2.3.1 Layered spherical defect

Consider an inhomogeneous sphere of porous medium embedded in a homogeneous host medium. We assume that the permeability of the sphere depends only on the radius and is a piecewise constant function, i.e., the sphere is a set of nested spherical layers. The permeability of the background medium is kð Þ<sup>o</sup> , that of the outermost layer is kð Þ<sup>1</sup> , and so on to the central sphere whose permeability is kð Þ<sup>i</sup> .

To calculate the perturbation of the incident pressure due to the sphere seen as a scatter, we proceed the following: the pressure field is calculated in each layer of the sphere. The pressure field in the layer number n is related to those in the layer number n þ 1 and n � 1 by the boundary conditions. It is assumed that the defect is a set of N concentric spherical layers in which the value of permeability is constant kð Þ <sup>n</sup> . Let κð Þ <sup>n</sup> denotes the ratio kð Þ <sup>n</sup> =η in the layer number n delimited by the spheres of radii rn and rnþ<sup>1</sup> such that rn >rnþ1. The core of the sphere has index i ¼ N (Figure 9). The determination of pressure and velocity in this type of inclusion consists in solving the Laplace equation <sup>Δ</sup>pð Þ <sup>n</sup> <sup>¼</sup> 0 in each layer and in connecting the solutions using the boundary conditions: continuity of fluid flow and that of the radial component of the stress.

In the layer number n, the pressure is noted:

$$p^{(n)} = A^{(n)} r \cos \theta + B^{(n)} r^{-2} \cos \theta. \tag{47}$$

The coefficients A and B of two consecutive layers are connected by the following conditions in r ¼ rnþ1:

$$
\phi^{(n)} v\_r^{(n)}(r = r\_{n+1}) = \phi^{(n+1)} v\_r^{(n+1)}(r = r\_{n+1}),\tag{48}
$$

$$
\tau\_{rr}^{(n)}(r = r\_{n+1}) = \tau\_{rr}^{(n+1)}(r = r\_{n+1}).\tag{49}
$$

Figure 9. Layered sphere.

function of κð Þ<sup>i</sup> =κð Þ<sup>o</sup> for different sets of the depolarization factors when incident

Hydraulic polarizability <sup>χ</sup> <sup>¼</sup> <sup>α</sup>x=<sup>V</sup> of an ellipsoidal inclusion vs. log <sup>κ</sup>ð Þ<sup>i</sup> <sup>=</sup>κð Þ<sup>o</sup> for different values of the depolarization factors (Nx, Ny, Nz) (�•�: Nx ¼ 0:1, Ny ¼ 0:1, Nz ¼ 0:8); (�⋄�: Nx ¼ 0:2, Ny ¼ 0:2, Nz <sup>¼</sup> <sup>0</sup>:6); (�◀�: Nx <sup>¼</sup> <sup>0</sup>:4, Ny <sup>¼</sup> <sup>0</sup>:4, Nz <sup>¼</sup> <sup>0</sup>:2); (�■�: Nx <sup>¼</sup> <sup>0</sup>:45, Ny <sup>¼</sup> <sup>0</sup>:45, Nz <sup>¼</sup> <sup>0</sup>:1); (–––:

<sup>κ</sup>ð Þ<sup>o</sup> <sup>r</sup> cos <sup>θ</sup> <sup>þ</sup>

This relation is similar to (29), differing from it only by the expression of the dipole moment. The fundamental difference between the spherical and ellipsoidal inclusions is that the pressure scattered by the sphere contains only a dipolar field, whereas in strictness, the ellipsoid also scatters high-order multipolar fields. We can then deduce from this remark that the more the shape of the inclusion is distant from that of the sphere, the more the scattered pressure contains high-order

The result obtained in (42) does not show these terms since the calculation of the integral F<sup>2</sup> is an approximate computation when ξ=r ≫ 1. When we move away from the ellipsoidal inclusion, the multipolar terms of order greater than 2 decrease very quickly, leaving only the contribution of the incident pressure and the dipole term

In fact, defects rarely exhibit homogeneous structure. The parameter that characterizes the defect (in our case the permeability) is generally a variable varying according to a law which depends on the way in which the defect develops. For the spherical defect, the simplest situation is the radial variation of the

permeability. The fundamental difference between homogeneous and

Pelli,a 4π

cos θ

<sup>r</sup><sup>2</sup> : (46)

pressure is along Ox.

Acoustics of Materials

Nx ¼ 1=3, Ny ¼ 1=3, Nz ¼ 1=3).

