2.3.1. Oscillatory fluid flow problem

From the conservation of momentum equation, one obtains that the leading-order oscillatory pressure is locally uniform, which is a classical result: $p^{(0)}(x,y) = p^{(0)}(x) = P$ . At $\epsilon ^{-1}$ , the mass conservation (2.46) yields $\boldsymbol{\nabla} _{\!y} \boldsymbol{\cdot }\boldsymbol{v}^{(0)} = 0$ , which shows that the leading-order oscillatory fluid velocity is divergence-free.

The boundary-value problem for $\boldsymbol{v}^{(0)}$ and the pressure corrector $p^{(1)}$ read

(2.52) \begin{equation} \eta \boldsymbol{\nabla} _{\!y}^{2} \boldsymbol{v}^{(0)} - \boldsymbol{\nabla} _{\!y} p^{(1)} - \textrm{j} \omega \rho _{e} \boldsymbol{v}^{(0)} - \boldsymbol{\nabla} _{\!x} P = \rho _{e} \big[ (\boldsymbol{v}_{0} \boldsymbol{\cdot }\boldsymbol{\nabla} _{\!y}) \boldsymbol{v}^{(0)} + \big(\boldsymbol{v}^{(0)} \boldsymbol{\cdot }\boldsymbol{\nabla} _{\!y}\big) \boldsymbol{v}_{0}\big ] \quad \mbox{in} \quad \varOmega _{\!f}, \end{equation}

(2.53) \begin{equation} \boldsymbol{\nabla} _{\!y}\boldsymbol{\cdot }\boldsymbol{v}^{(0)}=0 \quad \mbox{in} \quad \varOmega _{\!f}, \end{equation}

(2.54) \begin{equation} \boldsymbol{v}^{(0)}=\boldsymbol{0} \quad \mbox{on} \quad \varGamma _{s}. \end{equation}

The terms on the left-hand side of (2.52) correspond to those that appear in the oscillatory Stokes problem in classical rigid-frame permeable media, while those on the right-hand side correspond to linearised convective terms due to the background flow. Here, $\boldsymbol{v}_{0}$ is considered as in (2.17), i.e. $\boldsymbol{v}_{0} = \boldsymbol{v}_{0}^{(0)}$ . Hence, it is necessary to solve the local problem (2.14)–(2.16) to obtain $\boldsymbol{v}_{0}$ , which is then used to solve the local fluid flow problem in the dynamic regime given by (2.52)–(2.54) (see also Appendix C for the cell problems for $\hat {\boldsymbol{h}}$ and $\hat {\pi }_{0}$ ).

The oscillatory fluid flow problem (2.52)–(2.54) is linear, forced by the macroscopic oscillatory pressure gradient and parametrised by $\boldsymbol{v}_{0}(y,\varpi )$ . Therefore, its solution can be written as

(2.55) \begin{equation} \boldsymbol{v}^{(0)} = -\frac { \hat{\boldsymbol{\mathcal{K}}} (y,\omega ,\varpi )}{\eta }\boldsymbol{\cdot }\boldsymbol{\nabla} _{\!x}P, \end{equation}

where $\hat{\boldsymbol{\mathcal{K}}}(y,\omega , \varpi )$ is a frequency-dependent, complex-valued $y$ -periodic second-rank tensor that also depends on the geometry of the material and convective effects which are characterised by the background velocity field $\boldsymbol{v}_{0}$ , and thus by $\varpi$ . This field represents a local normalised velocity field in response to a unit forcing term. We shall stress that the form of this solution is classical and that, in the absence of rotation, $\hat {\boldsymbol{\mathcal{K}}} (y,\omega ,\varpi )$ reduces to the classical local field $\hat {\boldsymbol{\mathcal{K}}}(y,\omega )$ that determines the leading-order velocity in permeable media with its saturating fluid initially at rest.

The pressure corrector can be expressed as

(2.56) \begin{equation} p^{(1)} = - \hat {\boldsymbol{\pi }}(y,\omega ,\varpi ) \boldsymbol{\cdot }\boldsymbol{\nabla} _{\!x} P + \hat {p}^{(1)}(x). \end{equation}

Since $p^{(1)}$ is determined up to a constant independent of the local variable $y$ , namely $\hat {p}^{(1)}(x)$ , the vector $\hat {\boldsymbol{\pi }}(y,\omega ,\varpi )$ is taken to have a zero spatial mean. This standard condition (Auriault et al. Reference Auriault, Boutin and Geindreau2009) ensures that the spatial mean of $p^{(1)}$ equals $\hat {p}^{(1)}(x)$ . The cell problem for $\hat {\boldsymbol{\mathcal{K}}} (y,\omega ,\varpi )$ and $\hat {\boldsymbol{\pi }}(y,\omega ,\varpi )$ is presented in Appendix C.

The oscillatory fluid flow constitutive law is then obtained by applying the spatial mean operator (2.20)–(2.55), i.e.

(2.57) \begin{equation} \big\langle \boldsymbol{v}^{(0)}\big\rangle =-\frac {\boldsymbol{\mathcal{K}}(\omega ,\varpi )}{\eta }\boldsymbol{\cdot }\boldsymbol{\nabla} _{\!x}P, \end{equation}

where the dynamic permeability tensor is calculated as

(2.58) \begin{equation} \boldsymbol{\mathcal{K}}(\omega ,\varpi ) = \big\langle \hat {\boldsymbol{\mathcal{K}}}(y,\omega ,\varpi ) \big\rangle . \end{equation}

Rewriting the fluid flow constitutive law (2.57) in the form of Newton’s second law yields

(2.59) \begin{equation} \textrm{j} \omega \boldsymbol{\rho }(\omega ,\varpi ) \boldsymbol{\cdot } \big\langle \boldsymbol{v}^{(0)} \big\rangle = -\boldsymbol{\nabla} _{\!x}P, \end{equation}

where the second-rank effective dynamic density tensor $\boldsymbol{\rho }(\omega ,\varpi )$ is given by

(2.60) \begin{equation} \boldsymbol{\rho }(\omega ,\varpi ) =\frac {\eta }{\textrm{j} \omega }\boldsymbol{\mathcal{K}}^{-1} =\frac {{\unicode{x1D646}}^{-1}}{\textrm{j} \omega }, \end{equation}

where ${\unicode{x1D646}} = \boldsymbol{\mathcal{K}} / \eta$ is the dynamic conductivity tensor.

Since the underlying operator is a viscous Stokes-type problem with lower-order convective terms, it remains well posed, except possibly at particular resonance frequencies. This suggests that $\unicode{x1D646}$ remains invertible over the considered long-wavelength regime. However, as no formal proof of the invertibility of $\unicode{x1D646}$ is here provided, the existence of ${\unicode{x1D646}}^{-1}$ is retained as a working hypothesis throughout this work. Under this assumption, the dynamic density tensor $\boldsymbol{\rho }(\omega ,\varpi )$ is well defined.

Let us point out that Appendix B develops the weak formulation of the boundary-value problem (2.52)–(2.54). This will be used in § 3.2, where the properties of the effective parameters, namely $\boldsymbol{\mathcal{K}}(\omega ,\varpi )$ and ${\unicode{x1D646}}(\omega ,\varpi )$ , are analysed.

2.3.2. Oscillatory heat transfer problem

Performing coefficient identification in the energy conservation (2.49) yields, at leading order, the following convective heat transfer problem governing the leading-order oscillatory temperature $\tau ^{(0)}$ :

(2.61) \begin{equation} \kappa \boldsymbol{\nabla} _{\!y}^{2} \tau ^{(0)} - \textrm{j} \omega \rho _{e} c_{\!p} \tau ^{(0)} + \textrm{j} \omega P = \rho _{e} c_{\!p} (\boldsymbol{v}_{0} \boldsymbol{\cdot }\boldsymbol{\nabla} _{\!y}) \tau ^{(0)} \quad \mbox{in} \quad \varOmega _{\!f}, \end{equation}

(2.62) \begin{equation} \tau ^{(0)} = 0 \quad \mbox{on} \quad \varGamma _{s}. \end{equation}

Since this problem is linear in both $\tau ^{(0)}$ and the forcing term $\textrm{j} \omega P / \kappa$ , its solution is proportional to the forcing term and can be written as

(2.63) \begin{equation} \tau ^{(0)} = \frac {\hat {\theta }(y,\omega ,\varpi )}{\kappa } \textrm{j} \omega P. \end{equation}

Here, $\hat {\theta }$ represents a frequency-dependent normalised local $y$ -periodic temperature field in response to a unit forcing term. As in the fluid flow problem, the form of the solution (2.63) is classical, but $\hat {\theta }$ depends on frequency, the geometry of the material and, a priori, on the background velocity field $\boldsymbol{v}_{0}$ . In the absence of rotating motion, the local field reduces to its classical counterpart $\hat {\theta }(y,\omega )$ .

The respective constitutive law is obtained by applying the spatial mean operator (2.20)–(2.63), i.e.

(2.64) \begin{equation} \langle \tau ^{(0)} \rangle = \frac {\theta (\omega ,\varpi )}{\kappa } \textrm{j} \omega P, \end{equation}

where the dynamic ‘thermal permeability’ $\theta (\omega ,\varpi )$ is given by

(2.65) \begin{equation} \theta (\omega ,\varpi ) = \langle \hat {\theta }(y,\omega ,\varpi ) \rangle . \end{equation}

The cell problem for $\hat {\theta }(y,\omega ,\varpi )$ is presented in Appendix C, while the weak formulation of the convective heat transfer problem is developed in Appendix B and will be used in § 3.3 to analyse the properties of the effective parameter $\theta (\omega ,\varpi )$ .

