References
###### Abstract

We introduce and investigate the wellposedness of a model describing the self-propelled motion of a small abstract swimmer in the 3- incompressible fluid governed by the nonstationary Stokes equation, typically associated with the 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’s forces. In this paper we are attempting to extend the 2- version of this model, introduced in [18]-[20], to the 3- case. Models like this are of interest in biological and engineering applications dealing with the study and design of propulsion systems in fluids.

Wellposedness of a swimming model in the 3- incompressible fluid governed by the nonstationary Stokes equation.

A.Y. Khapalov 111This work was supported in part by NSF Grant DMS-1007981.

Department of Mathematics

Washington State University, Pullman, WA 99164-3113;

e-mail: khapala@math.wsu.edu

Key words: Swimming models, hybrid systems, nonstationary Stokes fluid.

1. Introduction and problem formulation. It seems that the first quantitative research in the area of swimming phenomenon was aimed at the biomechanics of specific biological species: Gray [10](1932), Gray and Hancock [11] (1951), Taylor [36] (1951), [37] (1952), Wu [41] (1971), Lighthill [25] (1975), and others. These efforts resulted in the derivation of a number of mathematical models (linked the size of Reynolds number) for swimming motion in the whole - or -spaces with the swimmer to be used as the reference frame, see, e.g., Childress [3] (1981) and the references therein. Such approach however requires some modification if one wants to track the actual position of swimmer in a fluid.

A different modeling approach was proposed by Peskin in the computational mathematical biology (see Peskin [30] (1975), Fauci and Peskin [6] (1988), Fauci [7] (1993), Peskin and McQueen [31] (1994) and the references therein), where a swimmer is modeled as an immaterial immersed boundary identified with the fluid, further discretized for computational purposes on some grid. In this case a fluid equation is to be complemented by a coupled infinite dimensional differential equation for the aforementioned “immersed boundary”.

In this paper we intend to deal with the swimming phenomenon in the framework of non-stationary PDE’s along the immersed body approach summarized in Khapalov [20] (2005-2010). Namely, in [18] (2005), inspired by the ideas of the above-cited Peskin’s method, we introduced a 2- model for “small” flexible swimmers assuming that their bodies are identified with the fluid occupying their shapes. This approach views such a swimmer as the already discretized aforementioned immersed boundary supported on the respective grid cells, see, e.g., Fig. 1 below. Our model offered two novel features: (1) it was set in a bounded domain with (2) governing equations to be a fluid equation coupled with a system of ODE’s describing the spatial position of swimmer within the space domain. We established the wellposedness of this model up to the contact of a swimmer at hand either with the boundary of space domain or with itself. The need of such type of models was motivated by the intention to investigate controllability properties of swimming phenomenon (see [20]).

Our goal in this paper is to investigate the wellposedness of a 3- version of this model.

Figure 1: 4-parallelepipeds swimmer with all elastic forces and rotation forces about only.

Further remarks on bibliography. It should be noted that, the classical mathematical issue of wellposedness of a swimming model as a system of PDE’s for the first time was apparently addressed by Galdi [8] (1999) for a model of swimming micromotions in (with the swimmer serving as the reference frame).

Another available approach to modeling of swimming motion (apparently, initiated by the work Shapere and Wilczeck [34] (1989)) exploits the idea that the swimmer’s body shape transformations during the actual swimming process can be viewed as a set-valued map in time. The respective models describe swimmer’s position via such maps, see [13] (1981), [33] (2008), [4] (2011) and the references therein. Some models treat these maps as a priori prescribed, in which case the crux of the problem is to identify which maps are admissible, i.e., compatible with the principle of self-propulsion of swimming locomotion. In the case when the aforementioned motion map is not a priori prescribed (i.e., it will be defined at each moment of time by swimmer’s internal forces and the interaction of its body with the resisting surrounding medium), the model will have to include extra equations, see, e.g., [40] in the framework of the immersed boundary method and the references therein.

More recently, a number of significant efforts, both theoretical and experimental, were made to study models of possible bio-mimetic mechanical devises which employ the change of their geometry, inflicted by internal forces, as the means for self-propulsion, see, e.g., S. Hirose [16] (1993), Mason and Burdick [26] (2000); McIsaac and Ostrowski [27] (2000); Martinez and J. Cortes [28] (2001); Trintafyllou et al. [39] (2000); Morgansen et al. [29] (2001); Fakuda et al. [5] (2002); Guo et al. [12] (2002); Hawthornee et al. [14] (2004), and the references therein. It was also recognized that sophistication and complexity of design of bio-mimetic robots give rise to control-theoretic methods, see, e.g., Koiller et al. [23] (1996); McIsaac and Ostrowski [27] (2000); Martinez and Cortes [28] (2001); Trintafyllou et al. [39] (2000); San Martin et al. [32](2007), Alouges et al. [1] (2008), Sigalotti and Vivalda [35] (2009), and the references therein. It should be noted however that the above-cited results deal with control problems in the framework of ODE’s only.

