Well-posedness of 2D and 3D swimming models in incompressible fluids governed by Navier--Stokes equations

TL;DR

This paper establishes the well-posedness of mathematical models describing the self-propelled motion of bio-mimetic swimmers in 2D and 3D incompressible fluids governed by Navier-Stokes equations, extending previous work with Stokes models.

Contribution

It introduces and analyzes the well-posedness of new models for self-propelled swimmers in Navier-Stokes fluids, incorporating elastic and rotational forces for shape change-based propulsion.

Findings

Proved existence and uniqueness of solutions for the models.

Extended previous Stokes-based models to Navier-Stokes equations.

Applicable to biological and engineering propulsion systems.

Abstract

We introduce and investigate the wellposedness of two models describing the self-propelled motion of a "small bio-mimetic swimmer" in the 2D and 3D incompressible fluids modeled by the Navier-Stokes equations. It is assumed that the swimmer's body consists of finitely many subsequently connected parts, identified with the fluid they occupy, linked by the rotational and elastic forces. The swimmer employs the change of its shape, inflicted by respective explicit internal forces, as the means for self-propulsion in a surrounding medium. Similar models were previously investigated in [15]-[19] where the fluid was modeled by the liner nonstationary Stokes equations. Such models are of interest in biological and engineering applications dealing with the study and design of propulsion systems in fluids and air.