2.3.3. Macroscopic description

From (2.47) and (2.48), the following equations are obtained at $\epsilon ^{0}$ :

(2.66) \begin{equation} \frac {\rho ^{(0)} }{\rho _{e}} = \frac {P}{P_{e}} - \frac {\tau ^{(0)}}{T_{e}} \quad \mbox{in} \quad \varOmega _{\!f}, \end{equation}

(2.67) \begin{equation} \frac {1}{\rho _{e}} \boldsymbol{\nabla} _{\!y} \rho ^{(0)} = \frac {1}{P_{e}} \big(\boldsymbol{\nabla} _{\!x} P + \boldsymbol{\nabla} _{\!y} p^{(1)}\big) - \frac {1}{T_{e}}\boldsymbol{\nabla} _{\!y} \tau ^{(0)} \quad \mbox{in} \quad \varOmega _{\!f}. \end{equation}

The leading-order conservation of mass equation is given by

(2.68) \begin{equation} \begin{aligned} \textrm{j} \omega \frac {\rho ^{(0)}}{\rho _{e}} + \boldsymbol{\nabla} _{\!x} \boldsymbol{\cdot }\boldsymbol{v}^{(0)} + \boldsymbol{\nabla} _{\!y}\boldsymbol{\cdot }\boldsymbol{v}^{(1)} + \frac {1}{\rho _{e}} \boldsymbol{\nabla} _{\!y} \rho ^{(0)} \boldsymbol{\cdot }\boldsymbol{v}_{0} = 0 \quad \mbox{in} \quad \varOmega _{\!f}. \end{aligned} \end{equation}

Substituting (2.66) and (2.67) into (2.68), and applying the mean operator (2.20) gives

(2.69) \begin{equation} \begin{aligned} \textrm{j} \omega \left\langle \frac {P}{P_{e}}-\frac {\tau ^{(0)}}{T_{e}}\right\rangle + \boldsymbol{\nabla} _{\!x} \boldsymbol{\cdot }\langle \boldsymbol{v}^{(0)}\rangle + \frac {\boldsymbol{\nabla} _{\!x} P}{P_{e}} \boldsymbol{\cdot }\langle \boldsymbol{v}_{0} \rangle = - \big\langle \boldsymbol{\nabla} _{\!y} \boldsymbol{\cdot }\boldsymbol{v}^{(1)} \big\rangle \\ - \left \langle \left (\frac {\boldsymbol{\nabla} _{\!y} p^{(1)}}{P_{e}} - \frac { \boldsymbol{\nabla} _{\!y} \tau ^{(0)}}{T_{e}} \right ) \boldsymbol{\cdot }\boldsymbol{v}_{0} \right \rangle \!. \end{aligned} \end{equation}

The first term on the right-hand side vanishes due to the no-slip condition $\boldsymbol{v}^{(1)} = 0$ on $\varGamma _{s}$ and periodicity conditions over $\varGamma$ , i.e. $\langle \boldsymbol{\nabla} _{\!y} \boldsymbol{\cdot }\boldsymbol{v}^{(1)}\rangle =0$ . The second term on the right-hand side also vanishes upon integration by parts, and using that $\boldsymbol{v}_{0}$ is periodic, divergence-free and satisfies $\boldsymbol{v}_{0}\boldsymbol{\cdot }\boldsymbol{n}=0$ on $\varGamma _{s}$ . Then, inserting (2.63) together with (2.64) leads to the following macroscopic mass balance equation:

(2.70) \begin{equation} \boldsymbol{\nabla} _{\!x} \boldsymbol{\cdot }\langle \boldsymbol{v}^{(0)} \rangle + \textrm{j} \omega \mathsf{C}(\omega ,\varpi )P = -\frac {\boldsymbol{\nabla} _{\!x}P}{P_{e}}\boldsymbol{\cdot }\langle \boldsymbol{v}_{0}\rangle =-\frac {\boldsymbol{\nabla} _{\!x}P}{P_{e}}\boldsymbol{\cdot }(\varpi a\boldsymbol{h}), \end{equation}

where the effective compressibility $\mathsf{C}(\omega ,\varpi )$ is defined as

(2.71) \begin{equation} \mathsf{C}(\omega ,\varpi ) = \frac {1}{P_{e}} \left \langle 1 -\frac {P_{e}}{T_{e}} \frac {\tau ^{(0)}}{P} \right \rangle = \frac {\phi }{P_{e}} \left (1-\frac {\gamma -1}{\gamma }\textrm{j}\omega \rho _{e}c_{\!p}\frac {\theta (\omega ,\varpi )}{\phi \kappa }\right )\!. \end{equation}

Here, we have used the thermodynamic identity $({(\gamma -1)}/{\gamma })\rho _{e}c_{\!p}=({P_{e}}/{T_{e}})$ , where $\gamma$ is the adiabatic constant of the saturating fluid, and it is recalled that $\theta (\omega ,\varpi )$ is given by (2.65).

In summary, the macroscopic description of acoustic wave propagation in permeable media composed of rotating cylindrical inclusions is given by (2.57) and (2.70). These correspond to a macroscopic fluid flow constitutive law and a macroscopic mass balance equation, respectively. This description is fundamentally different from that of classical permeable media and constitutes a central result of this paper. On the one hand, the fluid flow constitutive law has the same mathematical form as that of classical permeable media, but the effective parameter, namely the dynamic permeability (or conductivity), is affected by convective effects induced by the permanent velocity field generated by the rotating inclusions, in addition to the classical viscous and dynamic inertial effects. On the other hand, the spatial mean of such permanent velocity field explicitly appears in the macroscopic mass balance equation (see the term on the right-hand side of (2.70)). This is not the case for classical permeable media where such a term does not exist. As will be analysed in the next section, the presence of the permanent fluid flow and the morphology of the material lead to rather atypical acoustic behaviour.

2.3.4. Effective wavenumber, speed of sound and admittance

Eliminating the averaged oscillatory fluid velocity in (2.70) with the aid of (2.59), the following equation for the acoustic pressure $P$ is derived:

(2.72) \begin{equation} \boldsymbol{\nabla} _{\!x} \boldsymbol{\cdot }(- {\unicode{x1D646}} \boldsymbol{\cdot }\boldsymbol{\nabla} _{\!x} P) + \textrm{j} \omega P \mathsf{C} + \frac {\varpi a}{P_{e}} \boldsymbol{\nabla} _{\!x} P \boldsymbol{\cdot }\boldsymbol{h}(\varpi ) = 0. \end{equation}

Here, it is pertinent to mention that for all two-dimensional (2-D) materials with cylindrical inclusions, one has that $\boldsymbol{h} = \boldsymbol{0}$ if their cells have two (or more) symmetry planes.

To determine the in-plane effective wave vector $\boldsymbol{k}$ , let us consider a plane wave propagating in the medium, that is, $P(\boldsymbol{x},\omega , \varpi )=|A| e^{-\textrm{j}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}} e^{\textrm{j} \omega t}$ , where $\boldsymbol{k}= \text{k} \cos (\varTheta ) {\boldsymbol{e}}_{1} + \text{k} \sin (\varTheta ) {\boldsymbol{e}}_{2}$ , with $\varTheta$ the incidence angle. Inserting $P(\boldsymbol{x},\omega , \varpi )$ into (2.72) leads to a quadratic equation for $\text{k}$ whose solution reads as

(2.73) \begin{equation} \text{k}_{\pm }(\omega ,\varpi ,\varTheta ) = - \frac {\mathcal{B}}{2\mathcal{A}} \pm \frac {\sqrt {\mathcal{B}^{2}-4\mathcal{A} \mathcal{Q} }}{2\mathcal{A} } = - \frac {\mathcal{B}}{2\mathcal{A}} \pm \sqrt { \left ( - \frac {\mathcal{B}}{2 \mathcal{A}} \right )^{2} - \frac {\mathcal{Q}}{\mathcal{A} } }, \end{equation}

where $\mathcal{A}$ , $\mathcal{B}$ and $\mathcal{Q}$ are given by

(2.74) \begin{equation} \mathcal{A}(\omega ,\varpi ,\varTheta ) = K_{11} \cos ^{2}(\varTheta ) + K_{22} \sin ^{2}(\varTheta ) + (K_{12}+K_{21}) \sin {(\varTheta )} \cos {(\varTheta )}, \end{equation}

(2.75) \begin{equation} \mathcal{B}(\omega ,\varpi ,\varTheta ) = - \textrm{j} \frac {\varpi a}{P_{e}} (\cos (\varTheta ) h_{1} + \sin (\varTheta ) h_{2}), \end{equation}

(2.76) \begin{equation} \mathcal{Q}(\omega ,\varpi ) = \textrm{j} \omega \mathsf{C}. \end{equation}

The effective speed of sound $\mathbb{C}$ in the medium is then given by

(2.77) \begin{equation} \mathbb{C} = \frac {\omega }{\text{k}_{\pm }(\omega ,\varpi ,\varTheta )}, \end{equation}

while inserting $P(\boldsymbol{x},\omega , \varpi )$ into (2.57) leads to the effective characteristic admittance vector of the medium $\boldsymbol{\mathcal{Y}} = (\mathcal{Y}_{\!x}, \mathcal{Y}_{\!y})$ . Its components are given by

(2.78) \begin{eqnarray} \mathcal{Y}_{\!x}(\omega ,\varpi ,\varTheta ) = \textrm{j} \text{k}_{\pm } [K_{11} \cos {(\varTheta )} + {K_{12}} \sin {(\varTheta )}], \\[3pt] \nonumber \mathcal{Y}_{\!y}(\omega ,\varpi ,\varTheta ) = \textrm{j} \text{k}_{\pm } [K_{21} \cos {(\varTheta )} + K_{22} \sin {(\varTheta )}]. \end{eqnarray}

3.2.1. Visco-inertial regime

For negligible convective effects, i.e. $\varpi \to 0$ , (3.3) reduces to the dynamic conductivity of classical permeable media. Thus, the transition from viscosity-dominated to inertia-dominated fluid flow is classically given by the viscous characteristic frequency $\omega _{v}$ . This is implicitly defined as the frequency at which the magnitudes of the viscous and dynamic inertial powers are balanced, i.e.

(3.13) \begin{equation} |\widetilde {\mathcal{V}}(\omega _{v},0)| = |\omega _{v} \widetilde {\mathcal{M}}_{v}(\omega _{v},0)|. \end{equation}

A classical estimation of $\omega _{v}$ is then given by (Johnson et al. Reference Johnson, Koplik and Dashen1987; Venegas & Boutin Reference Venegas and Boutin2017)

(3.14) \begin{equation} \omega _{v} = \frac {\phi \eta }{ \mathcal{K}_{0} \rho _{e} \alpha _{\infty }} = \frac {\phi }{ K_{0}(\varpi =0) \rho _{e} \alpha _{\infty }}, \end{equation}

where $\mathcal{K}_{0}$ is the static viscous permeability in the absence of rotation and $\alpha _{\infty }$ is the tortuosity (as defined by Johnson et al. (Reference Johnson, Koplik and Dashen1987)).

The low- and high-frequency asymptotic behaviour of the dynamic conductivity is then classically given by (Allard & Atalla Reference Allard and Atalla2009; Auriault et al. Reference Auriault, Boutin and Geindreau2009)

(3.15) \begin{equation} K_{0} (\varpi =0) = K (\omega \ll \omega _{v},\varpi =0), \end{equation}

(3.16) \begin{equation} K_{ \infty } ( \varpi =0) = K (\omega \gg \omega _{v},\varpi =0) = - \textrm{j} \omega \widetilde {\mathcal{M}}_{v}(\omega \gg \omega _{v},\varpi =0) = \frac {\phi }{ \textrm{j} \omega \rho _{e} \alpha _{\infty }}. \end{equation}

3.2.2. Visco-convective regime

At low frequencies, i.e. $\omega \to 0$ , dynamic inertial effects can be neglected in a first approximation. The flow therefore approaches a quasi-steady laminar regime. In this limit, the fluid velocity becomes asymptotically real-valued, i.e. $\boldsymbol{v} \to \boldsymbol{v}_{R} = \text{Re} {(\boldsymbol{v})}$ , and the dynamic conductivity tends to (for a unit forcing term)

(3.17) \begin{equation} K \approx \widetilde {\mathcal{V}} + \widetilde {\mathcal{C}}_{vR} \quad \mbox{i.e.} \quad K \approx \eta \langle \boldsymbol{\nabla} _{\!y} \boldsymbol{v}_{R} : \boldsymbol{\nabla} _{\!y} \boldsymbol{v}_{R} \rangle + \rho _{e} \langle [ (\boldsymbol{v}_{R} \boldsymbol{\cdot }\boldsymbol{\nabla} _{\!y}) \boldsymbol{v}_{0} ] \boldsymbol{\cdot }\boldsymbol{v}_{R} \rangle . \end{equation}

This expression reveals the following dimensionless number

(3.18) \begin{equation} \mathcal{N}^{v}_{c} = \frac { |\widetilde {\mathcal{V}}|}{|\widetilde {\mathcal{C}}_{vR}|} = \frac { |\eta \langle \boldsymbol{\nabla} _{\!y} \boldsymbol{v}_{R} : \boldsymbol{\nabla} _{\!y} \boldsymbol{v}_{R} \rangle |}{|\rho _{e} \langle [ (\boldsymbol{v}_{R} \boldsymbol{\cdot }\boldsymbol{\nabla} _{\!y}) \boldsymbol{v}_{0} ] \boldsymbol{\cdot }\boldsymbol{v}_{R} \rangle |} = \mathcal{O}\left (\frac {\eta / \mathcal{K}_{0}}{\rho _{e} v_{0c} / (\ell -a) } \right ) = \mathcal{O}\left (\frac {\eta }{\rho _{e} \varpi \mathcal{K}_{0}} \frac {\ell - a}{a} \right )\!, \end{equation}

which shows that viscosity effects dominate over the convective effects when

(3.19) \begin{equation} \mathcal{N}^{v}_{c} \gt 1, \quad \mbox{i.e.} \quad \varpi \lt \varpi _{c} = \mathcal{O} \left ( \frac {\eta }{\rho _{e} \mathcal{K}_{0}} \frac {\ell - a}{a} \right )\!, \end{equation}

while viscosity effects play a negligible role in the acoustic flow when these inequalities are reversed. Moreover, a characteristic rotatory Mach number value, for which $\mathcal{N}^{v}_{c} = 1$ , i.e. viscosity and convective effects are of the same order, is identified as $M_{c} = \mathcal{O}((\ell -a) / \rho _{e} c_{0} K_{0} )$ .

An asymptotic analysis is now performed to derive an approximated expression for the fluid flow velocity at negligible frequency. We introduce the small parameter $ \alpha = (({\rho _{e} \varpi \mathcal{K}_{0}})/{\eta }) \ll 1$ , which shows that viscosity effects dominate over convective ones. Hence, the weak formulation (B1)–(B6) reduces to

(3.20) \begin{align} \forall {\boldsymbol{w}}\in \mathcal{H}^{1},\quad \langle \boldsymbol{\nabla} _{\!y} \boldsymbol{v} : \boldsymbol{\nabla} _{\!y} \boldsymbol{w} \rangle + \frac {\alpha }{\mathcal{K}_{0} \varpi } \langle [ (\boldsymbol{v}_{0} \boldsymbol{\cdot }\boldsymbol{\nabla} _{\!y}) \boldsymbol{v} + (\boldsymbol{v} \boldsymbol{\cdot }\boldsymbol{\nabla} _{\!y})\boldsymbol{v}_{0} ] \boldsymbol{\cdot }{\boldsymbol{w}} \rangle = - \frac {1}{\eta } \boldsymbol{\nabla} _{\!x} P \boldsymbol{\cdot }\langle \boldsymbol{w} \rangle . \end{align}

The fluid velocity is then written in the form $\boldsymbol{v} = \boldsymbol{v}^{[0]} +\alpha \boldsymbol{v}^{[1]}$ . At leading order, one obtains $\forall {\boldsymbol{w}}\in \mathcal{H}^{1}$ , $\mathcal{V}(\boldsymbol{v}^{[0]}, \boldsymbol{w}) = b(\boldsymbol{w})$ . This corresponds to the classical static fluid flow problem in a permeable medium with immobile inclusions. Its solution reads $\boldsymbol{v}^{[0]} = - \hat {{\unicode{x1D660}}}_{0}(y) \boldsymbol{\cdot }\boldsymbol{\nabla} _{\!x} P / \eta$ .

At the next order, the problem for $\boldsymbol{v}^{[1]}$ becomes $\forall {\boldsymbol{w}}\in \mathcal{H}^{1}$ , $\langle \boldsymbol{\nabla} _{\!y} \boldsymbol{v}^{[1]} : \boldsymbol{\nabla} _{\!y} \boldsymbol{w} \rangle = - ({1}/({\mathcal{K}_{0} \varpi })) \langle [ (\boldsymbol{v}_{0} \boldsymbol{\cdot }\boldsymbol{\nabla} _{\!y}) \boldsymbol{v}^{[0]} + (\boldsymbol{v}^{[0]} \boldsymbol{\cdot }\boldsymbol{\nabla} _{\!y})\boldsymbol{v}_{0}] \boldsymbol{\cdot }{\boldsymbol{w}} \rangle$ . Since the forcing term depends linearly on $\boldsymbol{v}^{[0]}$ , which in turn is driven by the pressure gradient, the spatial average of the fluid velocity corrector can be written as $\langle \boldsymbol{v}^{[1]} \rangle = {\unicode{x1D668}}_{c} \boldsymbol{\cdot }\boldsymbol{\nabla} _{\!x} P / \eta$ , where ${\unicode{x1D668}}_{c}=\langle \hat {{\unicode{x1D668}}}_{c} (y) \rangle$ is the effective response of the medium to the forcing convective term and its magnitude is $\mathcal{O}(\mathcal{K}_{0})$ . The spatially averaged velocity is then given by $\langle \boldsymbol{v} \rangle = \langle \boldsymbol{v}^{[0]} \rangle + \alpha \langle \boldsymbol{v}^{[1]} \rangle$ . Considering a given flow direction (e.g. x-direction when the system is excited in the same direction), one obtains

(3.21) \begin{equation} \langle \boldsymbol{v} \rangle \approx - \frac { \mathcal{K}_{0}}{\eta _{\textit{eff}}} \boldsymbol{\nabla} _{\!x} P = - K(\omega \to 0, \varpi ) \boldsymbol{\nabla} _{\!x} P \quad \mbox{with} \quad \eta _{\textit{eff}} = \eta \left (1 + \frac {\rho _{e} \varpi s_{c} }{\eta } \right )\!. \end{equation}

This result shows that the effective viscosity of the fluid is dependent on the rotation of the inner inclusions. However, the following dynamic conductivity ratio is obtained:

(3.22) \begin{equation} \frac {K (\omega \to 0, \varpi )}{K (\omega \to 0, \varpi =0)} = \frac {1}{1 + \frac {\rho _{e} \varpi s_{c} }{\eta }}, \end{equation}

which shows that it typically decreases as the angular rotation frequency $\varpi$ increases.

