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Meccanica

Open Access 06.04.2024 | Research

Breaking the left-right symmetry in fluttering artificial cilia that perform nonreciprocal oscillations

verfasst von: Ariel Surya Boiardi, Roberto Marchello

Erschienen in: Meccanica

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Abstract

Recent investigations on active materials have introduced a new paradigm for soft robotics by showing that a complex response can be obtained from simple stimuli by harnessing dynamic instabilities. In particular, polyelectrolyte hydrogel filaments actuated by a constant electric field have been shown to exhibit self-sustained oscillations as a consequence of flutter instability. Owing to the nonreciprocal nature of the emerging oscillations, these artificial cilia are able to generate flows along the stimulus. Building upon these findings, in this paper we propose a design strategy to break the left-right symmetry in the generated flows, by endowing the filament with a natural curvature at the fabrication stage. We develop a mathematical model based on morphoelastic rod theory to characterize the stability of the equilibrium configurations of the filament, proving the persistence of flutter instability. We show that the emerging oscillations are nonreciprocal and generate thrust at an angle with the stimulus. The results we find at the level of the single cilium open new perspectives on the possible applications of artificial ciliary arrays in soft robotics and microfluidics.

Ariel S. Boiardi and Roberto Marchello have contributed equally to this work.

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1 Introduction

Cilia are tiny active filaments ubiquitous in living systems, from single cell organisms to mammalian epithelial tissue. Both isolated, as is the case of flagella, or assembled in complex ciliary carpets, cilia serve a variety of functions, such as sensing [1], locomotion [2, 3], fluid motion [4] and transport [5]. The effectiveness of these structures in carrying out such a wide range of tasks has inspired the development of artificial replicas for biomimetic actuators, with possible applications in micro-robotics.

While self-sustained oscillations in biological cilia are realized internally by complex molecular motors [6], various external actuation principles have been proposed for artificial cilia, depending on the technologies and materials employed, such as light driven liquid crystals [7], electrically [8] and acoustically [9] actuated hydrogels, and a variety of magnetically responsive materials [10, 11]. Notably, in all mentioned examples, self-sustained oscillations were achieved by spatial or temporal modulation of the external stimuli.

Recent studies have proposed solutions to overcome this limitation by harnessing mechanical instabilities as a way to generate a time dependent response from a steady stimulus. Along this research line, notable examples comprise passive filaments driven by Quincke rotation [12], photoactive filaments exploiting the self-shadowing mechanism proposed in [13], and the emergence of flutter instability in polyelectrolyte (PE) hydrogels, as shown in [14]. An intriguing perspective on this approach comes from the natural world, where instabilities have been found to be responsible for many complex emergent behaviours: from flagellar beating [15, 16] to plants circumnutations [17‐19].

As for the work in [14], experiments on PE gel filaments have provided evidence that self-sustained, periodic oscillations can be achieved by means of a steady and uniform electric field, a behaviour that was interpreted as an instance of flutter instability emerging from the filament’s activity and its interaction with the viscous environment. Interestingly, the arising cyclic motion of such filaments is nonreciprocal, and indeed causes a net thrust in the surrounding fluid in the direction of the electric field, as shown in the left panel of Fig. 1.

Left-right (LR) symmetry breaking is an important function carried out by specialized ciliate cells during mammal embryo development [20]. Drawing inspiration from Nature, the possibility to generate and control complex fluid flows in lab-on-a-chip devices has become a prevalent research field. As shown in the central panel of Fig. 1, asymmetric flows can be generated with the model of [14] by simply reorienting the electric field. Clearly, this strategy would require the modulation in space of the external stimulus to generate a target flow field from an array of artificial cilia. Instead, one might want to achieve the same result by the material and geometric design of the cilia, while keeping the external control simple. This would allow to realize a system of cilia which, under a uniform and constant stimulus, may oscillate so as to generate complex flow fields in the fluid.

In this work, we present a way to tackle this point. In particular, we extend the theoretical results of [14] by considering active filaments which are designed to be naturally curved, i. e. whose stress-free configuration is curved. Results from the present study are anticipated in the right panel of Fig. 1, showing that it is indeed possible to control the emerging dynamics of an active filament by proper design.

The manuscript is organized as follows. We introduce in Sect. 2 the mathematical model for morphoelastic filaments endowed with natural curvature and interacting with a viscous fluid in the context of low Reynolds number hydrodynamics. Section 3 is devoted to the analysis of the equilibrium configurations of such active filaments and to the characterization of their stability. Numerical experiments are presented in Sect. 4 to explore the dynamics of self-oscillating filaments in the nonlinear regime. Section 5 closes the manuscript and presents future perspectives.

2 Mathematical model

In this section, we introduce a minimal mathematical model for an active filament with natural curvature. We describe the filament as an inextensible and unshearable rod [21] and account for its activity by exploiting the framework of morphoelastic rods [22]. Hence, the filament’s activity is encoded by an internal degree of freedom, namely the spontaneous curvature, which constitutively depends on the external stimulus, as later detailed. Similarly to a previous study [14], the filament is clamped at one extremity and is allowed to undergo planar motions while interacting with a viscous fluid.