A number of attempts were made along these lines to introduce various reduction techniques to convert swimming model equations into systems of ODE’s, namely, by making use of applicable analytical considerations, empiric observations and experimental data, see, e.g., Becker et al [2] (2003); Kanso et al. [17] (2005); San Martin et al. [32] (2007); Alogues et al. [1] (2008), and the references therein.

Problem formulation for 3- swimming model. We consider the following model, consisting of two coupled systems of equations: one is a PDE system– for the fluid, governed by the nonstationary 3- Stokes equation, and the other is an ODE system– for the position of the swimming object (or swimmer) in it:

 ∂y∂t=νΔy+F(z,v)−∇pinQT=Ω×(0,T),y=(y1,y2,y3)
 divy=0inQT,y=0inΣT=∂Ω×(0,T),y∣t=0=y0inΩ,
 dzidt=1mes{Si(0)}∫Si(zi(t))y(x,t)dx,zi(0)=zi0,i=1,…,n,n>2,

where for :

 z(t)=(z1(t),…,zn(t)),zi(t)∈R3,i=1,…,n,v(t)=(v1(t),…,vn−2(t))∈Rn−2,
 F(z,v)=n∑i=2[ξi−1(x,t)ki−1(∥zi(t)−zi−1(t)∥R3−li−1)∥zi(t)−zi−1(t)∥R3(zi(t)−zi−1(t))
 +ξi(x,t)ki−1(∥zi(t)−zi−1(t)∥R3−li−1)∥zi(t)−zi−1(t)∥R3(zi−1(t)−zi(t))]
 +n−1∑i=2vi−1(t)⎡⎣ξi−1(x,t)(Ai(zi−1(t)−zi(t)))(t)−ξi+1(x,t)∥zi−1(t)−zi(t)∥2R3∥zi+1(t)−zi(t)∥2R3(Bi(zi+1(t)−zi(t)))(t)⎤⎦
 −n−1∑i=2ξi(x,t)vi−1(t)⎡⎣(Ai(zi−1(t)−zi(t)))(t)−∥zi−1(t)−zi(t)∥2R3∥zi+1(t)−zi(t)∥2R3(Bi(zi+1(t)−zi(t)))(t)⎤⎦.

In the above, is a bounded domain in with boundary of class , and are respectively the velocity and the pressure of the fluid at point at time , while is a kinematic viscosity constant. Let us explain the terms in (1.1)-(1.3) in more detail.

Swimmer: The swimmer in (1.1)-(1.3) is modeled as a collection of bounded sets of non-zero measure (such as balls, parallelepipeds, etc.), identified with the fluid within the space which they occupy. These sets are assumed to be open bounded connected sets symmetric relative to the points ’s which are their centers of mass. The sets ’s are viewed as the given sets ’s (“0” stands of the origin) that have been shifted to the respective positions ’s without changing their orientation in space. Respectively,

 ξi(x,t)={1,ifx∈Si(zi(t)),0,ifx∈Ω∖Si(zi(t)),i=1,…,n.

Throughout the paper we assume that each lies in a “small” neighborhood of the origin of given radius , while denotes the set shifted to point . Denote by

 S0=maxi=1,…,n{mes{Si(0)}},S0=mini=1,…,n{mes{Si(0)}}.

Forces: We assume that these sets are subsequently linked by forces described by the term. No “actual” physical links between sets are assumed (i.e., they are assumed to be negligible in terms of affect on the resisting surrounding fluid). The forces in (1.3) are internal, relative to the swimmer – their sum is zero. We assume that a force applied to a set acts evenly upon all its points, and, as such, it creates an external force on the fluid surrounding .

The structural integrity of the swimming object is preserved by the elastic forces acting according to Hooke’s Law. They act along the lines connecting the respective adjacent centers ’s when the distances between any two adjacent points and deviate from the respective given values as described in the first sum in (1.3). The parameters characterize the rigidity of the links , . The matching pairs of these forces between and , and between and are shown on Fig. 1.