We now investigate asymptotically the regime where rotation-induced convective transport dominates viscous dissipation, while dynamic inertial effects are neglected. This corresponds to $\beta = \alpha ^{-1} = ({\eta }/({\rho _{e} \varpi \mathcal{K}_{0}})) \ll 1$ . In this regime, the weak formulation (B1)–(B6) becomes

(3.23) \begin{align} \forall {\boldsymbol{w}}\in \mathcal{H}^{1},\quad \beta \langle \boldsymbol{\nabla} _{\!y} \boldsymbol{v} : \boldsymbol{\nabla} _{\!y} \boldsymbol{w} \rangle + \frac {1}{\varpi \mathcal{K}_{0}} \langle [ (\boldsymbol{v}_{0} \boldsymbol{\cdot }\boldsymbol{\nabla} _{\!y}) \boldsymbol{v} + (\boldsymbol{v} \boldsymbol{\cdot }\boldsymbol{\nabla} _{\!y})\boldsymbol{v}_{0} ] \boldsymbol{\cdot }{\boldsymbol{w}} \rangle = - \frac {\boldsymbol{\nabla} _{\!x} P \boldsymbol{\cdot }\langle \boldsymbol{w} \rangle }{\rho _{e} \varpi \mathcal{K}_{0}}. \end{align}

The convective operator is now the principal part, whereas viscosity appears as a corrector. Following the same above-mentioned procedure, but considering $\boldsymbol{v} = \boldsymbol{v}^{[0]} + \beta \boldsymbol{v}^{[1]}$ , one obtains $\forall {\boldsymbol{w}}\in \mathcal{H}^{1}$ , $\mathcal{C}_{v}(\boldsymbol{v}_{0};\boldsymbol{v}^{[0]},{\boldsymbol{w}}) = b(\boldsymbol{w})$ (see (B6) and (B3)). Assuming the existence of the solution and taking the spatial average, one obtains $\langle \boldsymbol{v}^{[0]} \rangle = - \boldsymbol{\psi }_{c} \boldsymbol{\cdot }\boldsymbol{\nabla} _{\!x} P / (\rho _{e} \varpi )$ , where $\boldsymbol{\psi }_{c} = \langle \hat {\boldsymbol{\psi }}_{c}(y) \rangle$ is a dimensionless second-rank tensor. It is emphasised that the derived fluid flow constitutive law is purely convective and depends on $\rho _{e} \varpi$ . Letting $\boldsymbol{w} = \boldsymbol{v}^{[0]} = - \hat {\boldsymbol{\psi }}_{c} \boldsymbol{\cdot }\boldsymbol{G} / (\rho _{e} \varpi )$ , where $\boldsymbol{G} = \boldsymbol{\nabla} _{\!x}P$ , one obtains $\rho _{e} \varpi \mathcal{C}_{v}(\boldsymbol{v}_{0};\boldsymbol{v}^{[0]},\boldsymbol{v}^{[0]}) = b(\boldsymbol{v}^{[0]})$ . Using (B3), (B6) and (B12), this equation can be rewritten, by noting that $\mathfrak{C}(\boldsymbol{v}_{0};\boldsymbol{v}^{[0]},\boldsymbol{v}^{[0]}) = 0$ , as $\rho _{e}^{2} \varpi ^{2} \mathfrak{C}(\boldsymbol{v}^{[0]}; ({\boldsymbol{v}_{0}}/{\varpi }) ,\boldsymbol{v}^{[0]}) = \boldsymbol{G} \boldsymbol{\cdot }\boldsymbol{\psi }_{c} \boldsymbol{\cdot }\boldsymbol{G}$ . Recalling that $\boldsymbol{v}_{0} = a \varpi \hat {\boldsymbol{h}}$ (see (2.17)) and defining the dimensionless spatial variable $\boldsymbol{\xi } = \boldsymbol{y}/a$ , one obtains $ \langle [ ( (\hat {\boldsymbol{\psi }}_{c} \boldsymbol{\cdot }\boldsymbol{G}) \boldsymbol{\cdot }\boldsymbol{\nabla} _{\xi }) \hat {\boldsymbol{h}} ] \boldsymbol{\cdot }(\hat {\boldsymbol{\psi }}_{c} \boldsymbol{\cdot }\boldsymbol{G}) \rangle = \boldsymbol{G} \boldsymbol{\cdot }\boldsymbol{\psi }_{c} \boldsymbol{\cdot }\boldsymbol{G}$ , which expressed in indicial notation reads $ G_{k} G_{l} \langle (\hat {\psi }_{c})_{jk}( {\partial \hat {h}_{i} }/{ \partial \xi _{\!j}}) (\hat {\psi }_{c})_{il} \rangle = G_{k} (\psi _{c})_{kl} G_{l}$ . Since the problem for $\hat {\boldsymbol{h}}$ , given by (C1)–(C3), can be reformulated in terms of $\boldsymbol{\xi }$ after introducing the rescaled term $\tilde {\pi }_{0} = a \hat {\pi }_{0}$ , the left-hand side of the previous expression depends only on dimensionless quantities and is therefore independent of the local characteristic size $a$ . In turn, the quadratic form $G_{k} (\psi _{c})_{kl} G_{l}$ is independent of $a$ for every forcing vector $\boldsymbol{G}$ . It then follows that the homogenised response associated with $\boldsymbol{\psi }_c$ , namely $\boldsymbol{G} \boldsymbol{\cdot }\boldsymbol{\psi }_{c} \boldsymbol{\cdot }\boldsymbol{G}$ , is entirely determined by cell problems formulated in terms of $\boldsymbol{\xi }$ . Consequently, the effective tensor $\boldsymbol{\psi }_c$ is characterised solely by dimensionless cell problems and therefore does not depend on the absolute local characteristic size. This result is consistent with the estimate $|\boldsymbol{\psi }_{c}| = \psi _{c} = \mathcal{O}((\ell -a)/a)$ , which only characterises the order of magnitude of $\boldsymbol{\psi }_{c}$ , denoted by $\psi _{c}$ , and does not, by itself, imply independence from the local characteristic size.

The corrector $\boldsymbol{v}^{[1]}$ due to viscosity satisfies $\forall {\boldsymbol{w}}\in \mathcal{H}^{1}$ , $\mathcal{C}_{v}(\boldsymbol{v}_{0};\boldsymbol{v}^{[1]},{\boldsymbol{w}}) = -(({\rho _{e} \varpi \mathcal{K}_{0}})/{\eta }) \mathcal{V}(\boldsymbol{v}^{[0]},\boldsymbol{w})$ . Assuming the existence of the solution, its spatial average can be expressed as $\langle \boldsymbol{v}^{[1]} \rangle = {\unicode{x1D668}}_{v} \boldsymbol{\cdot }\boldsymbol{\nabla} _{\!x} P/\eta$ , with $ {\unicode{x1D668}}_{v} = \langle \hat {{\unicode{x1D668}}}_{v}(y) \rangle$ . Following a similar line of argument as for $\boldsymbol{\psi }_{c}$ , the cell problem for the dimensionless tensor $\hat {{\unicode{x1D668}}} = \rho _{e} \varpi \hat {{\unicode{x1D668}}}_{v}/\eta$ , evaluated at the solution $\boldsymbol{w} = \boldsymbol{v}^{[1]}$ , can be expressed as $G_{k} \mathcal{A}_{kl} G_{l} =({\mathcal{K}_{0}}/{ a^{2}}) G_{k} \mathcal{B}_{kl} G_{l}$ , where $ \mathcal{A}_{kl} = \langle \hat {s}_{jk} ({\partial \hat {h}_{i} }/{ \partial \xi _{\!j}}) \hat {s}_{il} \rangle$ and $ \mathcal{B}_{kl} = \langle (( {\partial (\hat {\psi }_{c})_{ik}})/{\partial \xi _{m}}) ({\partial \hat {s}_{il}}/{\partial \xi _{m}})) \rangle$ . Both tensors $\boldsymbol{\mathcal{A}}$ and $\boldsymbol{\mathcal{B}}$ are defined entirely in terms of the dimensionless variable $\boldsymbol{\xi }=\boldsymbol{y}/a$ and dimensionless fields. Therefore, they depend only on the dimensionless microstructure. Moreover, the ratio $\mathcal{K}_{0}/a^{2}$ is well known to be independent of the local characteristic size $a$ . Consequently, the cell problem defining $\hat {{\unicode{x1D668}}}$ is independent of $a$ , which is also the case for its solution. Since the effective tensor ${\unicode{x1D668}}_{v}$ is given by $ {\unicode{x1D668}}_{v} = \langle \hat {{\unicode{x1D668}}}_{v} \rangle = \eta \langle \hat {{\unicode{x1D668}}}\rangle / \rho _{e} \varpi$ , it follows that ${\unicode{x1D668}}_{v}$ depends only on the dimensionless microstructure and is therefore independent of the absolute local characteristic size. This result is consistent with the estimate $|{\unicode{x1D668}}_{v} | = \mathcal{O}(\psi _{c} \eta / \rho _{e} \varpi )$ , which characterises the order of magnitude of ${\unicode{x1D668}}_{v}$ , but does not, by itself, imply independence from the local characteristic size. Thus, the fluid velocity is approximated by $\langle \boldsymbol{v} \rangle = \langle \boldsymbol{v}^{[0]} \rangle + \beta \langle \boldsymbol{v}^{[1]} \rangle$ . Considering a given flow direction, one then has