We fix an orthonormal basis $\{\textbf{e}_1, \textbf{e}_2 \}$ such that the external stimulus ${\textbf{E} = E \textbf{e}_2}$ is in the direction of $\textbf{e}_2$. Since the rod is subject to large deformation, we identify its reference configuration with a segment in the direction of $\textbf{e}_2$ and we describe its current configuration through the motion $(s,t) \mapsto \textbf{r}(s,t)$, parameterized in arc-length $s \in [0,L]$ and time $t \ge 0$, see Fig. 2. Defining $\theta (s,t)$ as the positive, anticlockwise angle between the tangent to the filament and $\textbf{e}_{2}$, we introduce the unit tangent $\textbf{t}(s,t)$ and the unit normal $\textbf{n}(s,t)$ to the current configuration as

$$\begin{aligned} \begin{aligned} \textbf{t}&= - \sin \theta \, \textbf{e}_1 + \cos \theta \, \textbf{e}_2 , \\ \textbf{n}&= -\cos \theta \, \textbf{e}_1 - \sin \theta \, \textbf{e}_2 . \end{aligned} \end{aligned}$$

(1)

Since $\textbf{t}= \partial _s \textbf{r}$ and $\textbf{r}(0,t) = \textbf{0}$, noting that $\partial _t \textbf{t}= \textbf{n}\,\partial _t \theta$, we can express

$$\begin{aligned} \textbf{r}= \int _{0}^{s} \textbf{t},\quad \textbf{v}= \partial _t \textbf{r}= \int _{0}^{s} \textbf{n}\,\partial _t\theta , \end{aligned}$$

(2)

so that the kinematics of the filament is completely determined by $\theta$.

The dynamics is governed by the balance equations of linear and angular momentum: neglecting inertia these read

$$\begin{aligned} \partial _s \textbf{C}+ \textbf{d}= \textbf{0}, \quad \partial _s \textbf{M}+ \partial _s \textbf{r}\times \textbf{C}= \textbf{0}, \end{aligned}$$

(3)

where $\textbf{C}$ is the internal contact force, $\textbf{M}$ is the internal bending moment, and $\textbf{d}$ is the viscous drag force per unit length from the surrounding fluid.

Assuming low Reynolds number hydrodynamics, the viscous drag can be approximated by Resistive Force Theory (RFT) [3]. The fluid structure interaction is hence reduced to a local coupling with the filament velocity by two resistive coefficients $\mu _\parallel$ and $\mu _\perp$, so that

$$\begin{aligned} \begin{aligned} \textbf{d}&= - \left[ \mu _\parallel \textbf{t}\otimes \textbf{t}+ \mu _\perp \textbf{n}\otimes \textbf{n}\right] \textbf{v} = - \mu _\parallel v_\parallel \textbf{t}- \mu _\perp v_\perp \textbf{n}, \end{aligned} \end{aligned}$$

(4)

where $v_\parallel$ and $v_\perp$ are the tangent and normal components of the velocity, respectively. In the slender limit, it is known that $\mu _\perp = 2\mu _\parallel$, a relation that will be assumed next.

It is useful to express the balance equations (3) with respect to the local basis $\{\textbf{t}, \textbf{n}\}$. Having split the internal action in axial and shear components as $\textbf{C}= A\textbf{t}+ V\textbf{n}$, noting that $\partial _s \textbf{t}= \textbf{n}\,\partial _s\theta$ and that $\partial _s \textbf{n}= -\textbf{t}\, \partial _s\theta$, we can project the balance of linear momentum (3)$_1$ along $\textbf{t}$ and $\textbf{n}$, leading to

$$\begin{aligned} \begin{aligned} \partial _s A - V \partial _s \theta - \mu _\parallel v_\parallel&= 0, \\ \partial _s V + A \partial _s \theta - 2 \mu _\parallel v_\perp&= 0. \end{aligned} \end{aligned}$$

(5)

Looking back at (2)$_{2}$, we remark that this formulation of the balance of linear momentum is actually an integro-differential equation. Furthermore, since the problem is planar, the bending moment lies along $\textbf{e}_3 = \textbf{e}_1 \times \textbf{e}_2 = \textbf{t}\times \textbf{n}$, i. e. $\textbf{M}= M \textbf{e}_3$, so that the balance of angular momentum (3)$_2$ reduces to

$$\begin{aligned} \partial _s M + V = 0. \end{aligned}$$

(6)

For what concerns the contact force components A and V, the inextensibility and unshearability constraints prevent us from constitutively prescribing them. These are indeed reactive terms to the constraints above and need to be determined through balance equations (see [21]). We recall that the rod is modelled as morphoelastic and that it has a natural curvature $\kappa _N$, such that the constitutive law for the bending moment is

$$\begin{aligned} M = B \left( \partial _s \theta - \kappa - \kappa _N \right) , \end{aligned}$$

(7)

where B is the bending stiffness and $\kappa (s,t)$ is the spontaneous curvature, which encodes the active response of the filament to the external stimulus $\textbf{E}$. According to (7), the visible curvature $\partial _s\theta$ coincides with the sum of spontaneous and natural curvature in the absence of stress. While the natural curvature is a geometric feature introduced at the fabrication stage, the spontaneous curvature evolves according to the phenomenological law

$$\begin{aligned} \tau \partial _t \kappa + \kappa = -{\kappa }_E \sin \theta . \end{aligned}$$

(8)

This models the viscous relaxation with characteristic time $\tau$ to a target curvature $\kappa _E$, dependent on the intensity of the external stimulus modulated according to the current configuration of the filament with respect to the external field $\textbf{E}$. At points of the filament orthogonal to the external field, the spontaneous curvature evolves to $\kappa _E$, which is then the maximum curvature attainable by the filament at equilibrium. The value and sign of the target curvature depend on the external stimulus. In particular, $\kappa _E$ is positive (negative, respectively) if E is positive (negative, respectively), and higher values of E induce higher values of $\kappa _E$. This evolution law, reminiscent of the gravitropic response of plants in the gravity field [17, 23], proves to be sufficiently accurate to describe the activity of the materials under consideration, as shown in [14] by a prototypical model. Plugging (7) into (6), the balance of angular momentum reads

$$\begin{aligned} B \left( \partial _s^2 \theta - \partial _s \kappa \right) + V = 0 . \end{aligned}$$

(9)

The system of nonlinear equations (5), (8) and (9) is endowed with suitable boundary and initial conditions. Since the rod is clamped at the extremity $s=0$ and has the endpoint $s=L$ free, the boundary conditions are

$$ \begin{aligned} &\theta (0,t) = 0, \\ &A(L,t) = 0, \quad V(L,t)&= 0,\quad M(L,t) = 0, \end{aligned} $$

(10)

for any $t \ge 0$, while the initial conditions read

$$\begin{aligned} \theta (s,0) = \theta _0 (s) , \quad \kappa (s,0) = \kappa _0(s) , \end{aligned}$$

(11)

for any $s \in [0,L]$.

