2.3.4 Inclusion with continuously variable permeability

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 function Fð Þξ . Eq. 33 is then:

$$\frac{d}{d\xi}\left(\kappa(\xi)\left(\xi+a^2\right)R(\xi)\frac{dF}{d\xi}\right) + \frac{d\kappa(\xi)}{d\xi}\frac{R(\xi)}{2}F(\xi) - \frac{d\kappa(\xi)}{d\xi}\frac{R(\xi)}{2}E = 0.\tag{77}$$

It is easy to verify that when κ is constant, we find the case of the homogeneous inclusion, and that if we put a ¼ b ¼ c, then we find the result of spherical inclusion.

#### 2.4 Anisotropic defects

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 law is

$$
v\_i = -\frac{k\_{\vec{\eta}}}{\eta} \partial\_{\vec{p}} p \quad \text{where} \quad \partial\_{\vec{\eta}} p = \frac{\partial p}{\partial \mathfrak{x}\_{\vec{\jmath}}}.\tag{78}$$

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 aligned with the hydraulic gradient, in anisotropic media, this is true only along the principal directions of the tensor. It is therefore not surprising that the flow movement of the fluid is seriously disturbed by this type of defects.

In a 3D space, the permeability tensor has three principal directions perpendicular to each other and for which the permeability corresponds to the tensor eigenvalues. In the coordinates system defined by these directions, the permeability tensor is diagonal:

$$\mathbf{k} = \begin{pmatrix} k\_1 & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & k\_2 & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & k\_3 \end{pmatrix}. \tag{79}$$

Eq. (80) takes the form:

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

m, n Að Þ<sup>i</sup> m,nr nPm

The amplitudes Að Þ<sup>o</sup>

When <sup>r</sup> ! <sup>∞</sup>, <sup>p</sup>ð Þ<sup>o</sup> is:

P0

79

conditions at r ¼ a and when r ! ∞.

the incident pressure gradient), then

<sup>1</sup> ð Þ¼ cos <sup>θ</sup> <sup>P</sup>1ð Þ¼ cos <sup>θ</sup> cos <sup>θ</sup> and <sup>P</sup><sup>1</sup>

þ ∑ m, <sup>n</sup>

<sup>0</sup>, <sup>1</sup><sup>r</sup> cos <sup>θ</sup> <sup>þ</sup> <sup>A</sup>ð Þ<sup>o</sup>

Bð Þ<sup>o</sup> m,n rnþ<sup>1</sup> <sup>P</sup><sup>m</sup>

<sup>p</sup>ð Þ<sup>o</sup> ð Þ¼ <sup>r</sup>; <sup>θ</sup>; <sup>φ</sup> <sup>A</sup>ð Þ<sup>o</sup>

<sup>p</sup>ð Þ<sup>i</sup> ð Þ¼ <sup>r</sup>; <sup>θ</sup>; <sup>φ</sup> <sup>∑</sup>

<sup>κ</sup>0ð Þ<sup>i</sup> <sup>∂</sup><sup>2</sup>

m, n

m,n, Cð Þ<sup>o</sup>

terms are those for which n ¼ 1. Taking into account the relations

By identification with (86) with help of (87), we find:

Að Þ<sup>o</sup>

Cð Þ<sup>o</sup>

conservation of the fluid flow through the inclusion surface) are:

Að Þ<sup>o</sup>

þ ∑ m, <sup>n</sup>

In (84), only the finite terms at r ¼ 0 appear.

m,n, Bð Þ<sup>o</sup>

The solutions pð Þ<sup>i</sup> and pð Þ<sup>o</sup> , respectively, are:

Tortuosity Perturbations Induced by Defects in Porous Media

<sup>p</sup>ð Þ<sup>o</sup> ð Þ¼ <sup>r</sup>; <sup>θ</sup>; <sup>φ</sup> <sup>∑</sup>

pð Þ<sup>i</sup> ∂2 <sup>x</sup><sup>0</sup> <sup>þ</sup> ∂2 pð Þ<sup>i</sup> ∂2 <sup>y</sup><sup>0</sup> <sup>þ</sup>

<sup>n</sup> ð Þ cos θ cos mð Þþ φ ∑

Að Þ<sup>o</sup> m,nr n þ Bð Þ<sup>o</sup> m,n rnþ<sup>1</sup> !Pm

!

