2. Brinkman equation: functional setting

In this section, we present some results about the Brinkman equation. It was originally proposed in 5 to model a fluid flowing through a porous medium as a correction to Darcy’s law by the addition of a diffusive term. A rigorous mathematical derivation from the Navier–Stokes equation via homogenization can be found in 1 2.

In a Lipschitz domain , the Brinkman system reads The positive constant α takes into account the permeability properties of the porous matrix and the viscosity of the fluid, the constant ν is an effective viscosity of the fluid, while the third equation in the system is the no-slip boundary condition. The condition u=0 at infinity is significant, and necessary, only when the domain Ω is unbounded. From now on, we will get rid of the effective viscosity, upon a redefinition of α, by setting ν=1. A brief discussion on the constant ν can be found in Brinkman’s paper 5.

In order to cast Equation (2.1) in the weak form, we introduce the function spaces in which we will look for the weak solution. Define Both and are equipped with the standard H ¹ norm but we introduce this equivalent one the equivalence being a consequence of Korn’s inequality.

The weak formulation of Equation (2.1) is now given by where the boundary velocity is a given function , the solution being the unique minimum in of the strictly convex energy functional Here and henceforth the symbol Eu denotes the symmetric gradient of u, namely .

We call Ω an exterior domain with a Lipschitz boundary if Ω is an unbounded, connected open set whose boundary ∂Ω is bounded and Lipschitz [6, Section 2]. If we consider the term α² u as a forcing term f in system (2.1), we can invoke a classical existence and uniqueness result, see 7 21 or 23.

Theorem 2.1

Let . Then,  the following results hold.

a.	Let Ω be a bounded connected open subset of with the Lipschitz boundary. If there exists a unique solution u to problem (2.2). Moreover,  there exists p∈L 2(Ω),  such that Δu−∇p=f in .
b.	Let be an exterior domain with Lipschitz boundary. Then,  problem (2.2) has a solution. Moreover,  there exists with for every ρ>0,  such that Δu−∇p=f in .

The following density result is particularly useful when dealing with exterior domains.

Theorem 2.2 [(Density 10)]

Let be an exterior domain with Lipschitz boundary. Then,  the space is dense in for the H ¹ norm.

We now define some physically relevant quantities. The stress tensor associated with the velocity field u and the pressure p is given by The viscous force and torque are the resultant of the viscous forces and torques acting on the boundary ∂Ω, respectively, and are given by These definitions are valid under the condition that σn has a trace in . Since, in general, this assumption is not fulfilled, we have to define the viscous force and torque in a different way, namely by introducing σn as an element of . This will lead to a consistent definition of the power of the viscous force and torque. In order to do this, we introduce , the space of 3×3 symmetric matrices and recall that every can be orthogonally decomposed as σ=(1/3)trσI+σ_D where the deviatoric part σ_D is traceless.

We are now ready to give the following definition.

Definition 2.3

Let be an exterior domain with a Lipschitz boundary and let be such that and . The trace of σn,  still denoted by σn,  is defined as the unique element of satisfying the equality where ⟨⋅, ⋅⟩ _Ω denotes the duality pairing between and and v is any function in such that v=V on ∂Ω.

If there is no risk of misunderstanding, the subscript Ω will be dropped whenever the domain of integration is clear. Notice that if σ is sufficiently smooth then integrating Equation (2.5) by parts leads to the equality

In the general case, the right-hand side of Equation (2.5) is easily proved to be well defined, given the assumptions on σ. In fact, and make the first integral well defined, while the second one is also good since σ : Ev=σ_D : Ev, because of the symmetry of Ev, and both σ_D and Ev belong to . Lastly, the definition is independent of the choice of , since the right-hand side vanishes for every : this follows from the very same computation for the more regular case, by the density Theorem 2.2. It is easy to see that Equation (2.5) defines a continuous linear functional on by choosing an extension of V .

We now proceed in showing other useful properties of the duality pairing introduced in Definition 2.3. Let and let u be the solution to the Brinkman problem (2.2) with boundary datum U and let σ be the corresponding stress tensor. Since all the assumptions of Definition 2.3 are fulfilled, for any given , we have where v is an arbitrary element in such that v=V on ∂Ω. If we take, in particular, v to be the solution to problem (2.2) with boundary datum V , we recover the well-known reciprocity condition [9, Sections 3–5] with τ being the stress tensor associated with v. Moreover, by taking U=V in Equation (2.6), we obtain This equality allows us to show that the quadratic form ⟨σn, U⟩ is positive definite: if ⟨σn, U⟩=0, then it follows that u=0, and therefore U=0.