2.1 Nondimensional form of the governing equations

We now recast the governing equations in nondimensional form. For this purpose, we identify the length of the rod L as characteristic length and the relaxation time $\tau$ as characteristic time. Consequently we define $\bar{s}\in [0,1]$ and $\bar{t} \ge 0$ as dimensionless independent variables such that, $s = L \bar{s}$ and $t = \tau \bar{t}$. Likewise, we rescale curvatures by $L^{-1}$, velocities by $L\tau ^{-1}$ and forces by $BL^{-2}$. With a slight abuse of notation, in the following we denote dimensionless quantities by the same symbols as their dimensional counterparts. Upon introducing the three parameters

$$\begin{aligned} \eta _\parallel = \frac{\mu _\parallel L^4}{\tau B}, \quad \chi _E = L \kappa _E, \quad \chi _N = L \kappa _N, \end{aligned}$$

(12)

the dimensionless governing equations are

$$\begin{aligned} \begin{aligned} \partial _s A - V \partial _s \theta - \eta _\parallel v_\parallel&= 0 ,\\ \partial _s V + A \partial _s \theta - 2 \eta _\parallel v_\perp&= 0 ,\\ \partial ^2_s \theta - \partial _s \kappa + V&= 0 ,\\ \partial _t \kappa + \kappa + \chi _E \sin \theta&= 0 , \end{aligned} \end{aligned}$$

(13)

where the dimensionless bending moment reads as

$$\begin{aligned} M = \partial _s \theta - \kappa - \chi _N. \end{aligned}$$

(14)

Finally, the boundary conditions (10) become

$$\begin{aligned} \begin{aligned} &\theta (0,t)= 0,\\ & A(1,t) = 0, \quad V(1,t)&= 0, \quad M(1,t) = 0, \end{aligned} \end{aligned}$$

(15)

for any $t \ge 0$.

3 Stability analysis

Having in mind the work in [14], where nonreciprocal periodic oscillations in gel filaments were proved to be an instance of flutter instability, we are now interested in the stability of equilibria attained by the system governed by equations (13) and (15).

3.1 Equilibrium configurations

Since the external forces acting on the filament are of viscous nature, they are null at equilibrium. Hence, the balance of momenta together with the free-end boundary condition at $s=1$ imply that the internal force and the bending moment vanish at equilibrium, i. e. $A_{\textrm{eq}}= V_{\textrm{eq}}= 0$, and $M_{\textrm{eq}}= 0$. The latter condition means that the equilibrium angle $\theta _{\textrm{eq}}$ must satisfy the first order differential equation $\theta _{\textrm{eq}}'(s) - \kappa _{\textrm{eq}}(s) - \chi _N = 0$ for all $s \in [0,1]$, where the prime denotes differentiation with respect to s. On the other hand, the evolution equation (13)$_4$ gives $\kappa _{\textrm{eq}}= - \chi _E \sin \theta _{\textrm{eq}}$. Combining these relations, it is possible to determine the equilibrium configuration of the filament for different values of the angle $\varphi$ at the clamp. More specifically, the equilibrium configuration is obtained by solving the following initial value problem:

$$\begin{aligned} {\left\{ \begin{array}{ll} \theta _{\textrm{eq}}' + \chi _E \sin \theta _{\textrm{eq}}- \chi _N = 0 \quad \,\,\text {in } (0,1], \\ \theta _{\textrm{eq}}(0) = \varphi , &{} \end{array}\right. } \end{aligned}$$

(16)

where $\chi _E \in \mathbb {R}$, $\chi _N \ge 0$ and $\varphi \in [-\pi /2,\pi /2]$. The case with $\chi _N < 0$ may be easily deduced by symmetry considerations and will not be examined.

First of all, in the absence of the external stimulus, i. e. for $\chi _E = 0$, we easily get

$$\begin{aligned} \theta _{\textrm{eq}}(s) = \varphi + \chi _N s , \end{aligned}$$

(17)

since no internal active processes take place in the rod and its equilibrium configuration is simply an arc of circle (possibly with many loops). Furthermore, if also $\chi _N = 0$, the equilibrium coincides with a straight line.

If $\chi _E \ne 0$ and $\chi _N = 0$, the integration of Eq. (16) leads to the solution

$$\begin{aligned} \theta _{\textrm{eq}}(s) = 2 \arctan \left[ \tan \left( \frac{\varphi }{2} \right) e^{-\chi _E s} \right] , \end{aligned}$$

(18)

which coincides with a vertical straight line in the case of $\varphi = 0$.

