Micromotions of a Swimmer in the 3-D Incompressible Fluid Governed by the Nonstationary Stokes Equation

Abstract

In this paper we consider a model describing the self-propelled motion of a small abstract swimmer in the three-dimensional (3-D) incompressible fluid, governed by the nonstationary Stokes equation. Typically this fluid is associated with the low Reynolds numbers when inertia is viewed as negligible. It is assumed that the swimmer's body consists of finitely many subsequently connected parts, identified with the fluid they occupy, linked to each other by rotational and elastic Hooke's forces [A. Y. Khapalov and G. Trinh, Discrete Contin. Dyn. Syst. Ser. A, 33 (2012), pp. 1513--1544], [A. Y. Khapalov, Int. J. Appl. Math. Comput. Sci., 23 (2013), pp. 277--290]. Our goal is to derive a formula for asymptotically small motions of this swimmer in the 3-D incompressible fluid. These results can be useful for biological and engineering applications dealing with the study and design of propulsion systems in fluids to serve as a constructive approach to studying their steering (or controllability) properties [A. Y...