Elastohydrodynamical instabilities of active filaments, arrays, and carpets analyzed using slender-body theory

The rhythmic motions and wavelike planar oscillations in filamentous soft structures are ubiquitous in biology. Inspired by these, recent work has focused on the creation of synthetic colloid-based active mimics that can be used to move, transport cargo, and generate fluid flows. Underlying the functionality of these mimics is the coupling between elasticity, geometry, dissipation due to the fluid, and active force or moment generated by the system. Here, we use slender-body theory to analyze the linear stability of a subset of these—active elastic filaments, filament arrays and filament carpets—animated by follower forces. Follower forces can be external or internal forces that always act along the filament contour. The application of slender-body theory enables the accurate inclusion of hydrodynamic effects, screening due to boundaries, and interactions between filaments. We first study the stability of fixed and freely suspended sphere-filament assemblies, calculate neutral stability curves separating stable oscillatory states from stable straight states, and quantify the frequency of emergent oscillations. The results from the slender-body theory differ from that obtained using an approximate theory used often in the literature to study dynamics of filaments, referred to as the resistance force theory, in which the tangential and normal components of the fluid traction at a point on the filament are proportional to the tangential and normal components of the velocity of the filament. Next, we examine the onset of instabilities in a small cluster of filaments attached to a wall and examine how the critical force for onset of instability and the frequency of sustained oscillations depend on the number of filaments and the spacing between the filaments. Our results emphasize the role of hydrodynamic interactions in driving the system toward perfectly in-phase or perfectly out-of-phase responses depending on the nature of the instability. Specifically, the first bifurcation corresponds to filaments oscillating in-phase with each other. We then extend our analysis to filamentous (line) array and (square) carpets of filaments and investigate the variation of the critical parameters for the onset of oscillations and the frequency of oscillations on the interfilament spacing. The square carpet also produces a uniform flow at infinity and we determine the ratio of the mean-squared flow at infinity to the energy input by active forces. We conclude by analyzing the bending and buckling instabilities of a straight passive filament attached to a wall and placed in a viscous stagnant flow—a problem related to the growth of biofilms, and also to mechanosensing in passive cilia and microvilli. Taken together, our results provide the foundation for more detailed nonlinear studies on elastohydrodynamical instabilities in active filament systems.

Figure 1

(a) Schematic of the sphere-filament assembly (not to scale) that illustrates the geometry of the system, and orientation of relevant vectors. Material cross-sections along the filament are parametrized by an arc-length coordinate $s$. As one traverses the filament centerline, the set $(\mathbf{n},\mathbf{p})$ rotates around the tangent axis. (b) Schematic showing the active follower forces $-f_{\mathrm{a}}\mathbf{t}$, and viscous traction $\mathbf{f}$ acting on an infinitesimal length of the filament. Moments acting at $s$ and $s+ds$ are indicated.

Figure 2

Trial functions ${\mathrm{\Theta }}_n\left(s\right)$ used for the shape of the filament. The first four functions corresponding to $n=1,2,3,$ and 4 are shown.

Figure 3

(a) $f_1/U_1$ for the longitudinal motion of the sphere-filament assembly with $\epsilon =0.01$ and $A=0.2$. The results obtained with $N=15$ are indicated by open squares (blue, online); with $N=40$ by open circles (red, online); and with $N=40$ and the regularization scheme by a (blue) dashed line. (b) $f_1/U_1$ for the longitudinal motion of the sphere-filament assembly with $\epsilon =0.01,A=0.2$, and rounded end. The results obtained with $N=15$ and uniform filament radius are indicated by open circles (black) while those with $N=40$ and ${\epsilon }_{\mathrm{f}}\left(s\right)$ as given by Eq. (27) are indicated by the solid (red) line. The filament shape used for the latter is indicated by the dashed curve for ${\epsilon }_{\mathrm{f}}$.

Figure 4

(a) Local drag coefficient, $C_{\parallel }\equiv f_1/U_1$, for the longitudinal motion of the sphere-filament assembly with $\epsilon =0.01$. (b) The translational velocity $U_1$ of the sphere-filament assembly acted upon by the total force of unit magnitude along the filament axis. The open circles (red, online) and stars (blue, online) are the computed results for, respectively, $\epsilon$ = 0.01 and 0.001. The lines represent the approximate fit given by Eq. (29). (c) Local drag coefficient, $C_{\perp }\equiv f_2/U_2$, for the transverse motion of the sphere-filament assembly with $\epsilon =0.01$. (d) The translational velocity $U_2$ of the sphere-filament assembly acted upon by a force of unit magnitude perpendicular to the filament axis. The open circles (red, online) and stars (blue, online) are the computed results for, respectively, $\epsilon$ = 0.01 and 0.001. The lines represent the approximate fit given by Eq. (28).

Figure 5

Results from the linear stability analysis of the slender-body equations for the sphere-filament assembly. (a) Critical values of the nondimensional active force, $\beta$, above which sustained oscillations occur as a function of scaled radius $A\equiv a/l$ of the sphere. Filled circles (red, online) and stars (blue, online) correspond to a freely suspended sphere-filament assembly with $\epsilon$ equal to, respectively, 0.01 and 0.001 while the black squares correspond to the sphere held fixed and $\epsilon =0.01$. Curves/lines are shown to guide the eye. (b) Frequency of sustained oscillations $\omega$ as a function of $A$ at the critical value of $\beta$. Circles (red, online) and stars (blue, online) correspond to freely suspended sphere with $\epsilon$ equal to, respectively, 0.01 and 0.001 while the squares (black, online) correspond to the sphere held fixed and $\epsilon =0.01$. Circles (red, online) and stars (blue, online) correspond to freely suspended sphere-filament assembly with $\epsilon$ equal to, respectively, 0.01 and 0.001.

