol{r}) ) = - \kappa_{ss} + \sigma \kappa, \end{gather}

(2.20b)\begin{gather}\frac{\eta}{2} \hat{\boldsymbol{e}}_s \boldsymbol{\cdot} ( \dot{\boldsymbol{r}} - \hat{\boldsymbol{e}}_x (\hat{\boldsymbol{e}}_z \boldsymbol{\cdot} \boldsymbol{r}) ) = \left( \sigma + \frac{1}{2} \kappa^2 \right)_s, \end{gather}

where

(2.21)\begin{equation} \eta=\frac{4 {\rm \pi}\mu_0 (2a)^4 \dot{\gamma}}{E I \ln(\epsilon^{-1})}\end{equation}

is a dimensionless compliance. Using the same normalization of $EI$ as for the bead model, $EI/({\rm \pi} \mu _0 \dot {\gamma } L_0 (2a)^3)$, we can formally write $\eta = {4(2a)}/{A L_0\ln (\epsilon ^{-1})}$. A physical comparison between the elastica and bead models will be presented in § 4.3.

The dimensionless compliance $\eta$ is very similar to the elasto-viscous number $\bar {\eta }={8 {\rm \pi}\mu _0 L^4 \dot {\gamma }}/{E I \ln (\epsilon ^{-1})}$ (Becker & Shelley Reference Becker and Shelley2001; Tornberg & Shelley Reference Tornberg and Shelley2004; Wandersman et al. Reference Wandersman, Quennouz, Fermigier, Lindner and Du Roure2010; Nguyen & Fauci Reference Nguyen and Fauci2014; Liu et al. Reference Liu, Chakrabarti, Saintillan, Lindner and Du Roure2018; LaGrone et al. Reference LaGrone, Cortez, Yan and Fauci2019; du Roure et al. Reference du Roure, Lindner, Nazockdast and Shelley2019). The main difference is that $\bar {\eta }$ has the fibre's length $L$ as the typical length scale, while $\eta$ uses the fibre's radius.

4.1. Bending time from numerical simulations

In addition to bending, when in a shear flow, a fibre undergoes rotation (it tumbles). It is instructive to compare the bending time $\tau _b$, the curling time $\tau _c$ and the stretching time $\tau _s$ (see figure 3) with two indicators of a fibre's rotational motion: the flipping time $\tau _f$ and the turning time $\tau _m$ (see figure 5(a) and the insets indicating shapes for $\tau _f$ and $\tau _m$). We find for $A \in [1,10\,000)$ that $\tau _f < \tau _m$, which shows that the ends of the flexible fibre pass above each other earlier than the middle of the fibre rotates. As $A$ increases, both $\tau _m$ and $\tau _f$ acquire the interpretation of the half-tumbling time $\tau /2$ or the quarter period $T_J/4$ of the Jeffery orbit (Jeffery Reference Jeffery1922), which is understood here as a periodic motion of a certain ‘effective’ rigid elongated object in shear flow (Słowicka et al. Reference Słowicka, Wajnryb and Ekiel-Jeżewska2015, Reference Słowicka, Stone and Ekiel-Jeżewska2020) with the same period $T_J=2\tau$. The period $T_J$ of a Jeffery orbit is approximately proportional to the length of a fibre consisting of $n$ beads (Jeffery Reference Jeffery1922; Kim & Karrila Reference Kim and Karrila1991; Dhont & Briels Reference Dhont and Briels2007; Graham Reference Graham2018).

Figure 5. Characteristic time scales in the fibre motion. (a) Comparison of time scales of the fibre motion: the bending time $\tau _b$, the stretching time $\tau _s$, the curling time $\tau _c$, the flipping time $\tau _f$ and the turning time $\tau _m$ for $n=60$ and the model $\mathcal {M}_1$. Insets show the shapes for the times $\tau _f$ and $\tau _m$. (b,c) Bending time $\tau _b$ as a function of $A$ for different $n$ from the models $\mathcal {M}_1$ and $\mathcal {M}_2$, respectively. The solid lines show $\tau _b \propto A^{1/3}$. In (b) the best linear fit is $(0.335 \pm 0.002) \log _{10} A + 0.845 \pm 0.005$. (d,e) Bending time $\tau _b$ normalized by $n$ is an almost universal function of $A/n^3$, as confirmed by the numerical data from the models $\mathcal {M}_1$ and $\mathcal {M}_2$, respectively.

For fibres that are very flexible or long enough (e.g. $A=10$ and $n=300$), $\tau _c$ and $\tau _s$ are not defined, because the fibre does not straighten out again (Słowicka et al. Reference Słowicka, Wajnryb and Ekiel-Jeżewska2015), bending multiple times if $n$ is sufficiently large (figure 4a). In the limit of very stiff fibres, all the time scales defined above, $\tau _b$, $\tau _f$, $\tau _m$ and $\tau _c+\tau _s$, converge to $T_J/4$, as shown in figure 5(a) for large values of $A$. More details about the relation between different time scales can be found in appendix B.

The dynamics of bending changes systematically as a function of $A$. In figures 5(b) and 5(c) we show $\tau _b$ (see figure 3) as a function of $A$ for different $n$, using the models $\mathcal {M}_1$ and $\mathcal {M}_2$, respectively. Three regimes are visible. First, in the small $A$ regime ($A \lesssim 10$), the bending dynamics is dominated by large bending angles close to the excluded volume limit. Second, for intermediate $A$, $\tau _b$ does not depend on $n$ and it follows a single power law $\tau _b \propto A^{1/3}$. Third, in the regime of large $A$, the bending time systematically deviates from the power law and saturates at a constant value, where larger $n$ have larger limiting $\tau _b$, in agreement with $\tau _b \rightarrow T_J/4$.

Therefore, in figures 5(d) and 5(e) we replot the data from figures 5(b) and 5(c), respectively, using the rescaled bending time, $\tau _b/n$. To obtain the universal scaling, we also need to rescale the bending stiffness as $A/n^3$. Indeed, after such rescaling, we observe an almost universal curve in the whole range of values of $n$ and $A$.