The 2nd sum in (1.3) describes the rotation forces about any on the points which make the adjacent points to rotate about it perpendicular to the lines connecting the respective ’s. To satisfy the 3rd Newton’s Law, these forces lie in the same plane along with the matching counter-force given in the 3rd sum in (1.3). Respectively,

 Ai=Ai(zi−1(t),zi(t)),zi+1(t)):R3→span{zi−1(t)−zi(t),zi+1(t)−zi(t))},
 Bi=Bi(zi−1(t),zi(t)),zi+1(t)):R3→span{zi−1(t)−zi(t),zi+1(t)−zi(t)}

denote a nonlinear mappings, defined at each moment of time by three vectors , such that for :

•  (zi−1(t)−zi(t))′[(Ai(zi−1(t)−zi(t)))(t)]=0,(zi+1(t)−zi(t))′[(Bi(zi+1(t)−zi(t)))(t)]=0;
•  ∥(Ai(zi−1(t)−zi(t)))(t)∥R3=∥zi−1(t)−zi(t)∥R3,
 ∥(Bi(zi+1(t)−zi(t)))(t)∥R3=∥zi+1(t)−zi(t)∥R3;
• and the directions of vectors and are such that they correspond to either folding or unfolding motion of lines and relative to the point .

The magnitudes and directions of the rotation forces are determined by the given coefficients , . The choice of fractional coefficients at terms in (1.3) ensures that the momentum of swimmer’s internal forces is conserved at any (see calculations in [19] in the 2- case). A matching pair of rotation forces, generated by for the adjacent points, is shown on Fig. 1.

Swimmer’s motion. Dynamics of points are determined by the average motion of the fluid within their respective supports ’s as described in (1.2).

Local and global approach to solutions of (1.1)-(1.3). Note that, when the adjacent points in the swimmer’s body share the same position in space, the forcing term in (1.3) and hence model (1.1)-(1.3) become undefined. While such situation mathematically seems possible, it does not have to happen. First of all, one can address the issue of local existence of solutions to (1.1)-(1.3) on some “small” time-interval , assuming that initially model (1.1)-(1.3) is well-defined in the above sense. This is the primary subject of this paper (see the next section). Then the question of global existence can be viewed as the issue of suitable selection of coefficients ’s with the purpose to ensure that the aforementioned ill-posed situation is avoided.

In model (1.1)-(1.3) we chose the fluid governed by the nonstationary Stokes equation which, along with its stationary version, is a typical choice of fluid for micro-swimmers (the case of Low Reynolds numbers). The empiric reasoning behind this is that, due to the small size of swimmer, the inertia terms in the Navier-Stokes equation, containing the 1-st order derivatives in and , can be omitted, provided that the frequency parameter of the swimmer at hand is a quantity of order unity. However, it was noted that a microswimmer (e.g., a nano-size robot) may use a rather high frequency of motion, which may justify at least in some cases the need for the term in Stokes model equations. In general, it seems reasonable to suggest that the presence of this term (in a number of cases) can provide a better approximation of the Navier-Stokes equation than the lack of it. We also point out that in [6], [7], [31] the full-size Navier-Stokes equation is used for micro-swimmers. It also seems that the methods we use for the nonstationary Stokes equation (as opposed to stationary Stokes equation), may serve as a natural step toward the swimming models, based on the Navier-Stokes equation.

2. Main result: Local existence and uniqueness. Let denote the set of infinitely differentiable vector functions with values in which have compact support in and are divergence-free, i.e., in . Denote by the closure of this set in the -norm and by denote the orthogonal complement of in (see, e.g., [24], [38]). In introduce the scalar product

 [ϕ1,ϕ2]=∫Ω3∑j=13∑i=1ϕ1jxiϕ2jxidx,ϕ1(x)=(ϕ11,ϕ12,ϕ13),ϕ2(x)=(ϕ21,ϕ22,ϕ23).

Denote by the Hilbert space which is the completion of in the norm

 ∥ϕ1∥H(Ω)=  ⎷∫Ω3∑j=13∑i=1ϕ21jxidx.

Everywhere below we will assume the following two assumptions:

Assumption 2.1. For the given , defining the size of sets in (1,4), assume that

 li−1>2r,i=2,…,n;¯¯¯¯¯Si(zi(0))⊂Ω,∥zi,0−zj,0∥R3>2r,  i,j=1,…,n, i≠j;

and the sets are such that

 ∫(Si(0)∪Si(h))∖(Si(0)∩Si(h))dx=∫Ω|ξi(x)−ξi(x−h)|dx≤C∥h∥R3   ∀h∈Bh0(0)

for some positive constants and , where is the characteristic function of and .

Conditions (2.1) mean that at time , any two sets do not overlap, and that the swimmer lies in . Condition (2.2) is a regularity assumption of Lipschitz type regarding the shift of the set . It is satisfied, for instance, for balls and parallelepipeds.