Figure 6

Summary of results from two versions of the RFT-based minimal formulation described in the Appendix. The first version (Set I, dashed curves) ignores shadowing effects with parameters given by Eq. (A13) in the Appendix. The second version (Set II, solid curves and symbols) accounts for them in an approximate way with parameters given by Eq. (A14) in the Appendix. Here, red corresponds to $\epsilon ={10}^{-2}$ and blue to $\epsilon ={10}^{-3}$. (a) Critical value of $\beta$ for the onset of instability at which a single real eigenvalue or a complex conjugate pair crosses the real axis. We note that for set II that incorporates the effect of shadowing in an approximate manner, the critical value monotonically decreases with $A$ for both values of the aspect ratio. For set I, however, this curve for $\epsilon ={10}^{-3}$ does not follow this trend. (Inset) Focusing on set I, we find that the critical value of $\beta$ for instability is nonmonotonic in $A$ for $\epsilon \in (7\times {10}^{-3},{10}^{-3})$. (b) Plotting the imaginary component of the critical eigenvalue(s) for both sets I and II shows that the nonmonotonic nature arises due to a change in the nature of the bifurcation. For set I, there is a range of $A$ for which the critical eigenvalue is a single eigenvalue with zero imaginary component. In this parameter range the bifurcation is a divergence bifurcation (DB) and the emergent solutions are not oscillatory solutions. For set II, however, instability is always due to a complex conjugate pair and thus the bifurcating branch is a oscillatory solution. (Inset) Close-up of the small $A$ region showing the qualitative differences between the two models.

Figure 7

Displacement $w(s,t)$—to within a multiplicative constant—of the filament as a function of arc-length position $s$ for various times. (a) fixed sphere with $A=0.85$; (b) freely suspended sphere with $A=0.05$.

Figure 8

Here we show the (a) amplitude, and (b) the phase angle $\varphi \left(s\right)$, for the following cases: the freely suspended sphere with $A=0.05$, open squares (blue, online); $A=0.1$, open circles (red, online); $A=0.3$, open diamonds (cyan, online); and $A=0.8$, stars (green, online). The fixed sphere with $A=0.8$, is shown as the solid black line.

Figure 9

(a) Numerical results for the instability modes for two filaments ($N_{\mathrm{f}}=2$) attached to a wall. (i) The first (circles, blue-online) and second (squares, red-online) critical values of $\beta$ at which oscillatory instabilities emerge are plotted as a function of the spacing $D$ between the filaments. The first mode corresponds to the two filaments oscillating in-phase with each other, and the second mode corresponds to the out-of-phase oscillations. (ii) The frequencies of the emergent oscillatory solutions corresponding to these two modes. (b) Numerical results for more multiple filaments. The number of independent modes equals the number of filaments. Critical value of ${\beta }_s$ (i) and the corresponding frequency $\omega$ (ii) at criticality as a function of the number of filaments. There are $N_f$ critical modes and only the results for the first ones that occur as $\beta$ is increased from zero and the last ones are shown. The first mode corresponds to the mode in which all filaments oscillate in phase with each other. The interfilament spacing $D$ is 0.3 and $\epsilon =0.01$.

Figure 10

Amplitudes (to within a constant) and phase angles (divided by $2\pi$) as functions of arc-length $s$ for four hydrodynamically interacting filaments. The distance between the filaments is 0.3 and $\epsilon =0.01$. Panels (a) and (b) correspond to the first mode to become unstable while panels (c) and (d) correspond to the second mode to become unstable. The other two modes that become unstable at even larger values of $\beta$ are not shown. In panel (a) the lower two curves correspond to the two outer filaments while the upper two (almost indistinguishable) curves correspond to the middle two filaments; in panel (c), the lower two curves correspond to the middle two filaments.

Figure 11

Shown are (a) ${\beta }_s$ and (b) the oscillation frequency $\omega$ as functions of the interfilament spacing $D$. Closed circles (green, online) correspond to a square array while stars (blue, online) correspond to a row of filaments in the plane of oscillation. The result for a single filament attached to a wall is indicated by the (red) solid line. (c) Amplitude (to within a constant) and (d) phase angle (divided by $2\pi$) as functions of $s$ for square and line arrays of filaments at the onset of instability. The solid lines are for a square array with $D=0.3$; the dashed lines for a square array with $D=0.7$; the open (red, online) circles and (red, online) stars represent the results for line arrays with spacing equal to, respectively, 0.3 and 0.7.

Figure 12

Mean squared flow generated at infinity divided by the mean power input per unit area from the active force acting on the filaments in a periodic array.

Figure 13

Displacements for the two instabilities for the straight filament with $\epsilon =0.01$ attached to a wall placed in a compressive extensional flow. Displacement as a function of $s$ at the onset of bending instability at $\beta =122$ is shown in (a) and that at the onset of the buckling instability at $\beta =1400$ in (b).

Figure 14

The tension force $T\left(s\right)$ on the filament with $\epsilon =0.01$ attached to a wall in the presence of a compressive extensional flow. The numerical results are indicted by filled circles (red, online) while the approximate values given by Eqs. (48) and (49) are represented by, respectively, the dashed and the solid lines.

Figure 15

Critical values of $\beta$ as a function of $\epsilon$ for the onset of (a) bending and (b) buckling instabilities for a filament attached to a wall placed in a compressive extensional flow. The numerical results are indicated by the filled circles while the dashed lines correspond to the result obtained by Guglielmini et al. [59] who used leading order approximation for the resistivity. The dotted lines correspond to the prediction obtained by adding $O\left(1\right)$ constant to their leading $log\left(\epsilon \right)$ terms. The solid lines corresponding to the fit to the numerical results are given by Eq. (50).