4.2. A similarity solution at early times for the elastica

In figure 5(b–e) we show the dependence $\tau _b \propto A^{1/3}$, which can be argued with the help of the elastica model, as we will demonstrate in this and the next section. We observe numerically that, in the power-law regime, the bending time does not depend on the fibre length $n$, which suggests an analysis based on the model of a very long fibre, initially aligned with the flow, with a tip positioned at $S=0$. We assume small deflections from the straight line $\boldsymbol {R} = S \hat {\boldsymbol {e}}_x + U(S,T) \hat {\boldsymbol {e}}_z$, which leads to $K = U_{SS}$. Further, we assume that because of small deflections, $\hat {\boldsymbol {e}}_s = \hat {\boldsymbol {e}}_x$ and $\hat {\boldsymbol {e}}_n = \hat {\boldsymbol {e}}_z$. Under these assumptions, we rewrite the dimensional linearized equations (2.19)

(4.1a)\begin{gather} \dot{U}(S,T) = - \frac{E I \ln(\epsilon^{-1})}{4 {\rm \pi}\mu_0} U_{SSSS}(S,T) + \frac{\ln(\epsilon^{-1})}{4 {\rm \pi}\mu_0} F_E \delta(S) , \end{gather}

(4.1b)\begin{gather}U(S,T) = - \frac{\ln(\epsilon^{-1})}{2 {\rm \pi}\mu_0} \varSigma_S(S,T), \end{gather}

with the Dirac delta $\delta (S)$ in the additional term that introduces the hydrodynamic force $F_E=O(1)$, perpendicular to the fibre axis, acting on the tip of the fibre at $S=0$. This force results from the hydrodynamic interactions of the fibre beads in response to the shear flow (see §§ 4.3 and 2.1.4). Alternatively to the delta term, the constant tip force can be formally written as a boundary condition, $U_{SSS}(S\!=\!0,T)$=$F_E /(E I)$. We will use this approach to write (4.1) in the dimensionless form, corresponding to (2.20),

(4.2a)\begin{gather} \dot{u} = - \frac{1}{\eta} u_{ssss}, \end{gather}

(4.2b)\begin{gather}u = - \frac{2}{\eta} \sigma_s. \end{gather}

In order to solve (4.2a) we apply boundary conditions

(4.3a–c)\begin{equation} u \left(\infty,t \right) = 0, \quad u_{ss} \left(\infty,t \right) = 0, \quad \frac{1}{\eta} u_{sss}\left( 0, t \right) = \mathcal{F}, \end{equation}

where

(4.4)\begin{equation} \mathcal{F}=\frac{\ln(\epsilon^{-1})}{4} \,f_E \end{equation}

with $\,f_E = F_E / ({\rm \pi} \mu _0 \dot {\gamma } (2a)^2)$. We impose an initial condition

(4.5)\begin{equation} u \left(s,t=0 \right) = 0. \end{equation}

We seek a similarity solution (Barenblatt Reference Barenblatt1996; Duprat & Stone Reference Duprat and Stone2015; Eggers & Fontelos Reference Eggers and Fontelos2015) and account for the forcing as

(4.6)\begin{equation} u \left( s,t \right) = \eta^{1/4} \mathcal{F} t^{3/4} \mathcal{U} \left( \chi \right),\quad \textrm{where} \ \chi = \eta^{1/4} \frac{s}{t^{1/4}}, \end{equation}

which leads to the equation

(4.7)\begin{equation} 4 \mathcal{U}_{\chi\chi\chi\chi} + 3 \mathcal{U} - \chi \mathcal{U}_{\chi} = 0 \end{equation}

with the boundary conditions

(4.8a–c)\begin{equation} \mathcal{U} \left( \infty \right) = 0,\quad \mathcal{U}_{\chi\chi} \left( 0 \right) = 0,\quad \mathcal{U}_{\chi\chi\chi} \left( 0 \right) = 1.\end{equation}

The solution can be expressed with the help of special (hypergeometric) functions (e.g. use Mathematica) and the gamma $\varGamma (\cdot )$ function

(4.9)\begin{equation} \mathcal{U}(\chi)= \frac{\chi^3}{6} -\frac{2 \chi \,{}_1F_3\left(-\dfrac{1}{2};\dfrac{1}{2},\dfrac{3}{4},\dfrac{5}{4}; \dfrac{\chi^4}{256}\right)}{\sqrt{{\rm \pi} }} +\dfrac{\sqrt{2} \, {}_1F_3\left(-\dfrac{3}{4};\dfrac{1}{4},\dfrac{1}{2},\dfrac{3}{4}; \dfrac{\chi^4}{256}\right)}{\varGamma \left(\frac{7}{4}\right)}. \end{equation}

The function $\mathcal {U}(\chi )$ is shown in figure 6. From $\mathcal {U}(\chi )$ we calculate

(4.10)\begin{align} u(s,t) &= \mathcal{F} \Bigg( \frac{ \eta s^3}{6} - \left( \frac{4 \eta t}{\rm \pi} \right)^{1/2} s \, {}_1F_3 \left(-\frac{1}{2};\frac{1}{2},\frac{3}{4},\frac{5}{4};\frac{s^4 \eta }{256 t}\right) \nonumber\\ &\quad + \frac{16 \eta^{1/4}}{3 {\rm \pi}} t^{3/4} \varGamma \left(\frac{5}{4}\right) \, {}_1F_3 \left(-\frac{3}{4};\frac{1}{4},\frac{1}{2},\frac{3}{4};\frac{s^4 \eta }{256 t}\right) \Bigg). \end{align}

This result can be expanded around $s=0$

(4.11)\begin{equation} u(s,t) = \mathcal{F} \left( \frac{ 16 \eta^{1/4} t^{3/4}}{3 {\rm \pi}} \varGamma \left(\frac{5}{4}\right)- s \left( \frac{ 4 \eta t}{\rm \pi} \right)^{1/2} + \frac{ \eta s^3}{6} + \ldots \right) \end{equation}

and, in particular at $s=0$, the end moves according to

(4.12)\begin{equation} u(0,t) = (\eta t^{3})^{1/4} \mathcal{F} \frac{ 16 \varGamma \left(\frac{5}{4}\right)}{3 {\rm \pi}}. \end{equation}

The complete solution of the similarity ansatz has the magnitude of the deflection $u(0,t) \propto (\eta t^{3})^{1/4}$. The length scale on which the deflection occurs is $s \propto (t/\eta )^{1/4} \propto (u(0,t)/\eta )^{1/3}$. In time the ‘height’ of the deflection grows more rapidly than its ‘width’, which makes the tip more and more steep. The bending stiffness has an opposite effect and makes the deformation less steep with increasing $A \propto \eta ^{-1}$.

Figure 6. Similarity solution from the elastica model (4.7)–(4.8a–c), valid at early times, for a fibre initially aligned with the flow.

