A Appendix A: Solutions to uncoupled Navier–Stokes equations

Well-posedness of 2- and 3- swimming models in incompressible fluids governed by Navier–Stokes equations1

Abstract

We introduce and investigate the wellposedness of two models describing the self-propelled motion of a “small bio-mimetic swimmer” in the 2- and 3- 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.

Corresponding author: Alexandre Khapalov2 , Department of Mathematics,

Washington State University, USA (email: khapala@math.wsu.edu)

Piermarco Cannarsa, Department of Mathematics, University of Rome “Tor Vergata”, Italy

Fabio S. Priuli, Istituto per le Applicazioni del Calcolo “M. Picone” (C.N.R.), Rome, Italy

Giuseppe Floridia, Department of Mathematics, University of Rome “Tor Vergata”, Italy

Key words: Swimming models, Navier-Stokes equations, hybrid PDE/ODE systems


AMS subject classifications: 76D05, 35Q30, 76Z10, 70K40.

1 Introduction

The swimming phenomenon has been the subject of interest for many researchers in various areas of natural sciences for a long time, aimed primarily at understanding biomechanics of swimming locomotion of biological organisms, see Gray [12](1932), Gray and Hancock [13] (1951), Taylor [28] (1951), [29] (1952), Wu [32] (1971), Lighthill [22] (1975), and others. This research resulted in the derivation of a number of mathematical models for swimming motion in the (whole) - or -spaces with the swimmer to be used as the reference frame, see, e.g., Childress [5] (1981) and the references therein. In particular, based on the size of Reynolds number, it was suggested (for the purpose of simplification) to divide swimming models into three groups: microswimmers (such as flagella, spermatozoa, etc.) with “insignificant” inertia; “regular” swimmers (fish, dolphins, humans, etc.), whose motion takes into account both viscosity of fluid and inertia; and Euler’s swimmers, in which case viscosity is to be ’’neglected”.

It also appears that the following two, in fact, mutually excluding approaches were distinguished to model the swimming phenomenon (see, e.g., Childress, [5] (1981)). One, which we can call the “shape-transformation approach”, exploits the idea that the swimmer’s shape transformations during the actual swimming process can be viewed as a set-valued map in time (see the seminal paper by Shapere and Wilczeck [26] (1989)). The respective models describe the swimmer’s position in a fluid via the aforementioned maps, see, e.g., [6] (1981), [25] (2008), [7] (2011) and the references therein. Typically, such models consider these maps as a priori prescribed, in which case the question whether the respective maps are admissible, i.e., compatible with the principle of self-propulsion of swimming locomotion or not, remains unanswered. In other words, one cannot guarantee that the model at hand describes the respective motion as a self-propulsive, i.e., swimming process. To ensure the positive answer to this question one needs to be able to answer the question whether the a priori prescribed body changes of swimmer’s shape can indeed be a result of actions of its internal forces under unknown in advance interaction with the surrounding medium.

The other modeling approach (we will call it the “swimmer’s internal forces approach” or SIF approach) assumes that the available internal swimmer’s forces are explicitly described in the model equations and, therefore, they determine the resulting swimming motion. In particular, these forces will define the respective swimmer’s shape transformations in time as a result of an unknown-in-advance interaction of swimmer’s body with the surrounding medium under the action of the aforementioned forces. For this approach, we refer to Peskin [23] (1975), Fauci and Peskin [8] (1988), Fauci [9] (1993), Peskin and McQueen [24] (1994), Tytell, Fauci et al [31] (2010), Khapalov [17] and the references therein.

The original idea of Peskin’s approach is to view a “narrow” swimmer as an immaterial “immersed boundary”. Within this approach the swimming motion is defined at each moment of time by the explicit swimmer’s internal forces. Following the ideas of Peskin’s approach, Khapalov introduced the immersed body SIF modeling approach in which the bodies of “small” flexible swimmers are assumed to be identified with the fluid within their shapes, see [15]-[19] (2005-2014). Indeed, in the framework of Peskin’s method the swimmer is modeled as an immaterial curve, identified with the fluid, further discretized for computational purposes on some grid as a collection of finitely many “cells”, which in turn can be viewed as an immerse body, see Figures 1 and 2. The idea here is to try, making use of mathematical simplifications of such approach, to focus on the issue of macro dynamics of a swimmer. The simplifications (they seem to us to be legitimate within the framework of our interest) include the reduction of the number of model equations and avoiding the analysis of micro level interaction between a “solid” swimmer’s body surface and fluid. It should be noted along these lines that in typical swimming models dealing with “solid” swimmers, the latter are modeled as “traveling holes” in system’s space domain, that is, the aforementioned “micro level” surface interaction is not in the picture as well.