m,n, and Dð Þ<sup>o</sup>

<sup>p</sup>ð Þ<sup>o</sup> ! <sup>E</sup>∞<sup>r</sup> cos <sup>β</sup>, with <sup>E</sup><sup>∞</sup> ¼ � <sup>U</sup><sup>∞</sup>

where β is the angle between vector E<sup>∞</sup> and the direction of the observer OM ¼ r. If (θ, φ) resp. (θ0, φ0) are the angular coordinates of the observer (resp. of

The expressions (86) and (87) show that, in the expansion (85), only nonzero

<sup>1</sup>, <sup>1</sup>rsin <sup>θ</sup> cos <sup>φ</sup> <sup>þ</sup> <sup>C</sup>ð Þ<sup>o</sup>

The boundary conditions at r ¼ a (continuity of the stress component τrr and

�

<sup>p</sup>ð Þ<sup>o</sup> <sup>r</sup>¼<sup>a</sup> <sup>¼</sup> <sup>p</sup>ð Þ<sup>i</sup> � � �

<sup>n</sup> ð Þ cos θ cos mð Þþ φ ∑

Cð Þ<sup>o</sup> m,nr n þ Dð Þ<sup>o</sup> m,n rnþ<sup>1</sup>

� �

∂2 pð Þ<sup>i</sup> ∂2 z0

m, n Bð Þ<sup>i</sup> m,nr nPm

Pm

cos β ¼ sin θ sin θ<sup>0</sup> cos φ � φ<sup>0</sup> ð Þþ cos θ cos θ0: (87)

<sup>1</sup>ð Þ¼ cos θ sin θ, we obtain:

m, <sup>n</sup>

<sup>1</sup>, <sup>1</sup>rsin θ sin φ

<sup>0</sup>, <sup>1</sup> ¼ E<sup>∞</sup> cos θ0, (89)

<sup>1</sup>, <sup>1</sup> ¼ E<sup>∞</sup> sin θ<sup>0</sup> cos φ0, (90)

<sup>1</sup>, <sup>1</sup> ¼ E<sup>∞</sup> sin θ<sup>0</sup> sin φ0: (91)

Dð Þ<sup>o</sup> m,n rnþ<sup>1</sup> Pm

¼ 0: (83)

<sup>n</sup> ð Þ cos θ sin mð Þ φ , (84)

<sup>κ</sup>ð Þ<sup>o</sup> , (86)

<sup>n</sup> ð Þ cos θ sin mð Þ φ :

<sup>r</sup>¼<sup>a</sup> (92)

(85)

(88)

<sup>n</sup> ð Þ cos θ cos mð Þ φ

<sup>n</sup> ð Þ cos θ sin mð Þ φ :

m,n, are determined by the boundary

By its definition, mobility inherits properties of symmetry of permeability and is therefore a symmetric tensor such that κij ¼ kij=η.

The principal directions of the permeability tensors of the host medium and of the inclusion define coordinate systems which generally do not coincide. The system linked to the host environment is called the primary system, while that of the inclusion is called the secondary system.

In this part, the study of the effects of an anisotropic spherical inclusion in a porous medium explores three different configurations:


#### 2.4.1 Anisotropic defect in isotropic medium

In this configuration, the primary system is such that the incident pressure gradient is along the Oz axis and the secondary coordinate system is defined by the principal directions of the permeability tensor. Then, let (θ0, φ0) and (θ, φ) be the angular directions of the incident pressure gradient ∇pð Þ<sup>o</sup> and of the observation vector OM ¼ r in the primary system. We note β as the angle between these two directions.

The external pressure verifies the classical Laplace equation, while the internal one is solution of the following equation:

$$
\kappa\_1^{(i)} \frac{\partial p^{(i)}}{\partial \mathbf{x}^2} + \kappa\_2^{(i)} \frac{\partial p^{(i)}}{\partial \mathbf{y}^2} + \kappa\_3^{(i)} \frac{\partial p^{(i)}}{\partial \mathbf{z}^2} = \mathbf{0}.\tag{80}
$$