4.3. Comparing the numerical simulations with the similarity solution

In this section, we present results from the numerical simulations based on the model $\mathcal {M}_1$ and compare them with the predictions from the elastica model. According to the elastica similarity solution, the fibres have features of the shape that follow the scaling laws with $t$ and $\eta$ presented above. Therefore, we analyse the $z$ coordinate of the relative position of the first bead at time $t$ with respect to its initial position, $z_1(t) = \hat {\boldsymbol {e}}_z \boldsymbol{\cdot} (\boldsymbol {R}_1(t)-\boldsymbol {R}_1(0) )$, which is calculated from the bead-spring simulations $\mathcal {M}_1$ (figure 7a).We show the same data with the rescaled time $t/A^{1/3}$ in figure 7(b). This scaling is suggested by the elastica model, if we identify the height $z_1(t)$ of the fibre end with the deflection of the elastica tip $u(0,t)$, given by (4.12), and we remember that $\eta \propto 1/A$. We also fit a straight line to the numerical values of $\log _{10} z_1(t)$ as a function of $\log _{10} (t /A^{1/3})$, in the linear region $\log _{10} (t/ A^{1/3}) < -1$, where deformations are still small and no deviations from the power law are observed. While fitting, we used data from all the simulations where $n\ge 60$ and $A\ge 50$. The calculated slope $0.787$ is very close to $3/4$ theoretically predicted from the dynamics of the elastica. In order to further compare simulations with the theoretical results we will use the best fit of the tip height in the form $z_1(t) = C (t/A^{1/3})^{3/4}$, which is suggested by the elastica, that results with $C \approx 10^{-0.81}$.

Figure 7. Scalings in the numerical simulations for small deflections. (a) Height $z_1(t)$ of the fibre tip above the horizontal line passing through the centre of the fibre (from the $\mathcal {M}_1$ model). Fibres have $n=100$, except the fibre denoted with $^{*}$ that has $n=140$. (b) Height $z_1(t)$ from panel (a) as a function of the rescaled time $t/A^{1/3}$ (in log–log scale). The black solid line comes from a fit to all $\mathcal {M}_1$ data with $n\ge 60,\ A\ge 50$ for times $t/A^{1/3}<0.1$. (c) Shapes of fibres from the bead model $\mathcal {M}_1$ for $n=140$ and different $A$ at arbitrarily chosen times at the end of the regime of the similarity solution. (d) Shape of fibres from panel (c), scaled according to the similarity solution $z_1(t)\approx u(s,t)$, with $u$ given by (4.6), and translated to approximately overlay the left ends. The numerically obtained shapes are superimposed onto theoretically calculated shapes $a_{k}\mathcal {U}(\tilde {x} b_{k}),\ k = \mathrm {I,II}$, with ${\mathcal {U}}$ given by (4.9) and the coefficients $a_k,b_k$ given in terms of ${\mathcal {F}}$ and $\eta$ which result from approaches I and II to compare between the bead model and the elastica, given in (4.13)–(4.15) and (4.16), (4.17), respectively.

In figure 7(c) we present the numerical shapes of the fibres with different $A$ and $n$ taken at different times but still within the range of the similarity solution, with $t/A^{1/3} \lesssim 10^{0.6} \approx 4$. The ends of these shapes can be to a certain extend superimposed onto each other by scaling the coordinates as $\tilde {x}=x/(tA)^{1/4}$ and $\tilde {z}=z/(t^3/A)^{1/4}$, respectively, in accord with predictions from the elastica, and translating by a shift $x_0$, which is different for each fibre, as shown in figure 7(d). The rescaled shapes are plotted together with two plots of the similarity solution as a function $a_{k}\mathcal {U}(\tilde {x} b_{k}),\ k = \mathrm {I,II}$, which correspond to our two different approaches to compare the hydrodynamic forces, $\,f_{H1}$ and $\,f_E$, exerted on the fibre tip in the bead (§ 2.1) and elastica (§ 4.2) models, respectively. (Actually, we will be comparing the limiting value $\,f_H$ for $n \rightarrow \infty$ rather than $\,f_{H1}$.)

In both approaches, we assume the identical tip heights $z_1(t)$ and $u(0,t)$ in the bead and elastica models, respectively. In the first approach ($\mathrm {I}$), both forces are assumed to be the same, $\,f_H=\,f_E$. Therefore, ${\mathcal {F}}$ is related to $\,f_H$ by (4.4). On the other hand, $\eta \approx 4/(A \ln \epsilon ^{-1})$, as shown below (2.21), and (4.12) links ${\mathcal {F}}$ to the height $u(0,t)=z_1(t)$ of the fibre. Using the numerical fit of $z_1(t)$, shown in figure 7(b), we find $\mathcal {F}$, and from this result, using (4.4) we determine the magnitude $\,f_E$ of the dimensionless tip force in the elastica model. The approach (I) is given by the following equations:

(4.13)\begin{gather} \mathcal{F} = 10^{-0.81} \frac{3 {\rm \pi}(\ln \epsilon^{-1})^{1/4}}{16 \varGamma \left(\frac{5}{4}\right) \sqrt{2}} \approx 0.071 (\ln\epsilon^{-1})^{1/4}, \end{gather}

(4.14)\begin{gather}\,f_E = 0.284 (\ln\epsilon^{-1})^{-3/4}=\,f_H \approx 0.16, \end{gather}

(4.15)\begin{gather}\eta \approx 4/(A \ln \epsilon^{-1}). \end{gather}

From (4.14) we find $\epsilon ^{-1}\approx 9$, which is rather far from the typical aspect ratios used in the numerical simulations. We use the above values to compare shape of the fibre made of beads with the elastica. Starting from (4.6) we find that $a_{\mathrm {I}} = 0.1$ and $b_{\mathrm {I}} = 1.16$ and plot the corresponding elastica shape (solid line) in figure 7(d).

In the second approach ($\mathrm {II}$), we assume that velocities of the fibre segments are the same in the bead and elastica models. In this way, mobilities times forces are equal to each other. The mobility for the elastica comes from the SBT, while in the equations of motion for a fibre made of beads, there appear the single-sphere Stokes mobility. Therefore in approach (II), we match the elastica and bead quantities as follows:

(4.16)\begin{gather} \mathcal{F} \equiv \frac{\ln(\epsilon^{-1})}{4} \,f_E= \frac{1}{3} \,f_{H}, \end{gather}

(4.17)\begin{gather}\eta =\frac{3}{A}. \end{gather}

Then, using the numerical results, from (4.12) we obtain

(4.18)\begin{equation} f_H=10^{-0.81} \frac{9{\rm \pi}}{16\varGamma\left(\frac{5}{4}\right) 3^{1/4}} \approx 0.23, \end{equation}

which is not very far from the numerical value $\,f_H=0.16$ obtained for large $n$. In this approach the parameter $\epsilon ^{-1}$ is not used at all. With the use of this set of values and (4.6) we find that $a_{\mathrm {II}} = 0.1$ and $b_{\mathrm {II}} = 1.32$. We plot the corresponding elastica shape (dotted line) in figure 7(d).