If both $\chi _E$ and $\chi _N$ are not null, it is easy to show that the constant angles

$$\begin{aligned} \theta _{\textrm{eq}}(s) = {\left\{ \begin{array}{ll} \pi / 2 &{} \quad \text {if } \chi _E = \chi _N , \\ - \pi / 2 &{} \quad \text {if } \chi _E = -\chi _N , \\ \arcsin {\left( \chi _N /\chi _E \right) } &{} \quad \text {if } |\chi _E| > \chi _N , \\ \end{array}\right. } \end{aligned}$$

(19)

are solutions of Eq. (16) if they are compatible with the value $\varphi$ at the clamp, while there are no constant solutions if $|\chi _E| < \chi _N$. Moreover, if $\chi _E = \pm \chi _N$ the nonconstant equilibrium solutions read

$$\begin{aligned} \begin{aligned} \theta _{\textrm{eq}}(s) =&\mp \frac{\pi }{2} \\&+ 2 \arctan \left[ \tan \left( \frac{\varphi }{2} \pm \frac{\pi }{4} \right) + \chi _N s \right] . \end{aligned} \end{aligned}$$

(20)

Apart from the cases discussed above, the general solution of problem (16) reads

$$\begin{aligned} \begin{aligned}&\theta _{\textrm{eq}}(s) = 2 \arctan \Bigg \{ \frac{\chi _E}{\chi _N} - \sqrt{1 - \frac{\chi _E^2}{\chi _N^2}} \tan \Bigg [ \\&\quad - \frac{\chi _N s}{2} \sqrt{1 - \frac{\chi _E^2}{\chi _N^2}} + \arctan \frac{\frac{\chi _E}{\chi _N} - \tan \left( \frac{\varphi }{2} \right) }{\sqrt{1 - \frac{\chi _E^2}{\chi _N^2}}} \Bigg ] \Bigg \}. \end{aligned} \end{aligned}$$

(21)

It is worth noting that, despite appearance, this function is real-valued even in the case of $|\chi _E| > \chi _N$, as can be shown by trivial trigonometric identities. Clearly, equilibrium configurations exist for $s \in [0,1]$, but the above function is formally defined for all $s \ge 0$. In particular, it can be shown that for $|\chi _E| > \chi _N$ the limit of (21) for $s \rightarrow \infty$ exists

$$\begin{aligned} \theta _{\textrm{lim}} = \lim _{s\rightarrow \infty }\theta _{\textrm{eq}}(s) = \arcsin \left( \frac{\chi _N}{\chi _E}\right) . \end{aligned}$$

(22)

Having computed all the possible equilibrium angles, we now assume the boundary condition $\varphi = 0$, which is relevant to the problem at hand, and we show in Fig. 3 the corresponding equilibrium configurations for $\chi _N \in \{1,2,16\} \times 2\pi$ while varying $\chi _E > 0$. In the absence of the external stimulus, i. e. for $\chi _E = 0$, the filaments form 1, 2, and 16 loops, respectively. More in general, we notice that the spontaneous curvature, which is driven by $\chi _E$, competes with the natural curvature in determining the configuration of the filaments at equilibrium. As the stimulus magnitude is increased, the filaments unwind and, for sufficiently large value of $\chi _E$, become essentially straight and oriented as the limit angle $\theta _{\textrm{lim}}$, modulo a boundary layer at the clamp dictated by the boundary condition $\theta _{\textrm{eq}}(0) = \varphi =0$.

3.2 Linearization about a stationary solution

In order to study the stability of the equilibrium configurations, we linearize the equations (13) assuming incremental solutions of the form

$$\begin{aligned} \begin{aligned} A(s,t)&= \varepsilon A^\ell (s,t) , \\ V(s,t)&= \varepsilon V^\ell (s,t) , \\ \theta (s,t)&= \theta _{\textrm{eq}}(s) + \varepsilon \theta ^\ell (s,t) , \\ \kappa (s,t)&= - \chi _E \sin {\theta _{\textrm{eq}}(s)} + \varepsilon \kappa ^\ell (s,t) , \end{aligned} \end{aligned}$$

(23)

being $A_{\textrm{eq}}= 0$, $V_{\textrm{eq}}= 0$ and $\theta _{\textrm{eq}}$ the equilibrium angle discussed in Sect. 3.1, depending on $\chi _E$ and $\chi _N$, with $\varphi = 0$. Plugging the assumptions (23) into the governing equations (13) and retaining first order terms in $\varepsilon$, we get to the following linearized equations

$$\begin{aligned} \begin{aligned} \partial _s A^\ell - V^\ell \theta _{\textrm{eq}}' - \eta _\parallel v_\parallel ^\ell&= 0 , \\ \partial _s V^\ell + A^\ell \theta _{\textrm{eq}}' - 2 \eta _\parallel v_\perp ^\ell&= 0 , \\ \partial _s^2 \theta ^\ell - \partial _s \kappa ^\ell + V^\ell&= 0 , \\ \partial _t \kappa ^\ell + \kappa ^\ell + \chi _E \left( \cos {\theta _{\textrm{eq}}} \right) \theta ^\ell&= 0 , \end{aligned} \end{aligned}$$

(24)

where

$$\begin{aligned} \begin{aligned} v_\parallel ^\ell&= \sin \theta _{\textrm{eq}}\int _0^s \cos \theta _{\textrm{eq}}\partial _t \theta ^\ell \\&\quad - \cos \theta _{\textrm{eq}}\int _0^s \sin \theta _{\textrm{eq}}\partial _t \theta ^\ell \end{aligned} \end{aligned}$$

(25)

and

$$\begin{aligned} \begin{aligned} v_\perp ^\ell&= \cos \theta _{\textrm{eq}}\int _0^s \cos \theta _{\textrm{eq}}\partial _t \theta ^\ell \\&\quad + \sin \theta _{\textrm{eq}}\int _0^s \sin \theta _{\textrm{eq}}\partial _t \theta ^\ell \end{aligned} \end{aligned}$$

