An approximation of the axially symmetric flow through a pipe-like domain with a moving part of a boundary

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.

This paper is devoted to the perturbed Navier-Stokes equations in a cylindrical coordinate system assuming rotationally symmetric flow

with an artificial equation for the pressure

() in an a priori unknown domain

where

with the given

and ρ, μ, L, L^{+}, T, R^{-}, R^{+} are given positive constants, L^{-}\ge 0, R^{+}\le R^{-}.

Unknown are the velocity field (v_x,v_r), the pressure p and the interface η on [0,L]. We shall require from η to satisfy the following conditions:

for , ,

for any and, for convenience, also

Here a, b, c, f are given nonnegative numbers and d, e, E are supposed to be positive. Moreover, p_w=p_w(x,t) is a given function. Of course, (1.1) is to be complemented with additional boundary and initial conditions. We shall consider

for any , ,

for any -L^{-}\le x\le 0 and L\le x\le L^{+}, ,

for any , ,

for any , ,

for any , and, finally,

for any , . Here p_{\mathrm{in}} and p_{\mathrm{out}} are given functions of two variables (r,t) for 0\le r\le R^{-} and 0\le r\le R^{+}, respectively, 0\le t\le T.

The aim of this paper is to extend the existence result of [1] concerning the two-dimensional fluid-structure interaction problem to the radially symmetric case expressed in terms of the cylindrical coordinate system. Since we have not been able yet to handle this nonlinear problem modelling the flow of an incompressible viscous fluid in a cylinder with deformable wall, here nonlinear perturbed Navier-Stokes equations coupled with the artificial equation (1.2) and assuming e>0 in the equation of the wall (1.6) are studied. As it turns out, we have not been able to let

up to now due to the lack of regularity in the time variable.

We can give a justification of this limit, but only for a linearised perturbed Navier-Stokes equations and by including the additional term

in (1.6). Let us illustrate some arguments for the presence of this term in (1.6). To prove our existence result, we shall transform \mathrm{\Omega }(h) for h given by (1.4) on a fixed domain by setting

The incompressibility condition takes then the form

In the final stage of the proof, we are to send \epsilon \to 0 in an integral identity that includes also terms

where

({\mathit{u}}^{\epsilon },q^{\epsilon },{\eta }^{\epsilon }) solve an appropriate collection of approximating problems and {\mathit{\psi }}^{\epsilon } are test functions. The only way to overcome the lack of regularity for \{q^{\epsilon }\} is to consider divergence-free test functions, i.e.

This motivates us to find an explicit expression for a class of functions satisfying (1.18). For \varphi \in C^3(\overline{D}\times [0,T]), let us define

and the desired equality (1.18) is valid for {\mathit{\psi }}^{\epsilon }=({\psi }_1^{\epsilon },{\psi }_2^{\epsilon }). But now another difficulty arises, for, by inserting (1.19) into (1.17), the term

appears. The problem is that we cannot deduce more than weak convergence of the velocity gradient in L^2. What saves us are just the strong convergences of

This example of {\mathit{\psi }}^{\epsilon } implies that we in general need more regularity for h^{\epsilon }, and therefore we will assume (1.15). Section 7 provides a detailed analysis of a limit process as \epsilon \to 0 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 [5]. 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 [1] and references therein. See also [6–8] and [9]. The methods that we apply borrow some material from [5, 10–12] and [13]; 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. [14]) 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 \epsilon \to 0 in (1.2) but only for the linearised version of our problem, see (7.2).

Throughout the paper we will always assume that there are positive constants α, K, , such that

for all (x,t)\in [-L^{-},L^{+}]\times [0,T]. Note that at the end we have to prove the validity of (1.20).

Our approach is to consider an approximate problem first. Given numbers

and given a function h\in C^1([-L^{-},L^{+}]\times [0,T]) satisfying (1.20), consider the system

in

equipped with the following boundary and initial conditions:

for any , ,

for any -L^{-}\le x\le 0 and L\le x\le L^{+}, ,

for any , ,

for any , ,

for any , ,

for any , ,

for any , ,

for any , ,

for any and, finally,

for any . Note that in this section h is given and (v_x,v_r), p and η are to be found (h and η are therefore not related by (1.4)).