4.4. At early times beyond small deformations

In this section, we will focus on an early phase of bending for times $t \lesssim \tau _b$. We will analyse the simulations with the bead model $\mathcal {M}_1$ and compare them with the scaling laws following from the elastica model. In the early phase, a flexible fibre aligned with shear flow slowly starts to bend its ends while the middle part of the fibre remains straight. The characteristic length scale of the deformed fibre segments at both ends remains approximately constant in time until a significant, rapid bending is developed at the bending time $\tau _b$, associated with a fast increase of both the local curvature $\kappa$ (figure 3b) and the tip deflection $z_1(t)$ along $z$ (figure 7a,b). A corresponding sequence of consecutive fibre shapes, found numerically with the model $\mathcal {M}_1$, is shown in figure 8. The significant bending from the middle to the last shape occurs on a very short time scale.

Figure 8. A typical evolution of the shapes of a flexible fibre in an early stage of bending. Here, $n=140$ and $A=1000$. The consecutive time instants shown are $t=50, 66, 69, 70, 70.5$, with the last one approximately equal to $\tau _b$.

In figure 3(b), the bending time $\tau _b$ was defined as the time when the maximal local fibre curvature reaches one half of its largest value, $\kappa (\tau _b)=\kappa _b/2$. We now add a physical interpretation: at a time close to $\tau _b$, the deflection $z_1(t)$ of the fibre tip approaches the first local maximum, visible in figure 7(a,b). The corresponding fibre shape at $t=\tau _b$ is shown in figure 8 as the last one.

The linear regime of small deformations is limited to short times. For example, in figure 8, all shown fibres are already beyond this regime. However, to understand the dynamics in the early phase, it is worthwhile to begin the analysis from the linear regime where the universal scaling of shapes follows directly from the self-similar solution, specified by (4.6) and (4.9). One of characteristic features of the self-similar solution is that the same deflection $z_1(t)=u(0,t)$ of the fibre tip is reached at the same rescaled time $t A^{-1/3} \propto t \eta ^{1/3}$ (and with the same rescaled length $sA^{-1/3}$ along the fibre). The numerical results shown in figure 7(b) illustrate that $z_1(t)$ is determined by $t A^{-1/3}$ not only in the range of small deformations, but also beyond it. The tip deflection $z_1(t)$ remains a universal function of $t A^{-1/3}$ until its argument reaches value slightly smaller than, but very close to $t_{{max}} A^{-1/3} \approx 7.2$, with $t_{{max}}$ defined as the time when $z_1$ has a local maximum $u_{{max}}$. In the log–log scale, as shown in figure 7(b), the universal curve is close to a straight line for $t A^{-1/3} \lesssim 4$ (in the linear regime). For $t A^{-1/3} \gtrsim 4$, a significant deviation from the straight line is observed caused by nonlinear effects.

The deviations from the linear regime in the numerical simulations can be interpreted by the elastica evolution in the range where the nonlinear terms in (2.20) become important, i.e.

(4.19)\begin{equation} \kappa_{ss} \sim \sigma \kappa. \end{equation}

Next, we observe that the change in the elastica dynamics occurs away from the small deflection state so we can use the relation (4.2b), which implies the scaling,

(4.20)\begin{equation} \sigma \sim \eta u s \end{equation}

with the results from the similarity solution to show that

(4.21)\begin{equation} O(1) = O(\sigma s s) = O(\eta u s^3) \approx \eta^{1/2} t^{3/2} \Rightarrow O(1) = \eta t^{3}. \end{equation}

As $\eta \propto A^{-1}$, these balances suggest that the time scale $\tau _b$, when the nonlinear terms become important, follows the power law $\tau _b \propto A^{1/3}$, which is in a good agreement with results presented in figure 5(b–e).

For times close to $\tau _b$ and $t_{{max}}$, the tip deflection $z_1(t)$ leaves the universal curve, as shown in figure 7(a,b), and the maximum deflection $u_{{max}}$ depends on $A$. We observe that in the range of $A$, in which the evolution follows a power law, $u_{{max}} \propto A^{0.33}$, as determined numerically from the bead model $\mathcal {M}_1$ and shown in figure 9(a). This scaling might reflect a memory of the initial phase of the fibre movement as analysed using the elastica. A linearly deflected fibre changes its shape over a length scale $s \propto A^{1/4} t^{1/4}$ and time scale $t \propto A^{1/3}$ (4.6) thus, we find that $s \propto A^{1/3}$ at time $t$. However, as illustrated in figure 8, for early times $t\lesssim \tau _b$, the typical length scale of bending remains almost constant in time, what allows us to expect that $u_{{max}} \propto A^{1/3}$. The numerical results suggest that during the rapid bending the scaling in the $x$ direction becomes comparable to the scaling in the $z$ direction. To demonstrate this feature, we collect several fibre shapes for different values of $n$ and $A$, at times $t_{{max}}$ of the maximum deflection (figure 9b), and we replot them in figure 9(c) by rescaling both axes by $A^{1/3}$, which allows for the approximate overlapping of the shapes after translating them in the $x$ direction.

Figure 9. Maximum deflection $u_{{max}}$ of the fibre tip, defined as $z_1(t_{{max}})$ at the time $t_{{max}}$ of the first maximum, evaluated numerically from the bead model $\mathcal {M}_1$. (a) $u_{{max}}$ as a function of $A$ for different $n$, with the approximate scaling $\propto A^{1/3}$. (b) Shapes of fibres at $t_{{max}}$ for different $A$ and $n$. (c) Rescaled shapes of fibres from (b), translated to overlay the bending left ends.

The scalings with $A^{1/3}$, typical for the early phase of the bending process, do not depend on $n$. For $t \lesssim \tau _b$, bending of the fibre ends is a local process. However, the typical bending length scale increases with $A$, and therefore for a larger stiffness, the scalings are satisfied by a sufficiently long fibre only.

5.1. Bead model simulations

We now move on to discuss the dynamics for times $\tau _b \lesssim t \lesssim \tau _b+\tau _c$, when the fibre is significantly bent, with the maximum local bending curvature $\kappa _{b2}/2 \le \kappa (t)\le \kappa _{b2}$, where $\kappa _{b2}$ is the largest maximum local curvature during this time period (see figure 3). In this range, the main feature of the dynamics is its maximum local curvature $\kappa (t)$. Therefore, we first discuss if (and how) the characteristic features of the dynamics depend on a specific choice of the time instant when the curvature is determined.

