We present an analytical framework to study the motion of microswimmers in a viscous fluid. Our main result is that, under very mild regularity assumptions, the change of body shape uniquely determines the motion of the swimmer. We assume that the Reynolds number is very small, so that the velocity field of the surrounding infinite fluid is governed by the Stokes system and all inertial effects can be neglected. Moreover, we enforce the self propulsion constraint (no external forces and torques). Therefore, Newton’s equations of motion reduce to the vanishing of the viscous force and torque acting on the body. By exploiting an integral representation of viscous force and torque, the equations of motion can be reduced to a system of six ordinary differential equations. Variational techniques are used to prove the boundedness and measurability of this system’s coefficients, so that classical results on ordinary differential equations can be invoked to prove existence and uniqueness of the solution.

### An existence and uniqueness result for the motion of self-propelled micro-swimmers

#### Abstract

We present an analytical framework to study the motion of microswimmers in a viscous fluid. Our main result is that, under very mild regularity assumptions, the change of body shape uniquely determines the motion of the swimmer. We assume that the Reynolds number is very small, so that the velocity field of the surrounding infinite fluid is governed by the Stokes system and all inertial effects can be neglected. Moreover, we enforce the self propulsion constraint (no external forces and torques). Therefore, Newton’s equations of motion reduce to the vanishing of the viscous force and torque acting on the body. By exploiting an integral representation of viscous force and torque, the equations of motion can be reduced to a system of six ordinary differential equations. Variational techniques are used to prove the boundedness and measurability of this system’s coefficients, so that classical results on ordinary differential equations can be invoked to prove existence and uniqueness of the solution.
##### Scheda breve Scheda completa Scheda completa (DC)
2011
Settore ICAR/08 - Scienza delle Costruzioni
Dependence of solutions of PDEs on domain and boundary data; Low Reynolds number flows; Stokes equations; Swimming;
File in questo prodotto:
File
DM-DeS-Mor-SIMA-11.pdf

Accesso chiuso

Tipologia: Published version
Licenza: Non pubblico
Dimensione 519.65 kB
Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11384/84157`