This one is transformed into a Laplace equation as follows: at first, we dimensionalize the mobilities κ ð Þi <sup>j</sup> by introducing the scalar quantity <sup>κ</sup>0ð Þ<sup>i</sup> . Eq. (80) then becomes

$$
\kappa^{\prime(i)} \left( \kappa^{(i)}\_{r,1} \frac{\partial p^{(i)}}{\partial \mathbf{x}^2} + \kappa^{(i)}\_{r,2} \frac{\partial p^{(i)}}{\partial \mathbf{y}^2} + \kappa^{(i)}\_{r,3} \frac{\partial p^{(i)}}{\partial \mathbf{z}^2} \right) = \mathbf{0},\tag{81}
$$

where κ ð Þi r,j ¼ κ ð Þi <sup>j</sup> <sup>=</sup>κ0ð Þ<sup>i</sup> . Using the linear transformation:

$$\frac{\partial}{\partial \mathbf{x}\_i'} = \sqrt{\kappa\_{r,i}^{(i)}} \frac{\partial}{\partial \mathbf{x}\_i},\tag{82}$$

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

Eq. (80) takes the form:

aligned with the hydraulic gradient, in anisotropic media, this is true only along the principal directions of the tensor. It is therefore not surprising that the flow move-

In a 3D space, the permeability tensor has three principal directions perpendicular to each other and for which the permeability corresponds to the tensor eigenvalues. In the coordinates system defined by these directions, the permeability

> k<sup>1</sup> 0 0 0 k<sup>2</sup> 0 0 0 k<sup>3</sup>

By its definition, mobility inherits properties of symmetry of permeability and is

The principal directions of the permeability tensors of the host medium and of the inclusion define coordinate systems which generally do not coincide. The system linked to the host environment is called the primary system, while that of the

In this part, the study of the effects of an anisotropic spherical inclusion in a

In this configuration, the primary system is such that the incident pressure gradi-

ent is along the Oz axis and the secondary coordinate system is defined by the principal directions of the permeability tensor. Then, let (θ0, φ0) and (θ, φ) be the angular directions of the incident pressure gradient ∇pð Þ<sup>o</sup> and of the observation vector OM ¼ r in the primary system. We note β as the angle between these two directions. The external pressure verifies the classical Laplace equation, while the internal

> ð Þi 2

This one is transformed into a Laplace equation as follows: at first, we

ð Þi r, 2 ∂pð Þ<sup>i</sup> <sup>∂</sup>y<sup>2</sup> <sup>þ</sup> <sup>κ</sup>

<sup>j</sup> <sup>=</sup>κ0ð Þ<sup>i</sup> . Using the linear transformation:

∂ ∂x0 i ¼

� �

ffiffiffiffiffiffi κ ð Þi r,i q ∂ ∂xi

∂pð Þ<sup>i</sup> <sup>∂</sup>y<sup>2</sup> <sup>þ</sup> <sup>κ</sup>

ð Þi 3

> ð Þi r, 3 ∂pð Þ<sup>i</sup> ∂z<sup>2</sup>

∂pð Þ<sup>i</sup>

<sup>j</sup> by introducing the scalar quantity <sup>κ</sup>0ð Þ<sup>i</sup> . Eq. (80)

<sup>∂</sup>z<sup>2</sup> <sup>¼</sup> <sup>0</sup>: (80)

¼ 0, (81)

, (82)

1

CA: (79)

ment of the fluid is seriously disturbed by this type of defects.

k ¼

therefore a symmetric tensor such that κij ¼ kij=η.

porous medium explores three different configurations:

3. the host medium and the defect are anisotropic.

1. the host medium is isotropic and the defect is anisotropic,

2. the host medium is anisotropic and the defect is isotropic, and

inclusion is called the secondary system.

2.4.1 Anisotropic defect in isotropic medium

one is solution of the following equation:

dimensionalize the mobilities κ

then becomes

where κ

78

ð Þi r,j ¼ κ ð Þi κ ð Þi 1

κ0ð Þ<sup>i</sup> κ ð Þi r, 1 ∂pð Þ<sup>i</sup> <sup>∂</sup>x<sup>2</sup> <sup>þ</sup> <sup>κ</sup>

∂pð Þ<sup>i</sup> <sup>∂</sup>x<sup>2</sup> <sup>þ</sup> <sup>κ</sup>

ð Þi

0

B@

tensor is diagonal:

Acoustics of Materials

$$\kappa^{\prime(i)} \left( \frac{\partial^2 p^{(i)}}{\partial^2 \mathbf{x}^{\prime}} + \frac{\partial^2 p^{(i)}}{\partial^2 p^{\prime}} + \frac{\partial^2 p^{(i)}}{\partial^2 \mathbf{z}^{\prime}} \right) = \mathbf{0}. \tag{83}$$

