2.1 Variational principle for two-phase models

The first step in our modelling work aims at deriving a system of conservation laws equipped with a mathematical entropy evolution equation and checking that the convective structure of the system is hyperbolic. Following the lines of Berdichevsky (Reference Berdichevsky2009) for barotropic fluids, of Gavrilyuk & Saurel (Reference Gavrilyuk and Saurel2002) for a system of equations for two compressible fluids and two temperatures and of Caro et al. (Reference Caro, Coquel, Jamet and Kokh2006) for a homogeneous isothermal two-fluid model, we propose to use Hamilton’s principle of stationary action to derive the conservative structure of our two-phase model. Before going any further, we introduce a few notations: if $\boldsymbol{a}$ and $\boldsymbol{b}$ are two column vectors whose components are $a_{i}$ and $b_{i}$ , $1\leqslant i\leqslant d$ , then $\boldsymbol{a}^{T}$ is a row vector, $\boldsymbol{a}^{\text{T}}\boldsymbol{b}=\sum _{i}a_{i}b_{i}$ is a real number and $\boldsymbol{a}\boldsymbol{b}^{\text{T}}$ is a square matrix of size $d$ with $(\boldsymbol{a}\boldsymbol{b}^{\text{T}})_{i,j}=a_{i}b_{j}$ . If $\boldsymbol{x}\in \mathbb{R}^{d}\mapsto A$ is a field of square matrices of size $d$ then $\text{div}(\unicode[STIX]{x1D608})$ is a column vector of size $d$ , where $\text{div}(\unicode[STIX]{x1D63C})_{i}=\sum _{j}\unicode[STIX]{x2202}_{x_{j}}\unicode[STIX]{x1D608}_{ij}$ . Finally, we note by $D_{t}(\cdot )=\unicode[STIX]{x2202}_{t}(\cdot )+\boldsymbol{u}^{\text{T}}\unicode[STIX]{x1D735}(\cdot )$ the material derivative.

Now, we need to define the kinetic energy $E_{kin}$ and the potential energy $E_{pot}$ that are involved in the conservative transformations of our system. We postulate that the specific potential energy of our system is the free energy $f$ . Also, following Gavrilyuk & Saurel (Reference Gavrilyuk and Saurel2002), a key element of our study consists in considering a two-scale kinetic energy composed of a bulk kinetic energy $\unicode[STIX]{x1D70C}|\boldsymbol{u}|^{2}/2$ and a small-scale kinetic energy $\unicode[STIX]{x1D708}(D_{t}\unicode[STIX]{x1D6FC})^{2}/2$ , where $\unicode[STIX]{x1D708}>0$ is a function of the volume fraction $\unicode[STIX]{x1D6FC}$ only and whose physical interpretation and characterization is given later on. This allows a finer representation of general two-phase flows and the consequences of such a hypothesis will be further discussed in §§ 3 and 4.

Now, the Lagrangian ${\mathcal{L}}=E_{kin}-E_{pot}$ of our system is fully defined, namely,

(2.6) $$\begin{eqnarray}\displaystyle {\mathcal{L}}(\boldsymbol{u},\unicode[STIX]{x1D70C},Y,\unicode[STIX]{x1D6FC},D_{t}\unicode[STIX]{x1D6FC})={\textstyle \frac{1}{2}}\unicode[STIX]{x1D70C}|\boldsymbol{u}|^{2}+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D708}(D_{t}\unicode[STIX]{x1D6FC})^{2}-\unicode[STIX]{x1D70C}f(\unicode[STIX]{x1D70C},Y,\unicode[STIX]{x1D6FC}). & & \displaystyle\end{eqnarray}$$

Let $\mathscr{V}(t)$ be the volume occupied by a portion of the fluid at time $t\in [t_{0},t_{1}]$ and $\unicode[STIX]{x1D6FA}=\{(\boldsymbol{x},t)\in \mathbb{R}^{3}\times [t_{0},t_{1}]~|~\boldsymbol{x}\in \mathscr{V}(t)\}$ be the subset of all space–time points covered by $\mathscr{V}(t)$ during $t_{0}\leqslant t\leqslant t_{1}$ . The space variable $\boldsymbol{X}\in \mathscr{V}(t_{0})$ denotes the Lagrangian coordinates associated with the reference frame at instant $t=t_{0}$ . If $(\boldsymbol{X},t)\mapsto \unicode[STIX]{x1D753}$ is the mapping that gives the position $\unicode[STIX]{x1D753}(\boldsymbol{X},t)$ at instant $t$ of a fluid element that was located at $\boldsymbol{X}$ at time $t=t_{0}$ , then obviously $\unicode[STIX]{x1D6FA}=\{(\unicode[STIX]{x1D753}(\boldsymbol{X},t),t)~|~\boldsymbol{X}\in \mathscr{V}(t_{0}),~t_{0}\leqslant t\leqslant t_{1}\}$ .

From a pure Eulerian point of view, a transformation of the medium is fully characterized by the fields $(\boldsymbol{x},t)\mapsto (\unicode[STIX]{x1D70C},\boldsymbol{u},Y,\unicode[STIX]{x1D6FC})$ . Equivalently, we can say that a transformation of the medium is fully characterized by the Eulerian fields $(\boldsymbol{x},t)\mapsto (Y,\unicode[STIX]{x1D6FC})$ and the Lagrangian mapping $(\boldsymbol{X},t)\mapsto \unicode[STIX]{x1D753}$ under the hypothesis that $\unicode[STIX]{x1D753}$ is compliant with the mass conservation.

If $(\boldsymbol{x},t)\mapsto (Y,\unicode[STIX]{x1D6FC})$ and $(\boldsymbol{X},t)\mapsto \unicode[STIX]{x1D753}$ is a given transformation of the medium, we consider a family of transformations $(\boldsymbol{x},t,\unicode[STIX]{x1D701})\mapsto (\widehat{Y},\widehat{\unicode[STIX]{x1D6FC}})$ and $(\boldsymbol{X},t,\unicode[STIX]{x1D701})\mapsto \widehat{\unicode[STIX]{x1D753}}$ parametrized by $\unicode[STIX]{x1D701}$ which lies in the vicinity of $0$ , such that:

1. (i) $(\widehat{Y},\widehat{\unicode[STIX]{x1D6FC}})(\boldsymbol{x},t,\unicode[STIX]{x1D701}=0)=(Y,\unicode[STIX]{x1D6FC})(\boldsymbol{x},t)$ and $\widehat{\unicode[STIX]{x1D753}}(\boldsymbol{X},t,\unicode[STIX]{x1D701}=0)=\unicode[STIX]{x1D753}(\boldsymbol{X},t)$ ;

2. (ii) $\widehat{Y}$ and $\widehat{\unicode[STIX]{x1D753}}$ verify the mass and partial mass conservation (2.4) and (2.5), for $\unicode[STIX]{x1D701}$ close to $0$ .

We adopt the classic definition of the infinitesimal transformations that act on the medium by introducing the infinitesimal displacement $(\boldsymbol{x},t)\mapsto \unicode[STIX]{x1D743}$ , where

(2.7) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D743}(\unicode[STIX]{x1D753}(\boldsymbol{X},t),t)=\left({\displaystyle \frac{\unicode[STIX]{x2202}\widehat{\unicode[STIX]{x1D753}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}}}\right)_{\boldsymbol{X},t}(\boldsymbol{X},t,\unicode[STIX]{x1D701}=0), & & \displaystyle\end{eqnarray}$$

and by setting for any Eulerian field $(\boldsymbol{x},t,\unicode[STIX]{x1D701})\mapsto \widehat{b}$

(2.8) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}b(\boldsymbol{x},t)=\left({\displaystyle \frac{\unicode[STIX]{x2202}\widehat{b}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}}}\right)_{\boldsymbol{x},t}(\boldsymbol{x},t,\unicode[STIX]{x1D701}=0). & & \displaystyle\end{eqnarray}$$

Using the lines of Gavrilyuk & Saurel (Reference Gavrilyuk and Saurel2002), Berdichevsky (Reference Berdichevsky2009), Gavrilyuk (Reference Gavrilyuk, dell’Isola and Gavrilyuk2012), our hypotheses provide the infinitesimal variations of $\unicode[STIX]{x1D70C}$ , $Y$ , $u$ and $D_{t}\unicode[STIX]{x1D6FC}$ . We obtain indeed that

(2.9a-c ) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70C}=-\text{div}(\unicode[STIX]{x1D70C}\unicode[STIX]{x1D743}),\quad \unicode[STIX]{x1D6FF}\boldsymbol{u}=D_{t}\unicode[STIX]{x1D743}-(\unicode[STIX]{x1D743}^{\text{T}}\unicode[STIX]{x1D735})\boldsymbol{u},\quad \unicode[STIX]{x1D6FF}Y=-(\unicode[STIX]{x1D743}^{\text{T}}\unicode[STIX]{x1D735})Y,\end{eqnarray}$$

and

(2.9d ) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}(D_{t}\unicode[STIX]{x1D6FC})=D_{t}(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FC})+(\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC})^{\text{T}}[D_{t}\unicode[STIX]{x1D743}-\text{div}(\boldsymbol{u}\unicode[STIX]{x1D743}^{\text{T}})].\end{eqnarray}$$

Now, we can define the Hamiltonian action ${\mathcal{A}}(\unicode[STIX]{x1D701})$ by setting

(2.10) $$\begin{eqnarray}\displaystyle {\mathcal{A}}(\unicode[STIX]{x1D701})=\int _{\unicode[STIX]{x1D6FA}}{\mathcal{L}}(\widehat{\boldsymbol{u}},\widehat{\unicode[STIX]{x1D70C}},\widehat{Y},\widehat{\unicode[STIX]{x1D6FC}},\unicode[STIX]{x2202}_{t}\widehat{\unicode[STIX]{x1D6FC}}+\widehat{\boldsymbol{u}}^{\text{T}}\unicode[STIX]{x1D735}\widehat{\unicode[STIX]{x1D6FC}})\,\text{d}\boldsymbol{x}\,\text{d}t, & & \displaystyle\end{eqnarray}$$

and compute the infinitesimal variations of the action, $\unicode[STIX]{x1D6FF}{\mathcal{A}}=(\text{d}{\mathcal{A}}/\text{d}\unicode[STIX]{x1D701})(\unicode[STIX]{x1D701}=0)$ :

(2.11) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}{\mathcal{A}}=\displaystyle \int _{\unicode[STIX]{x1D6FA}}\left[\left({\displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}\boldsymbol{u}}}\right)^{\text{T}}\unicode[STIX]{x1D6FF}\boldsymbol{u}\!+\!{\displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70C}+{\displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}Y}}\unicode[STIX]{x1D6FF}Y\!+\!{\displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FC}\!+\!{\displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}(D_{t}\unicode[STIX]{x1D6FC})}}\unicode[STIX]{x1D6FF}(D_{t}\unicode[STIX]{x1D6FC})\right]\!(\boldsymbol{u},\unicode[STIX]{x1D70C},Y,\unicode[STIX]{x1D6FC},D_{t}\unicode[STIX]{x1D6FC})\,\text{d}\boldsymbol{x}\,\text{d}t. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

We make the classic assumption (Gavrilyuk & Gouin Reference Gavrilyuk and Gouin1999; Gavrilyuk & Saurel Reference Gavrilyuk and Saurel2002) that for our transformation family, $\unicode[STIX]{x1D743}$ and $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FC}$ vanish on $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ . Thanks to Green’s formula, we obtain after tedious calculations that