We are now in a position to define the viscous force and torque in a rigorous way, by means of the duality product introduced in Definition 2.3.

Definition 2.4

Let be an exterior domain with Lipschitz boundary,  let be the solution to the Brinkman problem (2.2) with boundary datum let σ be the corresponding stress tensor defined by Equation (2.4),  and let be the trace on ∂Ω defined according to Equation (2.5). The viscous force exerted by the fluid on the boundary ∂Ω is defined as the unique vector such that The torque exerted by the fluid on the boundary ∂Ω is defined as the unique vector such that where W _ω(x):=ω×x is the velocity field generated by the angular velocity ω.

Notice that this definition allows us to define two different physical quantities by means of the same mathematical object, namely the duality pairing defined in Equation (2.5).

3. Kinematics and the equations of motion

In this section, we describe the kinematics of the swimmer. The motion of a swimmer is described by a map t ↦ ϕ _t , where, for every fixed t, the state ϕ _t is an orientation preserving bijective C ² map from the reference configuration into the current configuration . Given a distinguished point x ₀∈A, for every fixed t, we consider the following factorization: where the position function r _t is a rigid deformation and the shape function s _t is such that We allow the map t ↦ s _t to be chosen in a suitable class of admissible shape changes and use it as a control to achieve propulsion as a consequence of the viscous reaction of the fluid. In contrast, t ↦ r _t is a priori unknown and it must be determined by imposing that the resulting ϕ _t =r _t ○s _t satisfies the equations of motion.

Since, as it is clear, the kinematics of the swimmer does not depend on the fluid the swimmer is surrounded by, we can adopt the same setting as in 6. For the reader’s convenience, we recall the results proved there and refer the reader to the above-mentioned paper and the references therein for a more detailed exposition.

The reference configuration of the swimmer is a bounded, connected open set of class C ². The time-dependent deformation of A from the point of view of an external observer is described by a function with the following properties: for every t; here and henceforth ∇ denotes the gradient with respect to the space variable. Under these hypotheses, A _t :=ϕ _t (A) is a bounded, connected open set of class C ² and We also assume that This assumption is technical and is made in order to prevent the change of topology in the swimmer and in the surrounding fluid.

Concerning the regularity in time, we require that

This condition implies that for almost every t, there exists , such that From this, the Eulerian velocity on the boundary ∂A _t , defined by belongs to with the Lipschitz constant independent of t.

We now introduce the description of the kinematics from the point of view of the swimmer. Let x ₀∈A be a distinguished point and let us look for a factorization of ϕ _t of the form (3.1). The function satisfies properties (3.2), in view of which it can be interpreted as a pure shape change from the point of view of an observer inertial with x ₀, and the rigid motion is written in the form with and R _t ∈SO(3), the set of orthogonal matrices with a positive determinant. This allows us to say that the deformation ϕ _t , from the point of view of an external observer, is decomposed into a shape change followed by a rigid motion.

From Equations (3.1), (3.3), and (3.5), the following properties of s _t can be inferred: for every t, where B _t :=s _t (A) (Figure 1). Note that Equation (3.6b) is a consequence of Equation (3.6a). Note also that B _t is a bounded connected open set of class C ² and that s _t (B _t )=A _t and s _t (∂B _t )=∂A _t . Moreover, since A is bounded and s _t is continuous, there exists a ball Σ_ρ centred at 0 with radius ρ, such that Lastly, Equation (3.4) implies that

By means of the polar decomposition theorem and factorization (3.1), it is possible to give explicit formulae for R _t and y _t that clearly show that the maps t  ↦  R _t and t ↦ y _t are Lipschitz continuous. Since , The third property in Equations (3.6a) and (3.8) implies that , with C independent of t. Moreover, condition (3.8) yields the existence of , such that

Other properties of s _t that are worth mentioning and whose full derivation can be found in [6, Section 3] are

To conclude the description of the kinematics of the swimmer, we give the form of the boundary velocity on the intermediate configuration B _t . It turns out that, if we define and , for every z∈∂B _t , an elementary computation shows that for almost every t∈[0, T]

We proceed now to the description of the motion of the swimmer. The motion t ↦ ϕ _t determines for almost every t∈[0, T] the Eulerian velocity U _t through the formula Notice that for almost every t∈[0, T]. By applying Theorem 2.1(b) with and, for almost every t∈[0, T], we obtain a unique solution u _t to the problem

Let F _{A t , U t} and M _{A t , U t} be the viscous force and torque determined by the velocity field U _t according to Equations (2.7) and (2.8). By neglecting inertia and imposing the self-propulsion constraint, the equations of motion reduce to the vanishing of the viscous force and torque, i.e.