The solutions pð Þ<sup>i</sup> and pð Þ<sup>o</sup> , respectively, are:

$$p^{(i)}(r,\theta,\varphi) = \sum\_{m\_1 n} A^{(i)}\_{m,n} r^n P^m\_n(\cos\theta) \cos(m\varphi) + \sum\_{m\_1 n} B^{(i)}\_{m,n} r^n P^m\_n(\cos\theta) \sin(m\varphi), \quad \text{(84)}$$

$$\begin{split} p^{(o)}(r,\theta,\rho) &= \sum\_{m,n} \left( A\_{m,n}^{(o)} r^n + \frac{B\_{m,n}^{(o)}}{r^{n+1}} \right) P\_n^m(\cos\theta) \cos\left(m\rho\right) \\ &+ \sum\_{m,n} \left( C\_{m,n}^{(o)} r^n + \frac{D\_{m,n}^{(o)}}{r^{n+1}} \right) P\_n^m(\cos\theta) \sin\left(m\rho\right) . \end{split} \tag{85}$$

In (84), only the finite terms at r ¼ 0 appear.

The amplitudes Að Þ<sup>o</sup> m,n, Bð Þ<sup>o</sup> m,n, Cð Þ<sup>o</sup> m,n, and Dð Þ<sup>o</sup> m,n, are determined by the boundary conditions at r ¼ a and when r ! ∞.

When <sup>r</sup> ! <sup>∞</sup>, <sup>p</sup>ð Þ<sup>o</sup> is:

$$p^{(o)} \to E\_{\infty} r \cos \beta, \quad \text{with} \quad E\_{\infty} = -\frac{U\_{\infty}}{\kappa^{(o)}},\tag{86}$$

where β is the angle between vector E<sup>∞</sup> and the direction of the observer OM ¼ r. If (θ, φ) resp. (θ0, φ0) are the angular coordinates of the observer (resp. of the incident pressure gradient), then

$$\cos \beta = \sin \theta \sin \theta\_0 \cos \left(\wp - \wp\_0\right) + \cos \theta \cos \theta\_0. \tag{87}$$

The expressions (86) and (87) show that, in the expansion (85), only nonzero terms are those for which n ¼ 1. Taking into account the relations P0 <sup>1</sup> ð Þ¼ cos <sup>θ</sup> <sup>P</sup>1ð Þ¼ cos <sup>θ</sup> cos <sup>θ</sup> and <sup>P</sup><sup>1</sup> <sup>1</sup>ð Þ¼ cos θ sin θ, we obtain:

$$\begin{split} p^{(o)}(r,\theta,\varphi) &= A\_{0,1}^{(o)}r\cos\theta + A\_{1,1}^{(o)}r\sin\theta\cos\varphi + C\_{1,1}^{(o)}r\sin\theta\sin\varphi\\ &+ \sum\_{m,n} \frac{B\_{m,n}^{(o)}}{r^{n+1}}P\_n^m(\cos\theta)\cos\left(m\varphi\right) + \sum\_{m,n} \frac{D\_{m,n}^{(o)}}{r^{n+1}}P\_n^m(\cos\theta)\sin\left(m\varphi\right). \end{split} \tag{88}$$

By identification with (86) with help of (87), we find:

$$A\_{0,1}^{(o)} = E\_{\infty} \cos \theta\_0. \tag{89}$$

$$A\_{1,1}^{(o)} = E\_{\infty} \sin \theta\_0 \cos \rho\_0. \tag{90}$$

$$C\_{\mathbf{1},\mathbf{1}}^{(o)} = E\_{\infty} \sin \theta\_0 \sin \varphi\_0. \tag{91}$$

The boundary conditions at r ¼ a (continuity of the stress component τrr and conservation of the fluid flow through the inclusion surface) are:

$$p^{(o)}|\_{r=a} = \left. p^{(i)} \right|\_{r=a} \tag{92}$$

$$\kappa^{(o)} \frac{\partial p^{(o)}}{\partial r} \Big|\_{r=a} = \left. \left( \kappa^{(i)}\_{11} \frac{\partial p^{(i)}}{\partial r} + \kappa^{(i)}\_{12} \frac{\partial p^{(i)}}{r \partial \theta} + \kappa^{(i)}\_{13} \frac{\partial p^{(i)}}{r \sin \theta \partial \rho} \right) \right|\_{r=a}. \tag{93}$$

2.4.2 Isotropic defect in anisotropic porous medium

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

Tortuosity Perturbations Induced by Defects in Porous Media

tensor κð Þ<sup>o</sup> , this equation is written as:

<sup>κ</sup>0ð Þ<sup>o</sup> <sup>κ</sup>

ð Þo x κ0ð Þ<sup>o</sup>

> ð Þo r,j ¼ κ ð Þo

<sup>x</sup><sup>0</sup> <sup>¼</sup> <sup>x</sup>

transformed into an ellipsoid with the semiaxes ax ¼ r=

ffiffiffiffiffiffiffi κ ð Þo r,x

<sup>∂</sup>jpð Þ<sup>i</sup> <sup>¼</sup> <sup>κ</sup>0ð Þ<sup>o</sup>

<sup>∂</sup>jpð Þ<sup>i</sup> <sup>¼</sup> <sup>κ</sup>

κ ð Þo

ds

� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s þ a<sup>2</sup> x � � <sup>s</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup>

In this equation, the depolarization factor Nj is

s þ a<sup>2</sup> j

<sup>κ</sup>0ð Þ<sup>o</sup> <sup>þ</sup> Nj <sup>κ</sup>ð Þ<sup>i</sup> <sup>=</sup><sup>κ</sup>

<sup>q</sup> , y<sup>0</sup> <sup>¼</sup> <sup>y</sup>

Eq. (101) becomes a Laplace equation. Correspondingly, the sphere is

coincide with the axes of the ellipsoid, for each direction j, we find, for each of the components of the pressure gradient, the result of the ellipsoidal inclusion (40).

∂2 pð Þ<sup>o</sup> ∂x<sup>2</sup> þ

equation:

where κ

and az ¼ r=

or

81

Nj <sup>¼</sup> axayaz 2

ð<sup>∞</sup> O

through the depolarization factor Nj.

ð Þo

scalar such that the ratio κ

transformation of coordinates:

ffiffiffiffiffiffiffi κ ð Þo r, z q

The internal pressure gradient is then:

Consider an isotropic sphere of radius r whose mobility is κð Þ<sup>i</sup> which is included in an anisotropic host medium with its own mobility κð Þ<sup>o</sup> . The incompressibility of the saturating fluid imposes that the outside pressure is the solution of the

In the system of Cartesian coordinate defined by the principal directions of the

∂2 pð Þ<sup>o</sup> ∂y<sup>2</sup> þ

<sup>j</sup> are the eigenvalues of the mobility tensor and <sup>κ</sup>0ð Þ<sup>o</sup> is an arbitrary

ffiffiffiffiffiffiffi κ ð Þo r, y

<sup>q</sup> , z<sup>0</sup> <sup>¼</sup> <sup>z</sup>

. Since the principal directions of the inside permeability κð Þ<sup>i</sup>

ð Þo

ð Þo j

z � � <sup>r</sup> for <sup>j</sup> <sup>¼</sup> x, y, z: (105)

ð Þo j

y � � <sup>s</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup>

Thus, the anisotropy induced in the sphere by the change of variables appears

<sup>j</sup> þ Nj κð Þ<sup>i</sup> � κ

!

κ ð Þo z κ0ð Þ<sup>o</sup> ∂2 pð Þ<sup>o</sup> ∂z<sup>2</sup>

� � <sup>¼</sup> <sup>0</sup>: (100)

<sup>j</sup> <sup>=</sup>κ0ð Þ<sup>o</sup> is a dimensionless quantity. Using the linear

ffiffiffiffiffiffiffi κ ð Þo r, z

ffiffiffiffiffiffiffi κ ð Þo r,x q

rj � <sup>κ</sup>0ð Þ<sup>o</sup> � � <sup>∂</sup>jpð Þ<sup>o</sup> , (103)

� � <sup>∂</sup>jpð Þ<sup>o</sup> : (104)

¼ 0, (101)

<sup>q</sup> , (102)

ffiffiffiffiffiffiffi κ ð Þo r, y q

,

, ay ¼ r=

∂<sup>i</sup> κ ð Þo ij <sup>∂</sup>jp

> κ ð Þo y κ0ð Þ<sup>o</sup>

In these equations, κ ð Þi nm are the components of the tensor κð Þ<sup>i</sup> in the spherical coordinates given in Appendix D. These relations lead to the following expressions of the pressure:

$$p^{(i)}(r,\theta,\varphi) = \frac{3A\kappa\_{(o)}}{2\kappa^{(o)} + \kappa\_3^{(i)}}r\cos\theta + \frac{3B\kappa\_{(o)}}{2\kappa^{(o)} + \kappa\_1^{(i)}}r\sin\theta\cos\varphi + \frac{3D\kappa\_{(o)}}{2\kappa^{(o)} + \kappa\_2^{(i)}}r\sin\theta\sin\varphi,\tag{94}$$

