Viscous Flow Around a Rigid Body Performing a Time-periodic Motion

The equations governing the flow of a viscous incompressible fluid around a rigid body that performs a prescribed time-periodic motion with constant axes of translation and rotation are investigated. Under the assumption that the period and the angular velocity of the prescribed rigid-body motion are compatible, and that the mean translational velocity is non-zero, existence of a time-periodic solution is established. The proof is based on an appropriate linearization, which is examined within a setting of absolutely convergent Fourier series. Since the corresponding resolvent problem is ill-posed in classical Sobolev spaces, a linear theory is developed in a framework of homogeneous Sobolev spaces.


Introduction
We investigate the fluid flow past a rigid body B that moves through an infinite three-dimensional liquid reservoir with prescribed velocity with respect to its center of mass x C . Here t ∈ R and x ∈ R 3 denote time and spatial variable, respectively, ξ := d dt x C is the translation velocity and η the angular velocity of B with respect to its center of mass. We only consider the case where the angular velocity η is constant, but the translation velocity ξ may depend on time. In a frame attached to the body, with origin at its center of mass x C , the motion of an incompressible Navier-Stokes fluid around B that adheres to B at the boundary is described by the equations (1.1) see [12,Section 1]. Here Ω := R 3 \ B is the exterior domain surrounding B, and R represents the time axis. The functions u : R × Ω → R 3 and p : R × Ω → R describe velocity and pressure fields of the fluid. The constants ρ > 0 and μ > 0 denote density and viscosity, respectively. For the sake of generality, we additionally consider an external body force f : R × Ω → R 3 .
In this paper, we investigate a configuration where the rigid body B translates periodically with some prescribed time period T > 0. More precisely, we assume the data to be T -time-periodic As the main theorem we show existence of a solution (u, p) to (1.1) that shares this time periodicity.
28 Page 2 of 23 T. Eiter and M. Kyed JMFM We consider a prescribed motion of B where the axes of translation and rotation do not vary over time and are parallel. Without loss of generality, both are directed along the x 1 -axis such that η= ω e 1 for some T -periodic function α : R → R and a constant ω ∈ R. Note that, at least in the case where ξ is time-independent, this assumption can be made without loss of generality as long as ξ · η = 0 due to the Mozzi-Chasles theorem. We assume that the mean translational velocity of the body over one time period is non-zero: The case of vanishing mean translational velocity shall not be treated here. Not only does the fluid flow exhibit different physical properties when (1.2) is not satisfied, due to the absence of a wake region in this case, but also the mathematical properties of the linearization of (1.1) differ significantly. If (1.2) is satisfied, the linearization of (1.1) is a time-periodic generalized Oseen system, for which we shall establish suitable L q estimates in order to show existence of a solution to (1.1). If (1.2) is not satisfied, the linearization of (1.1) is a time-periodic generalized Stokes system, for which similar estimates cannot be derived. In this case, problem (1.1) thus has to be approached in a different way, which has recently been done by Galdi [15].
Since the case η = 0 was treated in [18], we only consider the case η = 0 in the following. Observe that then η ∧ x · ∇ is a differential operator with unbounded coefficient. Therefore, the linearization of (1.1) cannot be treated as a lower-order perturbation of the time-periodic Oseen problem, even if η is "small". In particular, as we will see below, the corresponding resolvent problem also requires an analysis in a different functional setting. This observation reflects the properties of the corresponding stationary problem (see [13, Chapter VIII]), which can be regarded as a special case of the time-periodic problem. In order to find a framework in which the time-periodic generalized Oseen problem is well posed, we employ the idea from [16,17], where the steady-state problem corresponding to (1.1) was considered, and the rotation term η ∧ u − η ∧ x · ∇u was handled by a change of coordinates into a non-rotating frame. This procedure only yields suitable estimates for time-periodic solutions when the change of coordinates maintains the time periodicity of the involved functions. This is the case if the angular velocity ω is an integer multiple of the angular frequency 2π/T of the time-periodic data. For simplicity, we assume This condition means that during one period the rigid body completes one full revolution. In other words, the rotation and the time-periodic data, which may be regarded as two different sources of time-periodic forcing, have to be compatible. The equations governing the fluid flow around a rigid body that performs a prescribed rigid motion have been studied by many researchers during the last decades. The first successful attempts of a rigorous mathematical treatment date back to the fundamental works of Oseen [44], Leray [36,37] and Ladyžhenskaya [34,35]. The study of time-periodic Navier-Stokes flows was proposed in a short note by Serrin [47], which induced Prodi [45], Yudovich [56] and Prouse [46] to initiate the examination in bounded domains. Through the years, this investigation has been continued and extended to other types of domains and fluid-flow configurations by several authors; see for example [5,10,11,14,18,[21][22][23][28][29][30][31]33,[38][39][40][41][42][43]49,[51][52][53][54][55]. We refer to [19] for a more detailed overview. Concerning in particular time-periodic Navier-Stokes flows around rigid bodies, more specifically the three-dimensional exteriordomain configuration, we emphasize the fundamental work of Yamazaki [55], who introduced a setting of L 3,∞ (Ω) spaces to obtain time-periodic solutions in the case ξ = η = 0. The main estimates in [55] are based on well-known L p -L q estimates of the Stokes semigroup. If one replaces these estimates with the L p -L q estimates obtained by Shibata [48] for an Oseen semigroup with rotational effects, the approach in [55] also seems to yield existence of time-periodic solutions to (1.1) in an L 3,∞ (Ω) framework in the case of constant non-zero parameters ξ = 0, η = 0. This analysis was recently carried out by Geissert, Hieber and Nguyen [23], who introduced a general semigroup-based approach to show existence of mild solutions to time-periodic problems. Using a recent result by Hishida [27], who established L p -L q estimates for an evolution operator corresponding to a linearization of the Navier-Stokes equations in the case of time-dependent ξ(t) and η(t), the approach of Yamazaki [55] even leads to time-periodic solutions in an L 3,∞ (Ω) framework for general time-periodic ξ(t) and η(t). In this general case, Galdi and Silvestre [21] already established the existence of time-periodic solutions in an L 2 (Ω) setting via a Galerkin approach.
As the main novelty of the present paper, we establish existence of strong solutions to (1.1) in an L q (Ω) setting for a certain range of exponents q ∈ (1, ∞). In this setting, better information on the spatial decay of the solutions can be derived compared to the L 3,∞ (Ω) and L 2 (Ω) frameworks described above. Our approach is based on an analysis of the linearization of (1.1) and the associated resolvent for suitable s ∈ R and F ∈ L q (Ω) 3 , 1 < q < ∞. At first glance, it seems reasonable to regard (1.4) as a resolvent problem (is − A)v = F for a closed operator A on the space of solenoidal vector fields in L q (Ω) 3 . However, the spectral analysis in this setting, which was carried out by Farwig and Neustupa [7,8], reveals that is, s ∈ R, belongs to the spectrum of A when s ∈ ωZ, whereas well-posedness of the time-periodic problem requires invertibility of (1.4) for s ∈ ωZ. Therefore, we propose to investigate the problem in homogeneous Sobolev spaces instead. Although it is merely possible to derive the non-classical resolvent estimate (2.4) in this setting (see Theorem 2.1 below), we are nevertheless able to conclude a suitable solution theory for the linearization of (1.1). To this end, we shall employ a framework of functions with absolutely convergent Fourier series. Finally, a fixed-point argument yields the existence of a solution to the nonlinear problem (1.1) when the data f , ξ and η are "sufficiently small".

Main Results
In virtue of (1.2) we may assume λ > 0 without loss of generality, and by (1.3) we have ω = 2π/T > 0. To reformulate (1.1) in a non-dimensional way, we let the diameter d > 0 of B serve as a characteristic length scale. We introduce the Reynolds number λ := λρd/μ, the Taylor number ω := ωρd 2 /μ, and the non-dimensional time and spatial variables t = ωt and x = x/d. In particular, Ω is transformed to Ω := {x/d | x ∈ Ω}. We define α (t ) := α(t)ρd/μ and the non-dimensional functions which are time-periodic with period T = 2π and can thus be identified with functions on the torus group T = R/2πZ with respect to time. Expressing (1.1) in these new quantities and omitting the primes, we obtain the non-dimensional formulation (2.1) Our analysis of (2.1) is based on the study of the linear time-periodic problem for k ∈ Z. For the latter we shall derive the following well-posedness result.
Note that for k = 0 we recover the well-known L q theory for the corresponding stationary problem; see [13,Theorem VIII.8.1].
In order to transfer estimate (2.4) to the time-periodic setting without losing information on the dependencies of the constant C 1 , we work within spaces A(T; X) of absolutely convergent X-valued Fourier series for suitable Banach spaces X; see (3.1) below. We establish the following solution theory for the time-periodic problem (2.2).
In Sect. 6, we finally prove the following existence result on solutions to the nonlinear system (2.1).

Theorem 2.3.
Let Ω ⊂ R 3 be an exterior domain of class C 3 , and let q ∈ 12 11 , 4 3 , ρ ∈ 3q−3 q , 1 and θ > 0. Then there are constants κ > 0 and λ 0 > 0 such that for all Remark 2.4. The lower bound λ 2 θ ≤ ω on the angular velocity in (2.6) may seem strange in light of the underlying physics of the problem since from a physical point of view, the limit ω → 0 towards the case of a non-rotating body seems uncritical. The lower bound on ω in (2.6) is an artifact of the change of coordinates into the rotating frame of reference employed in the mathematical analysis of the problem, which leads to a priori estimates with constants exhibiting a singular behavior as ω → 0. As a consequence, a lower bound on ω is required in Theorem 2.3 to obtain existence of a solution via a fixed-point iteration. A similar observation was made in the investigation of a steady flow past a rotating and translating obstacle carried out in [6]. Therefore, it is not surprising to see the same effect appearing in the more general time-periodic case investigated here.

Preliminaries
We use capital letters to denote global constants, while constants in small letters are local to the respective proof. When we want to emphasize that a constant C depends on the quantities α, β, γ, . . . , we write C(α, β, γ, . . . ).
We denote points in T × R 3 by (t, x), where t and x = (x 1 , x 2 , x 3 ) are referred to as time and spatial variable. The symbol Ω always denotes an exterior domain, that is, Ω ⊂ R 3 is connected and the complement of a non-empty compact set. We always assume that the origin is not contained in Ω.
Inner and outer product of two vectors a, b ∈ R 3 are denoted by a · b and a ∧ b, respectively. For any radius R > 0 we set For q ∈ [1, ∞] and k ∈ N 0 , the symbols L q (D) and W k,q (D) denote usual Lebesgue and Sobolev spaces with associated norms · q = · q;D and · k,q = · k,q;D , respectively. Furthermore, W 1,q 0 (D) denotes the subset of functions in W 1,q (D) with vanishing boundary trace, and W −1,q (D) (with norm · −1,q;D ) is the dual space of W 1,q 0 (D) where 1/q + 1/q = 1 with the usual convention 1/∞ := 0. Moreover, L 2 σ (D) denotes the set of solenoidal vector fields in L 2 (D) 3 , that is, and P H is the corresponding Helmholtz projection that maps L 2 (D) 3 onto L 2 σ (D). We always identify 2π-periodic functions with functions on the torus group T := R/2πZ, which is usually represented by the set [0, 2π). We consider T and G := T × R 3 as locally compact abelian groups. The (normalized) Haar measure on T is given by and G is equipped with the corresponding product measure. Recall that the dual group of T can be identified with T = Z and that of G with G : is the Schwartz-Bruhat space of generalized Schwartz functions on H, and S (H) denotes the corresponding dual space of tempered distributions; see [1,4] for precise definitions. The Fourier transform on T and G and the respective inverses are given by and provided the Lebesgue measure dξ is correctly normalized. By duality, F T and F G extend to homeomorphisms F T : S (T) → S (Z) and F G : S (G) → S ( G), respectively. Furthermore, we introduce the Sobolev space denotes the space of smooth functions of compact support on T × D . Let X be a Banach space. We introduce the projections P and P ⊥ by Note that Pu ∈ X is time-independent, and we have the decomposition u = Pu + P ⊥ u into the steady-state part Pu and the purely periodic part P ⊥ u of u.
Our analysis of the time-periodic problems (2.1) and (2.2) will be carried out within spaces of functions with absolutely convergent Fourier series defined by , which embeds into the X-valued continuous functions on T. It is well known that the scalar-valued space A(T; R) is an algebra with respect to pointwise multiplication, the so-called Wiener algebra. One can exploit this property to derive estimates in the X-valued case. For example, one readily shows the following correspondences of Hölder's inequality and interpolation inequalities.
be an open set and p, q, r ∈ [1, ∞] such that 1/p + 1/q = 1/r. Moreover, let f ∈ A(T; L p (D)) and g ∈ A(T; L q (D)). Then fg ∈ A(T; L r (D)) and for elements (f k ) ∈ 1 (Z; L p (D)) and (g k ) ∈ 1 (Z; L q (D)). Then fg = F −1 where the last estimate is due to Hölder's inequality. 1], and let f ∈ A(T; L p (D)) ∩ A(T; L q (D)). Then f ∈ A(T; L r (D)) and ). The classical interpolation inequality for Lebesgue spaces yields where the last estimate follows from Hölder's inequality on Z.