In the above-cited works by Khapalov [15]-[19], the immerse body SIF approach was applied to the nonstationary Stokes equations in 2- and 3- dimensions with the goal to investigate the well-posedness of respective models and their controllability properties. In this paper our goal is to extend these results with respect to well-posedness to the case of Navier-Stokes equations in both the 2- and 3- incompressible fluids. To our knowledge, there were no previous publications investigating this issue within the SIF approach.

Figure 1: 2-D swimmer consisting of 4 rectangles.
Figure 2: 3-D swimmer consisting of 4 parallelepipeds.

Related references on well-posedness of swimming models. To our knowledge, in the context of PDE approach to swimming modeling, the classical mathematical issues of well-posedness were addressed for the first time by Galdi [10] (1999), [11] (2002) for a model of swimming micromotions in (with the swimmer as the reference frame). In [25] (2008) San Martin et al discussed the well-posedness of a 2- swimming model within the framework of the shape transformation approach for the fluid governed by the 2- Navier-Stokes equations.

Swimming models in the framework of ODE’s. A number of attempts were made to introduce various reduction techniques to convert swimming model equations into systems of ODE’s (e.g., by making use of empiric observations and experimental data, etc.), see, e.g., Becker et al [3] (2003); Kanso et al. [14] (2005); Alouges et al. [1] (2008), Dal Maso et al. [7] (2011) and the references therein.


The paper is organized as follows. In Section 2 we state our main results. In Section 3 we discuss in detail the modeling approach of this paper to swimming locomotion. In Section 4 we prove our main results after stating several auxiliary lemmas, proven further in Section 5. In Sections 6 and 7 (Appendices A and B) we remind the reader some classical results used in our proofs.

2 Main results

To formulate these results, we will need the following function spaces.


Let be a bounded domain with locally Lipschitz boundary . Below, we use the following classical notations:

  • denotes the dimension of the space domain, equal either to 2 or to 3;

  • denotes the space of infinitely many times differentiable functions with compact support in ;

  • denotes space of distributions in , i.e., the dual space of ;

  • , denote the Sobolev spaces over , i.e., the Banach spaces of functions in whose first (generalized) derivatives belong to ;

  • , and ;

  • denotes the subspace of consisting of functions vanishing on . denotes the dual space of .

Following [30], page 5, we also introduce the following -dimensional vector function spaces:

where the symbol stands for the closure with respect to the -norm, and – with respect to the -norm. The latter is induced by the scalar product

where is the differentiation operator with respect to . To simplify notations, below we will use the notation (resp. ) both for functions (resp. ) and for functions (resp. ).


Let and stand for the dual spaces respectively of and . Then, identifying with , we have

Our main results deal with the well-posedness of 2- and 3- swimming models (3), (4), (10)-(3.2), described in detail in the next section and visualized by Figures 1 and 2.

Theorem 1 (Well-posedness of the 2- swimming model)

Let and for some let , , . Assume that the assumptions (H1)-(H2) (given in Section 3) hold, and that

(1)

where is the constant in (H1). (Assumption (1) ensures that no parts of swimmer’s body overlap with each other and all lie within .) Then, there exists such that system (3), (4), (10),(3.2)/(3.3),(3.2) admits a unique solution in , and

(2)

The formula for , complementing the given pair in models (3), (4), (10)-(3.2), is given in Proposition A.1 below.

Theorem 2 (Well-posedness of the 3- swimming model)

Let , be of class and . Then the result stated in Theorem 1 holds for model (3), (4), (10),(3.2)/(3.3), (3.2) for a unique triplet such that and , where .

Theorem 3 (Additonal regularity)

Let be of class . If , then in Theorem 2 solution lies in . In turn, if , then in Theorem 1.

Here, . This result is an immediate consequence of Theorem 6 in Section 6 (Appendix A).

Remark 2.1

The duration of time (i.e., of existence of solutions) in Theorems 1-3 depends on the parameters of respective model ’s, ’s, ’s and the initial shape and position of the swimmer in . One can view as a new initial instant of time to apply these theorems again to further extend the interval of existence from to some , provided that the assumptions of the corresponding theorem hold at , and so on. Our proofs below indicate that in the 2- case one can extend this time up to the collision of swimmer with the boundary of or up to the moment when swimmer’s body parts will collide with each other, see conditions (1)-(2). In turn, these circumstances depend on or can be regulated by a suitable choice of swimmer’s internal forces, i.e., functions ’s (and ’s when they can vary, see the next section). To the contrary, in the 3- case, such interval of existence of solution will also depend on ’s, ’s, and ’s via an additional condition (48) in Theorem 5 (compare it to Theorem 4).