Figure 8.

multipolar terms.

72

of the scattered pressure.

2.3 Inhomogeneous defect

The pressure outside the inclusion is then:

<sup>p</sup>ð Þ<sup>o</sup> ð Þ¼� <sup>r</sup>; <sup>θ</sup> <sup>U</sup><sup>∞</sup>

Transfer matrix: The linearity of the problem makes it possible to write that the pairs of coefficients (An, Bn) and (Anþ1, Bnþ1) are linked by a matrix equation such that:

$$
\begin{pmatrix} A^{(n)} \\ B^{(n)} \end{pmatrix} = T\_{n, n+1} \begin{pmatrix} A^{(n+1)} \\ B^{(n+1)} \end{pmatrix},\tag{50}
$$

Sn,nþ<sup>1</sup> <sup>¼</sup> <sup>1</sup>

product. From

we get:

refer to [12].

relations:

75

where Sn,nþ<sup>2</sup> ¼ Sn,nþ<sup>1</sup>

the system of differential equations:

2.3.3 Layered ellipsoidal inclusion

κð Þ <sup>n</sup>þ<sup>1</sup> þ 2κð Þ <sup>n</sup>

DOI: http://dx.doi.org/10.5772/intechopen.84158

Tortuosity Perturbations Induced by Defects in Porous Media

<sup>κ</sup>ð Þ <sup>n</sup>þ<sup>1</sup> � <sup>κ</sup>ð Þ <sup>n</sup> � �r<sup>3</sup>

Bð Þ <sup>n</sup> Að Þ <sup>n</sup>þ<sup>1</sup> !

Bð Þ <sup>n</sup>þ<sup>1</sup> Að Þ <sup>n</sup>þ<sup>2</sup> !

> Bð Þ <sup>n</sup> Að Þ <sup>n</sup>þ<sup>2</sup> !

2.3.2 Spherical inclusion with continuously variable permeability

d dr

> d dr

<sup>a</sup><sup>2</sup> � <sup>a</sup><sup>2</sup>

The scattering matrix of two consecutive layers is given by their Redheffer

¼ Sn,nþ<sup>1</sup>

¼ Snþ1,nþ<sup>2</sup>

¼ Sn,nþ<sup>2</sup>

Redheffer star product. For more details about the Redheffer star product, we can

When κ is a continuous function of the variable r, the matrix Eq. (50) becomes

The generalization of the radial variation of the permeability of the spherical inclusion to the ellipsoidal requires that the permeability only depends on ξ. This is true in orthogonal directions at its surface. This condition entails that inside the ellipsoid, the strata are limited by confocal ellipsoidal surfaces ξ ¼ ξk, i.e., having the same foci as the surface ξ; hence, their semiaxes are related by the following

<sup>k</sup> <sup>¼</sup> <sup>b</sup><sup>2</sup> � <sup>b</sup><sup>2</sup>

Consider a porous inhomogeneous ellipsoidal inclusion having the permeability kð Þ<sup>i</sup> embedded in a background medium of homogeneous mobility κð Þ<sup>o</sup> . We assume that the mobility of the inclusion is stratified, i.e., it is a constant piecewise function and each layer has its own mobility κð Þ <sup>n</sup> . The mobility in the outermost layer is κð Þ<sup>1</sup> , that of the next layer is κð Þ<sup>2</sup> , etc. to the central layer whose the mobilty is κð Þ<sup>i</sup> . The layers are limited by the confocal surfaces ξ ¼ ξ<sup>k</sup> whose semiaxes obey (65) and are numbered from 1 to i ¼ N from the outside to the inside, such that the strata n and

<sup>k</sup> ¼ c

<sup>2</sup> � <sup>c</sup> 2

2

r<sup>3</sup>κ<sup>2</sup>dA=dr <sup>d</sup>κ=dr � � <sup>þ</sup> <sup>r</sup>

κ<sup>2</sup>dB=dr <sup>r</sup><sup>3</sup>dκ=dr � � � <sup>2</sup><sup>κ</sup>

<sup>n</sup>þ<sup>1</sup> <sup>3</sup>κð Þ <sup>n</sup>þ<sup>1</sup> <sup>3</sup>κð Þ <sup>n</sup> <sup>2</sup> <sup>κ</sup>ð Þ <sup>n</sup>þ<sup>1</sup> � <sup>κ</sup>ð Þ <sup>n</sup> � �r�<sup>3</sup>

!