By assuming that ϕ _t is factorized as ϕ _t =r _t ○s _t , where r _t is a rigid motion as in Equation (3.5) and t ↦ s _t is a prescribed shape function, our aim is to find t ↦ r _t so that the equations of motion (3.10) are satisfied. To this extent, we present Theorem 3.1, whose result is that Equation (3.10) is equivalent to a system of ordinary differential equations where the unknown functions are the translation t ↦ y _t and the rotation t ↦ R _t of the map t ↦ r _t .

The coefficients of these differential equations are defined starting from the intermediate configuration described by the sets B _t =s _t (A) introduced before and the 3×3 matrices K _t , C _t , J _t , depending only on the geometry of B _t , whose entries are defined by where , the duality product is given in Definition 2.3 by formula (2.5) and σ[W] denotes the stress tensor associated with the outer Brinkman problem in with boundary datum W. The notation σ[W] is chosen to emphasize the linear dependence of σ on W. Formula (2.6) shows that K _t and J _t are symmetric. The matrix is often called in the literature as the grand resistance matrix and is symmetric and invertible. It originally arises in the case of a Stokes system 9, but the adaptation to the Brinkman system is straightforward: it only shares the structure with the original one, while the values of the entries are computed with a different formula, namely Equation (2.6). Let be its inverse. For almost every t∈[0, T], we defined and let and be the viscous force and torque on ∂B _t determined by the boundary velocity field W _t . The components of and are given, according to Equations (2.7) and (2.8), by Consider now the linear operator that associates with every the only skew-symmetric matrix such that ; therefore, ω is the axial vector of . Finally, we define a vector b _t and a matrix Ω _t according to which depend on s _t and, most importantly on , via Equation (3.13) and the definition of W _t .

Theorem 3.1

Assume that the shape function t ↦ s _t satisfies Equations (3.6) and (3.8) and that the position function t ↦ r _t satisfies Equation (3.5) and is Lipschitz continuous with respect to time. Then,  the following conditions are equivalent:

i.	the deformation function t ↦ ϕ t :=r t °s t satisfies the equations of motion (3.10);
ii.	the functions t ↦ y t and t ↦ R t satisfy the system where b t and Ω t are defined in Equation (3.14).

The proof was given in 6 and need not be modified, so we skip it. It is developed by setting the problem in the intermediate configuration B _t , assuming the point of view of the coordinate system of the shape functions. Changing the variables according to y=r _t (z), , the velocity field is the solution to the problem where (Figure 2).

Denote by F _{B t , V t} and M _{B t , V t} the viscous force and torque on ∂B _t determined by the velocity field v _t according to Equations (2.7) and (2.8), with . A straightforward computation yields and , so that the equations of motion (3.10) reduce to Again by a simple manipulation, we obtain the following form of the equations of motion: which read, by means of Equation (3.14), as Equation (3.15).

Now, the standard theory of ordinary differential equations with possibly discontinuous coefficients 8 ensures that the Cauchy problem for Equation (3.15) has one and only one Lipschitz solution t ↦ R _t , t ↦ y _t , provided that the functions t ↦ Ω _t and t ↦ b _t are measurable and bounded. By Equations (3.12) and (3.14), this happens when the functions are measurable and bounded. The continuity of the first three functions will be proved in the last part of this section. The proof of the measurability and boundedness of the last two functions in Equation (3.17) requires some technical tools that will be developed in Section 4.

We need the following notion of set convergence: given a sequence of sets (S _k ) _k , we say that S _k converge to , , if for every ϵ>0 there exists m such that for every where and . The next lemma states a continuity property of the set-valued function t ↦ B _t .

Lemma 3.26

Let s _t satisfy Equation (3.8). Then,  if the sets B _t converge to the set in the sense of Equation (3.18).

Theorem 3.3

Let w _t be the solution to the exterior Brinkman problem (2.2) on with boundary datum W on ∂B _t ,  where W can be either a constant vector or the rotation W _ω :=ω×z, with . Define to be the extension Assume that t ↦ s _t satisfies Equation (3.8). Then,  the map is continuous from [0, T] into .

Proof

Let be a sequence that converges to . Lemma 3.2 ensures the convergence of the sets B _{t k} to in the sense of Equation (3.18).

Since w _{t k} are solutions to Brinkman problems, we have the bound , which, in turn, implies that Therefore, admits a subsequence that converges weakly to a function . By the convergence of the B _{t k} , it is easy to see that w*=W on . We now prove that solves the exterior Brinkman problem on . To see that, consider a test function . For k large enough, , so that This equality passes to the limit as , showing that is a solution to the Brinkman problem at . Therefore, , and we have proved that t ↦ w _t is strongly continuous from [0, T] into .