3 Swimming model

Following [17], [18]-[19], we describe the locomotion of a swimmer in a fluid by a hybrid nonlinear system of two sets of pde/ode equations:

(3)
(4)

System (3) describes the evolution of incompressible fluid due to the Navier-Stokes equations under the influence of the forcing term representing the actions of swimmer. Here, are respectively the velocity of the fluid and its pressure at point at time , and is the kinematic viscosity constant. In turn, system (4) describes the motion of the swimmer in , whose flexible body consists of sets within . These sets are identified with the fluid within the space they occupy at time and are linked between themselves by the rotational and elastic forces as illustrated on Figures 1 and 2. The points ’s represent the centers of mass of the respective parts of swimmer’s body. The instantaneous velocity of each part is calculated as the average fluid velocity within it at time .


We will now describe the assumptions on the parameters of model (3)-(4) in detail.

3.1 Swimmer’s body

Below, for simplicity of notations, we will denote the sets also as or . Throughout the paper we assume the following two main assumptions:

(H1) All sets are obtained by shifting the same set , i.e.,

(5)

where is open and lies in a ball of radius , and its center of mass is the origin.

Remark 3.1

The results of this paper will hold at no extra cost if we assume that the swimmer at hand consists of different body parts , in which case one will need to replace (5) with and add respective normalizing coefficients in the expressions for swimmer’s internal forces (10)-(3.2) to ensure that they satisfy the 3rd Newton’s Law. In particular, the swimmers on Figures 1 and 2 consist of identical sets each of which has its own orientation in space.


We will need the following concepts from  [2, Section 3.11] to formulate our second assumption on the geometry of swimmer in this paper, which will also be used in the respective proofs below.

Given a bounded set and a unit vector , denote by

respectively the orthogonal hyperplane to and the orthogonal projection of on . Then, for every we will call the set

(6)

the section of corresponding to . Accordingly, given a function and any , we define the function , called section of corresponding to , as

(7)

(H2) There exist positive constants and such that for any vector we can find a vector , which satisfies

(8)

where is the symmetric difference between and , i.e. .

Assumption (H2) means that size of the projection of on the hyperplane in perpendicular to vector changes uniformly Lipschitz continuously relative to the magnitude of the shift of the set in the direction of . This assumption is principally weaker than the respective assumption on the regularity of the shifts of in [17], [18]-[19], where was always selected to be (we will illustrate it in Example 3.1 below). In the case when , it is easy to verify that (H2) is satisfied for discs and rectangles in 2- and for balls and parallelepipeds in 3-.


Remark 3.2

In this paper we assume that all swimmer’s body parts are identical sets. One can choose these sets to be of distinct shapes and sizes, in which case, however, the respective normalizing coefficients should be added to the forcing terms to ensure that all swimmer’s forces are to be its internal forces.

We conclude this subsection with an example showing that there exist particular shapes of the set which require the presence of in (H2) instead of the straightforward choice .

Example 3.1. Fix a constant . We claim that there exist sets which satisfies (H2) for some , but cannot be selected to be co-linear with .

Figure 3: Set in Example 3.1.

Let us consider the set shown on Figure 3,

where the functions are defined as follows:

. Observe that the part of the boundary corresponding to is given by lines with slopes either or . If we introduce the notation () for the isosceles triangle of base in and height , and for the isosceles triangle , then we have

Denote by the center of mass of and set . We claim that, no matter what and we choose, if we shift by , for a suitable , and if we use , then we can always find such that

and, thus, (8) does not hold. On the other hand, by setting for all , we can prove that

(9)

so that  (8) is satisfied with and .

Since the Lebesgue measure is invariant with respect to translations, in the computations below we will always use and in place of and .

We start with the negative result. For every fixed and , let us consider large enough to have and . By choosing , we claim we can find such that , so that the choice is not suited for this set .

The projection on the –axis (which is the orthogonal line to ) is the segment . Thus, the value belongs to and (see Figures 3- 4)

i.e., consists of the union of the sections of the symmetric differences of the triangles in whose height exceeds . Hence, from which our claim follows.

Figure 4: The choice of section line in Example 3.1.

We now show that for , and , the sections of the symmetric difference for all satisfy (9).

Let . Consider, as a preliminary step, the symmetric difference for any fixed . It is not difficult to verify that for

as illustrated by Figure 5. To pass to the sections of , it is sufficient to observe that is the interval on the –axis and that or any , there holds:

Figure 5: The measure of the sections can be estimated by the height of the right triangle with base and hypotenuse parallel to the side of . Left: . Right: .