In figure 3(b) we have illustrated that there exists a typical plateau of the curvature, $\kappa _{b1}$, and the largest value $\kappa _{b2}$. Comparison with figures 4(b) and 4(d) indicates that both values vary systematically with $A$. We analyse this dependence in figure 10(a). On a log–log plot, $\kappa _{b1}$ is systematically below $\kappa _{b2}$, and the inset (for $n=140$) illustrates that the ratio $\kappa _{b1}/\kappa _{b2}$ slowly decays with increasing $A$, but this effect is not large. The numerical data show that, over a few decades of $A$, the ratio $\kappa _{b1}/\kappa _{b2}$ changes only by 30 %, and it tends to $\kappa _{b1}/\kappa _{b2} \sim 0.7$ for large $A$. This observation suggests that $\kappa _{b2}$ depends on $A$ in a similar way as $\kappa _{b1}$. The fibre is first in the state of a typical bend and tightens to a maximum curvature for a short time afterwards. However, the plateau in the $\kappa (t)$ that allows determination of $\kappa _{b1}$, for a given $n$, occurs only for a finite range of $A$, while $\kappa _{b2}$ is well defined for any value of $A$. For example, in case of $n=40$, in figure 10(a) there are no data points above $A=100$ indicating $\kappa _{b1}$ while for $n=140$, $\kappa _{b1}$ can be observed up to $A=2000$. Therefore, in the following we will focus on the analysis of $\kappa _{b2}$.

Figure 10. Fibre bending curvature as a function of $A$, evaluated from the bead models. (a) Difference between typical $\kappa _{b1}$ and maximum $\kappa _{b2}$ curvatures for two different lengths of fibre ($n=40$ and $n=140$), based on the model $\mathcal {M}_1$. The inset shows the ratio of bending curvatures $\kappa _{b1}/\kappa _{b2}$ as a function of $n$. Panels (b) and (c) show the scaling of the maximum curvature $\kappa _{b2}$ as a function of $A$, for different $n$, determined from the models $\mathcal {M}_1$ and $\mathcal {M}_2$, respectively. The solid lines correspond to $\kappa _{b2} \propto A^{-1/4}$, the dashed inclined lines show $\kappa _{b2} \propto A^{-1/3}$ and the horizontal dashed lines show the curvature based on EV. The schematics in (b) show the shape of a fibre with $n=20$ for $A=1,10$ and $100$.

Depending on the values of $A$, three regimes of fibre bending can be identified, as shown in figure 10(b,c) for the bead models $\mathcal {M}_1$ and $\mathcal {M}_2$, respectively. The schematics in figure 10(b) show three typical fibre shapes with $n=20$ for each of the regimes. First, in the regime of a very flexible fibre ($A \lesssim 10$), the maximum curvature is close to the excluded volume (EV) of the beads, with $\log _{10} \kappa _{b2} \lesssim \log _{10}(\sqrt {3}) \approx 0.24$, which is independent of $n$. Second, there is a regime $A \gtrsim 10$, where $\kappa _{b2}$ as a function of $A$ continues with a power-law dependence until it deviates from the slope, which happens for different $A$ depending on $n$. The larger $n$, the larger range of $A$ that exhibit the power-law dependence. Inside this regime, for a given $A$, all fibres that are long enough have the same $\kappa _{b2}$, which is independent of $n$. We interpret this response as local bending. Third, there is a large $A$ regime, which starts after $\kappa _{b2}$ departs from the power law. This corresponds to $\kappa _{b2}^{-1}$ comparable to or larger than the fibre length, which we interpret as global bending. This classification of $\kappa _{b2}$ is valid even for very long fibres (having multiple loops), and also for very stiff ones (with no pronounced plateau of the fibre curvature $\kappa (t)$). For each $n$, the regimes of $A$ where the power laws are observed for $\kappa _{b2}$ agree with the corresponding regimes identified for $\tau _b$ (compare the ranges of $A$ in figure 10(b,c) with the ranges in figure 5(b,c), respectively).

Comparison of figures 10(b) and 10(c) indicates that, if the bending stiffness ratio $A$ is not very small ($A \gtrsim 10$) and not too large (with the upper bound dependent on $n$), the power-law scalings of $\kappa _{b2}$ predicted by the ${\mathcal {M}}_1$ and ${\mathcal {M}}_2$ models are in a reasonable agreement with each other, and the curves only weakly depend on $n$. However, for $10 \lesssim A \lesssim 100$, the maximum curvature $\kappa _{b2}$ in the ${\mathcal {M}}_2$ model decays more rapidly with $A$ than in ${\mathcal {M}}_1$, with approximately $\kappa _{b2} \propto A^{-1/3}$ rather than $\kappa _{b2} \propto A^{-1/4}$, respectively (for the ${\mathcal {M}}_1$ model, $\kappa _{b2} \propto A^{-0.253 \pm 0.003}$ as determined numerically). For $A \lesssim 10$, the maximum curvature $\kappa _{b2}$ determined from the ${\mathcal {M}}_2$ model saturates at the EV value while in model ${\mathcal {M}}_1$ this effect is seen for more flexible fibres with $A \lesssim 1$. Although the treatment of hydrodynamic interactions is more precise within the bead model ${\mathcal {M}}_2$, it seems that the main reason for some differences between the maximum bending curvature $\kappa _{b2}$ obtained by the ${\mathcal {M}}_1$ and ${\mathcal {M}}_2$ models is the use of different expressions for the bending potential energy, as discussed in detail in appendix B.

5.2. Comparing with the elastica model

We are going to show now that the scaling of the fibre maximum curvature $\kappa _{b} \propto A^{-1/4}$, independent of $n$ and characteristic of the local bending, can be argued with the elastica model (2.20). We propose that in the local bending process there is only one length scale $\kappa _{b}^{-1}$ representative of the deformed fibre. This is consistent with the models of Harasim et al. (Reference Harasim, Wunderlich, Peleg, Kröger and Bausch2013), LaGrone et al. (Reference LaGrone, Cortez, Yan and Fauci2019) and Liu et al. (Reference Liu, Chakrabarti, Saintillan, Lindner and Du Roure2018) and with our findings that $\kappa (t)$ is in the curling motion close to a typical constant value $\kappa _{b1}$.

Next, from the linearity of shear flow, we argue that the magnitude of the flow velocity incident on the fibre and the fibre velocity scale linearly with the length scale $(\hat {\boldsymbol {e}}_x (\hat {\boldsymbol {e}}_z \boldsymbol{\cdot} \boldsymbol {r}) \!-\! \dot {\boldsymbol {r}} ) \propto \kappa _{b}^{-1}$. Comparing the magnitudes of terms in (2.20b), we find that the dimensionless tension scales as

(5.1)\begin{equation} \sigma \propto \eta \kappa_b^{-2} + \kappa_b^{2}. \end{equation}

This dependence, together with (2.20a), gives

(5.2)\begin{equation} \eta \kappa_b^{-1} \propto \kappa_b^{3} + \left(\eta \kappa_b^{-2} + \kappa_b^{2}\right) \kappa_b, \end{equation}

resulting in $\kappa _b \propto \eta ^{1/4} \propto A^{-1/4}$. It is also true that $\sigma \propto A^{-1/2}$ and $\sigma _s \propto A^{-3/4}$. Note that these arguments apply to the maximum curvature in the whole range of the curling motion with a large shape deformation, in particular for $\kappa _b=\kappa _{b1}$ and $\kappa _b=\kappa _{b2}$.