(3.24) \begin{equation} \langle \boldsymbol{v} \rangle \approx - \frac { \psi _{c}}{\rho _{e} \varpi } \left ( 1 - \frac {1}{\psi _{c}} \frac {s_{v}}{\mathcal{K}_{0}} \right ) \boldsymbol{\nabla} _{\!x} P = - K( \omega \to 0, \varpi ) \boldsymbol{\nabla} _{\!x} P. \end{equation}

This fluid flow constitutive law is of convective nature and shows that $K (\omega \to 0, \varpi ) = \mathcal{O}((\ell -a)/ \rho _{e} \varpi a)$ , which is fundamentally different from the dynamic conductivity of classical media. Here, the transport mechanism is driven by rotation-induced convection rather than by viscosity.

The following dynamic conductivity ratio, for the case of strong convective effects and weak viscosity effects, is then obtained,

(3.25) \begin{equation} \frac {K(\omega \to 0 , \varpi )}{K(\omega \to 0, \varpi =0)}=\frac {\eta \psi _{c}}{\rho _{e}\varpi \mathcal{K}_{0}}\left (1-\frac {s_v}{\psi _{c}\mathcal{K}_{0}}\right )\!. \end{equation}

This equation shows that a smaller dynamic conductivity ratio is obtained as the rotation frequency increases. Furthermore, materials with larger pores (i.e. higher $\mathcal{K}_{0}$ ) would tend to exhibit a smaller dynamic conductivity ratio for a given $\varpi$ .

It is emphasised that for sufficiently high rotation frequencies, the asymptotic developments derived earlier are no longer valid because the viscous term, which decreases as $\varpi$ increases, and the convective terms are comparable in magnitude. Indeed, when the convective power exceeds the viscous dissipated power, the dynamic conductivity can take negative values, resulting in oscillatory fluid flow directed towards increasing pressure gradients. This rather atypical behaviour, illustrated numerically later, resembles the fluid flow behaviour reported in active liquid suspensions where bacterial motion can overcome the viscous dissipation (López et al. Reference López, Gachelin, Douarche, Auradou and Clément2015). The latter phenomenon has been attributed by López et al. (Reference López, Gachelin, Douarche, Auradou and Clément2015) to a ‘negative apparent viscosity’, while in the present work, the physical origin is attributed to convective power counteracting the viscous dissipated power.

3.2.3. Convective-inertial regime

The influence of convective and dynamic inertial effects is now investigated by considering that viscosity effects have small or negligible influence on the acoustic flow. First, we consider the low-frequency regime and consequently assume that dynamic inertial effects remain small, while convective effects dominate over viscosity effects. Thus, one has that

(3.26) \begin{equation} K \approx \widetilde {\mathcal{C}}_{vR} - \textrm{j} \omega \widetilde {\mathcal{M}}_{v}. \end{equation}

This yields the following dimensionless number:

(3.27) \begin{equation} \mathcal{N}_{i}^{c} = \frac { |\textrm{j} \omega \widetilde {\mathcal{M}}_{v}|}{|\widetilde {\mathcal{C}}_{vR}|} = \mathcal{O} \left ( \frac {\omega }{\varpi } \frac {\ell -a}{a} \right )\!. \end{equation}

Defining $ \omega ^{\star } = ({\textrm{j} \omega }/{\varpi }) \ll 1$ , which means that convective effects dominate over dynamic inertial ones, and looking for an approximate fluid velocity in the form $\boldsymbol{v} = \boldsymbol{v}^{[0]} + \omega ^{\star } \boldsymbol{v}^{[1]}$ , one obtains, at leading order, that the weak formulation (B1)–(B6) becomes $\forall {\boldsymbol{w}}\in \mathcal{H}^{1}$ , $\mathcal{C}_{v}(\boldsymbol{v}_{0};\boldsymbol{v}^{[0]}, {\boldsymbol{w}}) = b( {\boldsymbol{w}})$ . As previously, the spatially averaged solution of this problem is expressed as $\langle \boldsymbol{v}^{[0]} \rangle = - \boldsymbol{\psi }_{c} \boldsymbol{\cdot }\boldsymbol{\nabla} _{\!x} P / \rho _{e} \varpi$ . The next order provides the problem for the corrector due to dynamic inertia, i.e. $\forall {\boldsymbol{w}}\in \mathcal{H}^{1}$ , $ \mathcal{C}_{v}(\boldsymbol{v}_{0};\boldsymbol{v}^{[1]}, {\boldsymbol{w}}) = - \rho _{e} \varpi \langle \boldsymbol{v}^{[0]} \boldsymbol{\cdot }\boldsymbol{w} \rangle$ . The spatially averaged solution can then be expressed as $\langle \boldsymbol{v}^{[1]} \rangle = + \boldsymbol{\psi }_{i} \boldsymbol{\cdot }\boldsymbol{\nabla} _{\!x} P / \rho _{e} \varpi$ , where $\boldsymbol{\psi }_{i} = \mathcal{O}(\boldsymbol{\psi }_{c}) = \mathcal{O}(1)$ . Then, the approximate spatially averaged fluid velocity is expressed, for a given flow direction, as

(3.28) \begin{equation} \langle \boldsymbol{v} \rangle \approx - \frac {\psi _{c}}{\rho _{e} \varpi } \left ( 1 - \frac {\textrm{j} \omega }{\varpi } \frac {\psi _{i}}{\psi _{c}} \right ) \boldsymbol{\nabla} _{\!x} P = - K (\omega \ll \varpi ,\varpi ) \boldsymbol{\nabla} _{\!x} P. \end{equation}

Hence, the dynamic conductivity ratio is given by

(3.29) \begin{equation} \frac {K (\omega \ll \varpi , \varpi )}{K (\omega =0,\varpi =0)} = \frac {\psi _{c}}{\rho _{e} \varpi K_{0} (\varpi =0)} \left ( 1 - \frac {\textrm{j} \omega }{\varpi } \frac {\psi _{i}}{\psi _{c}} \right )\!, \end{equation}

which shows that the corrector increases linearly with frequency.

As the frequency increases, the imaginary part of the convective power is no longer negligible. Recalling that the permanent velocity is real-valued and assuming, without loss of generality, that the macroscopic pressure gradient is of zero phase, the weak formulation (B1)–(B6) can then be decomposed, $\forall {\boldsymbol{w}}\in \mathcal{H}^{1}$ , as

(3.30) \begin{align} - \omega \mathcal{M}_{v}(\boldsymbol{v}_{I} ,\overline {\boldsymbol{w}}) + \mathcal{C}_{v}(\boldsymbol{v}_{0};\boldsymbol{v}_{R},\overline {\boldsymbol{w}}) = b(\overline {\boldsymbol{w}}) \quad \mbox{and} \quad \textrm{j} \omega \mathcal{M}_{v}(\boldsymbol{v}_{R} ,\overline {\boldsymbol{w}}) + \mathcal{C}_{v}(\boldsymbol{v}_{0}; \textrm{j} \boldsymbol{v}_{I},\overline {\boldsymbol{w}}) = 0. \end{align}

From the latter equation, one then has that $\boldsymbol{v}_{R} = - ({1}/{\omega }) [ (\boldsymbol{v}_{0} \boldsymbol{\cdot }\boldsymbol{\nabla} _{\!y}) \boldsymbol{v}_{I} + (\boldsymbol{v}_{I} \boldsymbol{\cdot }\boldsymbol{\nabla} _{\!y})\boldsymbol{v}_{0} ]$ . Inserting this in the left-hand side equation of (3.30), and by evaluating at the solution and considering a given flow direction, one obtains

(3.31) \begin{equation} \textrm{j}\omega \breve {\mathcal{M}}(\omega ,\varpi ) - \frac {\breve {\mathcal{E}}(\omega ,\varpi )}{\textrm{j} \omega } = - \textrm{j} K^{-1}, \end{equation}

with

(3.32) \begin{align} \breve {\mathcal{M}} (\omega ,\varpi ) = \frac {\mathcal{M}_{v}(\boldsymbol{v}_{I} ,\overline {\boldsymbol{v}})}{\langle \boldsymbol{v} \rangle \boldsymbol{\cdot }\langle \overline {\boldsymbol{v}} \rangle } \quad \mbox{and} \quad \breve {\mathcal{E}}(\omega ,\varpi ) = \frac {\mathcal{C}_{v}(\boldsymbol{v}_{0}; [ (\boldsymbol{v}_{0} \boldsymbol{\cdot }\boldsymbol{\nabla} _{\!y}) \boldsymbol{v}_{I} + (\boldsymbol{v}_{I} \boldsymbol{\cdot }\boldsymbol{\nabla} _{\!y})\boldsymbol{v}_{0} ],\overline {\boldsymbol{v}})} {\langle \boldsymbol{v} \rangle \boldsymbol{\cdot }\langle \overline {\boldsymbol{v}} \rangle }. \end{align}