3.2 Swimmer’s internal forces

In this subsection we give the precise description of the force term in (3). Due to the nature of the swimming motion as self-propulsion, all the forces in model (3)-(4) are internal relative to the swimmer, i.e., their sum is equal to zero and their torque is constant. In turn, these forces, acting between swimmer’s body parts, will create a pressure upon the surrounding fluid and, thus, will act as external forces upon it. We assume that all forces act through the immaterial links attached to the centers of mass ’s of the sets , and then are uniformly transmitted to all points in their respective supports.

Similar to [17], in this paper we consider two types of forces forming in (3): rotational forces and elastic forces, which we represent as

(10)

The 2nd term in (10) describes the forces acting as elastic links between any two subsequent sets ’s to preserve the integrity of swimmer’s structure. They act according to the 3rd Newton’s law and Hooke’s law with variable (positive) rigidity coefficients when the distances between any two adjacent points and , , deviate from the respective given values (see [17]):

(11)

Here denotes the characteristic function of , i.e.,

Remark 3.3

In the above structure we can also assume that can be of any sign and replace (3.2) with more general (and simpler) description of pairs of co-linear forces between ’s:

(12)

All the proofs in this paper are given for technically more complex case of elastic forces in (3.2).


The 1st term in (10) describes the forces that allow each point to rotate any pair of the adjacent points and about it in either folding or unfolding fashion. In turn, by the 3rd Newton’s law, these points will act back on with the respective countering force. The description of rotational forces requires principally different approaches for the 2- and 3- cases.

In the 2- case all the forces lie in the same plane at all times, and we can describe them by making use of the matrix

as follows (see [17]):

(13)

where functions characterize the strength and orientation (folding or unfolding) of respective pairs of rotational forces at time .

In turn, in the 3- case, to satisfy the 3rd Newton’s law, we need to make sure that the respective rotational forces acting on and lie in the same plane spanned by the vectors and . In order to achieve the continuity of these forces in time, in this paper we choose to reduce their magnitudes to zero, when the triplet approaches the aligned configuration (for other options see [19]). Indeed, such configuration admits infinitely many planes containing this triplet, which makes it an intrinsic point of discontinuity for the procedure of the choice of the rotational plane by means of the rotational forces whose magnitudes are strictly separated from zero. Respectively, we define the d rotational forces as follows:

(14)

where the scalar functions control the magnitudes of the rotational forces and determine whether they act in folding or unfolding fashion, and

Note that and for any when points converge to the aligned configuration.


Remark 3.4

The forcing term in (3), (10) can (more precisely) be denoted as . However, we will use a shorter notation as, typically, there is no ambiguity about the choices of and .

We can prove the following result.

Lemma 3.1

Let , and be fixed. Assume that for all there holds

with as in (H1). Then, the forcing term defined in (10) belongs to and there hold the following estimates

The above estimates yield that

(15)

where positive constant depends on and -norms of parameters ’s and ’s.

Remark 3.5

It is not difficult to see that the above-defined forces in (10)-(3.2) are internal relative to the swimmer at hand, i.e., their sum is zero and their angular momentums are constant (see, cf. [17, Chapter 12]).

4 Proofs of the main results

The proof of Theorem 1 will be based on Schauder’s fixed point theorem, and will follow from a series of lemmas. The general scheme of our proofs is similar to that introduced in [15], [16], [17] (Chapters 11-12), [19] for the case of the fluid governed by the linear nonstationary Stokes equations. In this paper we consider the full nonlinear Navier-Stokes equations, which will require some principal technical modifications of this scheme. Our arguments below are nearly the same for both 2- and 3- cases. However, in the part dealing with the Navier Stokes equations, the latter is traditionally more challenging.


Remark 4.1

In our our proofs given below we assume that all the parameters in Theorems 1 and 2 are fixed, namely: , the initial datum either in (if ) or in (if ), a vector as in (1) and functions , . We may also omit the explicit mentioning of dependence of some of the constants below from these parameters.


We will use the symbol to denote the length of the time-interval of existence of solutions to Navier Stokes equations in (3). For , due to Theorem 4 cited in Section 6, .


For , the size of depends on as cited in Theorem 5 in Section 6 (Appendix A).

More precisely, in this case we will need to identify the value of which, given , will work uniformly for any selection of used in the proofs of Theorems 1-2 below. Namely, we set:

(16)

where is from (15).

In subsection 4.1 we introduce necessary notations and will state the aforementioned lemmas, whose proofs will be given in section 5. In subsection 4.2 we will prove Theorems 1-2.

4.1 Formulation of auxiliary lemmas

Set . For any and , set

We will show below that, for suitable choice of pair , the following three nonlinear operators will be well defined.


Operator . Set

(17)

where are trajectories of system (4) with the aforementioned fixed initial state . We will show that for suitable pair they will satisfy (2).


Operator . For any , set: