An approximation of the axially symmetric flow through a pipe-like domain with a moving part of a boundary
© Filo and Pluschke; licensee Springer. 2013
Received: 6 August 2013
Accepted: 28 August 2013
Published: 18 November 2013
The purpose of this work is to study the existence of solutions for approximation of an unsteady fluid-structure interaction problem. We consider a perturbed Navier-Stokes equation in the cylindrical coordinate system assuming axially symmetric flow. A priori unknown part of the boundary (that interacts with the fluid) is governed with a linear equation of fifth order. We prove the existence of at least one weak solution as long as the boundary does not touch the axis of symmetry. An explicit expression for a class of divergence-free functions is given.
MSC:35Q30, 35Q35, 35D05, 74F10, 76D03.
and ρ, μ, L, , T, , are given positive constants, , .
for any , . Here and are given functions of two variables for and , respectively, .
up to now due to the lack of regularity in the time variable.
This example of implies that we in general need more regularity for , and therefore we will assume (1.15). Section 7 provides a detailed analysis of a limit process as in the linearised perturbed NS equations.
Our study was originally initiated by the papers of Quarteroni [2–4] and it is in some sense a continuation of the paper . Due to the fact that our main contribution is in introducing the cylindrical coordinate system to the fluid-structure interaction-like problem, we do not provide a representative list of references for 2-D problems. For a full summary of the research on solvability of the fluid-structure interaction problem, we refer the reader to the introduction in  and references therein. See also [6–8] and . The methods that we apply borrow some material from [5, 10–12] and ; nevertheless, their application to our problem seems to be not straightforward.
The paper is organised as follows. In Section 2 we regularise once more our problem by considering it on a domain without the axis of symmetry (see, e.g. ) and then we transform it on the fixed domain. The main result of this section is the identity (2.21). Sections 3 and 5 provide necessary a priori estimates including the delicate identity (3.12). Sections 4 and 6 present main results of the paper. In Section 4, using the Schauder fixed point theorem, we prove the existence of the solution for the unknown interface, and in Section 6 we go to a limit with our regularised parameters. Finally, in Section 7 we let in (1.2) but only for the linearised version of our problem, see (7.2).
for all . Note that at the end we have to prove the validity of (1.20).
for any . Note that in this section h is given and , p and η are to be found (h and η are therefore not related by (1.4)).
for any and for any .
The function spaces we use are rather familiar, and we adopt the notation of .
, , , and .
- (2)satisfies the system of differential equations, that is,(2.21)
for every , , .
for every test function with .
At the end of this section, we discuss the existence and uniqueness of weak solutions to problem (2.9)-(2.17) for given h, , and satisfying (1.20) and (2.20).
Proof of Theorem 2.1 By analogy with the approach taken in [, the proof of Theorem 6.3], we can construct our weak solution by the implicit time discretisation method. The paper  presents all the technicalities of the proof, therefore we omit the proof here. □
- (ii)Let now and fulfil the condition
Proof (i) The form of Nirenberg-Gagliardo inequality (2.26) can be found, e.g. in [, Theorem 2.2]. Then (2.27) follows from (2.26) for by integration over .
(ii) To prove (2.28), we define and transform the integrals on the domain D into integrals on the cylinder . Then the weighted interpolation inequality (2.28) is equivalent to an unweighted interpolation inequality in for given on , which yields the assertion (cf. also [, Theorem 19.9] for ). Inequality (2.29) again follows by integration of (2.28) over with . □
3 The first a priori estimate
Our ultimate goal is to show that a subsequence of solutions of the approximate problems (2.9)-(2.17) converges to a weak solution of (1.1)-(1.14). For this we will need some uniform estimates. Let us start with the following theorem.
Theorem 3.1 Let the hypotheses of Theorem 2.1 be satisfied, and let be the corresponding solution of problem (2.9)-(2.17).
The constant C does not depend on α, K, ε, κ and δ.
Before proving Theorem 3.1, we pause to introduce some of its consequences.
for any fixed and all ε, κ, δ satisfying (2.1).
and (3.5) follows easily. □
for any fixed .
Our plan now is to check the ellipticity condition that follows from the following two propositions.
