Internship : The Study of The N-Links Swimmer for Monoflagellate Micro-swimmers

[Blessed-joseph]

Introduction

Microrobotics is a field with diverse applications, such as environmentalmonitoring, medical procedures, and micro-assembly. Designing microrobots that can efficiently swim in fluids is challenging due to the dominance of viscosity over inertia at the microscale. Researchers have looked to biological microorganisms like bacteria and sperm cells, which have evolved effective swimming mechanisms for low-Reynolds number. Most microrobots rely on remote power sources, as implementing onboard energy sources at this scale is difficult. One of the pioneering works in this field taylor who established the mathematical settings for the problem of biological self-propulsion powered by thin undilating filaments.

Objectives

During my Intership at IRMA (Institut de Recherche Mathématiques Avancées), the aim of the internship was to study the 3D N-links model of micro-swimmers based on the thesis, then work on the validation of the model. The validation was carried out in two stages first, by studying planar motion , and then by examining non-planar motion , comparing the obtained results with those presented in the referenced articles used also conducing numerical simulations to analuse the factors influencing the micro-swimmer's movement.

Equation of Motion

The equation of motion of the swimmer, which relies on self-propulsion, is given by the balance of forces and torques applied to the head and the links, assuming no external forces or torques are present as:

$$ \begin{cases} F_h + \sum_{i=1}^{N} F_i = 0, \\ T_h + \sum_{i=1}^{N} T_i = 0. \end{cases} $$

The previous system can be rewritten matricially as:

$$ M(\Theta, \Phi) \begin{pmatrix} \dot{X} \\ \dot{\Theta} \\ \dot{\Phi} \end{pmatrix} = 0, $$

with $M$ a matrix of dimension 6 $\times$ (2N+6). The matrix $M (\Theta, \Phi)$ $\in \mathbb{R}^{6 \times (2N + 6)})$ can be subdivided into two sub-matrices $M_{X,\Theta}(\Theta, \Phi) \in \mathbb{R}^{6 \times 6}$ and $M_{\Phi}(\Theta, \Phi) \in \mathbb{R}^{6 \times 2N}$ such that

$$ M(\Theta, \Phi) = \left( M_{X,\Theta}(\Theta, \Phi) \mid M_{\Phi}(\Theta, \Phi) \right). $$

Using these sub-matrices in the equation above, and assuming that the values of $\dot{\Phi}(t)$ are prescribed, the position and orientation of a self-propelled swimmer can be obtained from the deformations of its tail by solving the following differential system:

$$ \begin{pmatrix} \dot{X} \\ \dot{\Theta} \end{pmatrix} = M_{X,\Theta}^{-1}(\Theta, \Phi) M_{\Phi}(\Theta, \Phi) \dot{\Phi}. $$

To summarize the modelization of the swimmer:

• The swimmer is described by $2N+6$ variables; 6 position variables and $2N$ shape variables.
• The position variables are $X$, the Cartesian coordinates of the swimmer's head, and $\Theta = (\theta_x^i, \theta_y^i, \theta_z^i))$, its orientation.
• The shape variables are the angles $\phi_y^i$ and $\phi_z^i$ of the (i)-th segment.
• The dynamics of the swimmer can be described by a linear control system where the control $u(t)$ represents the derivatives of $\Phi$:

Numericals Results

Planar frame

First we compute the displacement of a flagellum over the time given by this equation

$$ r(s, t) = \frac{L_{\text{head}}}{2} e_1(t) + \int_{0}^{s} \left[ \cos(\psi(u, t)) e_1(t) + \sin(\psi(u, t)) e_2(t) \right] , du, $$

where $\psi(s,t)$ expression is given by:

$$ \psi(s,t) = K_0 s + 2A_0 s \cos(\omega t - \frac{2 \pi s}{\lambda}) . $$

with $\lambda$ the wavelength of the flagellar wave, $\omega$ the angular frequency of the flagellar beat, $K_0$ a simple measure for the asymmetry of the mean shape of the flagellum, and $A_0$ the amplitude of curvature.
We obtain :

This Figure illustrates the displacement of a flagellum over time. The
flagellum oscillates periodically, which facilitates the movement of the micro-
swimmer. Each colored curve represents the position of the flagellum at
different moments in beat cycle.

The displacement of the head over the time :
Solving the equation giving the motion the swimmer, we obtain :

This figure shows the motion of the head of the swimmer in the frame $(e_x, e_y)$
with the flagellum beat

Non planar frame

In this case the equation of the motion of the flagellum is expressed by:

$$ \begin{align} \begin{cases} \frac{d\mathbf{r(s,t)}}{ds} &= \psi_1(s,t), \\ \frac{d\psi_1(s,t)}{ds} &= k_f \psi_3(s,t), \\ \frac{d\psi_2(s,t)}{ds} &= \tau_f \psi_3(s,t), \\ \frac{d\psi_3(s,t)}{ds} &= -k_f \psi_1(s,t) + \tau_f \psi_2(s,t). \end{cases} \end{align} $$

with $\psi_i \in \mathbb{R}^3$, $i \in {1, 2, 3}$, $k_f$ and $\tau_f$ are the flagellar curvature and twist, respectively. We assume a constant flagellar twist $\tau_f$ and a flagellar curvature $k_f$ given by a travelling bending wave,
$k_f(l, t) = K_0 + B \cos(\omega_0 t - \lambda l),$
where $K_0$ is the mean curvature, $B$ the amplitude, $\omega_0$ the angular frequency, and ${\lambda}$ the wavelength.

This figure shows the motion of the tail for three test cases for (N = 10).

• Top left ((K_0 = 35100), Twist = 0): The trajectory is smooth and controlled by the stiffness (K_0), with no influence from the twist. This results in a regular and predictable movement.
• Top right ((K_0 = 0), Twist = 4770.0): In the absence of stiffness, the high twist ((Twist = 4770.0)) leads to a more complex and varied trajectory.
• Bottom ((K_0 = 35100), Twist = 4770.0): This case combines both stiffness and twist, resulting in a trajectory that displays characteristics of both smoothness and complexity, reflecting the combined influence of these two parameters.

The displacement of the Head over the time

This model, designed to be easy to implement and solve, is well-suited for predicting the behavior
of microorganisms in low Reynolds number environments.

Future works :

The next challenge is about applying the multiple shooting method to study optimal control for the swimmer model, which could involve controlling the dynamics of the swimmer to achieve certain goals (e.g., maximizing distance traveled, minimizing energy expenditure, or following a specific trajectory).

For more : https://master-csmi.github.io/csmi-stages-2024/csmi-stages-2024/m1/_attachments/Assigbe-Komi-Joseph-Beni.pdf