The scalings obtained from the elastica model can be compared with the results from the bead-spring simulations with the model $\mathcal {M}_1$. The force $\boldsymbol {F}_i$ acting on each bead $i$ as the result of the elastic constitutive laws can be decomposed into the force components normal $\boldsymbol {N}_i$ and tangential $\boldsymbol {T}_i$ to the fibre centreline. In figure 11(a) we show shapes of locally bent fibres with $n=100$ for three different values of $A$. The colour-coded representations of $\boldsymbol {N}_i$ and $\boldsymbol {T}_i$ are included in the following way. Each bead is depicted by a hemisphere, which has an orientation that indicates the direction of $\boldsymbol {T}_i$ (inset), while the colour coding shows the ratio of the magnitudes of the forces normal and tangential to the fibre, $|\boldsymbol {N}_i|/|\boldsymbol {T}_i|$. In order to compare the simulation data quantitatively with the scalings deduced above from the elastica, it is sufficient to choose any time from the curling motion of the fibre. As we compare between different $A$, we introduce the transformation of the bead numbering $i' = (i-i_0) A^{-1/4}$ ($i$ is a discrete analogue of the arc length $s$), where a shift $i_0$ is chosen (for each fibre separately) to overlap the extrema. In the figure 11(b,c) we show the profile of local curvature $\kappa _i$ over half of the fibre ($i=1\ldots 50$) for the shapes presented in figure 11(a). In figure 11(b) the raw data are plotted and in figure 11(c) $\kappa _i$ is multiplied by $A^{1/4}$ to show the scaling suggested by the elastica model. From the bead-spring simulations we have direct access to $\boldsymbol {T}_i\boldsymbol{\cdot} \hat {\boldsymbol e}_s$ acting at the centre of bead $i$, which is the analogue of the derivative of the tension $\sigma _{s}(s)$ for the elastica. The value of $\boldsymbol {T}_i\boldsymbol{\cdot} \hat {\boldsymbol e}_s$ is shown in figure 11(d), and in figure 11(e) we demonstrate that the tangential forces scale as $A^{-3/4}$, which has been also suggested by the elastica model.

Figure 11. Instantaneous distribution of forces and curvatures on individual beads (fibre statics) for three locally bent fibres with $n=100$ and $A=100,500,2000$. (a) Shapes of the fibres. The colour coding shows the ratio of the normal $|\boldsymbol {N}_i|$ to the tangential $|\boldsymbol {T}_i|$ force components acting on each bead $i$, represented as a hemisphere. The orientation of hemispheres shows the direction of $\boldsymbol {T}_i$. (b) Curvature $\kappa _i$ on bead $i$ along the fibre. (c) Rescaled curvature as a function of rescaled position $i'=(i-i_0)A^{-1/4}$ along the fibre. (d) Tangential forces $\boldsymbol {T}_i\boldsymbol{\cdot} \hat {\boldsymbol e}_s$ acting on beads $i$ – the discrete analogue of the tension's derivative $\sigma _s$ for the elastica. (e) Rescaled $\boldsymbol {T}_i\boldsymbol{\cdot} \hat {\boldsymbol e}_s$ as a function of $i'$.

5.3. Curling velocity and curling time

In § 3 we introduced the curling motion and the associated curling time $\tau _c$ (see figure 3). During the curling motion, the first bead travels from left to right approximately over the distance $L=2na$ with respect to the fibre's centre. Snapshots illustrating the shape evolution are shown using the schematics at the top of figure 12(a) for the fibre with $n=100$ and $A=100$. We define the curling velocity $v_x(t)$ as the $x$-component of the velocity of the first bead. At the bottom of figure 12(a), we plot $v_x$ versus time, for the fibres with $A=100$ and different values of $n$. Initially, $v_x$ is close to zero. Then, it rises significantly and we observe the first peak, the plateau and the second peak. Numerical simulations show that for a given $A$, the profile of $v_x(t)$ in the initial phase of the motion is almost the same for different $n$. In figure 12(a), the plots of $v_x(t)$ for different $n$ are almost superimposed for a long time. The changes between $v_x(t)$ with different $n$ occur when the fibre stops to undergo curling motion due to its limited length. The peaks of $v_x$ are observed at the times of the steepest changes in $\kappa (t)$, as illustrated in figure 12(b). The first peak takes place at the time close to the bending time $\tau _b$, and the second one approximately after the curling time $\tau _c$ (compare with figure 3). Therefore, our definitions of $\tau _b$ and $\tau _c$ seem to well separate three different phases of the fibre dynamics.

Figure 12. The curling motion. Results from the bead model $\mathcal {M}_1$. (a) The $x$ component $v_x$ of the first bead velocity as a function of time for $A=100$ and different aspect ratios $n$. The enlarged black circles represent the fibre of aspect ratio $n=100$ (shown in figure 3). The schematics above the plot show the shapes for $n=100$ at times marked with vertical dashed lines; the first bead is marked with a dot. (b) Comparison of $v_x(t)$ (normalized with the maximum observed value $v_{x\textrm {m}}$) with $\kappa (t)$ (normalized with the maximum curvature $\kappa _{b2}$), for $A=100$ and $n=100$. (c) Scaling of the curling time $\tau _c$ as a function of $A/n^{3.5}$, found empirically.

We have found empirically that $\tau _c$ as a function of $A$ and $n$ can be collapsed on a single universal line when plotted versus $A/n^{3.5}$, as shown in figure 12(c). For small values of $A/n^{3.5}$, the curling time $\tau _c$ tends to a power law with the exponent $-1/3$, as determined empirically. Thus we observe that, in the limit of long fibres, we can approximate the curling time as $\tau _c \propto A^{-1/3} n^{1.17}$. That is, $\tau _c$ is almost linear in $n$. The deviations might be related to a larger average resistance during the curling motion of longer fibres. Indeed, figure 12(a) illustrates that, for longer fibres, the contribution to the average curling velocity from the initial and final peaks is smaller, and therefore the average curling velocity is smaller, which leads to the curling time increasing with $n$ a little faster than linearly.