Assumption 2.2. Assume that within some neighborhood of the initial datum in (1.2) the mappings and are Lipschitz for all in the following sense:

 ∥Ai(ai−1,ai,ai+1)(ai−1−ai)−Ai(bi−1,bi,bi+1)(bi−1−bi))∥R3
 ≤L{∥ai−1−bi−1)∥R3+∥ai−bi∥R3+∥ai+1−bi+1∥R3},
 ∥Bi(ai−1,ai,ai+1)(ai+1−ai)−Bi(bi−1,bi,bi+1)(bi+1−bi)∥R3
 ≤L{∥ai−1−bi−1)∥R3+∥ai−bi∥R3+∥ai+1−bi+1∥R3},

for any , where is a constant and ’s and ’s are defined as in the introduction by three respective vectors .

Assumption 2.2 can be satisfied if, e.g., the points are not on the same line and the mappings and are selected by making use of Gramm-Schmidt orthogonalization procedure for vectors and . Alternatively, we can define ’s and ’s, making use of the cross-product:

 (Ai(zi−1(t)−zi(t))(t)=e1(t)∥zi−1(t)−zi(t))∥R3,
 (Bi(zi+1(t)−zi(t))(t)=e2(t)∥zi+1(t)−zi(t))∥R3,ei(t)=vi(t)∥vi(t)∥R3,i=1,2,
 v1(t)=(zi−1(t)−zi(t))×[(zi−1(t)−zi(t))×(zi+1(t)−zi(t))],
 v2(t)=[(zi+1(t)−zi(t))(zi+1(t)−zi(t))]×(zi+1(t)−zi(t)).

Here is the main result of this paper.

Theorem 2.1.  Let ; ; , ; , ; and , , and let Assumptions 2.1 and 2.2 hold. Then there exists a , such that system (1.1) - (1.3) admits a unique solution on , . Moreover, , , where , and equations (1.1) and (1.2) are satisfied almost everywhere, while Assumptions 2.1 and 2.2 hold in .

Remark 2.1.

• The fact that conditions Assumptions 2.1 and 2.2 hold in implies that we are able to guarantee that within no parts of the swimmer’s body will “collide”, and simultaneously, that it stays strictly inside of . These conditions allow us to maintain the mathematical and physical wellposedness of model (1.1) - (1.3).

• As it will follow from the proof below, Theorem allows further extension of the solutions to (1.1) - (1.3) in time as long as Assumptions 2.1 and 2.2 continue to hold. This depends on the choice of parameters .

Our plan to prove Theorem is to proceed stepwise as follows:

• In Section we discuss the existence and uniqueness of the solutions to the decoupled version of (1.2).

• In Section 4 we will introduce three continuous mappings for the decoupled version of the system (1.1) - (1.3).

• In Section 5 we will apply a fixed point argument to prove Theorem 2.1.

In the proofs below we employ the methods introduced in [19] to investigate the wellposedness of the 2- version of model (1.1)-(1.3), modifying and extending them to the 3- case.

Without loss of generality, we will further assume that system (1.1)-(1.3) and all respective auxiliary systems below are considered on the time-intervals whose lengths are smaller than 1.

3. Preliminary results. Introduce the following decoupled version of system (1.2):

 dwidt=1mes(Si(0))∫Si(wi(t))u(x,t)dx,   wi(0)=zi,0,   i=1,…,n,

where is some given function. Denote .

Lemma .   Let and be given. Then there is a such that system (3.1) has a unique solution in satisfying Assumptions 2.1 and 2.2 with in place of , if they hold at time .

Proof. We will use the contraction principle to prove existence and uniqueness. Below the values of , are taken from (2.2).

Select to satisfy the following inequalities:

 0
 ≤mini=1,…,nmin⎧⎨⎩mes(Si(0))h204∥u∥2(L2(QT))3,(mes(Si(0)))2C2∥u∥2(L2(0,T;L∞(Ω)))3T,1⎫⎬⎭.

Let for any given :

 Bh0/2(p)={z∈C([0,T0];R3)∣∥z−p∥C([0,T0];R3)≤h02}⊂C([0,T0];R3).

For each , define a mapping by

 Di(wi(t))=zi,0+1mes(Si(0))∫t0∫Si(wi(τ))u(x,τ)dxdτ.

Then we can derive that:

 ∥Di(wi(t))∥R3≤ ∥zi,0∥R3+√T0√mes(Si(0))∥u∥(L2(QT))3 ∀ t∈[0,T0], i=1,…,n.

