Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Purcell swimmer near a wall

Enrico Micalizio, Marco Morandotti, Henry Shum, Marta Zoppello

Abstract

We study the effects of hydrodynamic interactions between a wall and the Purcell three-link swimmer in the two-dimensional case. After deriving the equations of motion in a low Reynolds number regime using Resistive Force Theory with suitably modified drag coefficients, we show, by means of criteria from Geometric Control Theory, that the system is controllable at configurations that are nearly parallel to the wall. Furthermore, we study configurations that are tilted, and we show net displacement with respect to the initial orientation. Some numerical experiments illustrate the analytical results.

Purcell swimmer near a wall

Abstract

We study the effects of hydrodynamic interactions between a wall and the Purcell three-link swimmer in the two-dimensional case. After deriving the equations of motion in a low Reynolds number regime using Resistive Force Theory with suitably modified drag coefficients, we show, by means of criteria from Geometric Control Theory, that the system is controllable at configurations that are nearly parallel to the wall. Furthermore, we study configurations that are tilted, and we show net displacement with respect to the initial orientation. Some numerical experiments illustrate the analytical results.
Paper Structure (6 sections, 1 theorem, 30 equations, 4 figures)

This paper contains 6 sections, 1 theorem, 30 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Mathematical model
  3. Controllability
  4. Swimmer tilted with respect to the wall
  5. Numerical simulations
  6. Conclusions

Key Result

Theorem 1

Let $L>0$ and let $L_{\pm1}=L$, $L_0=\lambda L$, for a certain $\lambda>0$. Then control system control_sys is locally controllable around an aligned configuration parallel to the wall, i.e., for $\theta_t\,,\alpha_t^{(\pm1)}\approx0$.

Figures (4)

  • Figure 1: Representation of the Purcell swimmer near the wall.
  • Figure 2: Numerically computed displacements $\Delta x$, $\Delta y$, and $\Delta\theta$ as functions of $\xi$ and corresponding theoretical values from the vector field $\mathbf{h}^3_t$. The initial state is $(x_0,y_0,\theta_0,\alpha^{(-1)}_0,\alpha^{(1)}_0) = (0,2,0,0,0)$. The geometrical parameters used are $L=1$, $\lambda = 1$, $r=0.01$.
  • Figure 3: Comparisons between numerically computed and theoretical displacements $||\Delta \mathbf{x}|| = \sqrt{\Delta x^2 + \Delta y^2}$ and $\Delta\theta$ as functions of the tilt angle $\theta_0$ with respect to the wall for controls corresponding to the vector field $\mathbf{h}^3_t$. The initial state is $(x_0,y_0,\theta_0,\alpha^{(-1)}_0,\alpha^{(1)}_0) = (0,2,\theta_0,0,0)$. The geometrical parameters used are $L=1$, $\lambda = 1$, $r=0.01$. The nonlinear numerical solutions use the nonlinear drag coefficients \ref{['coefficients']} whereas the linearized solution uses the approximation \ref{['136']}.
  • Figure 4: Comparisons between numerically computed and theoretical displacements $\Delta x$, $\Delta y$, and $\Delta\theta$ as functions of the link length ratio $\lambda$ for controls corresponding to the vector field $\mathbf{h}^3_t$ with amplitude $\xi=10^{-3}$. The initial state is $(x_0,y_0,\theta_0,\alpha^{(-1)}_0,\alpha^{(1)}_0) = (0,2,0,0,0)$. The common geometrical parameters used are $L=1$ and $r=0.01$.

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Remark 1
  • Remark 2