Að Þ <sup>n</sup> Bð Þ <sup>n</sup>þ<sup>1</sup> !

Að Þ <sup>n</sup>þ<sup>1</sup> Bð Þ <sup>n</sup>þ<sup>2</sup> !

Að Þ <sup>n</sup> Bð Þ <sup>n</sup>þ<sup>2</sup> !

<sup>∗</sup> Snþ1,nþ2. In this relation, the right-hand side is the

nþ1

, (60)

, (61)

, (62)

κA ¼ 0 (63)

<sup>r</sup><sup>4</sup> <sup>B</sup> <sup>¼</sup> <sup>0</sup>: (64)

<sup>k</sup>: (65)

: (59)

where Tn,nþ<sup>1</sup> is the transfer matrix between the two consecutive layers n and n þ 1, the entries of which are:

$$T\_{11} = \frac{\phi^{(n+1)}}{\phi^n} \left( \frac{k^{(n+1)}}{k^n} + 2 \frac{\mathbf{1} - \frac{k^{(n+1)}}{k^n}}{\mathbf{3} + \mathbf{1} 2k^n r\_n^{-2}} \right), \tag{51}$$

$$T\_{12} = 2\frac{\phi^{(n+1)}}{\phi^n} r\_n^{-3} \left( -\frac{k^{n+1}}{k^n} + \frac{\mathbf{1} + 2\frac{k^{(n+1)}}{k^n} + \mathbf{12}k^{(n+1)}r\_n^{-2}}{\mathbf{3} + \mathbf{12}k^n r\_n^{-2}} \right),\tag{52}$$

$$T\_{21} = \frac{\phi^{(n+1)}}{\phi^n} r\_n^3 \left( \frac{1 - \frac{k^{(n+1)}}{k^\*}}{3 + 12k^n r\_n^{-2}} \right), \tag{53}$$

$$T\_{22} = \frac{\phi^{(n+1)}}{\phi^n} \left( \frac{\mathbf{1} + 2\frac{k^{(n+1)}}{k^n} + \mathbf{1} 2k^{(n+1)}r\_n^{-2}}{\mathbf{3} + \mathbf{1} 2k^n r\_n^{-2}} \right). \tag{54}$$

From <sup>A</sup><sup>0</sup> ¼ �U∞=κð Þ<sup>o</sup> and <sup>B</sup>ð Þ<sup>i</sup> <sup>¼</sup> 0, it is possible to get the pressure in each layer of the sphere.

The effects of two consecutive layers are obtained by the product of the transfer matrices of each of these layers. So from the matrix equations:

$$
\begin{pmatrix} A^{(n)} \\ B^{(n)} \end{pmatrix} = \begin{pmatrix} T\_{n,n+1} \begin{pmatrix} A^{(n+1)} \\ B^{(n+1)} \end{pmatrix}, \\\\ \end{pmatrix} \tag{55}
$$

$$
\begin{pmatrix} A^{(n+1)} \\ B^{(n+1)} \end{pmatrix} = \begin{pmatrix} T\_{n+1,n+2} \\ B^{(n+2)} \end{pmatrix},\tag{56}
$$

we get:

$$
\begin{pmatrix} A^{(n)} \\ B^{(n)} \end{pmatrix} = T\_{n, n+2} \begin{pmatrix} A^{(n+2)} \\ B^{(n+2)} \end{pmatrix},\tag{57}
$$

where Tn,nþ<sup>2</sup> ¼ Tn,nþ<sup>1</sup>Tnþ1,nþ2.

Scattering matrix: Another way of linking the coefficients A and B of two consecutive layers is the use of the scattering matrix Sn,nþ1. The scattering matrix is sometimes more efficient than the transfer matrix for calculating the amplitudes of the waves reflected and transmitted by an object subjected to incident waves. It is such that:

$$
\begin{pmatrix} B^{(n)} \\ A^{(n+1)} \end{pmatrix} = \mathbb{S}\_{n, n+1} \begin{pmatrix} A^{(n)} \\ B^{(n+1)} \end{pmatrix}. \tag{58}
$$

Its entries are:

Tortuosity Perturbations Induced by Defects in Porous Media DOI: http://dx.doi.org/10.5772/intechopen.84158

$$\mathcal{S}\_{n,n+1} = \frac{\mathbf{1}}{\kappa^{(n+1)} + \mathbf{2}\kappa^{(n)}} \begin{pmatrix} (\kappa^{(n+1)} - \kappa^{(n)}) r\_{n+1}^3 & \mathbf{3}\kappa^{(n+1)} \\ \mathbf{3}\kappa^{(n)} & \mathbf{2} \left(\kappa^{(n+1)} - \kappa^{(n)}\right) r\_{n+1}^{-3} \end{pmatrix}. \tag{59}$$

The scattering matrix of two consecutive layers is given by their Redheffer product. From

$$
\begin{pmatrix} B^{(n)} \\ A^{(n+1)} \end{pmatrix} = \mathbb{S}\_{n, n+1} \begin{pmatrix} A^{(n)} \\ B^{(n+1)} \end{pmatrix},\tag{60}
$$

$$
\begin{pmatrix} \mathcal{B}^{(n+1)} \\ \mathcal{A}^{(n+2)} \end{pmatrix} = \mathcal{S}\_{n+1,n+2} \begin{pmatrix} \mathcal{A}^{(n+1)} \\ \mathcal{B}^{(n+2)} \end{pmatrix},\tag{61}
$$

we get:

Transfer matrix: The linearity of the problem makes it possible to write that the pairs of coefficients (An, Bn) and (Anþ1, Bnþ1) are linked by a matrix equation such

> Að Þ <sup>n</sup>þ<sup>1</sup> Bð Þ <sup>n</sup>þ<sup>1</sup> !

> > <sup>1</sup> � <sup>k</sup>ð Þ <sup>n</sup>þ<sup>1</sup> kn <sup>3</sup> <sup>þ</sup> <sup>12</sup>kn

> > > <sup>3</sup> <sup>þ</sup> <sup>12</sup>k<sup>n</sup>

r�<sup>2</sup> n

r�<sup>2</sup> n

Að Þ <sup>n</sup>þ<sup>1</sup> Bð Þ <sup>n</sup>þ<sup>1</sup> !

Að Þ <sup>n</sup>þ<sup>2</sup> Bð Þ <sup>n</sup>þ<sup>2</sup> !

> Að Þ <sup>n</sup> Bð Þ <sup>n</sup>þ<sup>1</sup> !

Að Þ <sup>n</sup>þ<sup>2</sup> Bð Þ <sup>n</sup>þ<sup>2</sup> !

!

<sup>1</sup> <sup>þ</sup> <sup>2</sup> <sup>k</sup>ð Þ <sup>n</sup>þ<sup>1</sup>

<sup>1</sup> � <sup>k</sup>ð Þ <sup>n</sup>þ<sup>1</sup> kn <sup>3</sup> <sup>þ</sup> <sup>12</sup>k<sup>n</sup>

<sup>k</sup><sup>n</sup> <sup>þ</sup> <sup>12</sup>kð Þ <sup>n</sup>þ<sup>1</sup>

!

<sup>3</sup> <sup>þ</sup> <sup>12</sup>k<sup>n</sup>

From <sup>A</sup><sup>0</sup> ¼ �U∞=κð Þ<sup>o</sup> and <sup>B</sup>ð Þ<sup>i</sup> <sup>¼</sup> 0, it is possible to get the pressure in each layer

The effects of two consecutive layers are obtained by the product of the transfer

¼ Tn,nþ<sup>1</sup>

¼ Tnþ1,nþ<sup>2</sup>

¼ Tn,nþ<sup>2</sup>

Scattering matrix: Another way of linking the coefficients A and B of two consecutive layers is the use of the scattering matrix Sn,nþ1. The scattering matrix is sometimes more efficient than the transfer matrix for calculating the amplitudes of the waves reflected and transmitted by an object subjected to incident waves. It is

¼ Sn,nþ<sup>1</sup>

!

!

r�<sup>2</sup> n

<sup>k</sup><sup>n</sup> <sup>þ</sup> <sup>12</sup>kð Þ <sup>n</sup>þ<sup>1</sup>

r�<sup>2</sup> n

r�<sup>2</sup> n

, (50)

, (51)

, (53)

, (55)

, (56)

, (57)

: (58)

: (54)

, (52)

r�<sup>2</sup> n

¼ Tn,nþ<sup>1</sup>

where Tn,nþ<sup>1</sup> is the transfer matrix between the two consecutive layers n and

kð Þ <sup>n</sup>þ<sup>1</sup> <sup>k</sup><sup>n</sup> <sup>þ</sup> <sup>2</sup>

kn þ

<sup>1</sup> <sup>þ</sup> <sup>2</sup> <sup>k</sup>ð Þ <sup>n</sup>þ<sup>1</sup>

<sup>ϕ</sup><sup>n</sup> <sup>r</sup> 3 n

Að Þ <sup>n</sup> Bð Þ <sup>n</sup> !

<sup>T</sup><sup>11</sup> <sup>¼</sup> <sup>ϕ</sup>ð Þ <sup>n</sup>þ<sup>1</sup> ϕn

<sup>T</sup><sup>22</sup> <sup>¼</sup> <sup>ϕ</sup>ð Þ <sup>n</sup>þ<sup>1</sup> ϕn

matrices of each of these layers. So from the matrix equations:

Að Þ <sup>n</sup> Bð Þ <sup>n</sup> !

Að Þ <sup>n</sup>þ<sup>1</sup> Bð Þ <sup>n</sup>þ<sup>1</sup> !

> Að Þ <sup>n</sup> Bð Þ <sup>n</sup> !

Bð Þ <sup>n</sup> Að Þ <sup>n</sup>þ<sup>1</sup> !

where Tn,nþ<sup>2</sup> ¼ Tn,nþ<sup>1</sup>Tnþ1,nþ2.

<sup>T</sup><sup>21</sup> <sup>¼</sup> <sup>ϕ</sup>ð Þ <sup>n</sup>þ<sup>1</sup>

ϕð Þ <sup>n</sup>þ<sup>1</sup> <sup>ϕ</sup><sup>n</sup> <sup>r</sup> �3 <sup>n</sup> � <sup>k</sup><sup>n</sup>þ<sup>1</sup>

that:

Acoustics of Materials

of the sphere.

we get:

such that:

74

Its entries are:

n þ 1, the entries of which are:

T<sup>12</sup> ¼ 2

$$
\begin{pmatrix} B^{(n)} \\ A^{(n+2)} \end{pmatrix} = \mathcal{S}\_{n,n+2} \begin{pmatrix} A^{(n)} \\ B^{(n+2)} \end{pmatrix},\tag{62}
$$

where Sn,nþ<sup>2</sup> ¼ Sn,nþ<sup>1</sup> <sup>∗</sup> Snþ1,nþ2. In this relation, the right-hand side is the Redheffer star product. For more details about the Redheffer star product, we can refer to [12].

### 2.3.2 Spherical inclusion with continuously variable permeability

When κ is a continuous function of the variable r, the matrix Eq. (50) becomes the system of differential equations:

$$\frac{d}{dr}\left(\frac{r^3\kappa^2dA/dr}{d\kappa/dr}\right) + r^2\kappa A = 0\tag{63}$$

$$\frac{d}{dr}\left(\frac{\kappa^2 dB/dr}{r^3 d\kappa/dr}\right) - \frac{2\kappa}{r^4} B = 0.\tag{64}$$

#### 2.3.3 Layered ellipsoidal inclusion

The generalization of the radial variation of the permeability of the spherical inclusion to the ellipsoidal requires that the permeability only depends on ξ. This is true in orthogonal directions at its surface. This condition entails that inside the ellipsoid, the strata are limited by confocal ellipsoidal surfaces ξ ¼ ξk, i.e., having the same foci as the surface ξ; hence, their semiaxes are related by the following relations:

$$a^2 - a\_k^2 = b^2 - b\_k^2 = c^2 - c\_k^2. \tag{65}$$

Consider a porous inhomogeneous ellipsoidal inclusion having the permeability kð Þ<sup>i</sup> embedded in a background medium of homogeneous mobility κð Þ<sup>o</sup> . We assume that the mobility of the inclusion is stratified, i.e., it is a constant piecewise function and each layer has its own mobility κð Þ <sup>n</sup> . The mobility in the outermost layer is κð Þ<sup>1</sup> , that of the next layer is κð Þ<sup>2</sup> , etc. to the central layer whose the mobilty is κð Þ<sup>i</sup> . The layers are limited by the confocal surfaces ξ ¼ ξ<sup>k</sup> whose semiaxes obey (65) and are numbered from 1 to i ¼ N from the outside to the inside, such that the strata n and

n þ 1 have the common boundary ξ ¼ ξnþ<sup>1</sup> (Figure 10). In each of these strata, the pressure is the solution of the Laplace equation given by:

$$p^{(j)}(\mathbf{r}) = -E\mathbf{x} \left[ A^{(j)} - \frac{B^{(j)}}{2} \int\_{\xi}^{\infty} \frac{d\sigma}{((\sigma + a^2)R^{(j)}(\sigma))} \right] \tag{66}$$