Embedding Theorem
This section deals with embedding properties of Sobolev spaces of time-periodic functions. The embedding theorem below is a refinement of [18, Theorem 4.1] adapted to the time-scaling employed in (2.1). Clearly, embeddings of the steady-state part Pu are independent of the actual period. Therefore, we only consider the case of purely periodic functions. For the sake of generality, we establish the following theorem in arbitrary dimension n ≥ 2.  ∈ (1, ∞). For α ∈ [0, 2] with αq < 2 and (2 − α)q < n let and for β ∈ [0, 1] with βq < 2 and (1 − β)q < n let Then the inequality holds for all u ∈ P ⊥ W 1,2,q (T × R n ) and a constant C 2 = C 2 (n, q, α, β) > 0.
Proof. Since the proof is analogue to [18, Proof of Theorem 4.1], we merely give a brief sketch here. Without restriction we may assume u ∈ S (G). Due to the assumption u = P ⊥ u, Utilizing the Fourier transform, we thus derive the identity where Employing the so-called transference principle for Fourier multipliers (see [3,4]) together with the Marcinkiewicz multiplier theorem, one readily verifies that M ω is an L q (G) multiplier for any q ∈ (1, ∞) such that with c 0 independent of ω. Moreover, when we choose [−π, π) as a realization of T, we obtain for some h ∈ C ∞ (T); see for example [24,Example 3.1.19]. In particular, this yields γ α ∈ L 1 1−α/2 ,∞ (T), so that Young's inequality implies that the mapping ϕ → γ α * ϕ extends to a bounded operator L q (T) → L r0 (T). Moreover, it is well known that the mapping ϕ → F −1 R n |ξ| α−2 * ϕ extends to a bounded operator where Minkowski's integral inequality is used in the second estimate. This is the asserted inequality for u. The estimate of ∇u follows in the same way.

Remark 4.2.
Note that the term on the right-hand side of (4.1) defines a norm equivalent to · 1,2,q on P ⊥ W 1,2,q (T × Ω) due to Poincaré's inequality on T.
Remark 4.3. Theorem 4.1 can be generalized to the setting of an exterior domain Ω ⊂ R n by means of Sobolev extensions. However, to maintain estimate (4.1), one has to construct a specific extension operator that respects the homogeneous second-order Sobolev norm. To this end, one can make use of results from [2].

Linear Theory
This section is dedicated to the investigation of the resolvent problem (2.3) and the linear time-periodic problem (2.2). After having shown Theorem 2.1, we establish Theorem 2.2 as an immediate consequence hereof.

The Whole Space
To study the problems (2.2) and (2.3) in an exterior domain, we first consider the case Ω = R 3 . In this whole-space setting one can namely change coordinates back to the non-rotating inertial frame and thereby reduce the study of (2.2) to an investigation of the time-periodic Oseen problem without rotation terms, which was analyzed in [18,32]. In this section, we set for appropriately fixed q.
is another solution to (5.1), then P ⊥ u = P ⊥ w, and Pu − Pw is a polynomial in each component, and p − q = p 0 , where p 0 (t, ·) is a polynomial for each t ∈ T.
Proof. We decompose (5.1) into two problems by splitting u = Pu + P ⊥ u =: u s + u p and p = Pp + P ⊥ p =: p s + p p . For the steady-state part (u s , p s ) we obtain the system which is the classical steady-state Oseen problem. The existence of a time-independent solution (u s , p s ) satisfying estimate (5.2) is well known; see for example [13,Theorem VII.4.1]. The remaining purely periodic part (u p , p p ) must solve (5.1), but with purely periodic right-hand side P ⊥ f . We define which leads to the system where λ = λω −1/2 . From [32, Theorem 2.1] we conclude the existence of a unique solution (U, P) that satisfies the estimate where c 0 is a polynomial in λ and can thus be bounded uniformly in λ ∈ (0, √ θ]. Estimate (5.3) with the asserted dependency of the constant C 4 follows after reversing the applied scaling.
The uniqueness statement is readily shown by means of the Fourier transform on G = T × R 3 . We consider (5.1) with f = 0 and apply the divergence operator to (5.1) 1 . This yields Δp = 0 and thus |ξ| 2 F R 3 [p(t, ·)] = 0 for all t ∈ T. Therefore, we obtain supp F R 3 [p(t, ·)] ⊂ {0}, so that p(t, ·) is a polynomial for all t ∈ T. Next we apply the Fourier transform to (5.1) 1 to deduce (iωk+|ξ| Multiplying with the symbol of the Helmholtz projection I − ξ ⊗ ξ/|ξ| 2 and utilizing div u = 0, we obtain , it follows that P ⊥ u = 0, and that each component of Pu is a polynomial. This completes the proof. Recalling Remark 4.2, we see that (5.2) and (5.3) can be formulated as for a constant C 6 = C 6 (q, θ) as long as λ 2 ≤ θω.
With Theorem 5.1 we now solve the linear problem (2.2) for Ω = R 3 and f ∈ L q (T × R 3 ) 3 .
and a constant