(2.12) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}{\mathcal{A}} & = & \displaystyle -\int _{\unicode[STIX]{x1D6FA}}\left\{-\unicode[STIX]{x1D70C}\unicode[STIX]{x1D735}\left({\displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}}\right)+\unicode[STIX]{x2202}_{t}\left({\displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}\boldsymbol{u}}}\right)+\text{div}\left({\displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}\boldsymbol{u}}}\boldsymbol{u}^{\text{T}}\right)+(\unicode[STIX]{x1D735}\boldsymbol{u})^{\text{T}}\left({\displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}\boldsymbol{u}}}\right)+\left({\displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}Y}}\right)\unicode[STIX]{x1D735}Y\right.\nonumber\\ \displaystyle & & \displaystyle \left.+\left[\unicode[STIX]{x2202}_{t}\left({\displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}(D_{t}\unicode[STIX]{x1D6FC})}}\right)+\text{div}\left({\displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}(D_{t}\unicode[STIX]{x1D6FC})}}\boldsymbol{u}\right)\right]\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}+{\displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}(D_{t}\unicode[STIX]{x1D6FC})}}\unicode[STIX]{x1D735}(D_{t}\unicode[STIX]{x1D6FC})\right\}^{\text{T}}\unicode[STIX]{x1D743}\,\text{d}\boldsymbol{x}\,\text{d}t\nonumber\\ \displaystyle & & \displaystyle -\,\int _{\unicode[STIX]{x1D6FA}}\left[-{\displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}}+\unicode[STIX]{x2202}_{t}\left({\displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}(D_{t}\unicode[STIX]{x1D6FC})}}\right)+\text{div}\left({\displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}(D_{t}\unicode[STIX]{x1D6FC})}}\boldsymbol{u}\right)\right]\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6FC}\,\text{d}\boldsymbol{x}\,\text{d}t.\end{eqnarray}$$

We follow the stationary action principle that boils down to the postulate that a physical transformation of the medium should extremize the Hamiltonian action  ${\mathcal{A}}$ . In our case, this yields

(2.13) $$\begin{eqnarray}\displaystyle & & \displaystyle -\unicode[STIX]{x1D70C}\unicode[STIX]{x1D735}\left({\displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}}\right)+\unicode[STIX]{x2202}_{t}\left({\displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}\boldsymbol{u}}}\right)+\text{div}\left({\displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}\boldsymbol{u}}}\boldsymbol{u}^{\text{T}}\right)+(\unicode[STIX]{x1D735}\boldsymbol{u})^{\text{T}}\left({\displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}\boldsymbol{u}}}\right)+\left({\displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}Y}}\right)\unicode[STIX]{x1D735}Y\nonumber\\ \displaystyle & & \displaystyle \quad +\left[\unicode[STIX]{x2202}_{t}\left(\frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}(D_{t}\unicode[STIX]{x1D6FC})}\right)+\text{div}\left({\displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}(D_{t}\unicode[STIX]{x1D6FC})}}\boldsymbol{u}\right)\right]\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}+{\displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}(D_{t}\unicode[STIX]{x1D6FC})}}\unicode[STIX]{x1D735}(D_{t}\unicode[STIX]{x1D6FC})\quad =\mathbf{0},\end{eqnarray}$$

(2.14) $$\begin{eqnarray}\displaystyle & & \displaystyle \quad -\,{\displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}}+\unicode[STIX]{x2202}_{t}\left({\displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}(D_{t}\unicode[STIX]{x1D6FC})}}\right)+\text{div}\left({\displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}(D_{t}\unicode[STIX]{x1D6FC})}}\boldsymbol{u}\right)\quad =0.\end{eqnarray}$$

For the choice of ${\mathcal{L}}$ expressed by (2.6), we have

(2.15a-c ) $$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}}={\displaystyle \frac{|\boldsymbol{u}|^{2}}{2}}-f-\unicode[STIX]{x1D70C}{\displaystyle \frac{\unicode[STIX]{x2202}f}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}},\quad {\displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}Y}}=-\unicode[STIX]{x1D70C}\frac{\unicode[STIX]{x2202}f}{\unicode[STIX]{x2202}Y},\quad {\displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}}={\displaystyle \frac{1}{2}}\unicode[STIX]{x1D708}^{\prime }(\unicode[STIX]{x1D6FC})\left(D_{t}\unicode[STIX]{x1D6FC}\right)^{2}-\unicode[STIX]{x1D70C}{\displaystyle \frac{\unicode[STIX]{x2202}f}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}},\end{eqnarray}$$

(2.15d,e ) $$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}(D_{t}\unicode[STIX]{x1D6FC})}}=\unicode[STIX]{x1D708}D_{t}\unicode[STIX]{x1D6FC},\quad {\displaystyle \frac{\unicode[STIX]{x2202}{\mathcal{L}}}{\unicode[STIX]{x2202}\boldsymbol{u}}}=\unicode[STIX]{x1D70C}\boldsymbol{u}.\end{eqnarray}$$

Relations (2.13) and (2.14) will respectively provide the evolution equations for the momentum and the volume fraction. Indeed, reinjecting (2.15) into (2.13)–(2.14) provides

(2.16a ) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}(\unicode[STIX]{x1D70C}\boldsymbol{u})+\text{div}(\unicode[STIX]{x1D70C}\boldsymbol{u}\boldsymbol{u}^{\text{T}})+\unicode[STIX]{x1D735}\left(\unicode[STIX]{x1D70C}^{2}{\displaystyle \frac{\unicode[STIX]{x2202}f}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}}+{\displaystyle \frac{1}{2}}\unicode[STIX]{x1D708}(D_{t}\unicode[STIX]{x1D6FC})^{2}\right)=\mathbf{0},\end{eqnarray}$$

(2.16b ) $$\begin{eqnarray}-\unicode[STIX]{x1D70C}{\displaystyle \frac{\unicode[STIX]{x2202}f}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}}=-{\textstyle \frac{1}{2}}(D_{t}\unicode[STIX]{x1D708})(D_{t}\unicode[STIX]{x1D6FC})+\unicode[STIX]{x2202}_{t}(\unicode[STIX]{x1D708}D_{t}\unicode[STIX]{x1D6FC})+\text{div}(\boldsymbol{u}\unicode[STIX]{x1D708}D_{t}\unicode[STIX]{x1D6FC}).\end{eqnarray}$$

We define the medium pressure $p$ and a new variable $w$ by setting

(2.17a,b ) $$\begin{eqnarray}p=\unicode[STIX]{x1D70C}^{2}{\displaystyle \frac{\unicode[STIX]{x2202}f}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}}\quad \text{and}\quad D_{t}\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D70C}Yw.\end{eqnarray}$$

Using the mass and partial mass conservation hypotheses, we obtain that the fluid transformations are governed by the following system of equations

(2.18a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D70C}+\text{div}(\unicode[STIX]{x1D70C}\boldsymbol{u})=0, & \displaystyle\end{eqnarray}$$

(2.18b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}(\unicode[STIX]{x1D70C}Y)+\text{div}(\unicode[STIX]{x1D70C}Y\boldsymbol{u})=0, & \displaystyle\end{eqnarray}$$

(2.18c ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}(\unicode[STIX]{x1D70C}\boldsymbol{u})+\text{div}(\unicode[STIX]{x1D70C}\boldsymbol{u}\boldsymbol{u}^{\text{T}})+\unicode[STIX]{x1D735}\left(p+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D708}(D_{t}\unicode[STIX]{x1D6FC})^{2}\right)=\mathbf{0}, & \displaystyle\end{eqnarray}$$

(2.18d ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D6FC}+\boldsymbol{u}^{\text{T}}\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D70C}Yw, & \displaystyle\end{eqnarray}$$

(2.18e ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}w+\boldsymbol{u}^{\text{T}}\unicode[STIX]{x1D735}w=-{\displaystyle \frac{1}{2}}{\displaystyle \frac{\unicode[STIX]{x1D708}^{\prime }(\unicode[STIX]{x1D6FC})}{\unicode[STIX]{x1D708}}}\unicode[STIX]{x1D70C}Yw^{2}-{\displaystyle \frac{1}{Y\unicode[STIX]{x1D708}}}\frac{\unicode[STIX]{x2202}f}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}. & \displaystyle\end{eqnarray}$$

Here, the stationary action principle only provides the conservative elements for our model: the momentum (2.18c ), the evolution equation for the volume fraction (2.18d ) and the small-scale pulsation evolutions (2.18e ). These equations supplement masses conservation, equations (2.18a )–(2.18b ), which has been postulated. Let us note that the evolution of $\unicode[STIX]{x1D6FC}$ is driven by (2.18d ) and (2.18e ) which involves more complex features than transport phenomena. Equation (2.18e ) can be interpreted as a small-scale momentum conservation equation. It is not surprising: the main role of the stationary action principle is to derive a momentum conservation equation from the data of kinetic energy and potential energy. In our case, we supplied kinetic energies for both the bulk and the small scales. Consequently, we recover momentum conservation equations for the bulk and the small scales that are coupled together. For the bulk-scale momentum, the small-scale kinematics accounts for small-scale inertia. In the following, $\unicode[STIX]{x1D708}$ will be referred to as micro-inertia.

We shall examine the dissipative structures of system (2.18) in the next section.

2.2 Dissipation and second principle of thermodynamics

While we are not interested in bulk dissipation phenomena, we aim at describing small-scale dissipation associated with the small-scale kinetic energy $\unicode[STIX]{x1D708}(D_{t}\unicode[STIX]{x1D6FC})^{2}/2$ , see equation (2.6). This way, the damping of small-scale pulsations of the interface due to various dissipative phenomena like the liquid viscosity can be taken into account.

Irreversible damping effects can be introduced in our two-phase system, by adding terms to (2.18) that are compatible with a mathematical entropy evolution equation. To do so, we first suppose that (2.18a )–(2.18d ) are valid, but we discard (2.18e ) and consider that $D_{t}w$ is now a quantity to be defined.

In the barotropic case, it is classic (Godlewski & Raviart Reference Godlewski and Raviart1996; Serre Reference Serre2007) to consider the total free energy of the system as a mathematical entropy

(2.19) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70C}\unicode[STIX]{x1D702}(\boldsymbol{u},\unicode[STIX]{x1D70C},Y,\unicode[STIX]{x1D6FC},w)={\textstyle \frac{1}{2}}\unicode[STIX]{x1D70C}|\boldsymbol{u}|^{2}+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D708}(\unicode[STIX]{x1D70C}Yw)^{2}+\unicode[STIX]{x1D70C}f(\unicode[STIX]{x1D70C},Y,\unicode[STIX]{x1D6FC}). & & \displaystyle\end{eqnarray}$$

We now seek for an entropy flux function $\boldsymbol{G}$ and a proper evolution principle for $D_{t}w$ , such that

(2.20) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}(\unicode[STIX]{x1D70C}\unicode[STIX]{x1D702})+\text{div}(\unicode[STIX]{x1D70C}\unicode[STIX]{x1D702}\boldsymbol{u}+\boldsymbol{G})\leqslant 0, & & \displaystyle\end{eqnarray}$$

or equivalently

(2.21) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70C}D_{t}\unicode[STIX]{x1D702}+\text{div}(\boldsymbol{G})\leqslant 0. & & \displaystyle\end{eqnarray}$$

Relation (2.21) reads

(2.22) $$\begin{eqnarray}\displaystyle \text{div}(\boldsymbol{G})+\unicode[STIX]{x1D70C}\left({\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}{\unicode[STIX]{x2202}\boldsymbol{u}}}\right)^{\text{T}}D_{t}\boldsymbol{u}+\unicode[STIX]{x1D70C}{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}}D_{t}\unicode[STIX]{x1D70C}+\unicode[STIX]{x1D70C}{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}{\unicode[STIX]{x2202}Y}}D_{t}Y+\unicode[STIX]{x1D70C}{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}}D_{t}\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D70C}{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}{\unicode[STIX]{x2202}w}}D_{t}w\leqslant 0. & & \displaystyle\end{eqnarray}$$

Using (2.18a )–(2.18d ) to express $D_{t}\boldsymbol{u}$ , $D_{t}\unicode[STIX]{x1D70C}$ , $D_{t}Y$ , $D_{t}\unicode[STIX]{x1D6FC}$ and

