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
43
1345
1368
Dal Maso, Gianni; De Simone, Antonio; Morandotti, M.
File in questo prodotto:
File
DM-DeS-Mor-SIMA-11.pdf

non disponibili

Licenza: Non specificato
Dimensione 519.65 kB
SIMA2011_Published.pdf

non disponibili

Dimensione 478.84 kB
Formato Unknown

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/20.500.11767/14630`
• ND
• 32
• 31