$$p^{(o)}(r,\theta,\varphi) = Ar\cos\theta + Br\sin\theta\cos\varphi + Dr\sin\theta\sin\varphi + \frac{Aa^3}{r^2}\frac{\kappa^{(o)} - \kappa\_3^{(i)}}{2\kappa^{(o)} + \kappa\_3^{(i)}}\cos\theta$$

$$+ \frac{Ba^3}{r^2}\frac{\kappa^{(o)} - \kappa\_1^{(i)}}{2\kappa^{(o)} + \kappa\_1^{(i)}}\sin\theta\cos\varphi + \frac{Da^3}{r^2}\frac{\kappa^{(o)} - \kappa\_2^{(i)}}{2\kappa^{(o)} + \kappa\_2^{(i)}}\sin\theta\sin\varphi,\tag{95}$$

where

$$A = E\_{\infty} \cos \theta\_0, \quad B = E\_{\infty} \sin \theta\_0 \cos \varphi\_0, \quad D = E\_{\infty} \sin \theta\_0 \sin \varphi\_0. \tag{96}$$

The first three terms of the right-hand side of (95) are due to the pressure gradient applied to the porous medium. The last three terms are the pressure induced by the hydraulic dipoles directed along the principal directions of the anisotropic sphere.

When E<sup>∞</sup> is along the Oz axis and for φ<sup>0</sup> ¼ 0, θ<sup>0</sup> ¼ 0 and κ ð Þi <sup>1</sup> ¼ κ ð Þi <sup>2</sup> ¼ κ ð Þi <sup>3</sup> <sup>¼</sup> <sup>κ</sup>ð Þ<sup>i</sup> , we find the internal and external pressures of isotropic spherical inclusions (19) and (20).

Moreover, from the relations (94) and (95), it is possible to obtain the directions of the pressure gradient and of the velocity field inside the defect.

The inside fluid velocity results from (94); its components are given by v ð Þi <sup>j</sup> ¼ �κ ð Þi <sup>j</sup> <sup>∂</sup>jpð Þ<sup>i</sup> , from which we obtain:

$$\mathbf{v}^{(i)} = -\frac{\mathbf{3}B\kappa\_1^{(i)}}{2\kappa^{(o)} + \kappa\_1}\mathbf{i} - \frac{\mathbf{3}D\kappa\_2^{(i)}}{2\kappa^{(o)} + \kappa\_2}\mathbf{j} - \frac{\mathbf{3}A\kappa\_3^{(i)}}{2\kappa^{(o)} + \kappa\_3}\mathbf{k}.\tag{97}$$

This is the generalization to the 3D case of the result obtained for the spherical inclusion when the pressure gradient is along the axis Ox (19).

The inner product of vð Þ<sup>i</sup> and of the incident field U∞, gives the angle γ whose the internal fluid velocity is deflected by the anisotropy of the inclusion:

$$\cos \gamma = \frac{\mathbf{U}\_{\infty} \cdot \mathbf{v}^{(i)}}{\|\mathbf{U}\_{\infty}\| \|\mathbf{v}^{(i)}\|},\tag{98}$$

$$=\frac{\frac{\sin^{2}\theta\_{0}\sin^{2}\varphi\_{0}\kappa\_{1}^{(i)}}{2\kappa^{(\epsilon)}+\kappa\_{2}^{(i)}}+\frac{\sin^{2}\theta\_{0}\cos^{2}\varphi\_{0}\kappa\_{2}^{(i)}}{2\kappa^{(\epsilon)}+\kappa\_{1}^{(i)}}+\frac{\cos^{2}\theta\_{0}\kappa\_{3}^{(i)}}{2\kappa^{(\epsilon)}+\kappa\_{2}^{(i)}}}{\sqrt{\frac{\sin^{2}\theta\_{0}\sin^{2}\varphi\_{0}\kappa\_{1}^{(i)}}{\left(2\kappa^{(\epsilon)}+\kappa\_{2}^{(i)}\right)^{2}}+\frac{\sin^{2}\theta\_{0}\cos^{2}\varphi\_{0}\kappa\_{3}^{(i)}}{\left(2\kappa^{(\epsilon)}+\kappa\_{1}^{(i)}\right)^{2}}}}}} . . \tag{99}$$