Equation (3.31) reveals a competition between dynamic inertia and convective effects, as in resonance phenomena. Thus, the balance between the convective term and the dynamic inertia of the fluid leads to a pseudo-resonance at a frequency which, in terms of order of magnitude, is estimated as

(3.33) \begin{equation} \omega _{ci} = \sqrt {\frac {\breve {\mathcal{E}}(\omega =\omega _{ci},\varpi )}{\breve {\mathcal{M}}(\omega =\omega _{ci},\varpi )}} = \mathcal{O} \left ( \frac {\varpi }{\sqrt {\alpha _{\infty }}} \frac {a}{\sqrt {\mathcal{K}_{0}}} \right ) = \mathcal{O} \left ( \frac {\varpi }{\sqrt {\alpha _{\infty }}} \frac {a}{(\ell - a)} \right )\!. \end{equation}

It is then clear that the magnitude of the dynamic conductivity takes large values at $\omega = \omega _{ci}$ , i.e. $|K(\omega = \omega _{ci},\varpi )| \to \infty$ . This also means that large averaged oscillatory velocities are observed in response to a finite pressure gradient excitation. Let us now consider the case $ ({1}/{\omega ^{\star }}) = ( {\varpi }/({\textrm{j} \omega })) \ll 1$ , meaning that dynamic inertial effects dominate over convective effects. Proceeding as previously, one has that the leading-order oscillatory velocity field is governed by the following boundary-value problem $\forall {\boldsymbol{w}}\in \mathcal{H}^{1}$ , $\textrm{j} \omega \mathcal{M}(\boldsymbol{v}^{[0]},\boldsymbol{w}) = b( {\boldsymbol{w}})$ , while the corrector $\boldsymbol{v}^{[1]}$ is a solution of $\forall {\boldsymbol{w}}\in \mathcal{H}^{1}$ , $ \mathcal{M}(\boldsymbol{v}^{[1]},\boldsymbol{w}) = -\mathcal{C}_{v}(\boldsymbol{v}_{0}/\varpi ;\boldsymbol{v}^{[0]}, {\boldsymbol{w}})$ . Thus, for a given flow direction, the averaged velocity field is given by

(3.34) \begin{equation} \langle \boldsymbol{v} \rangle = \big\langle \boldsymbol{v}^{[0]} \big\rangle + \frac {\varpi }{\textrm{j} \omega } \big\langle \boldsymbol{v}^{[1]} \big\rangle = - \frac {\phi }{\textrm{j} \omega \rho _{e} \alpha _{\infty }} \left(1 - \frac {\varpi }{\textrm{j} \omega } \psi _{\infty }\right) \boldsymbol{\nabla} _{\!x} P, \end{equation}

where $\psi _{\infty } = \mathcal{O}(a / (\ell - a))$ is a dimensionless effective parameter that accounts for the correction of dynamic inertial effects by convective ones. Equation (3.34) shows by construction that, within the framework of linear long-wavelength sound propagation, convective effects are negligible in the limit $\omega \to \infty$ .

3.2.4. Summary of asymptotic behaviours

Figure 2 provides an illustrative summary of the asymptotic behaviours investigated in §§ 3.2.1–3.2.3. For sufficiently small rotational frequencies $\varpi$ , the fluid flow is dominated by visco-inertial effects, as in classical porous media. When $\varpi \approx \varpi _{c}$ (see (3.19) for its estimation), the flow behaviour is governed by the interaction between visco-inertial terms and cross-convective terms. As $\varpi$ increases further, the flow transitions into a regime in which convection dominates over visco-inertial effects. The influence of the convective terms on the classical Biot frequency $\omega _{v}$ is also apparent: its value increases with increasing $\varpi$ (see red dashed line). An analogous trend is observed for the pseudo-resonance frequency $\omega _{ci}$ , which occurs when the imaginary part of the convective term balances the dynamic inertia of the fluid, and is proportional to $\varpi$ . Finally, for acoustic frequencies $\omega \gg \omega _{ci}$ , the behaviour is dominated by dynamic inertia.

Figure 2.

Illustrative summary of the asymptotic behaviours and theoretical estimates, and their dependence on the rotational angular frequency $\varpi$ and the acoustic angular frequency $\omega$ . The rotational frequency marking the transition from visco-inertial to convection-dominated regimes is denoted $\varpi _{c}$ (see (3.19)). The classical and enriched Biot frequencies are $\omega _{v}$ and $\omega _{B}$ , respectively. The pseudo-resonance frequency is $\omega _{ci}$ . The limit $\omega \rightarrow \infty$ corresponds to a high-frequency regime in which the flow behaviour is purely inertial.

Figure 2. Long description

A 2D coloured illustration displaying the quantitative effect of the rotation of the solid components on the conductivity. The vertical and horizontal axes represent the angular rotation and acoustic wave frequencies, respectively. Three pairs of coloured regions illustrate the possible dynamic regimes: the first pair, shown with a green-to-yellow gradient, represents the visco-inertial behaviour when the rotation frequency is small; the second pair, shown with an emerald-to-orange gradient, represents the visco-convective-inertial behaviour when the rotation frequency is near the critical regime; the last pair, corresponding to the cyan-to-red region, illustrates a convection-dominated flow. Red and blue dashed lines indicate the enriched Biot frequency and the pseudo-resonance frequency, respectively. Finally, a yellow region is superposed on all previous regions at high frequencies, illustrating that the behaviour converges towards its classical regime in the high-frequency limit.

3.2.5. Acoustic Hall effect

In the classical Hall effect (Hall Reference Hall1879), a transverse voltage arises across an electrical conductor when an electric current flows in the presence of a perpendicular magnetic field. A filtration analogue of the Hall effect has been reported for permanent fluid flow in a rotating porous medium (Auriault et al. Reference Auriault, Geindreau and Royer2002), where the permanent fluid flow induced by the rotation of the entire porous medium plays the role of the biasing magnetic field. This generates transverse fluid flow in response to a longitudinal macroscopic pressure gradient. In this work, we identify a similar effect in the oscillatory regime. Here, the permanent velocity field generated by the rotation of the inclusions biases the oscillatory fluid flow driven by an oscillatory pressure gradient applied along a given direction. As a result, oscillatory fluid flow emerges in the perpendicular direction, forming an acoustic analogue of the Hall effect.

Taking the difference between (B19) and (B20) (see also (B2)) yields

(3.35) \begin{eqnarray} K_{pq}(\omega ,\varpi ) - K_{qp}(\omega ,\varpi ) = +\overline {a}(\boldsymbol{v}_{0}(\varpi );\boldsymbol{k}^{p}(\omega ,\varpi ), \overline {\boldsymbol{k}}^{q}(\omega ,\varpi )) \\ \nonumber - \overline {a}(\boldsymbol{v}_{0}(\varpi ); \boldsymbol{k}^{q}(\omega ,\varpi ), \overline {\boldsymbol{k}}^{p}(\omega ,\varpi )), \end{eqnarray}

where $\boldsymbol{k}^{p}$ and ${\boldsymbol{k}}^{q}$ are the velocity fields corresponding to specific excitations, as detailed in Appendix B. Equation (3.35) shows that

(3.36) \begin{equation} K_{pq}(\omega ,\varpi ) = K_{qp}(\omega ,-\varpi ), \end{equation}

provided that

(3.37) \begin{equation} \overline {a}(\boldsymbol{v}_{0}(\varpi );\boldsymbol{k}^{p}(\omega ,\varpi ), \overline {\boldsymbol{k}}^{q}(\omega ,\varpi )) - \overline {a}(\boldsymbol{v}_{0}(-\varpi ); \boldsymbol{k}^{q}(\omega ,-\varpi ), \overline {\boldsymbol{k}}^{p}(\omega ,-\varpi )) = 0. \end{equation}

This condition holds exactly for materials with perfectly symmetric cells, while for some materials with non-symmetric cells, also studied in this work, it holds approximately. Therefore, the acoustic analogue of the Hall effect identified in this work is highly dependent on the morphology of the material.