Assume for a moment that (v_x,v_r,p,\eta ) is in fact a smooth solution of problem (2.2)-(2.6) above. Then

for , , and

for , solve the following problem:

in D\times (0,T), where D=(-L^{-},L^{+})\times (0,1),

in D\times (0,T),

for any , with the boundary and initial conditions listed below:

for any , , q_w(x,t)=p_w(x,t),

for any , , q_{\mathrm{out}}(y,t)=p_{\mathrm{out}}(R^{+}y,t),

for any , , q_{\mathrm{in}}(y,t)=p_{\mathrm{in}}(R^{-}y,t),

for any , ,

for any , and, finally,

for any and for any .

We continue this section by making precise the meaning of the solution of problem (2.9)-(2.17). To this end, we first define

where

and

Throughout this and the next sections, we assume

The function spaces we use are rather familiar, and we adopt the notation of [15].

Definition 2.1 We call (\mathit{u},q,z)\in L^2(0,T;V) a weak solution of the initial boundary value problem (2.9)-(2.17) if the following two properties are fulfilled:

1. (1)

   \mathit{u}\in L^{\mathrm{\infty }}(0,T;L^2{(D)}^2), q\in L^{\mathrm{\infty }}(0,T;L^2(D)), \frac{\partial (\ell q)}{\partial t}\in L^2(0,T;H^{-1}(D)), \frac{\partial (\ell \mathit{u})}{\partial t}\in {(L^2(0,T;\mathit{V})\cap L^4(0,T;L^4{(D)}^2))}^{\ast }=L^2(0,T;V^{\ast })+L^{4/3}(0,T;L^{4/3}{(D)}^2) and z\in L^{\mathrm{\infty }}(0,T;H_0^2(0,L))\cap H_{0-}^1(0,T;H_0^1(0,L)).

2. (2)

   (\mathit{u},q,z) satisfies the system of differential equations, that is,

   \begin{array}{r}{\int }_0^T\{\rho 〈\frac{\partial (\mathit{u}\ell )}{\partial t},\mathit{\psi }〉+\epsilon 〈\frac{\partial (q\ell )}{\partial t},\omega 〉\}dt\\ -\frac{\rho }{2}{\int }_0^T{\int }_D\{(\mathit{u}\cdot \mathit{\psi })\frac{\partial \ell }{\partial t}+[\frac{\partial \mathit{u}}{\partial y}\cdot \mathit{\psi }-\mathit{u}\cdot \frac{\partial \mathit{\psi }}{\partial y}][(h-\delta )y+\delta ]\frac{\partial h}{\partial t}y\}d(x,y)dt\\ -\frac{\epsilon }{2}{\int }_0^T{\int }_D\{q\omega \frac{\partial \ell }{\partial t}+[\frac{\partial q}{\partial y}\omega -q\frac{\partial \omega }{\partial y}][(h-\delta )y+\delta ]\frac{\partial h}{\partial t}y\}d(x,y)dt\\ +{\int }_0^T{\int }_D\{\frac{\rho }{2}[((\mathit{u}\cdot {\mathrm{\nabla }}_h)\mathit{u})\cdot \mathit{\psi }-((\mathit{u}\cdot {\mathrm{\nabla }}_h)\mathit{\psi })\cdot \mathit{u}]\ell \\ +\mu ([{\mathrm{\nabla }}_h\mathit{u}+{({\mathrm{\nabla }}_h\mathit{u})}^T]:{\mathrm{\nabla }}_h\mathit{\psi }+\frac{2{(h-\delta )}^2}{{\ell }^2}{u}_2{\psi }_2)\ell \\ +\epsilon ({\mathrm{\nabla }}_{h}q\cdot {\mathrm{\nabla }}_h\omega )\ell +(\omega {div}_h\mathit{u}-q{div}_h\mathit{\psi })\ell \}d(x,y)dt\\ +E{\int }_0^T{\int }_0^L\{\frac{\partial z}{\partial t}\xi +d\frac{{\partial }^{2}z}{\partial x^2}\frac{{\partial }^2\xi }{\partial x^2}+e\frac{{\partial }^{2}z}{\partial x\partial t}\frac{{\partial }^2\xi }{\partial x\partial t}+f\frac{{\partial }^{2}Z}{\partial x^2}\frac{{\partial }^2\xi }{\partial x^2}\\ +c\frac{\partial z}{\partial x}\frac{\partial \xi }{\partial x}+a\frac{\partial Z}{\partial x}\frac{\partial \xi }{\partial x}+bZ\xi \}dxdt\\ +{\int }_0^T{\int }_0^L\{q_w{\psi }_{2}h(h-\delta )+\kappa (u_2-z)({\psi }_2-\xi )\}dxdt\\ ={\int }_0^T{\int }_0^1\{q_{\mathrm{in}}(y,t)({\psi }_1\ell )(-L^{-},y,t)+q_{\mathrm{out}}(y,t)({\psi }_1\ell )(L^{+},y,t)\}dydt\end{array}