Uniqueness
Next we show a uniqueness result for the resolvent problem (2.3).
In particular, we can employ the divergence theorem to compute ΩR div (e 1 ∧x)|v| for any R > 0 with ∂B R ⊂ Ω. Passing to the limit R → ∞, we obtain Ω div (e 1 ∧x)|v| 2 dx = 0. (5.15) By the above integrability properties, we can further multiply (2.3) 1 by v and integrate over Ω. By utilizing (5.15) and integration by parts, we conclude This implies ∇v = 0. The imposed boundary conditions thus yield v = 0. Finally, (2.3) 1 leads to ∇p = 0, and the proof is complete.

A Priori Estimate
Next we establish an a priori estimate for the solution to the resolvent problem (2.3).
Since λ = 0, it follows from (5.24) that v ∈ L s1 (Ω). Lemma 5.6 thus implies v = ∇p = 0. iv. If ω j |k j | → ∞, we recall (5.20) and estimate for any ρ > R. Passing to the limit j → ∞, we thus obtain v = 0 on Ω ρ for each ρ > R, whence v = 0 on Ω. Hence, (5.22) 1 reduces to w + ∇p = 0. Clearly, we also have div w = 0 and w ∂Ω = 0, so that w + ∇p = 0 corresponds to the Helmholtz decomposition of 0 in L q (Ω). Since this decomposition is unique, we conclude w = ∇p = 0. Consequently, all four cases lead to w = v = ∇p = 0, which contradicts (5.23). This completes the proof in the case 1 < q < 3 2 . In the more general case q ∈ (1, 2), where we do not assert the constant C 1 to be independent of λ and ω, these parameters remain fixed in the contradiction argument above. Consequently, only the last two cases above have to be considered. The conclusion in both of these cases is valid for all q ∈ (1, 2), and we thus conclude the lemma.

Existence
To complete the proof of Theorem 2.1, it remains to show existence of a solution. For this purpose, recall the following property of the Stokes operator. Lemma 5.9. Let D ⊂ R 3 be a bounded domain with C 3 -boundary. Every u ∈ L 2 σ (D) ∩ W 1,2 0 (D) ∩ W 2,2 (D) satisfies ∇ 2 u 2 ≤ C 10 P H Δu 2 + ∇u 2 for a constant C 10 = C 10 (D) > 0 that does not depend on the "size" of D but solely on its "regularity". In particular, if D = Ω R for an exterior domain Ω with ∂Ω ⊂ B R , the constant C 10 is independent of R and solely depends on Ω.
Proof. Let R > 0 such that ∂B R ⊂ Ω, and take m ∈ N with m > 2R. Since the Stokes operator in the bounded domain Ω m is a positive self-adjoint invertible operator (see [50, Chapter III, Theorem 2.1.1]), there exist sequences (ψ j ) j∈N of (real-valued) eigenfunctions and (μ j ) j∈N ⊂ (0, ∞) of eigenvalues, that is, We show the existence of a function u = u m n ∈ X m n := span C ψ j j = 1, . . . , n satisfying Note that (5.26) is a resolvent problem for the skew-Hermitian matrix M , which is uniquely solvable. Existence of a unique solution u = u m n ∈ X m n to (5.25) thus follows. Next we need suitable estimates for u = u m n . Multiplication of both sides of (5.25) by the complex conjugate coefficient ξ * j and summation over j = 1, . . . , n yields ∇u 2 2 + Ωm ω(iku + e 1 ∧u − e 1 ∧x · ∇u) − λ∂ 1 u · u * dx = Ωm F · u * dx.