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Undulatory swimmer

Swimming bacteria, spermatozoa, or plankton are natural examples of self- propelled, active particles. These living microorganisms have the ability to deform or alter their internal features according to their environment in order to achieve a specific goal. Finding inspiration from such adaptive behaviors is key for the design of artificial devices used in medicine.[13] Such micro-robots are particularly appealing because they can reach selected locations, thus open- ing the way to an increased efficiency in targeted drug delivery and appearing as a minimally-invasive alternative to traditional surgery and large openings. complex interactions sketch don't hurt me no more

Problem formulation

Dynamics

We consider that the swimmers are elongated, flexible and inextensible. They are moreover very thin, meaning that their cross-section diameter \(d\) is much smaller than their length \(\ell\). This assumption allows describing their interactions with the fluid in terms of the slender-body theory. The swimmer is moreover embedded in an incompressible fluid flow whose unperturbed velocity field is denoted by \(u(\boldsymbol{x},t)\). The swimmer's conformation at time \(t\) is characterized by a curve \(X(s,t)\) parametrized by its arc-length \(s\in[-\ell/2,\ell/2]\). The dynamics of the fiber is given by the Cosserat equation, so that

\[\begin{eqnarray} \sigma\,\partial_t^2X &=& - \zeta\,\mathbb{R}\left[\partial_t X-u(X,t)\right] \nonumber +\ \partial_s(T\partial_s X) - K\,\partial_s^4 X + f(s,t). \label{eq:vel_fib-NA} \end{eqnarray}\]

The first force on the right-hand side is the force exerted by the fluid. It involves the drag coefficient \(\zeta = 8\pi\eta\rho_{\rm f}/[2\log(\ell/d)-1]\) (with \(\eta\) the fluid kinematic viscosity and \(\rho_{\rm p}\) its density) and the local Oseen's resistance tensor \(\mathbb{R} = \mathbb{I} -(1/2)\,\partial_sX\,\partial_sX^{\mathsf{T}}\). The second force in the right-hand side is the tension whose amplitude \(T\) is determined by the inextensibility constraint \(|\partial_s X(s,t)| = 1\), valid at all time along the swimmer. The third term is the bending elasticity force which depends upon the swimmer's flexural rigidity \(K\) (product of Young's modulus and inertia). The last term denoted \(f\) is an internal force that is prescribed to account for the so-called \emph{active behaviour} of the swimmer and that is responsible for its locomotion. Equation~(\ref{eq:vel_fib-NA}) is associated with the free-end boundary conditions \(\partial_s^2X(s,t) = 0\) and \(\partial_s^3X(s,t) = 0\) at the fiber's extremities \(s=\pm\ell/2\). The tension itself satisfies a second-order differential equation obtained by imposing \(\partial_t |\partial_sX|^2=0\) with the boundary conditions \(T(s,t) = 0\) at \(s=\pm\ell/2\).

In the limit of very small inertia, that is when the fiber's stopping time \(\sigma/\zeta\) is much sorter than any characteristic time associated to the outer fluid velocity field \(u\), the dynamics further simplifies. It follows the over-damped limit obtained by balancing the viscous drag to the other forces in the slender-body equation, so that

\[\begin{equation} \zeta\,\mathbb{R}\left[\partial_t X-u(X,t)\right] = \partial_s(T\partial_s X) - K\,\partial_s^4 X + f(s,t). \label{eq:vel_fib} \end{equation}\]

Active force

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