   (2.21)

for every (\mathit{\psi },\omega ,\xi )\in L^2(0,T;V), \mathit{\psi }\in L^4(0,T;L^4{(D)}^2), \xi \in H^1(0,T;H^1(0,L))\cap L^2(0,T;H_0^2(0,L)).

Remark 2.1 Note that

for every test function \mathit{\zeta }\in L^2(0,T;\mathit{V})\cap L^4(0,T;L^4{(D)}^2)\cap H^{1,1}(0,T;L^2{(D)}^2) with \mathit{\zeta }(T)=0.

At the end of this section, we discuss the existence and uniqueness of weak solutions (\mathit{u},q,z) to problem (2.9)-(2.17) for given h, q_w, q_{\mathrm{in}} and q_{\mathrm{out}} satisfying (1.20) and (2.20).

Theorem 2.1 Assume (2.1), (2.20), and let \overline{\eta }\in C^1([0,L]\times [0,T]) satisfying (1.7) and (1.8) be given. Hereafter h=h(\overline{\eta }) denotes h defined by (1.4) with the given \overline{\eta }, and we assume that (1.20) holds. Then there exists a unique solution

of problem (2.9)-(2.17) in the sense of Definition  2.1 such that

where the function ℓ in (2.23) is defined by (2.10) for h=h(\overline{\eta }), i.e.

Proof of Theorem 2.1 By analogy with the approach taken in [[5], the proof of Theorem 6.3], we can construct our weak solution by the implicit time discretisation method. The paper [5] presents all the technicalities of the proof, therefore we omit the proof here. □

We conclude this section by some interpolation inequalities which are needed especially for estimating the nonlinear item. For this purpose, we complete the notations of function spaces (2.18) and (2.19) by appropriate weighted spaces. Let

and

Proposition 2.1 (i) Let φ be any function in H^1(D) such that \phi =0 on S_w or \phi =0 on S_c. Then, for any p\ge 2 and for any number θ with

there exists a constant C=C(p,\theta ) such that

Moreover, if \phi \in L^2(0,T;H^1(D))\cap L^{\mathrm{\infty }}(0,T;L^2(D)) such that \phi =0 on S_w or \phi =0 on S_c for almost all t\in [0,T], then the following holds for any p\ge 2:

1. (ii)

   Let now p\ge 2 and \theta \le 1 fulfil the condition

   \frac{3}{2}\frac{p-2}{p}\le \theta \le 1.

Then there exists a constant C=C(p,\theta ) such that

If, moreover, \mathit{w}\in L^2(0,T;H_y^1{(D)}^2)\cap L^{\mathrm{\infty }}(0,T;L_y^2{(D)}^2) such that w_1=0 on S_c and w_2=0 on S_w for almost all t\in [0,T], then the following holds for any p\in [2,6]:

Proof (i) The form of Nirenberg-Gagliardo inequality (2.26) can be found, e.g. in [[16], Theorem 2.2]. Then (2.27) follows from (2.26) for \theta =(p-2)/p by integration over (0,T).

(ii) To prove (2.28), we define \tilde{\mathit{\psi }}(x,y_2,y_3):=\mathit{\psi }(x,\sqrt{y_2^2+y_3^2}) 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 {\mathbb{R}}^3 for \tilde{\mathit{\psi }} given on \tilde{D}, which yields the assertion (cf. also [[17], Theorem 19.9] for \theta =1). Inequality (2.29) again follows by integration of (2.28) over t\in (0,T) with \theta =3(p-2)/2p. □

Our ultimate goal is to show that a subsequence of solutions ({\mathit{u}}_{\epsilon ,\kappa ,\delta },q_{\epsilon ,\kappa ,\delta },z_{\epsilon ,\kappa ,\delta }) 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 \mathcal{L}(\overline{\eta })=(\mathit{u},q,z) be the corresponding solution of problem (2.9)-(2.17).

Then there exists a constant C such that