The dynamics analogous to the curling motion was investigated in the literature experimentally, numerically and by the elastica model, with and without the Brownian motion (Forgacs & Mason Reference Forgacs and Mason1959b; du Roure et al. Reference du Roure, Lindner, Nazockdast and Shelley2019). We have shown here that on the onset of curling motion the characteristic length scale (${\propto } A^{1/3}$) is different from the one observed later in the highly bent state (${\propto } A^{1/4}$), which leads to the analogous scalings for $v_x$ at earlier and later times, respectively. Therefore, we emphasize the importance of the time evolution during the curling motion but at the same time we benefit from the previous studies of Harasim et al. (Reference Harasim, Wunderlich, Peleg, Kröger and Bausch2013) and Liu et al. (Reference Liu, Chakrabarti, Saintillan, Lindner and Du Roure2018) who reported the linear dependence between the local radius of curvature of the bent tip and its track velocity (analogous to our curling velocity), both approximately constant in time.

We have found that, in the regime of the local bending, during the curling motion, the curling velocity and the local curvature are mostly determined by the bending stiffness $A$ and practically do not depend on the fibre aspect ratio $n$, This finding agrees well with the results of another numerical model of LaGrone et al. (Reference LaGrone, Cortez, Yan and Fauci2019) where only minute changes in the local radius of curvature of the fibre tip and its snaking (analogous to our curling) velocity have been observed in a wide range of relatively large fibre aspect ratios.

6.1. Shapes of fibres with different $n$ and $A$

The transition from the locally to globally bent fibres, observed for the increasing values of the bending stiffness $A$ and illustrated in figure 10(b,c), motivated us to search for $\kappa _{b2}n$ as a universal function of $A/n^{\gamma }$, with a certain value of the exponent $\gamma$. We use here $\kappa _{b2}n$ because in the global bending mode we expect bending along the whole fibre length. Indeed, in figure 13(a,b), plotted in log–log scale, we find the universal scaling of $\kappa _{b2}n$, based on the numerical simulations $\mathcal {M}_2$ (in a) and $\mathcal {M}_1$ (in b), respectively. We added to figure 13(a) also the results of the $\mathcal {M}_2$ simulations reported by Słowicka et al. (Reference Słowicka, Wajnryb and Ekiel-Jeżewska2015), with the parameters $n=10$, $L_0/(2a)=1.01$ and $k_s=2000$. From the numerical data for the model $\mathcal {M}_2$ we obtain the exponent $\gamma =3.25$ and we find the slopes $-$0.3 and $-$5 of the two straight lines for the local and global bending regimes, for $\log _{10}(A/n^{3.25})\lesssim -2.9$ and $\log _{10}(A/n^{3.25})\gtrsim -2.3$, respectively. The fits agree very well with the results of the $\mathcal {M}_2$ simulations, and reasonably well with the results of the $\mathcal {M}_1$ simulations, as shown in figures 13(a) and 13(b), respectively. The deviations from the universal curve are observed only owing to the EV effects seen for very flexible fibres, with the EV value of the maximum local curvature $\kappa _{b2} = \log _{10}\sqrt {3} \approx 0.24$. The deviations correspond to the first (small $A$) regime of the fibre bending described in § 5.1 and shown in figure 10. For the local bending, the relation $\log _{10}(\kappa _{b2}n) \sim -0.3 \log _{10}(A/n^{3.25})+0.48$, fitted to the $\mathcal {M}_2$ numerical data, gives the approximate scaling of the maximum curvature $\kappa _{b2} \sim A^{-0.3}$ independent of $n$, in agreement with the previous discussion of the local character of the dynamics of very elastic fibres. The exponent $-$0.3 is close but not identical to the $-$1/4 fitted to the $\mathcal {M}_1$ numerical data in figure 10(b). In the global bending regime, we find that $\kappa _{b2}n \sim (A/n^{3.25})^{-5}$.

Figure 13. Universal similarity scaling of fibre shapes, evaluated with the model $\mathcal {M}_2$, (a,c,e), and with the model $\mathcal {M}_1$, (b,d,f). The maximum curvature $\kappa _{b2}n$, scaled by the inverse of the fibre length, can be approximated as a universal function of $A/n^{3.25}$, as shown in (a,b) in the log–log scale. The regimes of local and global bending correspond to a more flat and a more steep straight line, respectively. As shown in figure 13(c–f), in both regimes the shapes of fibres are almost the same for approximately the same values of $A/n^{3.25}$ (as indicated). In (c) $(n,A)=(10,8.4), (20,79.4), (40,720.6), (60,2922.5)$, in (d) $(n,A)=(20,100), (40,1000)$, in (e) $(n,A)=(40,8.8), (60,29.4), (80,79.4)$ and in (f) $(n,A)=(60,100), (100,500)$.

The fitting of the exponent $\gamma$ in the relation $A/n^{\gamma }$ is based on the choice of $2an$ to represent the fibre length $L$, and it is sensitive to a choice of $L$. For example, $\gamma \approx 3$ if the fibre length $L=(n-1)L_0$ is chosen. Such a shorter fibre length was proposed by Farutin et al. (Reference Farutin, Piasecki, Słowicka, Misbah, Wajnryb and Ekiel-Jeżewska2016) as the result of comparing shapes of flexible fibres and deformable vesicles in Poiseuille flow and partially accounts for the rigidity of the beads at the fibre ends. On the other hand, in shear flow, a matching of the tumbling period with the half-period of the Jeffery orbit could be used to determine $L$. In the bead model $\mathcal {M}_2$ for stiffer fibres, the effective aspect ratio $L$ defined in this way is greater than $2an$, and it could lead to a $\gamma$ closer to 4, which means also closer to the scaling $\kappa _{b2} \sim A^{-1/4}$ of the local bending proposed in figure 10(b).

We expect that also the shape of the whole fibre is a universal function of $A/n^{3.25}$. Indeed, as illustrated in figure 13, for the models $\mathcal {M}_2$ (left) and $\mathcal {M}_1$ (right), the fibre shapes depend on $n$ and $A$ approximately through the ratio $A/n^{3.25}$. We show it separately for the global bending in figure 13(c,d) and for the local bending in figure 13(e,f). The corresponding values of $A/n^{3.25}$ are explicitly indicated below each fibre shape, with approximately the same values for all the similar shapes.

Comparison of the power-law scaling of the fibre shapes with an attempt to find another similarity solution, based on a logarithmic dependence on $n$ and a generalized elasto-viscous number, standard in the SBT and elastica approaches, is presented in appendix C. We show there that, although such a possibility cannot be excluded, it seems to be quite complicated to construct. Using simple arguments, we are able to find such a scaling function only for the local bending mode.

