REACTION OF THE FLUID FLOW ON TIME-DEPENDENT BOUNDARY PERTURBATION

. The aim of this paper is to investigate the eﬀects of time-dependent boundary perturbation on the ﬂow of a viscous ﬂuid via asymptotic analysis. We start from a simple rectangular domain and then perturb the upper part of its boundary by the product of a small parameter ε and some smooth function h ( x,t ). The complete asymptotic expansion (in powers of ε ) of the solution of the evolutionary Stokes system has been constructed. The convergence of the expansion has been proved providing the rigorous justiﬁcation of the formally derived asymptotic model.

1. Introduction. Fluid flow through time-dependent domain is an important subject that has been studied by numerous authors with different motivations and using different approaches. Problems of that kind naturally appear in the study of physiological flows (see e.g. [7,18]), flexible pipes [9], fishing nets [12], study of moving bodies in fluid [19], lubrication theory [20], etc.
The goal of the present paper is to study the effects of the moving boundary on the fluid flow in case when the displacement of the boundary is small. For the sake of simplicity, we take the rectangular domain and study the flow driven by the pressure drop from left to the right side of the domain. We modify the upper boundary with small perturbation εh(x, t) in vertical direction, causing the motion of the fluid in direction perpendicular to the pressure gradient as well as squeezing the fluid in the direction of the pressure gradient. It should be mentioned that introducing a small parameter as the perturbation quantity in the domain boundary makes analytical treatments very complicated. It is due to the fact that tedious change of a variable has to be performed. Consequently, not many analytical results on the subject can be found throughout the literature. In particular, the perturbation of the domain boundary has remained a rather neglected mathematical topic (see monograph [10]) and has been addressed mostly in the context of periodically corrugated boundaries (see e.g. [1], [11]), especially in the context of lubricated (thin-film) flows.
Before proceeding, we would like to refer to the works dealing with the analysis of the flows in a thin-film regime, being motivated by the engineering practice. Describing the boundary roughness by a periodic function, the flow of a viscous fluid between two very close surfaces has been investigated for various rugosity 1228 EDUARD MARUŠIĆ-PALOKA AND IGOR PAŽANIN profiles. The usual assumption where the size of the roughness is of the same small order as the film thickness, i.e. 0 < y < h ε (x) = εH x, x ε , 0 < ε 1 leads to the effective model in a form of the standard Reynolds approximation (see e.g. [3]). Consequently, to deduce the roughness-induced effects, one needs to compute the higher-order terms in the asymptotic expansion. As shown in [4], the similar result holds for h ε (x) = εH x, x ε β with β < 1. However, if one considers the specific rugosity profile given by x ε 2 , it turns out that an extra term appears in the limit modifying the Reynolds equation at the main order. For the details, we refer the reader to [6], [8], [17].
Recently, the authors of this work have been investigating the stationary fluid flow in case of time-independent boundary perturbation. We refer the reader to [13] for classical Newtonian flow, [14] for micropolar fluid flow, [15,16] for porous medium flow. The purpose of this paper is to generalize the techniques presented in those papers on the case of time-dependent boundary perturbation. Starting from evolutionary Stokes system, we manage to formally derive an asymptotic expansion of the flow giving an approximation of arbitrary order. The zero-order approximation, as expected, is not affected by the boundary perturbation. However, the correctors contain terms explicitly depending on the function h as well as on its derivatives (see Sections 4 and 5), allowing to closely study the reaction of the fluid flow on the motion of the boundary. Employing functional analysis techniques, we also prove the error estimate for the constructed asymptotic expansion in Section 6. By doing that, we provide the rigorous justification of our effective model and that represents our main contribution.
In the works [3,4,6,8,17] mentioned above, the oscillating part h describing the corrugations in H is assumed to be a periodic function, independent of time. As a consequence, the limit system allows to be rigorously justified by the notion of the two-scale convergence. Here, we cannot proceed in a similar manner to derive the error estimates, due to the fact that we take boundary perturbation function h to be an arbitrary (not necessarily periodic) function which, additionally, depends on time. For that reason, we strongly believe that the findings presented here, though not put in the lubrication context, will prove useful in the engineering practice as well.
The unknowns in the above system are u ε = (u ε x , u ε y ) and p ε representing the velocity and the pressure of the fluid, while p 0 (t) and p 1 (t) are prescribed pressures depending on time. We impose the following regularity and compatibility conditions on the given data: • ∂h ∂t (x, t) = 0 for x = 0, 1. Following the approach from [2], we can prove the existence of a solution (u ε , p ε ) ∈ L 2 (0, T ; Remark 1. In fact, in [2] the existence was proved for the Navier-Stokes problem and our case directly follows by taking Re = 0. The problem that they have considered is much more difficult since most of the proof is about handling the nonlinear inertial term. As in [13], we introduce the new variable with the basic difference that h now depends on x and t, and not only on x. Thus, the coordinate z is now moving in time but in some prescribed way determined by h. Obviously, It should be noted that now, in new variables (x, z), the domain Ω ε (t) becomes square Ω =]0, 1[ 2 . Indeed, introducing we deduce Ψ(Ω ε (t)) = Ω.
Taking into account the above change of variables, the partial derivatives are changing as follows: In order to get the form more appropriate for further asymptotic analysis, we multiply (5) by (1 − εh) 2 and (6) by 1 − εh. As a result, we get 4. Asymptotic expansion. To derive the complete asymptotic expansion, as usual, we assume that the given data h, u 0 and p k are as smooth as needed. For computation we need p k to be continuous, h to be a C 2 function in x and C 1 in t. As for the initial condition u 0 , since we use its Taylor expansion it has to be at least C k to compute the approximation of order k. We now seek for the solution of (5)- (7) in the form of the asymptotic expansion: with U k = (U k x (x, z, t), U k y (x, z, t)) and P k = P k (x, z, t). Plugging the above expansions in the governing system and collecting the terms with equal powers of ε give As a result, we obtain a recurrent system of problems for (U k , P k ). Hereinafter, For the zero-order approximation (U 0 , P 0 ) we have: First-order approximation (U 1 , P 1 ) is given by the equations: endowed with the following boundary and initial conditions: Finally, the higher-order terms (U k , P k ), k ≥ 2 satisfy the following system: For any T > 0, it is essential to observe that the right-hand side in the problems (16)-(20), (21)-(25) (the known part from the previous iteration) belongs to . The functions U k are not divergence free but, if both outlets of the domain are of the same size, i.e. if h(0, t) = h(1, t), we still have the equality of in and out flow, namely: Since u ε is divergence free, it holds i.e. it is independent of x. Consequently, Substituting the expansion (11) yields 5. Formal computation. Another approach would be to use the formal Taylor series approach. To keep the notation as simple as possible, we assume that h < 0 so that Ω ⊂ Ω ε . As a consequence, our solution (u ε , p ε ) is defined on Ω so we are in position to directly expand u ε in Taylor series with respect to y near the upper boundary (assuming that the velocity is analytic). Otherwise, we would have to extend the solution to Ω and contaminate the notation). It should be emphasized that this is just a technical assumption, i.e. the obtained results are valid for an arbitrary (smooth enough) function h. It is due to the fact that, by the end of this section, we are going to prove that the approximation derived in the sequel is asymptotically the same as the one that we derived in the previous section (by passing to an ε-independent domain). In view of that, we expand as follows Plugging an asymptotic expansion for the solution (u ε , p ε ) of the form and collecting the terms with equal powers of ε, we get using (28) and (29) −ε ∂h ∂t leading to the following effective boundary conditions: The problem for (V 0 , Q 0 ) now read: The higher-order terms (V k , Q k ), k ≥ 1 are given by: V k (x, y, 0) = 0.
By comparing (13)- (15) and (36)-(39), it is obvious that (U 0 , P 0 ) = (V 0 , Q 0 ). However, since (V 1 , Q 1 ) satisfy the following problem: it is clear that V 1 = U 1 . Nevertheless, from (16)- (20) and (44)-(47), it is straightforward to deduce Direct computation also yields whereas Employing the expansion In view of that, we conclude that two approximations are asymptotically the same, as emphasized at the beginning of this section.
For any T > 0 denoting Ω T =]0, T [×Ω, we have the following expressions for the right-hand side functions E(k, ε), e(k, ε), b(k, ε): for some ξ x , ξ y ∈]0, 1[. Thus, Before we proceed, we have to correct the divergence by constructing the function Such function exists and can be chosen in a way that d ε ∈ L 2 (0, T ; H 1 (Ω)) satisfying for some C > 0 independent on ε. We multiply (54) by and get the following integrals (abusing slightly the notations, because some of the integrals should be replaced by appropriate duality brackets due to the possible lack of regularity): 1. ) The second integral J 2 can be treated as whereas for J 1 we obtain:

EDUARD MARUŠIĆ-PALOKA AND IGOR PAŽANIN
For the sake of reader's convenience, we split the above integral in three parts and present the estimation process of each part in details: Finally, We have However, the last remaining term from J 32 1 , namely cannot be handled by partial integration since it only contains derivatives with respect to x and the two functions under the integral sign do not cancel for x = 0, 1. Therefore, we need the Lemma 1 from Appendix B, with We sum up the above computations to obtain Summing up the above we get with Integrating with respect to t, for ε < 1 2C1 , 1 |h| L ∞ we obtain |R(k, ε)| L 2 (0,T ;V ) ≤ Cε k+1 . (66) To estimate the pressure we go back to the original variables (x, y, t) and introduce Then for a.e. t ∈ [0, T ] where |M ε (·, t)| H −1 (Ω(t)) ≤ Cε k+1 , |B ε (·, t)| L 2 (Ω(t)) ≤ Cε k+1 for (a.e.) t ∈ [0, T ].
To solve that problem we use the Galerkin procedure. Let be the Hilbert space with H 1 norm and let V be it's dual space. Let (b j ) j∈N be a basis in V . We look for an approximation of the solution to the problem (73) in the form We choose the functions c mj such that for any k ∈ {1, . . . , m} where P m is an orthogonal projector on subspace span{b 1 , . . . , b m }. That is a linear system of ordinary differential equations that admits a unique solution. Due to the assumptions on f , p 0 , p 1 and a, we get that w m ∈ L 2 (0, T ; V ) ∩ L ∞ (0, T ; L 2 (Ω) 2 ). Furthermore, ∂w m ∂t ∈ L 2 (0, T ; V ). Now, we multiply the equation (73) with c mk and take the sum with respect to k from 1 to m. We obtain We easily deduce that Thus, ≤ C |f | L 2 (0,T ;V ) + |a| L 2 (0,T ;H 2 (0,1)) + ∂a ∂t L 2 (0,T ;L 2 (0,1)) .
Finally we need to recover the pressure. We define W(x, y, t) = t 0 w(x, y, s)ds, G = t 0 g(x, y, s)ds.

Appendix B:
Recursion related to the divergence corrector. In this Appendix, we prove a technical result needed to estimate the divergence truncation term e(k, ε) defined by (60).