(2.23a-d ) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}{\unicode[STIX]{x2202}\boldsymbol{u}}}=\boldsymbol{u},\quad {\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}}={\displaystyle \frac{1}{2}}\unicode[STIX]{x1D708}^{\prime }(\unicode[STIX]{x1D6FC})(\unicode[STIX]{x1D70C}Yw)^{2}+{\displaystyle \frac{\unicode[STIX]{x2202}f}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}},\quad {\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}}={\displaystyle \frac{1}{2}}\unicode[STIX]{x1D708}(Yw)^{2}+{\displaystyle \frac{\unicode[STIX]{x2202}f}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}},\quad {\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}{\unicode[STIX]{x2202}w}}=\unicode[STIX]{x1D708}\unicode[STIX]{x1D70C}Y^{2}w, & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

altogether with (2.17), we obtain that

(2.24) $$\begin{eqnarray}\displaystyle \text{div}\left[\boldsymbol{G}-\left(p+{\displaystyle \frac{1}{2}}\unicode[STIX]{x1D708}(\unicode[STIX]{x1D70C}Yw)^{2}\right)\boldsymbol{u}\right]+\unicode[STIX]{x1D70C}Yw\left(\unicode[STIX]{x1D708}\unicode[STIX]{x1D70C}YD_{t}w+\unicode[STIX]{x1D70C}{\displaystyle \frac{\unicode[STIX]{x2202}f}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}}+{\displaystyle \frac{1}{2}}(\unicode[STIX]{x1D70C}Yw)^{2}\unicode[STIX]{x1D708}^{\prime }(\unicode[STIX]{x1D6FC})\right)\leqslant 0. & & \displaystyle\end{eqnarray}$$

A simple choice for ensuring (2.24) consists in setting

(2.25a,b ) $$\begin{eqnarray}\displaystyle \boldsymbol{G}=\left(p+{\displaystyle \frac{1}{2}}\unicode[STIX]{x1D708}(\unicode[STIX]{x1D70C}Yw)^{2}\right)\boldsymbol{u}\quad \text{and}\quad {\displaystyle \frac{\unicode[STIX]{x2202}f}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}}+\unicode[STIX]{x1D708}YD_{t}w+{\displaystyle \frac{1}{2}}\unicode[STIX]{x1D70C}(Yw)^{2}\unicode[STIX]{x1D708}^{\prime }(\unicode[STIX]{x1D6FC})=-\unicode[STIX]{x1D700}Yw,\qquad & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D700}>0$ is a constant. Let us underline that this choice is guided by our knowledge of the non-dissipative structure of the system of equation resulting from the stationary action principle. This yields a definition of $\boldsymbol{G}$ and a new evolution equation for $w$ that reads

(2.26) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}w+\boldsymbol{u}^{\text{T}}\unicode[STIX]{x1D735}w=-{\displaystyle \frac{\unicode[STIX]{x1D700}}{\unicode[STIX]{x1D708}}}w-{\displaystyle \frac{1}{2}}{\displaystyle \frac{\unicode[STIX]{x1D708}^{\prime }(\unicode[STIX]{x1D6FC})}{\unicode[STIX]{x1D708}}}\unicode[STIX]{x1D70C}Yw^{2}-{\displaystyle \frac{1}{\unicode[STIX]{x1D708}Y}}{\displaystyle \frac{\unicode[STIX]{x2202}f}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}}. & & \displaystyle\end{eqnarray}$$

Consequently, the generic form of our two-phase flow system reads

(2.27a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D70C}+\text{div}(\unicode[STIX]{x1D70C}\boldsymbol{u})=0, & \displaystyle\end{eqnarray}$$

(2.27b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}(\unicode[STIX]{x1D70C}Y)+\text{div}(\unicode[STIX]{x1D70C}Y\boldsymbol{u})=0, & \displaystyle\end{eqnarray}$$

(2.27c ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}(\unicode[STIX]{x1D70C}\boldsymbol{u})+\text{div}(\unicode[STIX]{x1D70C}\boldsymbol{u}\boldsymbol{u}^{\text{T}})+\unicode[STIX]{x1D735}\left(p+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D708}(\unicode[STIX]{x1D70C}Yw)^{2}\right)=\mathbf{0}, & \displaystyle\end{eqnarray}$$

(2.27d ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D6FC}+\boldsymbol{u}^{\text{T}}\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D70C}Yw, & \displaystyle\end{eqnarray}$$

(2.27e ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}w+\boldsymbol{u}^{\text{T}}\unicode[STIX]{x1D735}w=-{\displaystyle \frac{\unicode[STIX]{x1D700}}{\unicode[STIX]{x1D708}}}w-{\displaystyle \frac{1}{2}}{\displaystyle \frac{\unicode[STIX]{x1D708}^{\prime }(\unicode[STIX]{x1D6FC})}{\unicode[STIX]{x1D708}}}\unicode[STIX]{x1D70C}Yw^{2}-{\displaystyle \frac{1}{Y\unicode[STIX]{x1D708}}}{\displaystyle \frac{\unicode[STIX]{x2202}f}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}}. & \displaystyle\end{eqnarray}$$

In order to complete the definition of our model, we need to specify the free energy of the medium. We consider that $f$ is the sum of a bulk mixture free energy and a compaction energy $\unicode[STIX]{x1D6FC}\mapsto e(\unicode[STIX]{x1D6FC})$ , where $e$ is a given function (Gavrilyuk & Saurel Reference Gavrilyuk and Saurel2002). We set

(2.28) $$\begin{eqnarray}\displaystyle f(\unicode[STIX]{x1D70C},Y,\unicode[STIX]{x1D6FC})=Yf_{1}\left({\displaystyle \frac{\unicode[STIX]{x1D70C}Y}{\unicode[STIX]{x1D6FC}}}\right)+(1-Y)f_{2}\left({\displaystyle \frac{\unicode[STIX]{x1D70C}(1-Y)}{1-\unicode[STIX]{x1D6FC}}}\right)+e(\unicode[STIX]{x1D6FC}). & & \displaystyle\end{eqnarray}$$

For this choice, granted that $\unicode[STIX]{x1D70C}_{k}^{2}\unicode[STIX]{x2202}f_{k}/\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}_{k}=p_{k}$ , $k=1,2$ , a straightforward calculation gives that

(2.29a,b ) $$\begin{eqnarray}\displaystyle p=\unicode[STIX]{x1D70C}^{2}\frac{\unicode[STIX]{x2202}f}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}=\unicode[STIX]{x1D6FC}p_{1}+(1-\unicode[STIX]{x1D6FC})p_{2}\quad \text{and}\quad \unicode[STIX]{x1D70C}{\displaystyle \frac{\unicode[STIX]{x2202}f}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}}=p_{2}-p_{1}+\unicode[STIX]{x1D70C}{\displaystyle \frac{\text{d}e}{\text{d}\unicode[STIX]{x1D6FC}}}, & & \displaystyle\end{eqnarray}$$

and system (2.27) reads here

(2.30a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D70C}+\text{div}(\unicode[STIX]{x1D70C}\boldsymbol{u})=0, & \displaystyle\end{eqnarray}$$

(2.30b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}(\unicode[STIX]{x1D70C}Y)+\text{div}(\unicode[STIX]{x1D70C}Y\boldsymbol{u})=0, & \displaystyle\end{eqnarray}$$

(2.30c ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}(\unicode[STIX]{x1D70C}\boldsymbol{u})+\text{div}(\unicode[STIX]{x1D70C}\boldsymbol{u}\boldsymbol{u}^{\text{T}})+\unicode[STIX]{x1D735}\left(p+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D708}(\unicode[STIX]{x1D70C}Yw)^{2}\right)=\mathbf{0}, & \displaystyle\end{eqnarray}$$

(2.30d ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D6FC}+\boldsymbol{u}^{\text{T}}\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D70C}Yw, & \displaystyle\end{eqnarray}$$

(2.30e ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}w+\boldsymbol{u}^{\text{T}}\unicode[STIX]{x1D735}w=-{\displaystyle \frac{\unicode[STIX]{x1D700}}{\unicode[STIX]{x1D708}}}w-{\displaystyle \frac{1}{2}}{\displaystyle \frac{\unicode[STIX]{x1D708}^{\prime }(\unicode[STIX]{x1D6FC})}{\unicode[STIX]{x1D708}}}\unicode[STIX]{x1D70C}Yw^{2}+\frac{1}{\unicode[STIX]{x1D70C}Y\unicode[STIX]{x1D708}}\left(p_{1}-p_{2}-\unicode[STIX]{x1D70C}{\displaystyle \frac{\text{d}e}{\text{d}\unicode[STIX]{x1D6FC}}}\right). & \displaystyle\end{eqnarray}$$

$w$

Let us conclude this section by stating well-posedness properties of our two-phase model with micro-inertia $\unicode[STIX]{x1D708}$ . We consider the sole convective part of system (2.27) for one-dimensional problems by discarding the source terms. The resulting system is hyperbolic and its characteristic velocities are

(2.31a-c ) $$\begin{eqnarray}\displaystyle u-c,\quad u,\quad u+c, & & \displaystyle\end{eqnarray}$$

where

(2.32a,b ) $$\begin{eqnarray}\displaystyle c^{2}=c_{\mathit{Frozen}}^{2}+\unicode[STIX]{x1D70C}\unicode[STIX]{x1D708}(Yw)^{2}\quad \text{and}\quad c_{\mathit{Frozen}}^{2}=Yc_{1}^{2}+(1-Y)c_{2}^{2}. & & \displaystyle\end{eqnarray}$$

Details related to the eigenstructure of system (2.27) are presented in appendix B.

2.3 Submodels and traversing the hierarchy

Based on our assumptions, the richest model (2.30) involves both parameters $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D708}$ . We examine two limit flow regimes, obtained for vanishing values of the parameters $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D708}$ . First, we study the case of a negligible micro-inertia compared to the internal dissipation effects, i.e. $\unicode[STIX]{x1D708}\rightarrow 0$ and $\unicode[STIX]{x1D700}=O(1)$ . Second, we consider the case when both micro-inertia and internal dissipation tend to zero, with $\unicode[STIX]{x1D708}\rightarrow 0$ and $\unicode[STIX]{x1D700}\rightarrow 0$ . Both cases allow to recover the two-phase systems presented in (Chanteperdrix et al. Reference Chanteperdrix, Villedieu and Vila2002). These systems are both composed of a conservative part and a (possibly null) stiff source term.

2.3.1 A 4-equation model for $\unicode[STIX]{x1D708}\rightarrow 0$ and $\unicode[STIX]{x1D700}=O(1)$

We consider system (2.30) and suppose here that $\unicode[STIX]{x1D708}\rightarrow 0$ for a fixed value of $\unicode[STIX]{x1D700}$ . We also assume here that $\unicode[STIX]{x1D708}^{\prime }(\unicode[STIX]{x1D6FC})/\unicode[STIX]{x1D708}$ is proportional to $\unicode[STIX]{x1D6FC}^{-1}$ . This assumption will be justified in § 3.1 where an identification of $\unicode[STIX]{x1D708}$ is proposed. Then, we can see that (2.30e ) provides that

(2.33) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70C}Yw={\displaystyle \frac{1}{\unicode[STIX]{x1D700}}}\left(p_{1}-p_{2}-\unicode[STIX]{x1D70C}{\displaystyle \frac{\text{d}e}{\text{d}\unicode[STIX]{x1D6FC}}}\right). & & \displaystyle\end{eqnarray}$$

By using equation (2.30d ), we obtain that the limit regime is governed by

(2.34a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D70C}+\text{div}(\unicode[STIX]{x1D70C}\boldsymbol{u})=0, & \displaystyle\end{eqnarray}$$

(2.34b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}(\unicode[STIX]{x1D70C}Y)+\text{div}(\unicode[STIX]{x1D70C}Y\boldsymbol{u})=0, & \displaystyle\end{eqnarray}$$

(2.34c ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}(\unicode[STIX]{x1D70C}\boldsymbol{u})+\text{div}(\unicode[STIX]{x1D70C}\boldsymbol{u}\boldsymbol{u}^{\text{T}})+\unicode[STIX]{x1D735}p=\mathbf{0}, & \displaystyle\end{eqnarray}$$

(2.34d ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D6FC}+\boldsymbol{u}^{\text{T}}\unicode[STIX]{x1D735}\unicode[STIX]{x1D6FC}=\frac{1}{\unicode[STIX]{x1D700}}\left(p_{1}-p_{2}-\unicode[STIX]{x1D70C}\frac{\text{d}e}{\text{d}\unicode[STIX]{x1D6FC}}\right). & \displaystyle\end{eqnarray}$$

In the specific case $\text{d}e/\text{d}\unicode[STIX]{x1D6FC}=0$ , we recover the relaxation system studied in Chanteperdrix et al. (Reference Chanteperdrix, Villedieu and Vila2002). The conservative part of system (2.34) is a hyperbolic system whose characteristic velocities are $\{u-c_{\mathit{Frozen}},u,u+c_{\mathit{Frozen}}\}$ and that is equipped with an entropy inequality

(2.35) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}\left({\textstyle \frac{1}{2}}\unicode[STIX]{x1D70C}|\boldsymbol{u}|^{2}+\unicode[STIX]{x1D70C}g(\unicode[STIX]{x1D70C},Y,\unicode[STIX]{x1D6FC})\right)+\text{div}\left(\left[{\textstyle \frac{1}{2}}\unicode[STIX]{x1D70C}|\boldsymbol{u}|^{2}+\unicode[STIX]{x1D70C}g(\unicode[STIX]{x1D70C},Y,\unicode[STIX]{x1D6FC})+p\right]\boldsymbol{u}\right)\leqslant 0. & & \displaystyle\end{eqnarray}$$

The stiff source term in (2.34d ) drives all the dissipation effects within system (2.34).

2.3.2 A 3-equation model for $\unicode[STIX]{x1D708}\rightarrow 0$ and $\unicode[STIX]{x1D700}\rightarrow 0$

The last model of our hierarchy can be obtained either by considering model (2.30) and the two vanishing coefficients $\unicode[STIX]{x1D708}\rightarrow 0$ and $\unicode[STIX]{x1D700}\rightarrow 0$ , or by taking $\unicode[STIX]{x1D700}\rightarrow 0$ in (2.34). Formally, we obtain

(2.36a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D70C}+\text{div}(\unicode[STIX]{x1D70C}\boldsymbol{u})=0, & \displaystyle\end{eqnarray}$$

(2.36b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}(\unicode[STIX]{x1D70C}Y)+\text{div}(\unicode[STIX]{x1D70C}Y\boldsymbol{u})=0, & \displaystyle\end{eqnarray}$$

(2.36c ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}(\unicode[STIX]{x1D70C}\boldsymbol{u})+\text{div}(\unicode[STIX]{x1D70C}\boldsymbol{u}\boldsymbol{u}^{\text{T}})+\unicode[STIX]{x1D735}p=\mathbf{0}, & \displaystyle\end{eqnarray}$$

(2.36d ) $$\begin{eqnarray}\displaystyle & \displaystyle p_{1}\left({\displaystyle \frac{\unicode[STIX]{x1D70C}Y}{\unicode[STIX]{x1D6FC}}}\right)-p_{2}\left({\displaystyle \frac{\unicode[STIX]{x1D70C}(1-Y)}{1-\unicode[STIX]{x1D6FC}}}\right)-\unicode[STIX]{x1D70C}{\displaystyle \frac{\text{d}e}{\text{d}\unicode[STIX]{x1D6FC}}}(\unicode[STIX]{x1D6FC})=0. & \displaystyle\end{eqnarray}$$

This system is fully conservative. Also, relation (2.36d ) implies that $\unicode[STIX]{x1D6FC}$ is no longer an independent variable: the volume fraction has become a function of $\unicode[STIX]{x1D70C}$ and $Y$ .

In the specific case $\text{d}e/\text{d}\unicode[STIX]{x1D6FC}=0$ , we recover the classic partial pressure equilibrium closure relation $p_{1}=p_{2}$ that was studied in Chanteperdrix et al. (Reference Chanteperdrix, Villedieu and Vila2002): the resulting system is strictly hyperbolic as it possesses three distinct real-valued characteristic velocities $\{u-c_{\mathit{Wood}},u,u+c_{\mathit{Wood}}\}$ , where $c_{\mathit{Wood}}$ is defined by

(2.37) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{1}{c_{\mathit{Wood}}^{2}}}={\displaystyle \frac{\unicode[STIX]{x1D6FC}^{2}}{Yc_{1}^{2}}}+{\displaystyle \frac{(1-\unicode[STIX]{x1D6FC})^{2}}{(1-Y)c_{2}^{2}}}. & & \displaystyle\end{eqnarray}$$

Concerning the mathematical properties of (2.36), we refer the reader to Chanteperdrix et al. (Reference Chanteperdrix, Villedieu and Vila2002). One can note that for both relations (2.32) and (2.37), the subcharacteristic condition is verified since $c_{\mathit{Wood}}\leqslant c_{\mathit{Frozen}}\leqslant c$ . Let us remark that these characteristic velocities do not match the propagation velocities of acoustic waves as the latter depend on the the wave frequency. This will be made clear in § 3.3.

3.1 Connection with the Rayleigh–Plesset equation

Let us consider a bubbly fluid which is monodisperse and characterized as follows: at each position and instant $(\boldsymbol{x},t)$ , the distribution of the number of bubbles is defined by the density function $n(\boldsymbol{x},t)$ and all bubbles are spherical with a radius $R(\boldsymbol{x},t)$ . Given the phasic gas density $\unicode[STIX]{x1D70C}_{1}(\boldsymbol{x},t)$ , all the bubbles have the same mass ${\mathcal{M}}_{b}(\boldsymbol{x},t)=4/3\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}_{1}(\boldsymbol{x},t)R^{3}(\boldsymbol{x},t)$ . Then, the gas volume fraction $\unicode[STIX]{x1D6FC}$ and partial mass $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70C}_{1}$ can be related to the flow structure parameters by

(3.2a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FC}(\boldsymbol{x},t)={\displaystyle \frac{4\unicode[STIX]{x03C0}}{3}}R^{3}(\boldsymbol{x},t)n(\boldsymbol{x},t),\quad \unicode[STIX]{x1D6FC}(\boldsymbol{x},t)\unicode[STIX]{x1D70C}_{1}(\boldsymbol{x},t)=n(\boldsymbol{x},t){\mathcal{M}}_{b}(x,t). & & \displaystyle\end{eqnarray}$$

Hypothesis (H5) boils down to $D_{t}{\mathcal{M}}_{b}=0$ . Using the conservation of the partial mass $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70C}_{1}=\unicode[STIX]{x1D70C}Y$ for the gas (2.30b ) and $n=\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70C}_{1}/{\mathcal{M}}_{b}$ , we obtain the following conservation law for $n$

(3.3) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}n+\text{div}(n\boldsymbol{u})=0. & & \displaystyle\end{eqnarray}$$

The conservation of the partial mass $(1-\unicode[STIX]{x1D6FC})\unicode[STIX]{x1D70C}_{2}=\unicode[STIX]{x1D70C}(1-Y)$ for the surrounding fluid and hypothesis (H6) imply that $\text{div}(\boldsymbol{u})$ is constrained by $D_{t}\unicode[STIX]{x1D6FC}$ through the relation

(3.4) $$\begin{eqnarray}\displaystyle D_{t}\unicode[STIX]{x1D6FC}+(\unicode[STIX]{x1D6FC}-1)\text{div}(\boldsymbol{u})=0. & & \displaystyle\end{eqnarray}$$

Now, let us express $D_{t}\unicode[STIX]{x1D6FC}$ , $w$ and $D_{t}w$ in terms of $R(t)$ and its material derivatives. We have $\unicode[STIX]{x1D6FC}=(4\unicode[STIX]{x03C0}/3)R^{3}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70C}_{1}/{\mathcal{M}}_{b}$ , which yields

(3.5) $$\begin{eqnarray}\displaystyle D_{t}\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FC}\left(-\text{div}(\boldsymbol{u})+3{\displaystyle \frac{D_{t}R}{R}}\right). & & \displaystyle\end{eqnarray}$$

Using (3.4), we obtain

(3.6) $$\begin{eqnarray}\displaystyle D_{t}\unicode[STIX]{x1D6FC}=3\unicode[STIX]{x1D6FC}(1-\unicode[STIX]{x1D6FC}){\displaystyle \frac{D_{t}R}{R}}. & & \displaystyle\end{eqnarray}$$

Expressing $m_{1}$ thanks to (3.2) and using (2.30d ) leads to

(3.7) $$\begin{eqnarray}\displaystyle w=\frac{4\unicode[STIX]{x03C0}}{{\mathcal{M}}_{b}}(1-\unicode[STIX]{x1D6FC})R^{2}D_{t}R. & & \displaystyle\end{eqnarray}$$

Relations (3.7) and (3.5) provide

(3.8) $$\begin{eqnarray}\displaystyle D_{t}w=\frac{4\unicode[STIX]{x03C0}}{{\mathcal{M}}_{b}}\left[(2-5\unicode[STIX]{x1D6FC})R(D_{t}R)^{2}+(1-\unicode[STIX]{x1D6FC})R^{2}D_{tt}^{2}R\right]. & & \displaystyle\end{eqnarray}$$

Finally, combining (3.8) and (2.30e ) gives the evolution equation for $R$ ,

(3.9) $$\begin{eqnarray}\displaystyle p_{1}-p_{2}-\unicode[STIX]{x1D70C}{\displaystyle \frac{\text{d}e}{\text{d}\unicode[STIX]{x1D6FC}}} & = & \displaystyle \unicode[STIX]{x1D700}{\displaystyle \frac{3\unicode[STIX]{x1D6FC}(1-\unicode[STIX]{x1D6FC})}{R}}D_{t}R+3\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D708}\left[(2-5\unicode[STIX]{x1D6FC})+{\displaystyle \frac{3}{2}}\unicode[STIX]{x1D6FC}(1-\unicode[STIX]{x1D6FC})^{2}{\displaystyle \frac{\unicode[STIX]{x1D708}^{\prime }(\unicode[STIX]{x1D6FC})}{\unicode[STIX]{x1D708}}}\right]\nonumber\\ \displaystyle & & \displaystyle \times \,\left[{\displaystyle \frac{D_{t}R}{R}}\right]^{2}+3\unicode[STIX]{x1D6FC}(1-\unicode[STIX]{x1D6FC})\unicode[STIX]{x1D708}{\displaystyle \frac{D_{tt}^{2}R}{R}}.\end{eqnarray}$$

We see that the two-phase model (2.30), supplemented by our bubbly flow structure assumptions (H5–H6), provide the evolution (3.9) for a spatial distribution of bubble radii.

In fact, we clearly see that the bubble radii evolve only along the trajectories of the flow. Therefore, by means of the following notation: ${\dot{R}}=D_{t}R$ and $\ddot{R}=D_{tt}^{2}R$ , we obtain an ODE of the radii along the trajectory of the flow

(3.10) $$\begin{eqnarray}\displaystyle p_{1}-p_{2}-\unicode[STIX]{x1D70C}{\displaystyle \frac{\text{d}e}{\text{d}\unicode[STIX]{x1D6FC}}} & = & \displaystyle \unicode[STIX]{x1D700}{\displaystyle \frac{3\unicode[STIX]{x1D6FC}(1-\unicode[STIX]{x1D6FC})}{R}}{\dot{R}}+3\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D708}\left((2-5\unicode[STIX]{x1D6FC})+{\displaystyle \frac{3}{2}}\unicode[STIX]{x1D6FC}(1-\unicode[STIX]{x1D6FC})^{2}{\displaystyle \frac{\unicode[STIX]{x1D708}^{\prime }(\unicode[STIX]{x1D6FC})}{\unicode[STIX]{x1D708}}}\right)\nonumber\\ \displaystyle & & \displaystyle \times \,\left({\displaystyle \frac{{\dot{R}}}{R}}\right)^{2}+3\unicode[STIX]{x1D6FC}(1-\unicode[STIX]{x1D6FC})\unicode[STIX]{x1D708}{\displaystyle \frac{\ddot{R}}{R}}.\end{eqnarray}$$

This ODE is analogous to a nonlinear oscillator with damping and forcing terms. This analogy also shows that the micro-inertia $\unicode[STIX]{x1D708}$ is indeed connected to inertial effects, while $\unicode[STIX]{x1D700}$ is related to damping and will be referred to as ‘micro-viscosity’. Now, the evolution of $R(t)$ along the trajectories can be compared with other existing models that account for bubble vibrations in specific flow regimes, in order to provide estimates for $\unicode[STIX]{x1D708}$ and $\unicode[STIX]{x1D700}$ in (2.30).

First, we propose to proceed with the Rayleigh–Plesset equation (A 4g ) of the Cheng, Drew and Lahey (CDL) system (A 4). To do so, we examine a flow regime involving a bubble radius and a volume fraction small enough such that $\unicode[STIX]{x1D6FC}\ll 1$ in (2.30) and $kR\ll 1$ in (A 4). We neglect the surface tension in (A 4g ) by setting $\unicode[STIX]{x1D70E}=0$ and relate $p_{1}-p_{2}-\unicode[STIX]{x1D70C}\,\text{d}e/\text{d}\unicode[STIX]{x1D6FC}$ in (3.9) to $p_{1i}-p_{2}$ in (A 4g ), recalling the hypothesis of uniform pressures (H7). By identifying the terms in ${\dot{R}}$ , $\ddot{R}$ and ${\dot{R}}^{2}$ , we respectively obtain

(3.11a-c ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D700}{\displaystyle \frac{3\unicode[STIX]{x1D6FC}}{R}}={\displaystyle \frac{4\unicode[STIX]{x1D707}_{2}}{R}},\quad {\displaystyle \frac{3\unicode[STIX]{x1D6FC}}{R}}\unicode[STIX]{x1D708}=\unicode[STIX]{x1D70C}_{2}R\quad \text{and}\quad \frac{3\unicode[STIX]{x1D6FC}}{R^{2}}\unicode[STIX]{x1D708}\left({\displaystyle \frac{3}{2}}\unicode[STIX]{x1D708}^{-1}\unicode[STIX]{x1D708}^{\prime }(\unicode[STIX]{x1D6FC})\unicode[STIX]{x1D6FC}+2\right)={\displaystyle \frac{3}{2}}\unicode[STIX]{x1D70C}_{2}.\quad & & \displaystyle\end{eqnarray}$$

The first two relations of (3.11) allow us to identify respectively $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D708}$ as

(3.12a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D700}_{\mathit{RP}}={\displaystyle \frac{4\unicode[STIX]{x1D707}_{2}}{3\unicode[STIX]{x1D6FC}}}\quad \text{and}\quad \unicode[STIX]{x1D708}_{\mathit{RP}}={\displaystyle \frac{\unicode[STIX]{x1D70C}_{2}R^{2}}{3\unicode[STIX]{x1D6FC}}}={\displaystyle \frac{\unicode[STIX]{x1D70C}_{2}(3\unicode[STIX]{x1D6FC})^{-1/3}}{(4\unicode[STIX]{x03C0}n)^{2/3}}}. & & \displaystyle\end{eqnarray}$$

The third relation of (3.11) is redundant but compatible with the definition of $\unicode[STIX]{x1D708}$ expressed in (3.12. Thus we see that for this specific flow regime, it is possible to insert a monodisperse bubble flow model into our two-phase model, for which the dynamics of the bubble radii degenerates to the Rayleigh–Plesset equation. Also, the resulting values of $\unicode[STIX]{x1D708}$ and $\unicode[STIX]{x1D700}$ match the results of Gavrilyuk & Saurel (Reference Gavrilyuk and Saurel2002) derived in the context of a Baer–Nunziato-type two-fluid model.

Next, let us remark that in the context of bubbly fluids, similar models for micro-inertia are available. In Temkin (Reference Temkin2005), a pulsational energy is considered in the form: $E_{puls}={\textstyle \frac{1}{2}}M_{puls}{\dot{R}}^{2}$ and $M_{puls}=4\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}_{2}R^{3}$ . An alternative approach that allows us to incorporate small-scale bubble velocity is also proposed in Lhuillier, Theofanous & Liou (Reference Lhuillier, Theofanous and Liou2010), by accounting for pseudo-turbulent kinetic energies generated by particle pulsations $K_{c}={\textstyle \frac{1}{2}}Q(\unicode[STIX]{x1D6FC})(D_{t}R)^{2}$ , where $Q(\unicode[STIX]{x1D6FC})$ is proportional to $3\unicode[STIX]{x1D6FC}$ in the dilute limit of the dispersed phase. In Gavrilyuk (Reference Gavrilyuk, dell’Isola and Gavrilyuk2012), a micro-scale kinetic energy of the form $3\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70C}_{2}(D_{t}R)^{2}$ is used for modelling vibrations of bubbles within a bubbly fluid.

3.2 Connection with the linearized Rayleigh equation

Further, we examine the bubbly fluid model of § 3.1 by considering the regime of small variations of the bubbles radius. Let us assume that

(3.13a-d ) $$\begin{eqnarray}\displaystyle R=\overline{R}(1+rz),\quad p_{2}=\overline{p_{2}}+r\unicode[STIX]{x1D6FF}p_{2},\quad n=\overline{n},\quad {\mathcal{M}}_{b}=\overline{{\mathcal{M}}_{b}}, & & \displaystyle\end{eqnarray}$$

where $\overline{R}$ , $\overline{p_{2}}$ , $\overline{{\mathcal{M}}_{b}}$ , $\overline{n}$ are constant values and $0<r\ll 1$ is a small parameter. If one notes $\overline{\unicode[STIX]{x1D6FC}}=4\unicode[STIX]{x03C0}\overline{R}^{3}\overline{n}/3$ , $\overline{\unicode[STIX]{x1D70C}_{1}}=\overline{n}\overline{{\mathcal{M}}_{b}}/\overline{\unicode[STIX]{x1D6FC}}$ , $\overline{p_{1}}=p_{1}(\overline{\unicode[STIX]{x1D70C}_{1}})$ , $\overline{c_{1}}=c_{1}(\overline{\unicode[STIX]{x1D70C}_{1}})$ , $\unicode[STIX]{x1D708}(\overline{\unicode[STIX]{x1D6FC}})=\overline{\unicode[STIX]{x1D708}}$ and $\overline{\unicode[STIX]{x1D70C}}=\overline{\unicode[STIX]{x1D6FC}}\overline{\unicode[STIX]{x1D70C}_{1}}+(1-\overline{\unicode[STIX]{x1D6FC}})\overline{\unicode[STIX]{x1D70C}_{2}}$ then (3.13) yield

(3.14a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FC}=\overline{\unicode[STIX]{x1D6FC}}(1+3rz)+O(r^{2}),\quad p_{1}=\overline{p_{1}}-3\overline{\unicode[STIX]{x1D70C}_{1}}\,\overline{c_{1}}^{2}rz+O(r^{2}), & \displaystyle\end{eqnarray}$$

(3.14b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D708}=\overline{\unicode[STIX]{x1D708}}+O(r),\quad \unicode[STIX]{x1D70C}{\displaystyle \frac{\text{d}e}{\text{d}\unicode[STIX]{x1D6FC}}}(\overline{\unicode[STIX]{x1D6FC}})=\overline{\unicode[STIX]{x1D70C}}{\displaystyle \frac{\text{d}e}{\text{d}\unicode[STIX]{x1D6FC}}}(\overline{\unicode[STIX]{x1D6FC}})+3\overline{\unicode[STIX]{x1D6FC}}z\left[\overline{\unicode[STIX]{x1D70C}}{\displaystyle \frac{\text{d}^{2}e}{\text{d}\unicode[STIX]{x1D6FC}^{2}}}(\overline{\unicode[STIX]{x1D6FC}})-\overline{\unicode[STIX]{x1D70C}_{2}}{\displaystyle \frac{\text{d}e}{\text{d}\unicode[STIX]{x1D6FC}}}(\overline{\unicode[STIX]{x1D6FC}})\right]r+O(r^{2}).\quad \qquad & \displaystyle\end{eqnarray}$$

(3.15) $$\begin{eqnarray}\displaystyle & & \displaystyle \overline{p_{1}}-\overline{p_{2}}-r\unicode[STIX]{x1D6FF}p_{2}-\overline{\unicode[STIX]{x1D70C}}{\displaystyle \frac{\text{d}e}{\text{d}\unicode[STIX]{x1D6FC}}}(\overline{\unicode[STIX]{x1D6FC}})-3\overline{\unicode[STIX]{x1D6FC}}z\left[\overline{\unicode[STIX]{x1D70C}}{\displaystyle \frac{\text{d}^{2}e}{\text{d}\unicode[STIX]{x1D6FC}^{2}}}(\overline{\unicode[STIX]{x1D6FC}})-\overline{\unicode[STIX]{x1D70C}_{2}}{\displaystyle \frac{\text{d}e}{\text{d}\unicode[STIX]{x1D6FC}}}(\overline{\unicode[STIX]{x1D6FC}})\right]r\nonumber\\ \displaystyle & & \displaystyle \quad =3r\left(\overline{\unicode[STIX]{x1D70C}_{1}}\,\overline{c_{1}}^{2}z+\overline{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D700}{\dot{z}}+\overline{\unicode[STIX]{x1D6FC}}\,\overline{\unicode[STIX]{x1D708}}\ddot{z}\right)+O(r^{2}).\end{eqnarray}$$

Identifying same-order terms with respect to $r$ yields

(3.16) $$\begin{eqnarray}\displaystyle \overline{p_{1}}-\overline{p_{2}}-\overline{\unicode[STIX]{x1D70C}}\frac{\text{d}e}{\text{d}\unicode[STIX]{x1D6FC}}(\overline{\unicode[STIX]{x1D6FC}})=0, & & \displaystyle\end{eqnarray}$$

and

(3.17) $$\begin{eqnarray}\displaystyle 3\overline{\unicode[STIX]{x1D6FC}}\,\overline{\unicode[STIX]{x1D708}}\ddot{z}+3\overline{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D700}{\dot{z}}+3\left[\overline{\unicode[STIX]{x1D70C}_{1}}\,\overline{c_{1}}^{2}+\overline{\unicode[STIX]{x1D6FC}}\,\overline{\unicode[STIX]{x1D70C}}{\displaystyle \frac{\text{d}^{2}e}{\text{d}\unicode[STIX]{x1D6FC}^{2}}}(\overline{\unicode[STIX]{x1D6FC}})-\overline{\unicode[STIX]{x1D6FC}}\overline{\unicode[STIX]{x1D70C}_{2}}{\displaystyle \frac{\text{d}e}{\text{d}\unicode[STIX]{x1D6FC}}}(\overline{\unicode[STIX]{x1D6FC}})\right]z=-\unicode[STIX]{x1D6FF}p_{2}. & & \displaystyle\end{eqnarray}$$

Equation (3.17) is a second-order linear ODE in $z$ that is consistent with the evolution of a linear harmonic oscillator with damping and forcing terms. This type of equation is classic in the literature for describing the motion of vibrating bubbles in the linear regime. Indeed, following Prosperetti (Reference Prosperetti1977), Cheng, Drew & Lahey (Reference Cheng, Drew and Lahey1983), we have

(3.18) $$\begin{eqnarray}\displaystyle \overline{\unicode[STIX]{x1D70C}_{2}}\overline{R}^{2}\ddot{z}+2\unicode[STIX]{x1D6FE}\,\overline{\unicode[STIX]{x1D70C}_{2}}\overline{R}^{2}{\dot{z}}+\overline{\unicode[STIX]{x1D70C}_{2}}\overline{R}^{2}\unicode[STIX]{x1D714}_{0}^{2}z=-\unicode[STIX]{x1D6FF}p_{2}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FE}_{vis}+\unicode[STIX]{x1D6FE}_{th}+\unicode[STIX]{x1D6FE}_{ac}$ drives the damping intensity of the system. The coefficient $\unicode[STIX]{x1D6FE}_{vis}$ pertains to viscous effects due to the surrounding liquid and is defined by $\unicode[STIX]{x1D6FE}_{vis}=2\unicode[STIX]{x1D707}_{2}/(\overline{\unicode[STIX]{x1D70C}_{2}}\,\overline{R}^{2})$ , $\unicode[STIX]{x1D6FE}_{th}$ is related to thermal exchanges between the gas and the liquid and $\unicode[STIX]{x1D6FE}_{ac}$ concerns acoustic scattering by the bubbles. Expressions for $\unicode[STIX]{x1D6FE}_{th}$ and $\unicode[STIX]{x1D6FE}_{ac}$ are available in Prosperetti (Reference Prosperetti1977), Cheng et al. (Reference Cheng, Drew and Lahey1983), and references from previous studies therein. They involve several intermediate parameters but also characteristic parameters of the forcing term, like the frequency of the perturbation $\unicode[STIX]{x1D6FF}p_{2}$ . Identifying terms in (3.17) and (3.18) yields the following definitions for $\unicode[STIX]{x1D708}$ and $\unicode[STIX]{x1D700}$

(3.19a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D700}_{\mathit{Lin}}={\displaystyle \frac{4\unicode[STIX]{x1D707}_{2}}{3\overline{\unicode[STIX]{x1D6FC}}}}+{\displaystyle \frac{2\overline{\unicode[STIX]{x1D70C}_{2}}\overline{R}^{2}}{3\overline{\unicode[STIX]{x1D6FC}}}}\left(\unicode[STIX]{x1D6FE}_{th}+\unicode[STIX]{x1D6FE}_{ac}\right),\quad \unicode[STIX]{x1D708}_{\mathit{Lin}}={\displaystyle \frac{\overline{\unicode[STIX]{x1D70C}_{2}}\overline{R}^{2}}{3\overline{\unicode[STIX]{x1D6FC}}}}. & & \displaystyle\end{eqnarray}$$

Thus we see that the above analysis and the resulting relation (3.19) provide a definition for $\unicode[STIX]{x1D700}$ that is different from (3.12). More specifically, as $\unicode[STIX]{x1D700}_{\mathit{Lin}}>\unicode[STIX]{x1D700}_{\mathit{RP}}$ we can see that (3.19) yields greater damping than (3.12). The discrepancy between $\unicode[STIX]{x1D700}_{\mathit{Lin}}$ and $\unicode[STIX]{x1D700}_{\mathit{RP}}$ can be explained by simplifying hypotheses at the core of model (2.30) and our simple bubbly fluid model. Indeed, (H3) does not allow us to describe thermal exchange in the fluids. Also, we have completely neglected pressure fluctuations within the bubbles although they make an important contribution to damping effects.

In the following, we will rely on the following relations:

1. (i) Pfriem’s expression that can be found in Cheng et al. (Reference Cheng, Drew and Lahey1983) for $\unicode[STIX]{x1D6FE}_{th}$ ,

   (3.20) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FE}_{th}^{P40}(\unicode[STIX]{x1D714})=\unicode[STIX]{x1D714}_{n}\,\frac{3(\unicode[STIX]{x1D6FE}_{1}-1)\sqrt{2a_{1}}}{2\sqrt{\unicode[STIX]{x1D714}}R}, & & \displaystyle\end{eqnarray}$$
   where $a_{1}$ is the thermal diffusivity of the gas and $\unicode[STIX]{x1D6FE}_{1}$ its ratio of specific heats.

2. (ii) The natural frequency $\unicode[STIX]{x1D714}_{n}$ is given by the relation from Cheng et al. (Reference Cheng, Drew and Lahey1983),

   (3.21) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}_{n}^{2}={\displaystyle \frac{3\unicode[STIX]{x1D705}_{1}p_{1}}{\unicode[STIX]{x1D70C}_{1}R^{2}}}, & & \displaystyle\end{eqnarray}$$
   with $\unicode[STIX]{x1D705}_{1}$ being the polytropic gas exponent.

3. (iii) For $\unicode[STIX]{x1D6FE}_{ac}$ , we consider the relation found in Prosperetti (Reference Prosperetti1977),

   (3.22) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FE}_{ac}^{P77}(\unicode[STIX]{x1D714})=0.5\,{\displaystyle \frac{\unicode[STIX]{x1D714}^{2}Rc_{2}}{c_{2}^{2}+(\unicode[STIX]{x1D714}R)^{2}}}. & & \displaystyle\end{eqnarray}$$

Finally, for practical purposes, we want to get rid of the dependence of these damping parameters on the frequency of the considered acoustic perturbation. A way to get simple and constant values for these damping effects is to evaluate expressions (3.20) and (3.22) at the natural frequency (3.21),

(3.23a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FE}_{th}=\unicode[STIX]{x1D6FE}_{th}^{P40}(\unicode[STIX]{x1D714}_{n})\quad \text{and}\quad \unicode[STIX]{x1D6FE}_{ac}=\unicode[STIX]{x1D6FE}_{ac}^{P77}(\unicode[STIX]{x1D714}_{n}). & & \displaystyle\end{eqnarray}$$

3.3 Dispersion relations for a plane and monochromatic wave

In order to test the relevance of our models and of the identification made for $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D708}$ , we compare the behaviour of systems (2.30), (2.34) and (2.36) in the acoustic regime, to experimental measures of sound waves dispersion in bubbly fluids. The linearization of systems (2.30), (2.34) and (2.36) and their acoustic properties are presented hereafter. The practical comparison with experimental data and reference dispersion relation will be given in § 4.

Considering smooth solutions of one-dimensional problems, all these systems can be expressed using the generic quasilinear form

(3.24) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}\boldsymbol{W}+\boldsymbol{A}(\boldsymbol{W})\unicode[STIX]{x2202}_{x}\boldsymbol{W}=\boldsymbol{S}(\boldsymbol{W}). & & \displaystyle\end{eqnarray}$$

Following standard lines (Whitham Reference Whitham1974; Burman & Sainsaulieu Reference Burman and Sainsaulieu1995), we seek a monochromatic wave solution of (3.24) by writing $\boldsymbol{W}$ in the form

(3.25a,b ) $$\begin{eqnarray}\displaystyle \boldsymbol{W}(x,t)=\boldsymbol{W}^{(0)}+r\boldsymbol{W}^{(1)}(x,t)+O(r^{2}),\quad \boldsymbol{W}^{(1)}(x,t)=\widehat{\boldsymbol{W}}^{(1)}\exp (\text{i}\unicode[STIX]{x1D714}t-\text{i}k(\unicode[STIX]{x1D714})x),\quad & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D714}$ is the angular frequency, $k$ the wavelength and $r$ is a small amplitude parameter. The states $\widehat{\boldsymbol{W}}^{(1)}$ and $\boldsymbol{W}^{(0)}$ are both constant. The fluid parameters involved with $\boldsymbol{W}^{(0)}$ are noted with the superscript $^{(0)}$ and for the sake of simplicity, we suppose that $\boldsymbol{W}^{(0)}$ is always a rest state, i.e.  $u^{(0)}=0$ .

Injecting (3.25) into (3.24) and identifying terms with respect to the powers of $r$ yields

(3.26a,b ) $$\begin{eqnarray}\displaystyle \boldsymbol{S}(\boldsymbol{W}^{(0)})=0,\quad \widehat{\boldsymbol{W}}^{(1)}\in \ker (\text{i}\unicode[STIX]{x1D714}\mathit{Id}-\text{i}k(\unicode[STIX]{x1D714})\boldsymbol{A}(\boldsymbol{W}^{(0)})-\boldsymbol{S}^{\prime }(\boldsymbol{W}^{(0)})). & & \displaystyle\end{eqnarray}$$

Consequently $\unicode[STIX]{x1D714}$ and $k(\unicode[STIX]{x1D714})$ are bound by the so-called dispersion relation

(3.27) $$\begin{eqnarray}\displaystyle \det (\text{i}\unicode[STIX]{x1D714}\mathit{Id}-\text{i}k(\unicode[STIX]{x1D714})\boldsymbol{A}(\boldsymbol{W}^{(0)})-\boldsymbol{S}^{\prime }(\boldsymbol{W}^{(0)}))=0, & & \displaystyle\end{eqnarray}$$

which allows us to defined the phase velocity and the spatial attenuation of the acoustic wave respectively by $\text{Re}[\unicode[STIX]{x1D714}/k(\unicode[STIX]{x1D714})]$ and $\text{Im}[k(\unicode[STIX]{x1D714})]$ . Now, let us detail the results for each system of our hierarchy. Let us note

(3.28) $$\begin{eqnarray}\displaystyle \mathsf{H}(\unicode[STIX]{x1D70C},Y,\unicode[STIX]{x1D6FC})=\unicode[STIX]{x1D70C}{\displaystyle \frac{Y(1-Y)c_{1}^{2}(\unicode[STIX]{x1D70C}_{1})c_{2}^{2}(\unicode[STIX]{x1D70C}_{2})}{\unicode[STIX]{x1D6FC}^{2}(1-\unicode[STIX]{x1D6FC})^{2}}}. & & \displaystyle\end{eqnarray}$$

1. (i) For the two-phase model with micro-inertia (2.30), the dispersion relation, the associated phase velocity $c_{\mathit{Phase}}^{\unicode[STIX]{x1D700},\unicode[STIX]{x1D708}}$ and the spatial attenuation $\unicode[STIX]{x1D6FD}^{\unicode[STIX]{x1D700},\unicode[STIX]{x1D708}}$ read

   (3.29a ) $$\begin{eqnarray}\displaystyle \left({\displaystyle \frac{k^{\unicode[STIX]{x1D700},\unicode[STIX]{x1D708}}(\unicode[STIX]{x1D714})}{\unicode[STIX]{x1D714}}}\right)^{2}={\displaystyle \frac{\unicode[STIX]{x1D708}\unicode[STIX]{x1D714}^{2}-\text{i}\unicode[STIX]{x1D700}\unicode[STIX]{x1D714}-c_{\mathit{Wood}}^{-2}\mathsf{H}(\unicode[STIX]{x1D70C}^{(0)},Y^{(0)},\unicode[STIX]{x1D6FC}^{(0)})}{\unicode[STIX]{x1D708}c_{\mathit{Frozen}}^{2}\unicode[STIX]{x1D714}^{2}-\text{i}\unicode[STIX]{x1D700}\,c_{\mathit{Frozen}}^{2}\unicode[STIX]{x1D714}-\mathsf{H}(\unicode[STIX]{x1D70C}^{(0)},Y^{(0)},\unicode[STIX]{x1D6FC}^{(0)})}}, & & \displaystyle\end{eqnarray}$$
   (3.29b,c ) $$\begin{eqnarray}\displaystyle c_{\mathit{Phase}}^{\unicode[STIX]{x1D700},\unicode[STIX]{x1D708}}(\unicode[STIX]{x1D714})=\text{Re}\left[{\displaystyle \frac{\unicode[STIX]{x1D714}}{k^{\unicode[STIX]{x1D700},\unicode[STIX]{x1D708}}(\unicode[STIX]{x1D714})}}\right],\quad \unicode[STIX]{x1D6FD}^{\unicode[STIX]{x1D700},\unicode[STIX]{x1D708}}(\unicode[STIX]{x1D714})=\text{Im}[k^{\unicode[STIX]{x1D700},\unicode[STIX]{x1D708}}(\unicode[STIX]{x1D714})]. & & \displaystyle\end{eqnarray}$$

2. (ii) For the micro-inertia free model (2.34) obtained by the limit $\unicode[STIX]{x1D708}\rightarrow 0$ , $\unicode[STIX]{x1D700}=O(1)$ , we get the dispersion relation, phase velocity $c_{\mathit{Phase}}^{\unicode[STIX]{x1D700}}$ and attenuation $\unicode[STIX]{x1D6FD}^{\unicode[STIX]{x1D700}}$ defined by

   (3.30a,b ) $$\begin{eqnarray}\displaystyle \left({\displaystyle \frac{k^{\unicode[STIX]{x1D700}}(\unicode[STIX]{x1D714})}{\unicode[STIX]{x1D714}}}\right)^{2}={\displaystyle \frac{\text{i}\unicode[STIX]{x1D700}\unicode[STIX]{x1D714}+c_{\mathit{Wood}}^{-2}\mathsf{H}(\unicode[STIX]{x1D70C}^{(0)},Y^{(0)},\unicode[STIX]{x1D6FC}^{(0)})}{\text{i}\unicode[STIX]{x1D700}c_{\mathit{Frozen}}^{2}\unicode[STIX]{x1D714}+\mathsf{H}(\unicode[STIX]{x1D70C}^{(0)},Y^{(0)},\unicode[STIX]{x1D6FC}^{(0)})}},\quad c_{\mathit{Phase}}^{\unicode[STIX]{x1D700}}(\unicode[STIX]{x1D714})=\text{Re}\left[{\displaystyle \frac{\unicode[STIX]{x1D714}}{k^{\unicode[STIX]{x1D700}}(\unicode[STIX]{x1D714})}}\right],\qquad \quad & & \displaystyle\end{eqnarray}$$
   (3.30c ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FD}^{\unicode[STIX]{x1D700}}(\unicode[STIX]{x1D714})=\text{Im}[k^{\unicode[STIX]{x1D700}}(\unicode[STIX]{x1D714})]. & & \displaystyle\end{eqnarray}$$

3. (iii) Finally, for the full equilibrium model (2.36) when $\unicode[STIX]{x1D708}\rightarrow 0$ , $\unicode[STIX]{x1D700}\rightarrow 0$ , the dispersion relation reads

   (3.31a-c ) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{k(\unicode[STIX]{x1D714})^{2}}{\unicode[STIX]{x1D714}^{2}}}={\displaystyle \frac{1}{c_{\mathit{Wood}}^{2}}},\quad c_{\mathit{Phase}}(\unicode[STIX]{x1D714})=c_{\mathit{Wood}},\quad \unicode[STIX]{x1D6FD}=0. & & \displaystyle\end{eqnarray}$$

We recall that $c_{\mathit{Wood}}$ and $c_{\mathit{Frozen}}$ are defined by (2.37) and (2.32).