(26)

are the incremental tangent and normal components of the filament’s velocity, while the linearized bending moment is $M^\ell = \partial _s \theta ^\ell - \kappa ^\ell$. Similarly, the boundary conditions (15) become

$$ \begin{aligned} &\theta ^\ell (0,t)= 0, \\& A^\ell (1,t) = 0 , \quad V^\ell (1,t)&= 0 , \quad M^\ell (1,t) = 0 . \end{aligned} $$

(27)

By looking at the linear equations (24), it is worth noting that in the case of a constant equilibrium angle (see Sect. 3.1) one retrieves the model presented in [14]. However, self-weight of the filament is not accounted for in the present study and the forcing term in the evolution law for the spontaneous curvature is modulated by $\cos \theta _{\textrm{eq}}$, to account for the curved equilibrium configuration of the filament. One may argue that, for given values of $\chi _N$ and $\eta _\parallel$, the stability threshold of $\chi _E$ can be estimated from the results reported in [14].

Interestingly, the system of linear, integro-differential equations (24) can be reduced to a single PDE of order 6 with the (incremental) tangential displacement as the only unknown. Similar equations arise in general from the linear analysis of elastic curved rods, as discussed in [24]. However, the numerical solution of the resulting high order PDE would require $\mathcal {C}^2$ continuity, in the case of the Finite Element Method, or the exploitation of other, more involved techniques. Furthermore, as pointed out in Fig. 3, large values of $\chi _E$ lead to equilibrium configurations in which the filament is almost straight, apart from a boundary layer at the clamp, so that the formulation of the problem in terms of the (incremental) tangential displacement may not be viable in the present context. For these reasons, we have resorted to equations (24) for the linear stability analysis of the filament’s equilibrium configurations.

We now recast the linear governing equations in the following form. By removing the superscript ‘$\ell$’ for the sake of simplicity, we define the vector of the incremental unknowns $\textbf{X}= (A,\,V,\,\theta ,\,\kappa )^\textsf{T}$ and write the equations (24) in matrix form as

$$\begin{aligned} \mathbb {D} \, \partial _t \textbf{X}= \mathbb {K} \, \textbf{X}, \end{aligned}$$

(28)

where

$$\begin{aligned} \mathbb {D} = \begin{bmatrix} 0 &{} 0 &{} \eta _\parallel J_\parallel &{} 0 \\ 0 &{} 0 &{} 2 \eta _\parallel J_\perp &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \\ \end{bmatrix} \end{aligned}$$

(29)

is the damping matrix, with $J_\parallel$ and $J_\perp$ being the integral operators corresponding to Eqs. (25) and (26), such that $J_\parallel \partial _t \theta = v_\parallel$ and $J_\perp \partial _t \theta = v_\perp$, and

$$\begin{aligned} \mathbb {K} = \begin{bmatrix} \partial _s &{} -\theta _{\textrm{eq}}' &{} 0 &{} 0 \\ \theta _{\textrm{eq}}' &{} \partial _s &{} 0 &{} 0 \\ 0 &{} 1 &{} \partial _s^2 &{} -\partial _s \\ 0 &{} 0 &{} -\chi _E (\cos {\theta _{\textrm{eq}}}) &{} -1 \\ \end{bmatrix} \end{aligned}$$

(30)

is the stiffness matrix.

In order to study the stability of the equilibrium configurations, we seek for time-harmonic solutions in the form of $\textbf{X}(s,t) = e^{\omega t} \hat{\textbf{X}}(s)$, such that Eq. (28) leads to the generalized eigenvalue problem

$$\begin{aligned} \omega \, \mathbb {D} \, \hat{\textbf{X}} = \mathbb {K} \, \hat{\textbf{X}} . \end{aligned}$$

(31)

The above problem cannot be solved in closed form, so we resort to a numerical approach by means of the Finite Element Method, as discussed in Appendix B.

3.3 Results

Denoting with $\omega _{\max } = \alpha + i \beta$ the eigenvalue solution of (31) with maximum real part, we distinguish between the regime of stability where $\alpha < 0$, and the regime of flutter instability, where $\alpha \ge 0$ and $\beta \ne 0$. In the regime of flutter, any perturbation of the equilibrium causes oscillations of increasing amplitude. After a transient, nonlinear effects drive the system towards a limit cycle [25], such that the transition from stability to flutter is an instance of a supercritical Hopf bifurcation [26].

We show in Fig. 4 the results of the stability analysis for $\chi _E \in [-50, 150]$ and $\eta _\parallel \in [0.1,2.5]$, by comparing the results of [14], i. e. for $\chi _N = 0$, with those from our model for a small value of the natural curvature, $\chi _N = 0.01 \times 2 \pi$. We notice that the behaviour of the two systems is essentially the same for $\chi _E > 0$, while they significantly differ for $\chi _E < 0$. In particular, we find that the flutter region predicted by the model in [14] (left panel) is suppressed by a however small imperfection introduced by the natural curvature (right panel). We speculate that the limit for $\chi _N\rightarrow 0$ might be discontinuous and restrict any further analysis to the case of $\chi _E > 0$.

We next explore the dependence of the region of flutter instability on the model parameter $\chi _N$. In particular, we report in Fig. 5 the results of the stability analysis for $\chi _E \in [90,300]$, $\eta _\parallel \in [0.1,2.5]$, and $\chi _N \in \{0, 8, 16, 24\} \times 2\pi$. We notice that, by increasing the value of $\chi _N$, the critical threshold of $\chi _E$ increases for all $\eta _\parallel$. This is reasonable in view of the equilibrium configurations shown in Fig. 3. In fact, a larger value of $\chi _N$ entails a larger value of the external stimulus needed to completely unwind the filaments and hence a larger value of $\chi _E$ to cause flutter. Not surprisingly, our analysis predicts that only an unrolled filament (i. e. for $\chi _E > \chi _N$) may flutter.

Furthermore, the colormaps in Fig. 5 show the dependence on the relevant parameters of the oscillation frequency $\beta / 2 \pi$ inside the instability regions. In all panels, the frequency decreases as the value of $\eta _\parallel$ increases, since the latter is a dimensionless measure related to the damping of the system. Besides, fixing a pair $(\eta _\parallel , \chi _E)$ that falls inside the instability regions for all $\chi _N$, we qualitatively notice that the oscillation frequency decreases for increasing $\chi _N$.

4 Numerical experiments from the nonlinear model

Table 1

Quantitative results of the numerical simulations shown in Fig. 6. We report the norm of the averaged reaction force on the clamp $\Vert \bar{\textbf{F}}\Vert$ and its angle $\phi _{\bar{\textbf{F}}}$, that correspond to the magenta dashed lines on the right panels of Fig. 6. We report also the limit angle $\theta _{\textrm{lim}}$ defined in Eq. (22), the period $2 \pi / \beta$ predicted by the linear stability analysis, and the actual period T of the nonlinear oscillations. All values are dimensionless

$\chi _N$	$\Vert \bar{\textbf{F}}\Vert$	$\phi _{\bar{\textbf{F}}}-\pi$	$\theta _{\textrm{lim}}$	2 $\pi / \beta$	T
8 $\times$ 2 $\pi$	1.0938	0.2900	0.2541	0.2555	0.2500
16 $\times$ 2 $\pi$	0.8045	0.5600	0.5267	0.2754	0.2750
24 $\times$ 2 $\pi$	0.2286	0.8683	0.8541	0.3301	0.3400

As previously discussed, the linear stability analysis predicts the emergence of blowing-up oscillations for specific combinations of the model parameters (regime of flutter, see Sect. 3.3). We now explore the dynamics of the system after the initial transient, when nonlinear effects become prevalent. We do so by means of numerical experiments on the nonlinear model equations (13). In particular, we discretize the governing equations via the Finite Element Method; see Appendix A for details on the implementation.

An aspect of particular interest in the present investigation is the total force that the oscillating filament exerts on the surrounding fluid, and its dependence on the natural curvature $\chi _N$. Such dynamic quantity can be easily computed by integration of Eq. (3)$_1$ and by exploiting the free-end boundary condition at $s=1$, leading to

$$\begin{aligned} \textbf{F}(t) := - \textbf{C}(0,t) = \int _0^1 \textbf{d}(s,t) \, \,\textrm{d}{s} , \end{aligned}$$

(32)

a force, function of time, for which we denote by $\phi _\textbf{F}(t)$ the angle it forms with $\textbf{e}_2$, assumed positive if anticlockwise.

Guided by the stability analysis, we fix $\chi _E = 200$ and $\eta _\parallel = 1$ in the numerical simulations, such that the model parameters fall inside the flutter regions shown in Fig. 5 for all the considered values of $\chi _N = \{8, 16, 24\} \times 2 \pi$ (notice the black dots in the figure). The results of the numerical computations are summarized in Fig. 6.

In all the cases, as predicted by the linear stability analysis, spontaneous oscillations of increasing amplitude bring the system on a closed orbit of period denoted by T. From a theoretical standpoint, such periodic solution (limit cycle) is attained asymptotically, but is well approximated for all practical purposes by the motion of the filament at finite time. Over one period, the reaction force at the clamp spans the cones delimited by the green arrows reported in the left panels. To quantify the net effect of this oscillating force on the fluid, we introduce $\bar{\textbf{F}}$ as the time average of $\textbf{F}(t)$ over one cycle of shape change, and $\phi _{\bar{\textbf{F}}}$ the associated, positive anticlockwise angle with $\textbf{e}_{2}$. While the reaction force is oriented for most of the time in a direction close to the orthogonal to the filament (graphs of $\phi _\textbf{F}(t)$ on the right panels), the average force $\bar{\textbf{F}}$, which is responsible for the net thrust exerted by the filament on the fluid, is almost aligned with the tangent to the filament (dashed lines for $\phi _{\bar{\textbf{F}}}$ on the right panels). Its norm $\Vert \bar{\textbf{F}}\Vert$, however, is typically much smaller than $\Vert \textbf{F}(t)\Vert$ over a cycle.

Increasing the value of $\chi _N$, we notice that the angle $\phi _{\bar{\textbf{F}}}$ increases, while the amplitude of the oscillations decreases, as shown also by the tip trajectories in the central panels. It is known that the area spanned by the state variables of a system undergoing a periodic motion measures its nonreciprocity [27, 28], which in turn is linked to the norm of $\bar{\textbf{F}}$ in our system. Indeed, a reciprocal shape change cannot generate a net thrust in the low Reynolds number regime. From the right panels of Fig. 6, we notice that the value of $\Vert \bar{\textbf{F}}\Vert$, even if small compared to $\Vert \textbf{F}(t)\Vert$, is not zero and decreases while increasing $\chi _N$. We gather in Table 1 the values of the reaction force averaged over a period, showing the dependence mentioned above. We also report the values of the period T itself, showing a strong agreement with the period $2\pi / \beta$ predicted by the linear stability analysis of Fig. 5.

From the results shown in Fig. 6 and in Table 1 it is clear that it is possible to increase the horizontal component of the net thrust, for fixed $\eta _\parallel$ and $\chi _E$, by increasing the value of the natural curvature $\chi _N$. On the other hand, we notice that such increase in the angle $\phi _{\bar{\textbf{F}}}$ occurs at the expense of the magnitude of the net force $\Vert \bar{\textbf{F}}\Vert$ that the filament is able to generate. Furthermore, having in mind the evolution of the region of instability in Fig. 5, increasing $\chi _N$ too much might bring the fixed value of $\chi _E$ in the stability regime, not allowing the filament to perform self-sustained oscillations and hence generate thrust.

To further investigate the dependence of the reaction force on the natural curvature of the filament, we show in Fig. 7 the norm and the angle of $\bar{\textbf{F}}$ as functions of $\chi _N \in [0,200]$. These results confirm that an increase in the natural curvature leads to an increase of the angle of the net force, but to a decrease of its magnitude.

We remark a significant agreement between the angle $\phi _{\bar{\textbf{F}}}$ and the limit angle $\theta _{\textrm{lim}}$, meaning that the filament pulls indicatively along the direction to which the tip points at equilibrium. Increasing the natural curvature above the value $\chi _N \approx 164$ leads the system out of the flutter regime. Hence, self-sustained oscillations disappear and the filament ceases to move the fluid and produce a net thrust.

5 Conclusions and perspectives

In this work, we have extended the model presented in [14] by endowing an active filament with a natural curvature at the fabrication stage. We have identified the different equilibrium configurations of the filament while varying the relevant model parameters. We have found that, under a minimal external stimulus of sufficiently large magnitude, a naturally curved morphoelastic filament performs nonreciprocal oscillations that break the LR symmetry while interacting with a viscous fluid. Such nonreciprocal oscillations are shown to arise as a consequence of flutter instability, in agreement with the results of [14].

For sufficiently large values of the external stimulus, the equilibrium configuration of the filament is almost straight and forms a characteristic angle with respect to the vertical direction. As flutter is initiated, oscillations of increasing amplitude take place around such equilibrium, as predicted by the linear stability analysis. Moreover, we have numerically simulated the time evolution of the system and we have shown that the periodic oscillations that arise as a consequence of the instability are nonreciprocal and allow the filament to generate a net thrust on the fluid environment. Interestingly, the direction of the thrust generated by the self-oscillating filament depends on its natural curvature. More precisely, we have shown that, fixing the value of the external stimulus, an increase in the angle at which the filament pulls on the fluid can be obtained at the expense of the magnitude of the drag force by increasing the natural curvature.

The purpose of this study is to identify suitable design principles to fabricate artificial cilia whose net effect on the fluid environment can be controlled by geometric and material parameters. In this respect, an interesting application of our results may be the study of the collective behaviour of artificial ciliary carpets, in which each cilium can be properly engineered so as to have different natural curvature and generate thrust at different angles, while fluttering under the same constant and uniform external stimulus. Lastly, it would be interesting to investigate formulations of the linear stability analysis alternative to that presented in Sect. 3.2.

Acknowledgements

Both authors are members of the ‘Gruppo Nazionale di Fisica Matematica’ (GNFM) of the ‘Istituto Nazionale di Alta Matematica’ (INdAM). The authors are grateful to Prof. Giovanni Noselli for his guidance in this work.

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Appendix

A Details on the numerical simulations

All numerical experiments were performed by Finite Element simulations using the commercial software COMSOL Multiphysics^® 6.1 (build 357).

The balance of linear momentum in the form of Eq. (3)$_1$, along with the zero force boundary condition at the endpoint $s=1$, allows to compute the contact forces as

$$\begin{aligned} \textbf{C}(s,t) = \int ^{1}_{s} \textbf{d}(\sigma ,t)\,\textrm{d}{\sigma }, \end{aligned}$$

(A1)

where $\textbf{C}$ and $\textbf{d}$ are dimensionless.

For the formulation of the Finite Element model, the two remaining equations, namely the balance of angular momentum and the evolution law of the spontaneous curvature, were recast in weak form. Starting with the balance of angular momentum, we multiply (13)$_3$ by a suitable test function $\hat{\theta }$ and apply integration by parts; exploiting the zero moment boundary condition (15)$_{4}$, we arrive at the following equation for the unknown $\theta$

$$\begin{aligned} \int _{0}^{1} \left[ \left( \partial _s\theta - \kappa -\chi _N\right) \partial _s\hat{\theta } - V\hat{\theta } \right] \,\textrm{d}{s} = 0. \end{aligned}$$

(A2)

Similarly, testing Eq. (13)$_{4}$ by a suitable function $\hat{\kappa }$, we obtain the weak form for the evolution of the spontaneous curvature:

$$\begin{aligned} \int _{0}^{1} \left[ \left( \partial _t \kappa + \kappa + \chi _E \sin \theta \right) \hat{\kappa } \right] \,\textrm{d}{s} = 0. \end{aligned}$$

(A3)

The parametrization interval [0, 1] was decomposed with a nonuniform mesh of 60 elements, which was finer near the clamp in order to better represent the boundary layer of some equilibrium solutions discussed in Sect. 3.1. For the semi-discretization in space, $\theta$ and $\kappa$ were approximated by quadratic and by linear Lagrange polynomials, respectively.

The temporal discretization was performed by backward finite differences using using the Time Dependent Solver of COMSOL with a step size $\Delta t = 0.005$.

A.1 Reconstruction of velocity field in the fluid

Having numerically solved the nonlinear governing equations for the configuration of the filament and for the distribution of the drag forces acting upon it, one can reconstruct the velocity field induced in the surrounding fluid (shown in Fig. 1) by the motion of the filament via a boundary integral approach, as detailed in the following.

Let us recall the fundamental solution of the Stokes equations associated with a point force $\textbf{f}\in \mathbb {R}^2$: the so-called Stokeslet. It is known [29] that the velocity field induced in a viscous fluid by a point force $\delta (\varvec{\xi })\textbf{f}$ concentrated at $\varvec{\xi } \in \mathbb {R}^2$ is given by

$$\begin{aligned} \textbf{u}(\textbf{x}) = \frac{1}{8\pi \mu } \mathbb {G}(\textbf{x};\varvec{\xi }) \textbf{f} , \end{aligned}$$

(A4)

where $\textbf{x}\in \mathbb {R}^2$, $\mu$ is the viscosity of the fluid, and the Oseen tensor $\mathbb {G}$ reads

$$\begin{aligned} \mathbb {G} (\textbf{x};\varvec{\xi }) = \frac{1}{R} \mathbb {I} + \frac{1}{R^3}\textbf{R}\otimes \textbf{R}, \end{aligned}$$

(A5)

with $\textbf{R} = \textbf{x} - \varvec{\xi }$ and $R = \Vert \textbf{R} \Vert$. By exploiting such fundamental solution, the effect of the filament on the surrounding fluid can be approximated through the drag force distribution $-\textbf{d}(s)$ in Eq. (4), so that at any place $\textbf{x}$ not on the filament the velocity reads

$$\begin{aligned} \textbf{u}\left( \textbf{x}\right) = - \frac{1}{8\pi \mu } \int _{0}^{L} \mathbb {G}\left( \textbf{x};\textbf{r}(s)\right) \textbf{d}(s)\,\textrm{d}{s}. \end{aligned}$$

(A6)

This approximation, valid again in the slender limit, is only local as hydrodynamic self-interactions of the filament are not considered.

From this approximation of the velocity field at all points in the fluid domain, we can compute path lines, that are the trajectories of tracer particles in Fig. 1, as solutions of the differential equation

$$\begin{aligned} \dot{\textbf{x}}(t) = \textbf{u}\left( \textbf{x}(t),t \right) . \end{aligned}$$

(A7)

The numerical solution of this equation was obtained using Matlab’s ode113 solver.

A.2 Parameters for the simulations in Fig. 1

The results in Fig. 1 were obtained by simulations of the nonlinear equations in nondimensional form (13), as detailed in the previous sections of this Appendix. The nondimensional parameters were set as follows: $\eta _{\parallel } = 1$, $\chi _E = 200$ and $\chi _N = 24\times 2\pi$. For the central panel, the electric field was rotated by an angle 0.8541, corresponding to the limit equilibrium angle (22) of the filament with natural curvature in the right panel for the selected parameters.

B Numerical solution of the generalized eigenvalue problem

In this section we give some details on the approximation of the generalized eigenvalue problem (31) implemented in COMSOL Multiphysics^® 6.1.

The internal forces were computed by integration of the drag distribution on the filament, as in (A1). However, in the linear case the drag distribution is computed with the linear counterparts of the velocity components (25) and (26).

Equations (24)$_{3}$ and (24)$_{4}$ were rewritten in weak form and discretized as their nonlinear counterparts as discussed in Appendix A.

With this semi-discretization in space, the damping and stiffness matrices can be assembled, leading to the generalized eigenvalue problem

$$\begin{aligned} \omega \mathbb {D}_{h} \hat{\textbf{X}}_{h} = \mathbb {K}_{h} \hat{\textbf{X}}_{h} , \end{aligned}$$

(B8)

where $\textbf{X}_{h}$ is the vector of nodal unknowns and $\mathbb {D}_{h}$ and $\mathbb {K}_{h}$ are the discrete damping and stiffness matrices, respectively. Once the discrete system (B8) has been assembled, the eigenvalues and eigenfunctions are determined by the solver using the QZ algorithm [30].

We remark that computing the internal actions as (A1) instead of imposing their balance in weak form removes empty columns in the discrete damping matrix. These empty columns are caused by the absence of time derivatives of $A^\ell$ and $V^\ell$ in the balance Eq. (24) and can compromise the reliability of the eigenvalue solver. The weak balance Eq. (A2) does not contain time derivatives explicitly, but the integral formulation of $V^\ell$ does contain time-derivatives of $\theta ^\ell$, through the velocity components in Eqs. (25) and (26).

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Titel: Breaking the left-right symmetry in fluttering artificial cilia that perform nonreciprocal oscillations
verfasst von: Ariel Surya Boiardi
Roberto Marchello
Publikationsdatum: 06.04.2024
Verlag: Springer Netherlands
Erschienen in: Meccanica
Print ISSN: 0025-6455
Elektronische ISSN: 1572-9648
DOI: https://doi.org/10.1007/s11012-024-01765-7

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\(\chi _N\)	\(\Vert \bar{\textbf{F}}\Vert\)	\(\phi _{\bar{\textbf{F}}}-\pi\)	\(\theta _{\textrm{lim}}\)	2 \(\pi / \beta\)	T
8 \(\times\) 2 \(\pi\)	1.0938	0.2900	0.2541	0.2555	0.2500
16 \(\times\) 2 \(\pi\)	0.8045	0.5600	0.5267	0.2754	0.2750
24 \(\times\) 2 \(\pi\)	0.2286	0.8683	0.8541	0.3301	0.3400

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Abstract

Publisher's Note

1 Introduction

2 Mathematical model

2.1 Nondimensional form of the governing equations

3 Stability analysis

3.1 Equilibrium configurations

3.2 Linearization about a stationary solution

3.3 Results

4 Numerical experiments from the nonlinear model

5 Conclusions and perspectives

Acknowledgements

Publisher's Note

Appendix

A Details on the numerical simulations

A.1 Reconstruction of velocity field in the fluid

A.2 Parameters for the simulations in Fig. 1

B Numerical solution of the generalized eigenvalue problem

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