6.2. Phase diagram of the dynamical modes

The analysis of the fibre dynamics can be summarized on a phase diagram in the space of the fibre aspect ratio $n$ and the bending stiffness $A$. In figure 14 we show the numerical results, with essentially the same features for the bead models $\mathcal {M}_1$ and $\mathcal {M}_2$. The elastic fibre initially aligned with the shear flow has three characteristic modes of motion, depending on values of $n$ and $A$:

1. (Mode 1) The fibre does not straighten out again. The curvature $\kappa$ does not return to zero after the first bending event.

2. (Mode 2) The fibre bends locally, curls and then stretches; correspondingly curvature grows, reaches a plateau and then returns to zero in a periodic way.

3. (Mode 3) The fibre periodically bends globally along the whole length. Curvature maxima are observed but the plateau vanishes.

Figure 14. Diagram of three dynamical modes in the phase space of the parameters $n$ and $A$, for the bead models $\mathcal {M}_1$ (filled symbols) and $\mathcal {M}_2$ (open symbols). The dynamical modes of the fibres initially aligned with the flow are the following: the fibres that are coiled and do not straighten out (mode 1, triangles); the fibres that straighten out along the flow while tumbling periodically and bend locally (mode 2, rhombus) or globally (mode 3, circles). A sharp transition between the fibres that straighten out while tumbling and the fibres that stay coiled is marked by a dashed line and the stars, taken from Słowicka et al. (Reference Słowicka, Wajnryb and Ekiel-Jeżewska2015), for the $\mathcal {M}_2$ model and by a solid line for the $\mathcal {M}_1$ model. In contrast, the transition between fibres bent locally and globally is gradual (grey area). The sizes of symbols for the $\mathcal {M}_2$ model discriminate between data from this work with $l_0=1.02$ and $k_s=1000$ (large open symbols) and the data of Słowicka et al. (Reference Słowicka, Wajnryb and Ekiel-Jeżewska2015) with $l_0=1.01$ and $k_s=2000$ (small open symbols). The symbols $+$ indicate such points that $\tau_{b2}=\tau_f$.

At early times, the fibre is bent only at the ends. During the curling motion, as shown in figures 3 and 12 and earlier by Harasim et al. (Reference Harasim, Wunderlich, Peleg, Kröger and Bausch2013), Liu et al. (Reference Liu, Chakrabarti, Saintillan, Lindner and Du Roure2018) and LaGrone et al. (Reference LaGrone, Cortez, Yan and Fauci2019), the range of the most curved segments shifts towards the central part of the fibre. The fibre ends become straight and almost aligned with the flow, and the length of straight ends increases with time. Therefore, in general, we might expect that the end of such a long fibre will behave in a similar way as a fibre of a comparable length aligned with the flow. Therefore, if the curling continues long enough, with $\tau _c \gtrsim \tau _b$, the fibre may bend its end again, even several times, and it will not straighten out. Indeed, such a scenario sometimes happens for very long or very flexible fibres, as shown in figure 4, and earlier by Nguyen & Fauci (Reference Nguyen and Fauci2014) and LaGrone et al. (Reference LaGrone, Cortez, Yan and Fauci2019). Using our scalings, $\tau _b \propto A^{1/3}$ and $\tau _c \propto A^{-1/3}n^{1.17}$, a dynamical transition could be expected around $A \propto n^{1.75}$. However, the physical origin of the transition between the coiled and straightening out modes is more complicated. Shorter fibres cannot bend several times, but still they do not straighten out along the flow when their bending stiffness is small enough.

The transition between the coiled and locally bent modes for shorter fibres with $n \le 40$ and a wide range of values of the bending stiffness $A$ has been analysed by Słowicka et al. (Reference Słowicka, Wajnryb and Ekiel-Jeżewska2015). The dynamics of flexible fibres was evaluated over a long time, starting from the initial configuration aligned with the flow. A characteristic value $A_{CS}(n)$ was found for the transition between the fibres that remain coiled and the fibres that straighten out along the flow while tumbling, with $A_{CS}(n) \propto n^{3/2}$. Moreover, the dynamics was shown to be very sensitive to a small change of $A$ close to $A_{CS}$. For $A$ slightly below the critical value, fibres often straightened out a smaller or larger number of times before changing to the coiled mode. Słowicka et al. (Reference Słowicka, Wajnryb and Ekiel-Jeżewska2015) sorted the data for the modes 1 and 2 based on the long-time behaviour. We present in figure 14 (small open symbols) some of the results obtained by Słowicka et al. (Reference Słowicka, Wajnryb and Ekiel-Jeżewska2015). The inset illustrates high precision of the critical values $A_{CS}$ determined there and marked by stars in figure 14. The results of the model ${\mathcal {M}}_2$ applied in this work (large open symbols) also support the $A_{CS}(n) \propto n^{3/2}$ scaling of the transition between the coiled and straightening out modes. The numerical simulations in the $\mathcal {M}_1$ model also agree well with the above scaling, with a different factor which could be interpreted as the result of different bending potentials in both models.

In contrast to the transition between the modes 1 and 2, the transition between the modes 2 and 3 is not sharp. It takes place in a range of the phase space $(n,A)$ marked in grey in figure 14. This stripe corresponds to $-2.3 \lesssim \log _{10}(A/n^{3.25})\lesssim -2.9$, i.e. the range between the local and global bending found numerically with the bead model $\mathcal {M}_2$ and shown in figure 13(a), in agreement with the findings of the model $\mathcal {M}_1$ presented in figure 13(b). The different symbols are the locally and globally bent fibres that indicate just an approximation, based on a comparison of the time instant of the maximum $\kappa _b$ to the flipping time $\tau _f$, as described in appendix D. In the regime of the local bending, for a smaller $A$ or larger $n$, the bending time scales as $\tau _b \propto A^{1/3}$ and the maximum local curvature scales as $\kappa _b \propto A^{-1/4}$, independently of $n$. In the regime of the global bending, $\tau _b \propto n$ independently of $A$, and $\kappa _b n\propto (A/n^{3.25})^{-5}$.

The transition between the local and global bending could be interpreted as a competition between bending and rotation. If the fibre bends before it manages to rotate in shear flow, it belongs to the local bending mode while if it rotates before it bends, it belongs to the global bending mode. Approximating the rotation time as $T_J/4 \propto n$, and equating $\tau _b \approx T_J/4$, we obtain $A \propto n^{3}$, which is an approximation of the transition between the local and global bending shown in figure 14. Another way of looking at the transition between the local and global bending is to compare the typical length scales. The length of the bent fibre end at $\tau _b$ scales as $A^{1/3}$. In the local bending mode, it needs to be smaller than half of the fibre length, proportional to $n$, which again estimates the transition roughly as $A \propto n^{3}$.