Let us first remark a distinctive behaviour of the acoustic waves for the system (2.30) equipped with both damping and micro-inertia: in the case of low internal dissipation i.e. small values of $\unicode[STIX]{x1D700}$ , the dispersion relation (3.29) leads to resonance in the vicinity of the frequency

(3.32) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}_{\mathit{res}}=\frac{1}{c_{\mathit{Wood}}}\sqrt{{\displaystyle \frac{\mathsf{H }(\unicode[STIX]{x1D70C}^{(0)},Y^{(0)},\unicode[STIX]{x1D6FC}^{(0)})}{\unicode[STIX]{x1D708}}}}. & & \displaystyle\end{eqnarray}$$

We observe that when one accounts for internal damping with $\unicode[STIX]{x1D700}>0$ and with micro-inertia (respectively without micro-inertia), the phase velocity of the acoustic wave $c_{\mathit{Phase}}^{\unicode[STIX]{x1D700},\unicode[STIX]{x1D708}}$ (respectively $c_{\mathit{Phase}}^{\unicode[STIX]{x1D700}}$ ) is not equal to the sound velocity $c$ (respectively $c_{\mathit{Frozen}}$ ) issued from the characteristic velocities when one discards the source terms in the system. This underlines the fact that the micro-inertia and the damping source terms have a substantial influence on the phase velocity of the acoustic waves.

Figure 1.

Phase velocity and spatial attenuation for the 5-equation model and influence of the value of $\unicode[STIX]{x1D708}$ .

Let us now examine the variations of the dispersion relations across the hierarchy. As $\unicode[STIX]{x1D708}$ and $\unicode[STIX]{x1D700}$ reach their asymptotic limit, the transition from one model to another materializes through the dispersion relations (3.29), (3.30) and (3.31). Indeed, we see that

(3.33a-c ) $$\begin{eqnarray}\displaystyle \lim _{\unicode[STIX]{x1D708}\rightarrow 0}c_{\mathit{Phase}}^{\unicode[STIX]{x1D700},\unicode[STIX]{x1D708}}=c_{\mathit{Phase}}^{\unicode[STIX]{x1D700}},\quad \lim _{\substack{ \unicode[STIX]{x1D708}\rightarrow 0 \\ \unicode[STIX]{x1D700}\rightarrow 0}}c_{\mathit{Phase}}^{\unicode[STIX]{x1D700},\unicode[STIX]{x1D708}}=c_{\mathit{Phase}},\quad \lim _{\unicode[STIX]{x1D700}\rightarrow 0}c_{\mathit{Phase}}^{\unicode[STIX]{x1D700}}=c_{\mathit{Phase}}. & & \displaystyle\end{eqnarray}$$

Let us note that these limits are not uniform over all frequencies: this is illustrated in figure 2. Indeed, we can observe that, whatever the value of $\unicode[STIX]{x1D700}$ , there is a frequency above which $c_{\mathit{Phase}}^{\unicode[STIX]{x1D700}}$ will be close to $c_{\mathit{Frozen}}$ . However, this critical frequency increases when $\unicode[STIX]{x1D700}$ decreases. On the contrary, the transition from $c_{\mathit{Phase}}^{\unicode[STIX]{x1D700},\unicode[STIX]{x1D708}}$ to $c_{\mathit{Phase}}^{\unicode[STIX]{x1D700}}$ is more uniform, as illustrated in figure 1. Indeed, when $\unicode[STIX]{x1D708}$ decreases, the damping effects due to $\unicode[STIX]{x1D700}$ prevail: although the resonant frequency only depends on $\unicode[STIX]{x1D708}$ , the effects of resonance are completely attenuated because of $\unicode[STIX]{x1D700}$ . One can also note that the asymptotic behaviour for spatial attenuation at low frequencies does not depend on the value of $\unicode[STIX]{x1D708}$ , but depends on the value of $\unicode[STIX]{x1D700}$ , see figure 2 for the 4-equation model.

The model hierarchy also shows through when one spans frequency values $\unicode[STIX]{x1D714}$ (and can also be noticed in figures 1 and 2). Indeed, if one considers acoustic waves at low frequencies $\unicode[STIX]{x1D714}\ll 1$ , then the dispersion relations (3.29), (3.30) and (3.31) yield

(3.34) $$\begin{eqnarray}\displaystyle \left({\displaystyle \frac{k^{\unicode[STIX]{x1D700},\unicode[STIX]{x1D708}}(\unicode[STIX]{x1D714})}{\unicode[STIX]{x1D714}}}\right)^{2}=\left({\displaystyle \frac{k^{\unicode[STIX]{x1D700}}(\unicode[STIX]{x1D714})}{\unicode[STIX]{x1D714}}}\right)^{2}+O(\unicode[STIX]{x1D714}^{2})={\displaystyle \frac{1}{c_{\mathit{Wood}}^{2}}}+O(\unicode[STIX]{x1D714}). & & \displaystyle\end{eqnarray}$$

In terms of phase velocities and attenuation we obtain

(3.35a,b ) $$\begin{eqnarray}\displaystyle c_{\mathit{Phase}}^{\unicode[STIX]{x1D700},\unicode[STIX]{x1D708}}=c_{\mathit{Phase}}^{\unicode[STIX]{x1D700}}+O(\unicode[STIX]{x1D714}^{2})=c_{\mathit{Wood}}+O(\unicode[STIX]{x1D714}),\quad \unicode[STIX]{x1D6FD}^{\unicode[STIX]{x1D700},\unicode[STIX]{x1D708}}=\unicode[STIX]{x1D6FD}^{\unicode[STIX]{x1D700}}+O(\unicode[STIX]{x1D714}^{4})=O(\unicode[STIX]{x1D714}). & & \displaystyle\end{eqnarray}$$

In the limit $\unicode[STIX]{x1D714}\rightarrow 0$ , the phase velocity of the acoustic waves for all models tends to $c_{\mathit{Phase}}=c_{\mathit{Wood}}$ and the spatial attenuation vanishes. Let us now turn to high frequencies $\unicode[STIX]{x1D714}\gg 1$ . From (3.29), (3.30) and (3.31) we have

(3.36) $$\begin{eqnarray}\displaystyle \left({\displaystyle \frac{k^{\unicode[STIX]{x1D700},\unicode[STIX]{x1D708}}(\unicode[STIX]{x1D714})}{\unicode[STIX]{x1D714}}}\right)^{2}=\left({\displaystyle \frac{k^{\unicode[STIX]{x1D700}}(\unicode[STIX]{x1D714})}{\unicode[STIX]{x1D714}}}\right)^{2}+O\left({\displaystyle \frac{1}{\unicode[STIX]{x1D714}^{2}}}\right)={\displaystyle \frac{1}{c_{\mathit{Frozen}}^{2}}}+O\left({\displaystyle \frac{1}{\unicode[STIX]{x1D714}}}\right). & & \displaystyle\end{eqnarray}$$

Thus the acoustic waves of both systems equipped with internal damping are such that

(3.37) $$\begin{eqnarray}\displaystyle c_{\mathit{Phase}}^{\unicode[STIX]{x1D700},\unicode[STIX]{x1D708}}=c_{\mathit{Phase}}^{\unicode[STIX]{x1D700}}+O\left({\displaystyle \frac{1}{\unicode[STIX]{x1D714}^{2}}}\right)=c_{\mathit{Frozen}}+O\left({\displaystyle \frac{1}{\unicode[STIX]{x1D714}}}\right). & & \displaystyle\end{eqnarray}$$

Consequently, in the limit $\unicode[STIX]{x1D714}\rightarrow +\infty$ , the phase velocities of the acoustic waves associated with (2.30) and (2.18) tend to $c_{\mathit{Frozen}}$ , while it remains constant and equal to $c_{\mathit{Wood}}$ for (2.36). In all cases the spatial attenuation tends to $0$ .

Figure 2.

Phase velocity and spatial attenuation for the 4-equation model and influence of the value of $\unicode[STIX]{x1D700}$ .

4.1 Influence of micro-viscosity and micro-inertia in the acoustic regime

We shall examine the behaviour of the models when $\unicode[STIX]{x1D714}$ spans the possible frequencies and distinguish three main ranges of frequency for characterizing the phase velocity and the spatial attenuation. Then we will focus specifically on the near-resonance frequencies.

4.1.1 Comparison across the whole spectrum of frequencies $\unicode[STIX]{x1D714}$

We consider a set of measures from Silberman (Reference Silberman1957) that involves a flow characterized by $R=2.5\,\text{mm}$ and $\unicode[STIX]{x1D6FC}=5.84\times 10^{-4}$ . Using relations (3.12) and (3.19), we obtain the following values for the parameters of the 5-equation model (2.30) and the 4-equation model (2.34)

(4.4a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D700}_{\mathit{RP}}=2.28,\quad \unicode[STIX]{x1D708}_{\mathit{RP}}=3.57, & & \displaystyle\end{eqnarray}$$

(4.5a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D700}_{\mathit{Lin}}=1.61\times 10^{3},\quad \unicode[STIX]{x1D708}_{\mathit{Lin}}=3.57. & & \displaystyle\end{eqnarray}$$

Figure 3 displays both phase velocity and spatial attenuation for all models of the hierarchy, for the model of Cheng, Drew and Lahey (CDL model) (Cheng et al. Reference Cheng, Drew and Lahey1985), superimposed on the experimental results (Silberman Reference Silberman1957).

Figure 3.

Dispersion relations for the CDL model and the different models of the hierarchy (numbered lines) and Silberman’s measures (symbols) for radii of bubbles around $R=2.0\times 10^{-3}~\text{m}$ , $\unicode[STIX]{x1D6FC}=5.84\times 10^{-4}$ . The model parameters are thus: $\unicode[STIX]{x1D708}=3.6$ , $\unicode[STIX]{x1D700}_{\mathit{RP}}=2.3$ , $\unicode[STIX]{x1D700}_{\mathit{Lin}}=1.6\times 10^{3}$ .

1. (i) Range $\unicode[STIX]{x1D714}\ll \unicode[STIX]{x1D714}_{\mathit{res}}^{\mathit{ref}}$ . For low frequencies, the pressure perturbation is very slow and we can thus expect the bubbles of the system to remain at an equilibrium state with respect to both mechanics and thermodynamics. Little internal dissipation is involved with this regime, which is visible through the measures that show a low spatial attenuation. The evaluation of $c_{\mathit{Phase}}^{\mathit{ref}}$ in the experiment provides values that are close to $c_{\mathit{Wood}}$ . All the models of the hierarchy show a good agreement with these data, as seen in figure 3. As $\unicode[STIX]{x1D714}$ increases, the spatial attenuation $\unicode[STIX]{x1D6FD}^{\mathit{ref}}$ also increases. This trend is correctly followed by the models of the hierarchy that account for internal dissipation. Nevertheless, Prosperetti (Reference Prosperetti1977) underlines that, in this regime and up to a certain frequency, the thermal dissipation is the dominant internal dissipation effect. For all models, $\unicode[STIX]{x1D6FD}^{\unicode[STIX]{x1D700},\unicode[STIX]{x1D708}}$ and $\unicode[STIX]{x1D6FD}^{\unicode[STIX]{x1D700}}$ are lower than experimental data. However, if the 5-equation model (2.30) importantly underestimates $\unicode[STIX]{x1D6FD}^{\unicode[STIX]{x1D700},\unicode[STIX]{x1D708}}$ when the damping is identified with Rayleigh–Plesset $(\unicode[STIX]{x1D700}=\unicode[STIX]{x1D700}_{\mathit{RP}})$ , the match with the reference data is very good when $\unicode[STIX]{x1D700}=\unicode[STIX]{x1D700}_{\mathit{Lin}}$ .

