The well-posedness of a swimming model in the 3-D incompressible fluid governed by the nonstationary Stokes equation

Abstract

We introduce and investigate the well-posedness of a model describing the self-propelled motion of a small abstract swimmer in the 3-D incompressible fluid governed by the nonstationary Stokes equation, typically associated with low Reynolds numbers. It is assumed that the swimmer's body consists of finitely many subsequently connected parts, identified with the fluid they occupy, linked by rotational and elastic Hooke forces. Models like this are of interest in biological and engineering applications dealing with the study and design of propulsion systems in fluids.

Addendum to “The well-posedness of a swimming model in the 3-D incompressible fluid governed by the nonstationary Stokes equation”

“…The above change of the forcing term in (3) to (4) is essentially a typing error, which does not affect the rest of our previous work (Khapalov, 2013).…”

“…In this addendum we address some unintentional omission in the description of the swimming model in our recent paper (Khapalov, 2013) .…”

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In this addendum we address some unintentional omission in the description of the swimming model in our recent paper (Khapalov, 2013) .Keywords: swimming models, coupled PDE/ODE systems, nonstationary Stokes equation.In our recent work (Khapalov, 2013) we introduced the following mathematical model for a swimmer whose body consists of finitely many subsequently connected parts S i (z i (t) linked by rotational and elastic Hooke forces:In the above, Ω is a bounded domain in R 3 with boundary ∂Ω of class C 2 , y(x, t) and p(x, t) are respectively the velocity and the pressure of the fluid at point x = (x 1 , x 2 , x 3 ) ∈ Ω at time t, while ν is the kinematic viscosity constant.The swimmer in (1)- (3) is modeled as a collection of n open connected bounded sets S i (z i (t)), i = 1, . .

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“…Namely, the forces described in this term are A. Khapalov intended to be internal relative to the swimmer. However, the form of (3) satisfies this condition in terms of forces applied to z i (t) as the centers of mass of S i (z i (t)) only if the sets S i (0)'s have identical measure (for details, see Khapalov, 2010;2013).…”

“…In our recent work (Khapalov, 2013) we introduced the following mathematical model for a swimmer whose body consists of finitely many subsequently connected parts S i (z i (t) linked by rotational and elastic Hooke forces:…”