Scattering matrix: The scattering matrix of the ellipsoidal inclusion is:

<sup>κ</sup>ð Þ <sup>n</sup> Nx,nþ1ð Þ <sup>1</sup> � Nx,nþ<sup>1</sup> <sup>κ</sup>ð Þ <sup>n</sup>þ<sup>1</sup> � <sup>κ</sup>ð Þ <sup>n</sup> � �V�<sup>1</sup>

Dipole moment: When we are only interested in the scattered far field, the inclusion can be replaced by an equivalent dipole. When r is large in front of the

ξ≃ r

2

The component Pelli,a of dipole moment along the direction of the fluid flow is obtained by identification of the terms in r�<sup>2</sup> in relations (46) and (75), namely:

3

For ellipsoidal inclusion, we assume that mobility depends only on the variable ξ. From (33), the problem comes down to finding of the differential equation of the

> Rð Þξ 2

It is easy to verify that when κ is constant, we find the case of the homogeneous

Often the defects occurring in porous media are anisotropic, i.e., some of their physical parameters like permeability are no longer scalar quantities but are tensors. For an anisotropic porous medium, assuming the Einstein convention, the Darcy's

The permeability is then defined by nine components kij, i.e., it has different values in different directions of the space. Liakopoulos [13] had shown that the permeability is a symmetric tensor of second rank. This leads to great simplifications for the study of such porous media. If in isotropic media the fluid velocity is

<sup>∂</sup>jp where <sup>∂</sup>jp <sup>¼</sup> <sup>∂</sup><sup>p</sup>

∂xj

<sup>F</sup>ð Þ� <sup>ξ</sup> <sup>d</sup>κ ξð Þ dξ

ð<sup>∞</sup> r2

dσ <sup>σ</sup><sup>5</sup>=<sup>2</sup> <sup>¼</sup> <sup>B</sup>ð Þ<sup>o</sup>

! (73)

nþ1

2 :

<sup>3</sup>r<sup>3</sup> (75)

E ¼ 0: (77)

: (78)

<sup>2</sup> (74)

U∞Bð Þ<sup>o</sup> : (76)

Rð Þξ 2

Sn,nþ<sup>1</sup> <sup>¼</sup> <sup>1</sup>

where Vn ¼ anbncn.

and the dipolar term is

function Fð Þξ . Eq. 33 is then:

2.4 Anisotropic defects

<sup>d</sup><sup>ξ</sup> κ ξð Þ <sup>ξ</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup> � �Rð Þ<sup>ξ</sup> dF

� �

d

inclusion.

law is

77

Bð Þ<sup>o</sup> 2

ð<sup>∞</sup> ξ

2.3.4 Inclusion with continuously variable permeability

dξ

vi ¼ � kij η

þ dκ ξð Þ dξ

inclusion, and that if we put a ¼ b ¼ c, then we find the result of spherical

κð Þ <sup>n</sup> þ Nx,nþ<sup>1</sup> κð Þ <sup>n</sup>þ<sup>1</sup> � κð Þ <sup>n</sup> ð Þ

Tortuosity Perturbations Induced by Defects in Porous Media

DOI: http://dx.doi.org/10.5772/intechopen.84158

<sup>κ</sup>ð Þ <sup>n</sup>þ<sup>1</sup> � <sup>κ</sup>ð Þ <sup>n</sup> � �Vnþ<sup>1</sup> <sup>κ</sup>ð Þ <sup>n</sup>þ<sup>1</sup>

lengths of the axes of the ellipsoid (r ≫ a, b, c), ξ is of the order of r

dσ <sup>σ</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup> ð ÞRð Þ <sup>σ</sup> <sup>≃</sup> <sup>B</sup>ð Þ<sup>o</sup>

<sup>P</sup>elli,a <sup>¼</sup> <sup>4</sup><sup>π</sup>

where

$$R^{(j)}(
\sigma) = \left(a\_j^2 + \sigma\right) \left(b\_j^2 + \sigma\right) \left(c\_j^2 + \sigma\right). \tag{67}$$