2. (ii) Range $\unicode[STIX]{x1D714}$ close to $\unicode[STIX]{x1D714}_{\mathit{res}}^{\mathit{ref}}$ . Near resonance, $\unicode[STIX]{x1D6FD}^{\mathit{ref}}$ increases with $\unicode[STIX]{x1D714}$ and becomes very large. On the contrary, $c_{\mathit{Phase}}^{\mathit{ref}}$ decreases as $\unicode[STIX]{x1D714}$ increases. It reaches $c_{\mathit{Phase}}^{\mathit{ref}}=0$ for some frequency $\unicode[STIX]{x1D714}_{\mathit{ext}}^{\mathit{ref}}<\unicode[STIX]{x1D714}_{\mathit{res}}^{\mathit{ref}}$ . For $\unicode[STIX]{x1D714}\in [\unicode[STIX]{x1D714}_{\mathit{ext}}^{\mathit{ref}},\unicode[STIX]{x1D714}_{\mathit{res}}^{\mathit{ref}}]$ , there is a good agreement between $c_{\mathit{Phase}}^{\unicode[STIX]{x1D700},\unicode[STIX]{x1D708}}$ , $c_{\mathit{Phase}}^{\unicode[STIX]{x1D700}}$ and $c_{\mathit{Phase}}^{\mathit{ref}}$ . On the contrary, there is an important discrepancy between the phase velocity predicted by the CDL model and $c_{\mathit{Phase}}^{\unicode[STIX]{x1D700},\unicode[STIX]{x1D708}}$ , $c_{\mathit{Phase}}^{\unicode[STIX]{x1D700}}$ for this range of frequencies. It is worth noting that $c_{\mathit{Phase}}^{\unicode[STIX]{x1D700},\unicode[STIX]{x1D708}}$ is much closer to the reference value for $\unicode[STIX]{x1D700}=\unicode[STIX]{x1D700}_{\mathit{Lin}}$ than for $\unicode[STIX]{x1D700}=\unicode[STIX]{x1D700}_{\mathit{RP}}$ . Concerning the spatial attenuation, we can see that the models of the hierarchy that account for micro-inertia, namely (2.30) and (2.18) fit quite well the reference results. In contrast, models (2.34) and (2.36) yield a poor estimate of the spatial attenuation. This suggests that the source terms related to $\unicode[STIX]{x1D708}$ in (2.27d ) and (2.27e ) play a key role in the system behaviour when $\unicode[STIX]{x1D714}\approx \unicode[STIX]{x1D714}_{\mathit{res}}^{\mathit{ref}}$ . Moreover it also hints that our estimate for $\unicode[STIX]{x1D708}$ in (3.12) and (4.5) is coherent. Finally, the results suggest that the source terms related to $\unicode[STIX]{x1D700}$ in (2.18) do not have a great influence on the values of $\unicode[STIX]{x1D6FD}^{\unicode[STIX]{x1D700},\unicode[STIX]{x1D708}}$ in this range of frequencies.

3. (iii) Range $\unicode[STIX]{x1D714}\gg \unicode[STIX]{x1D714}_{\mathit{res}}^{\mathit{ref}}$ . For high frequencies, very few experimental data are available and thus the main comparison elements are given by the CDL model. In this regime, one can presume that acoustic radiation effects cannot be neglected. Indeed, the bubbles start emitting acoustic waves that are transmitted to the liquid. This process will remove energy from the bubbles and therefore will become the main damping effect of the system. For all systems of our hierarchy except (2.36), the phase velocity will tend to $c_{\mathit{Frozen}}$ , which agrees with the behaviour of the CDL model. The spatial attenuation coefficients of the model hierarchy do not match well the reference data of the CDL model in this range of frequencies: (2.34) clearly overestimates damping (purple number 2 line), when model (2.30) with matching coefficients $\unicode[STIX]{x1D708}_{\mathit{RP}}$ and $\unicode[STIX]{x1D700}_{\mathit{RP}}$ provides too low a dissipation (red number 3 line). For the same model, the micro-viscosity choice $\unicode[STIX]{x1D700}_{\mathit{Lin}}$ clearly increases the damping effect but still at a much lower level than the dissipation of the CDL model.

4.1.2 Finer comparison near resonance

In the experiment of Leroy et al. (Reference Leroy, Strybulevych, Page and Scanlon2008), the set of measures is very dense for $\unicode[STIX]{x1D714}$ close to $\unicode[STIX]{x1D714}_{\mathit{res}}^{\mathit{ref}}$ . In this paragraph we discard both the 4-equation model (2.34) and the 3-equation model (2.36) as they cannot produce resonant behaviour. The bubbles in Leroy et al. (Reference Leroy, Strybulevych, Page and Scanlon2008) are smaller than those of Silberman (Reference Silberman1957). Thus, we consider different values of $(R,\unicode[STIX]{x1D6FC})$ by setting $R\approx 8.1\times 10^{-5}~\text{m}$ and $\unicode[STIX]{x1D6FC}=1.5\times 10^{-4}$ . Thanks to (3.12) and (3.19) we obtain

(4.6a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D700}_{\mathit{RP}}=8.89\times 10^{-1},\quad \unicode[STIX]{x1D708}_{\mathit{RP}}=1.46\times 10^{-3}. & & \displaystyle\end{eqnarray}$$

(4.7a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D700}_{\mathit{Lin}}=87.3,\quad \unicode[STIX]{x1D708}_{\mathit{Lin}}=1.46\times 10^{-3}. & & \displaystyle\end{eqnarray}$$

The results we obtain with this set of parameters is consistent with the previous comparison. Indeed, in figure 4 we can see that for $\unicode[STIX]{x1D714}$ close to $\unicode[STIX]{x1D714}_{\mathit{res}}^{\mathit{ref}}$ the phase velocity $c_{\mathit{Phase}}^{\unicode[STIX]{x1D700},\unicode[STIX]{x1D708}}$ is clearly overestimated for $\unicode[STIX]{x1D700}_{\mathit{RP}}$ but the match with experimental data for $\unicode[STIX]{x1D700}_{\mathit{Lin}}$ seems more accurate. Regarding the attenuation $\unicode[STIX]{x1D6FD}^{\unicode[STIX]{x1D700},\unicode[STIX]{x1D708}}$ , the choice of $\unicode[STIX]{x1D700}_{\mathit{Lin}}$ gives clearly a better match with the reference data than $\unicode[STIX]{x1D700}_{\mathit{RP}}$ .

Figure 4.

Dispersion relations for the different models from the hierarchy (numbered lines) and the Leroy et al. measures (symbols) for radii of bubbles around $R=8.1\times 10^{-5}~\text{m}$ , $\unicode[STIX]{x1D6FC}=1.5\times 10^{-4}$ . The model parameters are thus: $\unicode[STIX]{x1D708}=1.5\times 10^{-3}$ , $\unicode[STIX]{x1D700}_{\mathit{RP}}=8.9\times 10^{-1}$ , $\unicode[STIX]{x1D700}_{\mathit{Lin}}=8.7\times 10^{1}$ .

4.2 On the evaluation of the micro-viscosity and the micro-inertia

We succeeded in identifying the micro-inertia $\unicode[STIX]{x1D708}=\unicode[STIX]{x1D708}_{\mathit{Lin}}=\unicode[STIX]{x1D708}_{\mathit{RP}}$ and the micro-viscosity $\unicode[STIX]{x1D700}=\unicode[STIX]{x1D700}_{\mathit{RP}}$ (respectively $\unicode[STIX]{x1D700}=\unicode[STIX]{x1D700}_{\mathit{Lin}}$ ) in the 5-equation model (2.30) thanks to a comparison with the Rayleigh–Plesset equation (3.10) (respectively the linear radius evolution equation (3.17)). In this sense the 5-equation model (2.30) can be considered as a host for different bubbly fluid models that will characterize the micro-inertia and micro-viscosity.

In our case, the values of $\unicode[STIX]{x1D700}_{\mathit{RP}}$ and $\unicode[STIX]{x1D700}_{\mathit{Lin}}$ are significantly different and for a given $\unicode[STIX]{x1D714}\geqslant \unicode[STIX]{x1D714}_{\mathit{res}}^{\mathit{ref}}$ , $\unicode[STIX]{x1D700}_{\mathit{RP}}$ and $\unicode[STIX]{x1D700}_{\mathit{Lin}}$ yield different acoustic behaviours for system (2.30). Nevertheless, the agreement with reference data is better for $\unicode[STIX]{x1D700}=\unicode[STIX]{x1D700}_{\mathit{Lin}}$ . We believe that this discrepancy comes from the over-simplifying hypotheses of § 3.1: (H6) and (H7) do not allow us to account for pressure fluctuations within the bubble, neither for pressure fluctuations in the liquid that act as inertial terms with respect to the bubble motion. These phenomena are the grounds for thermal and acoustic damping effects, that are respectively driven by $\unicode[STIX]{x1D6FE}_{ac}$ and $\unicode[STIX]{x1D6FE}_{th}$ (Prosperetti Reference Prosperetti1977; Cheng et al. Reference Cheng, Drew and Lahey1983). Indeed, what we observed in § 4.1 is consistent with the analysis of Prosperetti (Reference Prosperetti1977): for the flows settings we chose, both thermal and acoustic damping are the main damping effects involved in the bubble vibrations. According to Prosperetti (Reference Prosperetti1977), when the radius is close to the value of Silberman’s experiment, $R=1~\text{mm}$ , one can estimate that $\unicode[STIX]{x1D6FE}_{th}/\unicode[STIX]{x1D6FE}_{vis}\in [5,5000]$ for $\unicode[STIX]{x1D714}\leqslant 10^{5}~\text{s}^{-1}$ and $\unicode[STIX]{x1D6FE}_{ac}/\unicode[STIX]{x1D6FE}_{vis}\in [5,5000]$ for $\unicode[STIX]{x1D714}\geqslant 5\times 10^{3}~\text{s}^{-1}$ . When $R=8.1\times 10^{-2}~\text{mm}$ , a value close to the bubbles radii in the Leroy et al. experiment, we have that $\unicode[STIX]{x1D6FE}_{th}/\unicode[STIX]{x1D6FE}_{vis}\in [10,500]$ for $\unicode[STIX]{x1D714}\leqslant 10^{6}~\text{s}^{-1}$ and $\unicode[STIX]{x1D6FE}_{ac}/\unicode[STIX]{x1D6FE}_{vis}\in [3,500]$ for $\unicode[STIX]{x1D714}\geqslant 10^{5}~\text{s}^{-1}$ .

Finally, we saw that, by increasing the value of $\unicode[STIX]{x1D6FD}^{\unicode[STIX]{x1D700},\unicode[STIX]{x1D708}}$ , it is possible to enforce an equivalent damping $\unicode[STIX]{x1D6FD}^{\unicode[STIX]{x1D700},\unicode[STIX]{x1D708}}\approx \unicode[STIX]{x1D6FD}_{th}$ that better fits the reference data in this range of frequencies, especially close to resonance. Thus, this enables an alternate means for determining $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D708}$ : one could tune $(\unicode[STIX]{x1D700},\unicode[STIX]{x1D708})$ in a heuristic way to better fit $c_{\mathit{Phase}}^{\unicode[STIX]{x1D700},\unicode[STIX]{x1D708}}$ and $\unicode[STIX]{x1D6FD}^{\unicode[STIX]{x1D700},\unicode[STIX]{x1D708}}$ with respect to reference data, and thus use the acoustic regime behaviour of model (2.30) as an evaluation tool.