We can now prove the following continuity result for the elements of the grand resistance matrix by means of Theorem 3.3.

Proposition 3.4

Assume that s _t satisfies Equations (3.6) and (3.8). Then,  the functions and consequently t ↦ H _t , t ↦ D _t , t ↦ L _t ,  are continuous.

Proof

Formulae (3.11) and (2.6) provide us with an explicit form for the elements of the grand resistance matrix where and are the functions defined in Equation (3.19) with W=e _i and W=e _i ×z, respectively. We prove the result for K _t only, since the others are similar. We write where and are the extensions considered in Equation (3.19). By Theorem 3.3, the first two integrals are continuous with respect to t. The continuity of the last integral is guaranteed by Lemma 3.2.

The proof of the measurability and boundedness of and is a delicate issue. The difficulty arises from the fact that both the domains B _t and the boundary data depend on time. Moreover, since it is meaningful and interesting to consider boundary values W _t that might be discontinuous with respect to t, we cannot expect the functions and to be continuous.

To prove the measurability, we start from an integral representation of and , similar to Equation (3.21). As is not necessarily zero, we will not be able to compute integrals over the whole space , so we will have to work in the complement of an open ball . Since, in general, this inclusion holds only locally in time, we first fix t ₀∈[0, T] and z ⁰∈B _{t 0} and select δ>0 and ϵ>0 so that the open ball of radius ϵ centred at z ⁰ satisfies This is possible thanks to the continuity properties of t ↦ s _t listed in the first part of this section.

Next, we consider the solution w _t to the problem In order for the flux condition (2.3) to be fulfilled by w _t on , we choose

Finally, putting together Equations (3.13) and (2.6), we obtain the following explicit integral representation of and : where and have been defined in the proof of Proposition 3.4 and . We deduce from Theorem 3.3 and Lemma 3.2 that the functions and are continuous from I _δ(t ₀) into . Therefore, the measurability and boundedness of and will be proved once t ↦ w _t is proved to be measurable. We first show that t ↦ w _t is measurable and bounded from I _δ(t ₀) into and eventually we will prove that the function is continuous with respect to time. These two results are proved in the next section.

4. Extensions of boundary data and main result

In order to prove the main result, some work is still to be done to prove the regularity property of the coefficients of the equations of motion (3.15). To this aim, results concerning the extension of boundary data are needed to be able to use standard variational techniques to solve the relevant minimum problem of Theorem 4.4. The following result has been proved in 6.

Proposition 4.1 (Solenoidal extension operators)

Assume that s _t satisfies Equations (3.6) and (3.8), and let t ₀ ∈[0, T] and z ⁰ ∈B _{t 0} . Let δ>0 and ϵ>0 be such that Equation (3.22) holds true. Then,  there exists a uniformly bounded family of continuous linear operators such that

i.	for all t∈I δ(t 0) and for all ,
ii.	for every the map is continuous from I δ (t 0 ) into .

In particular,  the following estimate holds where the constant C is independent of t and Φ.

Proposition 4.2

Assume that s _t satisfies Equations (3.6),  (3.7),  and (3.8). Let t ₀ ∈[0, T] and z ⁰ ∈B _{t 0} ,  and let and I _δ(t ₀) be as in Equation (3.22). Suppose,  in addition,  that for every t∈I _δ(t ₀),  there exists a C ² diffeomorphism coinciding with the identity on Σ _ρ ∖Σ _ρ−1 ,  such that on B _t . Let the map t ↦ Φ _t belongs to . Let w _t be the solution to the problem where . Then,  t ↦ w _t belongs to .

Proof

The proof can be easily adapted from that of [6, Proposition 6.1]; the following important estimate provides a uniform bound for the norms of the w _t ’s in that will also be useful in the proof of Proposition 4.3 where is defined by and is the function provided by Proposition 4.1 and extended on , C is the constant in Equation (4.1), and M>0 is a uniform upper bound of , whose existence is guaranteed by the fact that t ↦ Φ _t belongs to .

Proposition 4.3

Under the hypotheses of Proposition 4.2,  recalling that the maps where is the solution to the minimum problem (4.2) as in Proposition 4.2,  are continuous with respect to time in I _δ(t ₀).

Proof

We check the continuity with the definition Here, χ _Q denotes the characteristic function of the set Q, Δ is the symmetric difference operator, and CM is the uniform (with respect to t) upper bound coming from Equation (4.3). The continuity for the second map is achieved in the same way.