In the layers n and n þ 1, the solutions of the Laplace equations are:

$$p^{(n)}(\mathbf{r}) = -E\mathbf{x} \left[ A^{(n)} - \frac{B^{(n)}}{2} \int\_{\xi}^{\infty} \frac{d\sigma}{((\sigma + a^2)\mathcal{R}^{(n)}(\sigma))} \right],\tag{68}$$

$$p^{(n+1)}(\mathbf{r}) = -E\mathbf{x} \left[ A^{(n+1)} - \frac{B^{(n+1)}}{2} \int\_{\xi}^{\infty} \frac{d\sigma}{((\sigma + a^2)R^{(n+1)}(\sigma))} \right] \tag{69}$$

with the boundary condition at ξ ¼ ξ<sup>n</sup>þ1:

$$\phi^{(n)}\frac{\mathbf{1}}{h\_{\xi(n)}}\frac{\partial p^{(n)}}{\partial \xi} = \phi^{(n+1)}\frac{\mathbf{1}}{h\_{\xi(n+1)}}\frac{\partial p^{(n+1)}}{\partial \xi},\tag{70}$$

and

$$\phi^{(n)}\left(-p^{(n)} + 2\eta \frac{\partial v\_{\xi}^{(n)} }{h\_{\xi(n)} \partial \xi}\right) = \phi^{(n+1)}\left(-p^{(n+1)} + 2\eta \frac{\partial v\_{\xi}^{(n+1)} }{h\_{\xi(n+1)} \partial \xi}\right). \tag{71}$$

By proceeding in the same way as for the spherical cavity, one finds the transfer matrix and the scattering matrix.

Transfer matrix: The transfer matrix of the ellipsoidal inclusion is:

$$T\_{n,n+1} = \frac{1}{\kappa^{(n)}} \begin{pmatrix} \kappa^{(n)} + N\_{\mathbf{x},n+1} (\kappa^{(n+1)} - \kappa^{(n)}) & \frac{N\_{\mathbf{x},n+1} (\mathbf{1} - N\_{\mathbf{x},n+1})}{a\_{n+1} b\_{n+1} b\_{n+1}} \left(\kappa^{(n+1)} - \kappa^{(n)}\right) \\\ a\_{n+1} b\_{n+1} c\_{n+1} (\kappa^{(n+1)} - \kappa^{(n)}) & \kappa^{(n+1)} + N\_{\mathbf{x},n+1} (\kappa^{(n)} - \kappa^{(n)}) \end{pmatrix}. \tag{72}$$

Figure 10. Layered ellipsoidal inclusion. Scattering matrix: The scattering matrix of the ellipsoidal inclusion is:

$$\begin{aligned} S\_{n,n+1} &= \frac{1}{\kappa^{(n)} + N\_{\mathbf{x},n+1}(\kappa^{(n+1)} - \kappa^{(n)})} \\ &\quad \left( \left( \kappa^{(n+1)} - \kappa^{(n)} \right) V\_{n+1} & \kappa^{(n+1)} \\ &\quad \kappa^{(n)} & N\_{\mathbf{x},n+1} (1 - N\_{\mathbf{x},n+1}) \left( \kappa^{(n+1)} - \kappa^{(n)} \right) V\_{n+1}^{-1} \end{aligned} \tag{73}$$

where Vn ¼ anbncn.

n þ 1 have the common boundary ξ ¼ ξnþ<sup>1</sup> (Figure 10). In each of these strata, the

2

b2 <sup>j</sup> þ σ � �

> ð<sup>∞</sup> ξ

" #

ð<sup>∞</sup> ξ

hξð Þ <sup>n</sup>þ<sup>1</sup>

anþ<sup>1</sup>bnþ<sup>1</sup>bnþ<sup>1</sup>

<sup>¼</sup> <sup>ϕ</sup>ð Þ <sup>n</sup>þ<sup>1</sup> �pð Þ <sup>n</sup>þ<sup>1</sup> <sup>þ</sup> <sup>2</sup><sup>η</sup>

By proceeding in the same way as for the spherical cavity, one finds the transfer

Transfer matrix: The transfer matrix of the ellipsoidal inclusion is:

<sup>κ</sup>ð Þ <sup>n</sup> <sup>þ</sup> Nx,nþ<sup>1</sup> <sup>κ</sup>ð Þ <sup>n</sup>þ<sup>1</sup> � <sup>κ</sup>ð Þ <sup>n</sup> � � Nx,nþ<sup>1</sup>ð Þ <sup>1</sup> � Nx,nþ<sup>1</sup>

anþ<sup>1</sup>bnþ<sup>1</sup>cnþ<sup>1</sup> <sup>κ</sup>ð Þ <sup>n</sup>þ<sup>1</sup> � <sup>κ</sup>ð Þ <sup>n</sup> � � <sup>κ</sup>ð Þ <sup>n</sup>þ<sup>1</sup> <sup>þ</sup> Nx,nþ<sup>1</sup> <sup>κ</sup>ð Þ <sup>n</sup> � <sup>κ</sup>ð Þ <sup>n</sup> � �

" #

2

2

<sup>∂</sup><sup>ξ</sup> <sup>¼</sup> <sup>ϕ</sup>ð Þ <sup>n</sup>þ<sup>1</sup> <sup>1</sup>

<sup>j</sup> þ σ � �

In the layers n and n þ 1, the solutions of the Laplace equations are:

ð<sup>∞</sup> ξ

" #

dσ <sup>ð</sup> <sup>σ</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup> ð ÞRð Þ<sup>j</sup> ð Þ <sup>σ</sup>

> c 2 <sup>j</sup> þ σ � �

dσ <sup>ð</sup> <sup>σ</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup> ð ÞRð Þ <sup>n</sup> ð Þ <sup>σ</sup>

∂pð Þ <sup>n</sup>þ<sup>1</sup>

!

dσ <sup>ð</sup> <sup>σ</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup> ð ÞRð Þ <sup>n</sup>þ<sup>1</sup> ð Þ <sup>σ</sup>

> ∂v ð Þ nþ1 ξ <sup>h</sup>ξð Þ <sup>n</sup>þ<sup>1</sup> <sup>∂</sup><sup>ξ</sup>

(66)

(69)

: (67)

, (68)

: (71)

1

CA: (72)

<sup>∂</sup><sup>ξ</sup> , (70)

<sup>κ</sup>ð Þ <sup>n</sup>þ<sup>1</sup> � <sup>κ</sup>ð Þ <sup>n</sup> � �

pressure is the solution of the Laplace equation given by:

where

Acoustics of Materials

and

Tn,nþ<sup>1</sup> <sup>¼</sup> <sup>1</sup>

Figure 10.

76

Layered ellipsoidal inclusion.

κð Þ <sup>n</sup>

<sup>p</sup>ð Þ<sup>j</sup> ð Þ¼� <sup>r</sup> Ex Að Þ<sup>j</sup> � <sup>B</sup>ð Þ<sup>j</sup>

<sup>R</sup>ð Þ<sup>j</sup> ð Þ¼ <sup>σ</sup> <sup>a</sup><sup>2</sup>

<sup>p</sup>ð Þ <sup>n</sup> ð Þ¼� <sup>r</sup> Ex Að Þ <sup>n</sup> � <sup>B</sup>ð Þ <sup>n</sup>

<sup>p</sup>ð Þ <sup>n</sup>þ<sup>1</sup> ð Þ¼� <sup>r</sup> Ex Að Þ <sup>n</sup>þ<sup>1</sup> � <sup>B</sup>ð Þ <sup>n</sup>þ<sup>1</sup>

∂pð Þ <sup>n</sup>

with the boundary condition at ξ ¼ ξ<sup>n</sup>þ1:

!

<sup>ϕ</sup>ð Þ <sup>n</sup> �pð Þ <sup>n</sup> <sup>þ</sup> <sup>2</sup><sup>η</sup>

matrix and the scattering matrix.

0 B@ <sup>ϕ</sup>ð Þ <sup>n</sup> <sup>1</sup> hξð Þ <sup>n</sup>

> ∂v ð Þ n ξ <sup>h</sup>ξð Þ <sup>n</sup> <sup>∂</sup><sup>ξ</sup>

Dipole moment: When we are only interested in the scattered far field, the inclusion can be replaced by an equivalent dipole. When r is large in front of the lengths of the axes of the ellipsoid (r ≫ a, b, c), ξ is of the order of r 2 :

$$
\xi \simeq r^2 \tag{74}
$$

and the dipolar term is

$$\frac{B^{(o)}}{2} \int\_{\xi}^{\infty} \frac{d\sigma}{(\sigma + a^2)R(\sigma)} \simeq \frac{B^{(o)}}{2} \int\_{r^2}^{\infty} \frac{d\sigma}{\sigma^{5/2}} = \frac{B^{(o)}}{3r^3} \tag{75}$$

The component Pelli,a of dipole moment along the direction of the fluid flow is obtained by identification of the terms in r�<sup>2</sup> in relations (46) and (75), namely:

$$\mathcal{P}\_{ellli,a} = \frac{4\pi}{3} U\_{\infty} B^{(o)}. \tag{76}$$