Similarly, in view of (3.2):

 ∥Di(wi(t))−zi,0∥C([0,T0];R3)≤√T0√mes(Si(0))∥u∥(L2(QT))3

Thus, maps into itself for each , where is treated as a constant function.

Let and denote the characteristic function of a set . Then, making use of (2.2), we obtain:

 ∥Di(w(1)i(t))−Di(w(2)i(t))∥R3=
 1mes(Si(0))∥∥∥∫t0∫Si(w(1)i(t))u(x,τ)dxdτ−∫t0∫Si(w(2)i(τ))u(x,τ)dxdτ∥∥∥R3
 =1mes(Si(0))∥∥∥∫t0∫Ωu(x,τ)(ξ(x,Si(w(1)i(t)))−ξ(x,Si(w(2)i(τ))))dxdτ∥∥∥R3
 ≤C√T0mes(Si(0))∥u∥(L2(0,T;L∞(Ω)))3∥w(1)i(t)−w(2)i(t)∥C([0,T0];R3), ∀ t∈[0,T0], i=1,…,n.

Therefore, after maximizing the left-hand side of (3.4) over , we conclude:

 ∥Di(w(1)i(t))−Di(w(2)i(t))∥C([0,T0];R3)≤C√T0mes(Si(0))∥u∥(L2(0,T;L∞(Ω)))3∥w(1)i−w(2)i∥C([0,T0];R3).

Hence, in view of (3.2),

 C√T0mes(Si(0))∥u∥(L2(0,T;L∞(Ω)))3<1,

it follows from (3.5) that is a contraction mapping on for each . Therefore, there exist unique , such that , i.e.,

 wi(t)=zi,0+1mes(Si(0))∫t0∫Si(wi(τ))u(x,τ)dxdτ, i=1,…,n,

which yields (3.1).

On restrictions (2.3): Estimates (3.3a-b) imply that we may select a such that for any , all ’s will stay “close enough” to their initial values ’s to guarantee that Assumptions 2.1 and 2.2 holds for , . This ends the proof of Lemma 3.1.

4. Decoupled solution mappings. Let denote a closed ball of radius (its value will be selected in Section ) with center at the origin in the Banach space endowed with the norm of :

 Bq(0)={ϕ∈L2(0,T;Jo(Ω))⋂L2(0,T;(H2(Ω))3) | ∥ϕ∥L2(0,T;(H2(Ω))3)≤q},

where .

Note that is continuously embedded into , and thus is continuously embedded into . This yields the estimate

 ∥ϕ∥(L2(0,T;L∞(Ω)))3≤K∥ϕ∥L2(0,T;(H2(Ω))3)   for some K>0.

This implies that Lemma holds for any .

4.1. Solution mapping for . We now intend to show that the operator

 A:Bq(0)⟶[C([0,T];R3)]n,     Au=w=(w1,…,wn),

where the ’s solve (3.1), is continuous and compact if is sufficiently small.

Continuity.  Let with in place of , where satisfies assumptions in the proof of Lemma 3.1 with in place of . Define for . To show is continuous, we will evaluate

 ∥Au(1)−Au(2)∥[C([0,T];R3)]n

term-by-term. To this end, similar to (3.4), we have the following estimate:

 ∥w(1)i(t)−w(2)i(t)∥R3
 =∥∥∥1mes(Si(0))∫t0∫S(w(1)i(τ))u(1)(x,τ)dxdτ−1mes(Si(0))∫t0∫S(w(2)i(τ))u(2)(x,τ)dxdτ∥∥∥R3
 =1mes(Si(0))∥∫t0∫S(w(1)i(τ))u(1)(x,τ)dxdτ−∫t0∫S(w(1)i(τ))u(2)(x,τ)dxdτ
 +∫t0∫S(w(1)i(τ))u(2)(x,τ)dxdτ−∫t0∫S(w(2)i(τ))u(2)(x,τ)dxdτ∥R3
 ≤1mes(Si(0))(∥∥∥∫t0∫Ω(u(1)(x,τ)−u(2)(x,τ))ξ(x,S(w(1)i(τ)))dxdτ∥∥∥R3
 +∥∥∥∫t0∫Ωu(2)(x,τ)(ξ(x,S(w(1)i(τ)))−ξ(x,S(w(2)i(τ))))dxdτ∥∥∥R3)
 ≤√T1√mes(Si(0))∥u(1)−u(2)∥(L2(QT1))3+C√T1mes(Si(0))∥u(2)∥(L2(0,T1;L∞(Ω)))3∥w(1)i−w(2)i∥C([0,T1];R3
 ≤√T1√mes(S0)∥u(1)−u(2)∥(L2(QT1))3+C√T1mes(S0)∥u(2)∥(