4.3.1. Influence of the rotation angular frequency $\varpi$

As previously, the material is a staggered array of cylinders with $b=400$ ${\unicode{x03BC}}\textrm{m}$ and $\phi = 0.90$ . Figure 6 shows the magnitude and phase of the normalised dynamic conductivity $K^{\star }(\omega ,\varpi ) = K^{\star }_{11}(\omega ,\varpi ) = |[K(\omega,\varpi)]|_{11}/K_{0}(\varpi=0)$ for several Mach numbers $M_{\!p}$ . In the absence of rotation, the dynamic conductivity exhibits classical visco-inertial behaviour (see black curve). When the cylinders rotate, the low-frequency conductivity is reduced in magnitude as $M_{\!p}$ increases. For very small $M_{\!p}$ , however, the visco-inertial behaviour remains essentially unchanged. In contrast, in this example, for $M_{\!p}$ ranging from $M_{\!p} = 8.7\times 10^{-3}$ to $M_{\!p} = 1.75\times 10^{-2}$ , a peak in the magnitude emerges, while the positive low-frequency conductivity drops by up to nearly two orders of magnitude compared with immobile cylinders. At the Mach number $M_{\!p}=2.33\times 10^{-2}$ (see blue curve), a drastic change in behaviour is observed. A low conductivity magnitude is observed in the low-frequency range and the conductivity phase tends to $\pi$ , indicating that the real part of the dynamic conductivity is negative. This unusual behaviour arises because convective effects dominate over viscosity effects, while both are of comparable magnitude. Specifically, the viscous dissipated power $\widetilde {\mathcal{V}}$ is counteracted by the real part of the convective power $\widetilde {\mathcal{C}}_{vR}$ , leading to oscillatory fluid flow directed towards increasing pressure gradients. This phenomenon is absent in conventional permeable media. As the frequency increases, the fluid flow is controlled by the positive imaginary part of the dynamic conductivity, with its phase tending to $\pi /2$ . Around the pseudo-resonance frequency $\omega _{ci}$ , an abrupt phase shift from $\pi /2$ to $-\pi /2$ occurs, coinciding with the peak of the magnitude. It is also worth mentioning that the increase in $\omega _{ci}$ with $\varpi$ is consistent with the estimate in (3.33).

As shown in figure 5, further increasing $\varpi$ leads to a positive normalised static conductivity ratio, which is preserved in the viscous-convective regime of the dynamic conductivity (see yellow curve in figure 6). Thus, high rotation speeds do not necessarily result in convective dominance over viscous effects in the real part of the dynamic conductivity. Moreover, it is emphasised that the observed alternate behaviour occurs due to the dominance of either convective over viscous effects or vice versa. It is, however, recalled that the results are valid for $M_{\!p}=\mathcal{O}(\varepsilon )$ . We also note that the behaviour of the dynamic conductivity at high frequencies is classical, as discussed in § 3.

The theoretical and numerical results show that interesting phenomena emerge at $M_{\!p} = 2.33\times 10^{-2}$ (see blue curve in figure 6). Hence, it is of fundamental interest to assess the specific influence of viscous, convective and dynamic inertial effects on the normalised dynamic conductivity.

Figure 7 shows the normalised magnitude, phase, and real and imaginary parts of $K_{11}^{\star }(\omega ,\varpi )$ , together with the contributions from the viscous, convective and dynamic inertial terms (see (3.3)). It is clear that classical behaviour is not observed, except at very high frequencies. Thus, visco-convective and convective-inertial regimes can be recognised based on the characteristic frequencies identified in § 3. In the visco-convective regime, i.e. $\omega \ll \omega _{B}$ , the oscillatory flow is governed by both $\widetilde {\mathcal{V}}$ and $\widetilde {\mathcal{C}}_{vR}$ . Viscous dissipation decreases in magnitude due to the rotating motion of the inclusions. Furthermore, $\widetilde {\mathcal{C}}_{vR}$ counteracts $\widetilde {\mathcal{V}}$ , i.e. convective effects dominate viscous effects, and its magnitude is slightly greater than $\widetilde {\mathcal{V}}$ . Hence, a negative $\text{Re} (K(\omega ,\varpi ))$ with small magnitude is observed. It is noted that throughout this regime, such a counteracting effect is determined by $\rho _{e} \mathfrak{C}(\boldsymbol{v}_{R};\boldsymbol{v}_{0},\boldsymbol{v}_{R})$ because $|\mathfrak{C}(\boldsymbol{v}_{R};\boldsymbol{v}_{0},\boldsymbol{v}_{R}) |\gg |\mathfrak{C}(\boldsymbol{v}_{I};\boldsymbol{v}_{0},\boldsymbol{v}_{I})|$ (see also (B15)). As discussed in § 3.2, the transition from the viscous-convective regime to the convective-inertial regime is determined by $\omega _{B}$ . In the convective-inertial regime, i.e. for $\omega \gg \omega _{B}$ , $\omega \widetilde {\mathcal{M}}_{v}$ and $\widetilde {\mathcal{C}}_{vI}$ govern the conductivity behaviour, and the salient feature is that $\widetilde {\mathcal{C}}_{vI}$ dominates over $\omega \widetilde {\mathcal{M}}_{v}$ , resulting in $\text{Im} (K(\omega ,\varpi ))\gt 0$ below the pseudo-resonance frequency $\omega _{ci}$ . It is recalled that $\omega _{ci}$ is implicitly defined as the frequency at which the imaginary part of the convective term is matched by the dynamic inertia of the fluid. For frequencies $\omega \gg \omega _{ci}$ , the classical dynamic inertial regime is reached, which is in agreement with the theoretical developments.

Figure 7.

Normalised (a) magnitude and (b) phase of the dynamic conductivity, and normalised magnitude of the local velocity field for the (c) viscous-convective, (d) inertial-convective and (e) inertial regimes. The real and imaginary parts of $K^{\star }(\omega ,\varpi )$ are shown with yellow lines in panels (f) and (g), respectively, together with the contribution of the viscous, convective and dynamic inertial terms to $K^{\star }(\omega ,\varpi )$ (see (3.3)). The dark (respectively light) grey circles represent the frequencies for which $\text{Re} (K^{\star }(\omega ,\varpi ))$ or $\text{Im} (K^{\star }(\omega ,\varpi ))$ are negative (respectively positive). The dashed magenta line shows the positive-valued contribution of $\mathcal{C}_{vR}$ . The inset in panel (g) shows $|\widetilde {\mathcal{C}}_{vI}| / | \omega \widetilde {\mathcal{M}}_{v}|$ .

Figure 7. Long description

This figure contains seven plots. Plots (a) and (b) show the normalised magnitude and phase of the dynamic conductivity $K_{11}(\omega,\varpi)$, respectively. The vertical axis represents the normalised quantity, and the horizontal axis represents the acoustic wave frequency. Red and blue curves correspond to weak and strong rotational effects, respectively. Plots (c)–(e) show the normalised local velocity field in the pore space for the dynamic regimes highlighted in Fig. 2, in the high-rotation regime. Plots (f) and (g) show the real and imaginary parts of the dynamic conductivity. In these plots, the vertical axis represents the real part and negative of the imaginary part of the normalised conductivity. Black and blue curves represent the contributions associated with the visco-inertial and convective terms, respectively. Yellow lines indicate the magnitude of each contribution, and grey open circles indicate their cumulative effect (e.g. viscous plus convective contributions).

As can be seen from the local visco-convective velocity field plot shown in figure 7(c), when the system is excited in the positive horizontal direction, the dynamic fluid flow is directed towards the vertical direction, whereas above $f_{ci}$ , the fluid flows towards the horizontal direction (see figure 7 e), as in the classical case. The former is an atypical behaviour reminiscent of the Hall effect. To further analyse this analogous behaviour, the cylinders are considered to rotate with permanent Mach number $M_{\!p}=\pm 2.33\boldsymbol{\cdot }10^{-2}$ .

Figure 8 shows the normalised magnitude and phase of the dynamic conductivity tensor. The results confirm that $K_{12}(\omega ,\varpi ) = K_{21}(\omega ,-\varpi )$ , in agreement with the condition identified in § 3.2.5, thereby demonstrating an acoustic analogue of the Hall effect. In other words, the oscillatory fluid flow is redirected due to the bias introduced by the permanent velocity field generated by the rotating cylinders. For cells with in-plane asymmetry, the Hall-like effect is suppressed due to the asymmetric fluid response to different macroscopic excitations, as illustrated in figure 8(c,d).

Figure 8.

Normalised magnitude (left) and phase (right) of the anisotropic conductivity tensor. The cylinders are rotating, at $M_{\!p}=2.33\boldsymbol{\cdot }10^{-2}$ , clockwise (positive $\varpi$ ) or counter-clockwise (negative $\varpi$ ). The cell is symmetric (top) or asymmetric (bottom), with the central cylinder centred at $d^{\star } = 2/3$ . The inset plots show the normalised phase of $K_{21}(\omega ,+\varpi )$ .

Figure 8. Long description

This figure contains two pairs of plots. Each pair shows the normalised magnitude and phase of the dynamic conductivity tensor $|[K(\omega,\varpi)]_{ij}|/K_{0}(\varpi = 0)$. The vertical axis represents the normalised quantity (magnitude or phase), and the horizontal axis represents the acoustic wave frequency. The first pair (top-left and top-right) corresponds to symmetric cells, while the second pair (bottom-left and bottom-right) corresponds to asymmetric cells. Each pair contains four curves (blue, black, red, and gray), corresponding to the four entries of the dynamic conductivity tensor $[K(\omega,\varpi)]_{ij}$. The gray curve corresponds to $[K(\omega,-\varpi)]_{21}$, computed using clockwise rotation. The plots indicate differences in tensor symmetry between symmetric and asymmetric configurations, with a Hall-like response observed in the symmetric case and a reduced effect in the asymmetric case.

