Homogenization of the fluid–structure interaction in acoustics of porous media perfused by viscous fluid

Abstract

This paper aims to clarify the homogenization results of the fluid–structure interaction in porous structures under the quasi-static and dynamic loading regimes. In the latter case, the acoustic fluctuations yield naturally a linear model which can be introduced in the configuration deformed as the consequence of the steady permanent flow. We consider a Newtonian slightly compressible fluid under the barotropic acoustic approximation. In contrast with usual simplifications, the advection phenomenon of the Navier–Stokes equations is accounted for. The homogenization results are based on the periodic unfolding method combined with the asymptotic expansion technique which provide a straight procedure leading the local problems for corrector functions yielding the effective model parameters and the macroscopic model. We show that the local problems for the solid and fluid parts are decoupled even in the dynamic interactions including the wall shear stress on the periodic interfaces. The dynamic permeability depends on the fluid flow properties including the advection effects associated with an assumed stationary perfusion of the porous structure.

Appendices

Appendix A: Basics of the homogenization by the unfolding operator method

The homogenization results were obtained by the periodic unfolding method, see [16]. To introduce the periodic unfolding operator, a domain is needed, containing the “entire” periods $\varepsilon Y$ only:

$$\begin{aligned} \hat{\varOmega }^\varepsilon = \text{ interior } \bigcup _{\zeta \in \varXi ^\varepsilon } \overline{Y}^{\varepsilon ,\zeta }, \quad Y^{\varepsilon ,\zeta }= \varepsilon ({Y} + \zeta ) , \end{aligned}$$

(A.1)

where $\varXi ^\varepsilon = \{\zeta \in {\mathbb {Z}}^3\,|\; Y^{\varepsilon ,\zeta } \subset \varOmega \}$, By the construction of the periodic lattice $\hat{\varOmega }^\varepsilon$, a small boundary neighbourhood $\varLambda ^\varepsilon =\varOmega {\setminus }\hat{\varOmega }^\varepsilon$ vanishes with $\varepsilon \rightarrow 0$, For all $z \in {\mathbb {R}}^3$, let [z] be the unique integer such that $z- [z] \in Y$. Since $z = [z]+\{z\}$ for all $z\in {\mathbb {R}}^3$, for all $\varepsilon >0$, the unique decomposition holds,

$$\begin{aligned} x = \varepsilon \left( \left[ \frac{x}{\varepsilon }\right] + \left\{ \frac{x}{\varepsilon }\right\} \right) = \xi + \varepsilon y \quad \forall x \in {\mathbb {R}}^3,\quad \xi = \varepsilon \left[ \frac{x}{\varepsilon }\right] . \end{aligned}$$

(A.2)

Based on this decomposition, the periodic unfolding operator ${\mathcal {T}}_{\varepsilon }\,{}: L^2(\varOmega ;{\mathbb {R}}) \rightarrow L^2(\varOmega \times Y;{\mathbb {R}})$ is defined as follows: for any function $v \in L^1(\varOmega ;{\mathbb {R}})$, extended to $L^1({\mathbb {R}}^3;{\mathbb {R}})$ by zero outside $\varOmega$, i.e. $v=0$ in ${\mathbb {R}}^3 {\setminus } \varOmega$,

$$\begin{aligned} {{\mathcal {T}}}_{\varepsilon }{\left( {v}\right) }(x,y) =\left\{ \begin{array}{ll} v\left( \varepsilon \displaystyle \left[ \frac{x}{\varepsilon }\right] + \varepsilon y \right) , &{}\quad x \in \hat{\varOmega }^\varepsilon , y \in Y, \\ 0 &{}\quad \text{ otherwise } . \\ \end{array}\right. \end{aligned}$$

By virtue of (A.2), ${{\mathcal {T}}}_{\varepsilon }{\left( {\nabla u}\right) } = \varepsilon ^{-1}\nabla _y {{\mathcal {T}}}_{\varepsilon }{\left( {u}\right) }$. For product of any u and v the unfolding yields ${{\mathcal {T}}}_{\varepsilon }{\left( {uv}\right) }={{\mathcal {T}}}_{\varepsilon }{\left( {u}\right) }{{\mathcal {T}}}_{\varepsilon }{\left( {v}\right) }$. The following integration formula holds:

$$\begin{aligned} \int \limits _{\hat{\varOmega }^\varepsilon } v\,\mathrm{d}x = \frac{1}{|Y|}\int \limits _{\varOmega \times Y} {{\mathcal {T}}}_{\varepsilon }{\left( {v}\right) }\,\mathrm{d}y\,\mathrm{d}x\quad \forall v \in L^1(\varOmega ), \end{aligned}$$

which is employed to treat the variational formulations. Further details, namely the convergence results, further properties and various extensions have been reported in [17].

Appendix B: Limit mass conservation

Using the truncated expansions (4.11) we provide limit expressions of selected terms involved in the weak formulation (4.7)–(4.8). In the mass conservation (4.7)$_2$, the advection-related term $q^\varepsilon \bar{{\textit{\textbf{w}}}}^\varepsilon \cdot \nabla \widetilde{p}^\varepsilon = \varepsilon \bar{{\textit{\textbf{w}}}} \cdot (\nabla _x p^0 + \nabla _y p^1) (q^0 + \varepsilon q^1)$ vanishes with $\varepsilon \rightarrow 0$. Recalling $\partial \varOmega _f^\varepsilon = \partial \varOmega \cup \varGamma ^\varepsilon$, we integrate by parts the divergence term, which then converges, as follows:

(B.1)

Above, the last two integrals are obtained due to

(B.2)

Using (B.1), in the limit $\varepsilon \rightarrow 0$, the mass conservation (4.7)$_2$ yields

(B.3)

where, in the last row, expressions involving $\dot{{\textit{\textbf{u}}}}^0$ and $q^1$ cancel; hence the limit mass conservation (4.17)$_{2,3}$ is proved.