Propositions 4.2 and 4.3 combined together give the continuity of and with respect to time, in the case of regular boundary data on ∂B _t , where the map t ↦ Φ _t belongs to . The next results will prove that when the boundary data on ∂B _t are given by , then the maps and are measurable and bounded.

Theorem 4.4

Assume that s _t satisfies Equations (3.6)–(3.8). Let t ₀ ∈[0, T] and z ⁰ ∈B _{t 0} ,  and let and I _δ(t ₀) be as in Equation (3.22). Suppose,  in addition,  that for every t∈I _δ(t ₀),  there exists a C ² diffeomorphism coinciding with the identity on Σ _ρ ∖Σ _ρ−1 ,  such that on B _t . Let w _t be the solution to the problem Then,  the function t ↦ w _t is measurable and bounded from I _δ(t ₀) into . Moreover,  also the functions (4.4) considered in Proposition 4.3 are measurable and bounded in I _δ(t ₀).

Proof

It suffices to convolve the boundary datum with a suitable regularizing kernel and to apply Propositions 4.2 and 4.3. By passing to the limit, the continuity is lost but the functions turn out to be measurable and bounded.

Proposition 3.4 and Theorem 4.4 give the regularity result for b _t and Ω _t in Equation (3.14), as stated in the following result.

Theorem 4.5

Assume that t ↦ s _t satisfies Equations (3.6)–(3.8). Then,  the vector b _t and the matrix Ω _t in Equation (3.14) are bounded and measurable with respect to t. If,  in addition,  the function t ↦ s _t belongs to then t ↦ (b _t , Ω _t ) belongs to .

We are now in a position to state the existence, uniqueness, and regularity result for the equations of motion (3.15).

Theorem 4.6

Assume that t ↦ s _t satisfies Equations (3.6)–(3.8). Let and R*∈SO(3). Then,  Equation (3.15) has a unique absolutely continuous solution t ↦ (y _t , R _t ) defined in [0, T] with values in such that y ₀ =y* and R ₀ =R*. In other words,  there exists a unique rigid motion t ↦ r _t (z)=y _t +R _t z such that the deformation function t ↦ ϕ _t =r _t ○s _t satisfies the equations of motion (3.10).

Moreover,  this solution is Lipschitz continuous with respect to t. If,  in addition,  the function t ↦ s _t belongs to then the solution t ↦ (y _t , R _t ) belongs to .

Proof

The existence and uniqueness of the solution of the Cauchy problem for Equation (3.15) follow immediately from Theorem 4.5, by standard results on ordinary differential equations with bounded measurable coefficients [8, Theorem I.5.1]. The assertion concerning the deformation function t ↦ ϕ _t and the equation of motion (3.10) follows from the equivalence Theorem 3.1. The Lipschitz continuity of the solution follows from the boundedness of the right-hand sides of Equation (3.15).

If, in addition, the function t ↦ s _t belongs to , then Theorem 4.5 ensures that the coefficients of the equations in Equation (3.15) are continuous with respect to t, and therefore the solutions are of class C ¹.

5. Conclusions and future work

We have shown that the framework for modelling the motion of a deformable body in a viscous fluid that we presented in 6 also fits in the case of a particulate system for which the Brinkman equation is assumed to model the fluid phase of the surrounding medium. A suitable functional setting has been developed and the solution to the Brinkman system has been found by solving a minimum problem for the associated functional. Some extra terms appeared, with respect to the Stokes case, due to the presence of the −α² u term in the Brinkman system. Nonetheless, the corresponding integrals, depending on time both in the integrand function and in the domain of integration, have been proved to be continuous with respect to time, thus allowing the coefficients of the equations of motion to be regular enough.

Another noteworthy feature of our work is that the infinite-dimensional control t ↦ s _t is coupled with and determines a finite-dimensional function to describe the position of the swimmer. In previous works 3 18 19, only swimmers with a finite number of shape parameters were dealt with. Here, we have been able to extend the study to the case of a more complex deformation.

In our model, we neglected the interactions between the solid particles and the swimmer, considering only the body–fluid phase viscous interaction. We think this is a reasonable approximation for using a simple model such as the Brinkman equation. Also, the mathematical model to describe the experiments in 12 is the same, and in that case, the elastic and adhesive interactions between the nematode and the surrounding particles are neglected as well. Nevertheless, we think it can be interesting to develop more complex models to take into account also that kind of contact forces, and this could be the objective of a future study.

Even though it has not been addressed in this work, we also expect our model to be able to predict, on the basis of an energy comparison, whether swimming in a particulate medium is more efficient than swimming in a plain viscous fluid; this would be an interesting theoretical check of the thesis advanced by Jung on the basis of his experimental results that C. elegans swims more efficiently in a particulate medium.