4.3.2. Influence of porosity and cell size

It is of interest to evaluate the influence of convective effects on the dynamic conductivity for different porosities and cell sizes. Figure 9 shows the magnitude and phase of ${K^{\star }}(\omega ,\varpi )$ for different porosities and cell sizes $b=400$ ${\unicode{x03BC}}\textrm{m}$ and $b=4$ $\textrm{cm}$ , with a permanent Mach number $M_{\!p}=2.33\times10^{-2}$ . For the smaller cell size ( $b=400$ ${\unicode{x03BC}}\textrm{m}$ ), the atypical acoustic behaviour is favoured for high-porosity materials, whereas in low-porosity materials, visco-inertial effects tend to dominate over convective effects. However, the major effect that an increase in cell size has on the behaviour of $K(\omega ,\varpi )$ is a general shift towards lower frequencies of the characteristic regimes that govern the behaviour of the oscillatory fluid flow. This is in line with the classical behaviour of $K$ , as an increase in cell size yields a progressively lower Biot frequency $\omega _{B}$ . The aforementioned features open up the possibility of tailoring the (pseudo)-resonant behaviour of the material by tuning the cell size and porosity, together with the rotation rate of the cylinders.

Figure 9.

Normalised magnitude (left) and phase (right) of the dynamic conductivity tensor for different porosities and cell sizes. The cylinders rotate at $M_{\!p}=2.33\times 10^{-2}$ . The cell size is (a,b) $b=400$ ${\unicode{x03BC}}\textrm{m}$ and (c,d) $b=4$ $\textrm{cm}$ .

4.4.1. Symmetric cells

Figure 10 shows the normalised magnitude and phase of the effective wavenumber $[k_{c}(\omega ,\varpi )]_{11} = \text{k}(\omega ,\varpi ) \to k_{c}(\omega ,\varpi )$ for permanent Mach numbers $M_{\!p}=2.33 \times 10^{-2}$ and $M_{\!p}=0$ . The cylinder radius and porosity of the array are $a=50$ ${\unicode{x03BC}}\textrm{m}$ and $\phi =0.90$ , respectively. The system is excited in the positive horizontal direction ( $\varTheta =0$ ).

Figure 10.

Normalised (a) magnitude and (b) phase of the effective wavenumber for $M_{\!p}=2.33\times 10^{-2}$ and $M_{\!p}=0$ . The excitation is in the positive horizontal direction. The material cell is symmetric, with cylinder radius $a=50$ ${\unicode{x03BC}}\textrm{m}$ and porosity $\phi =0.90$ . The vertical lines represent characteristic frequencies as indicated in the legend.

Figure 10. Long description

This figure contains two plots showing the magnitude (a) and phase (b) of the effective wave number $k_{c}(\omega,\varpi)$. The vertical axis represents the normalised quantity and the horizontal axis represents the acoustic wave frequency. As in Fig. 7(a,b), red and blue curves correspond to rotation cases characterised by $M_{p} = 0$ and $M_{p} = 2.33 \times 10^{-2}$, respectively. In addition, black and grey vertical dashed lines indicate the enriched Biot and pseudo-resonance frequencies, $f_{B}$ and $f_{ci}$, respectively.

The behaviour of $k_{c}(\omega ,\varpi = 0)$ is classical, with the phase equal to $-\pi /4$ , indicating attenuated progressive waves. For $M_{\!p}=2.33\times 10^{-2}$ , the rotation of the inclusions produces a positive $\text{Im} (k_{c}(\omega ))$ , indicating that waves propagating in the medium are amplified. In the inertial-convective regime, $k_{c}(\omega ,\varpi )$ is purely imaginary, with the phase approaching $\pi /2$ . This indicates that dissipative regressive waves are supported. A phase jump occurs near the frequency $f_{u}$ , with the phase approaching $-\pi /2$ from the third quadrant of the polar chart. Hence, the waves are being attenuated, but propagate backwards. At the pseudo-resonance frequency $f_{ci}$ , the magnitude of $k_{c}$ reaches a minimum, with the phase velocity exceeding the nominal sound speed in air, which indicates supersonic wave propagation. Finally, in the dynamic inertial regime, $k_{c}$ approaches the classical high-frequency limiting value of $\sqrt {\alpha _{\infty }}\omega /c_{0}$ . We recall that this study is conducted in the long-wavelength regime, which means that no scattering effects are accounted for.

4.4.2. Asymmetric cells

To illustrate the effect of cell asymmetry, figure 11 shows the normalised magnitude and phase of the effective wave number for a material with an asymmetric cell, porosity $\phi =0.90$ , cell size $b=400$ ${\unicode{x03BC}}\textrm{m}$ and $M_{\!p}=2.33\times 10^{-2}$ . The central cylinder is centred at $d^{\star } = 2/3$ (see the left-hand side plot in figure 4 b). Two different wave propagation directions are considered: $\varTheta = 0$ and $\varTheta = \pi /2$ . Breaking the cell symmetry reduces the atypical effects observed in symmetric configurations. Consequently, effects such as regressive or amplified waves may not emerge. This is reflected by the phase of $k_{c}$ , which is similar to that for an array of immobile cylinders, except in a frequency region just below the pseudo-resonance (damped) frequency $\omega _{ci}$ (see the peak in the plots in figure 8 at the bottom). However, the rotating motion of the cylinders still affects the magnitude of the wavenumber, producing higher values along one propagation direction than along the other.

Figure 11.

Normalised (a) magnitude and (b) phase of the effective wavenumber for $M_{\!p}=2.33\times 10^{-2}$ and two wave propagation directions. The material cell is asymmetric, with the central cylinder, of radius $a=50$ ${\unicode{x03BC}}\textrm{m}$ , being located at $d^{\star } = 2/3$ , and porosity $\phi =0.90$ .

4.5.1. Symmetric cells

Figure 12 shows the magnitude and phase of the normalised effective admittance vector, i.e. $\rho _{e} c_{0} [\mathcal{Y}(\omega ,\varpi )]_{i}$ , for a material with a symmetric cell of size $b=400$ ${\unicode{x03BC}}\textrm{m}$ , porosity $\phi =0.90$ and $M_{\!p}=2.33\times 10^{-2}$ . In the absence of rotation, $\angle \mathcal{Y}(\omega , \varpi =0)$ is bounded within $[0,\pi /4]$ , which reflects the visco-inertial behaviour of the oscillatory fluid flow. Furthermore, the flow in the direction perpendicular to the applied pressure gradient is negligible (i.e. the non-diagonal entries of the dynamic conductivity are negligible). When the cylinders rotate, however, the transverse admittance becomes orders of magnitude larger than the classical admittance in the longitudinal direction, demonstrating the redirection capability of acoustic waves – a signature of the acoustic Hall effect. The phase of the effective admittance exhibits atypical behaviour, producing negative real and imaginary parts of the admittance (negative acoustic conductance and susceptance, respectively). These are closely linked to the phenomenon of regressive waves and wave amplification supported in the studied media. At high frequencies, the effective admittance along the applied pressure gradient direction approaches the classical inertial limit, i.e. $\mathcal{Y}_{i}(\omega ,\varpi )\rightarrow \phi \sqrt {\alpha _{\infty }}\rho _{e}c_{0}/\alpha _{\infty }$ for $\varTheta =0$ and $i=1$ or $\varTheta =\pi /2$ and $i=2$ .

Figure 12.

Normalised (a) magnitude and (b) phase of the effective admittance for horizontal and vertical wave propagation. The material cell is symmetric, with cylinder radius $a=50$ ${\unicode{x03BC}}\textrm{m}$ and porosity $\phi =0.90$ . The permanent Mach number is $M_{\!p}=2.33\times 10^{-2}$ .

Figure 12. Long description

This figure contains two plots showing the (a) magnitude and (b) phase of the normalised effective admittance. The vertical axis represents the normalised quantity and the horizontal axis represents the acoustic wave frequency. Blue, red, black, and grey curves are shown. The blue and red curves correspond to the horizontal and vertical components of the effective admittance vector for a wave incidence angle $\Theta = 0$. The black and grey curves correspond to the same components but for a wave incidence angle $\Theta = \pi/2$.

Figure 13.

Normalised (a) magnitude and (b) phase of the effective admittance for different propagation directions. The material cell is horizontally symmetric, but vertically asymmetric ( $d^{\star } = 2/3$ ). The cylinder radius is $a=50$ ${\unicode{x03BC}}\textrm{m}$ and the porosity is $\phi =0.90$ , while the permanent Mach number is $M_{\!p}=2.33\times 10^{-2}$ .

4.5.2. Asymmetric cells

Figure 13 shows the magnitude and phase of the effective admittance vector for a material possessing a horizontally symmetric cell of size $b=400$ ${\unicode{x03BC}}\textrm{m}$ and $d^{\star } = 2/3$ (see the left-hand side plot in figure 4 b). The porosity and permanent Mach number are $\phi =0.90$ and $M_{\!p}=2.33\times 10^{-2}$ , respectively. As discussed previously, breaking the local symmetry of the staggered array prevents the emergence of a clear acoustic Hall effect. The fluid response to an applied oscillatory pressure gradient varies significantly for different solicitations. For example, the phase of $\mathcal{Y}_{2}(\omega ,\varpi ,\varTheta =\pi /2)$ shows no atypical behaviour (see grey curve), while an increased transverse admittance, i.e. $\mathcal{Y}_{2}(\omega ,\varpi ,\varTheta =0)$ , is observed over a broad frequency range for a given longitudinal solicitation. These rather atypical behaviours are attributed to the effects of local dynamic convection induced by the rotation of the inclusions.