Now, let us focus on the first term in the second line of (3.13).
and (3.14) follows easily. The validity of (3.14) for is a consequence of the approximation argument. □
We apply now Lemma 3.3 to the first term on the second line of (3.13) and the validity of (3.12) can be verified by direct computations. Let us note, however, that it is rather tedious. □
The proof of the next proposition can be found in [, Lemma 4.3].
that due to (3.16) and (3.17) implies (3.1). □
4 A free boundary value problem
Our plan is hereafter to formulate it as a fixed point problem.
(note that is increasing in T), we have the following theorem.
provided is sufficiently small. Moreover, satisfies (1.20) with the given α and K.
Note that smallness of (cf. Theorem 3.1) can be achieved by small outer pressure or a small time interval, but from now on, let T be fixed.
The second estimate in (1.20) obviously follows from (4.4) by the definition of h and the choice of K. □
Now, let be the solution of problem (2.9)-(2.17) according to Definition 2.1 with , and let us define a mapping .
Corollary 4.2 is well defined on and maps into itself.
Proof Note that fulfils (1.20) if in view of Theorem 4.1. Then Theorem 2.1 implies that the mapping is well defined and unique. Moreover, due to Theorem 3.1, belongs to with the finite norm defined by (4.2). Finally, estimate (4.4) in Theorem 4.1 yields . □
An application of the Schauder fixed point theorem requires the continuity of our mapping . This means that we need continuous dependence of solutions on the data.
Unfortunately, ι depends on the regularising parameter δ in such a way that as .
Proof of Theorem 4.3 Due to tedious calculations, we postpone the proof to the Appendix of this paper. □
Together with the embedding inequality (4.3) this yields continuity of the operator on . Hence, we arrive at the following result.
on the time interval . This solution is unique.
where denotes the norm (4.2) with integration over instead of . For small enough such that , finally, we obtain , which is a contradiction to the choice of τ. □
Note that our solution with depends on δ, ε, κ; however, the maximal time interval given by (4.7) does not depend on these regularising parameters.
5 The second a priori estimate
For compactness in time, the following second a priori estimate is essential.
Before starting the proof of the theorem, let us prepare some estimations of the nonlinear items.
under the same condition on λ.
where the first integral of the product on the right-hand side is bounded owing to (5.9) and the second integral is bounded by our assumption. □
- (i)In order to demonstrate the technique, we start with the first item occurring in the third line of (5.10) which isIn view of (5.2), (5.1) and , we may estimate
for any . In the last steps here, we have used inequality (5.11) and boundedness of the norm due to Theorem 3.1.
- (ii)We proceed with the estimation of the integral arising from the nonlinear items, cf. the forth line of (2.21). Recall the definition (2.10) of , after inserting into (2.21) and integration over , the items to be integrated are(5.12)and(5.13)withObserving (5.1), (5.3) and, moreover,we have to show boundedness of integrals of the form(5.14)
where and is the Steklov average defined after (5.10). Because of , we can assume . Moreover, due to inequality (5.11), we may omit the Steklov averages if the factors under the integrals are separated by Hölder’s inequality. Just this is the case in the proof of Lemma 5.2. Hence, we can apply Lemma 5.2 to the integrals (5.14). Since the assumptions of Lemma 5.2 are fulfilled due to Theorem 3.1, the integrals over the items (5.12) and (5.13) are estimated for by cτ with a constant c independent of κ, δ, ε.
- (iii)The integrals over the items and are estimated in a similar way as in (ii). We consider now(5.15)and(5.16)withif . Hence, in this case, we need boundedness of the integrals(5.17)This may be obtained for again by means of Hölder’s inequality, (5.11) and the estimates of Theorem 3.1. However, since
the integrals over items (5.15) and (5.16) can only be estimated by .
- (iv)Since the integrals over the derivatives of z are obviously bounded due to the estimates of Theorem 3.1, finally we have a look at the last item on the left-hand side of (2.21),(5.18)
due to Theorem 3.1 with c independent of κ. Since the estimation of the right-hand side of (2.21) is obvious, this concludes the proof. □
Then from Theorems 3.1 and 5.1 we can derive the following estimates, which yield compactness of w in .