D. Tensor

Let κ be a tensor of rank 2. We denote by κr, its expression in the system of rectangular coordinates defined by its principal directions, and κs, its expression in the corresponding spherical coordinates system. So we have

$$\begin{aligned} \mathbf{x}\_{r} &= \begin{pmatrix} \kappa\_{1} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \kappa\_{2} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \kappa\_{3} \end{pmatrix} \qquad \mathbf{x}\_{\mathfrak{t}} = \begin{pmatrix} \kappa\_{11} & \kappa\_{12} & \kappa\_{13} \\ \kappa\_{21} & \kappa\_{22} & \kappa\_{23} \\ \kappa\_{31} & \kappa\_{32} & \kappa\_{33} \end{pmatrix} \end{aligned} \tag{172}$$

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

with

one deduces

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or

r

D. Tensor

92

r

<sup>0</sup> ¼ rΔ òu Δ ¼

From (164) and (165), one has:

By eliminating φ, one finds:

<sup>0</sup><sup>2</sup> <sup>¼</sup> <sup>x</sup>0<sup>2</sup> <sup>þ</sup> <sup>y</sup>0<sup>2</sup> <sup>þ</sup> <sup>z</sup>0<sup>2</sup>

<sup>2</sup> sin <sup>2</sup>θ cos <sup>2</sup>φ κ1

s

þ

sin <sup>2</sup>θ cos <sup>2</sup>φ κ1

sin <sup>2</sup>θ sin <sup>2</sup>φ κ2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

sin <sup>2</sup>θ sin <sup>2</sup>φ κ2

κ1

κ2

s

� �

þ

cos <sup>φ</sup><sup>0</sup> <sup>¼</sup> <sup>r</sup>

sin <sup>φ</sup><sup>0</sup> <sup>¼</sup> <sup>r</sup>

avec δ ¼

Δ

Similar relationships between angles φ and φ<sup>0</sup> are deduced from (164) and (165):

δ

Let κ be a tensor of rank 2. We denote by κr, its expression in the system of rectangular coordinates defined by its principal directions, and κs, its expression in

κ<sup>s</sup> ¼

0

B@

1

CA

the corresponding spherical coordinates system. So we have

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

κ<sup>r</sup> ¼

0

B@

cos θ ffiffiffiffi κ3

sin φ ffiffiffiffi κ2

cos φ ffiffiffiffi κ1

cos <sup>θ</sup><sup>0</sup> <sup>¼</sup> <sup>1</sup>

sin <sup>φ</sup><sup>0</sup> <sup>¼</sup> <sup>1</sup> δ

cos <sup>φ</sup><sup>0</sup> <sup>¼</sup> <sup>1</sup>

þ

cos <sup>2</sup>θ κ3

þ

sin θ cos φ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos <sup>2</sup>φ κ1

þ

sin <sup>2</sup>φ κ2

p : (169)

p , (170)

p : (171)

1

CA (172)

κ<sup>11</sup> κ<sup>12</sup> κ<sup>13</sup> κ<sup>21</sup> κ<sup>22</sup> κ<sup>23</sup> κ<sup>31</sup> κ<sup>32</sup> κ<sup>33</sup>

sin θ sin φ:

cos <sup>2</sup>θ κ3

: (167)

: (168)

¼ x2 κ1 þ y2 κ2 þ z2 κ3

¼ r

<sup>x</sup><sup>0</sup> <sup>¼</sup> <sup>x</sup> κ1 ) r 0 sin θ<sup>0</sup>

<sup>y</sup><sup>0</sup> <sup>¼</sup> <sup>y</sup> κ2 ) r 0 sin θ<sup>0</sup>

sin <sup>θ</sup><sup>0</sup> <sup>¼</sup> sin <sup>θ</sup> <sup>δ</sup>

In the same way, from (166), one can establish

Δ

$$\kappa\_{11} = \kappa\_1 \sin^2 \theta \cos^2 \rho + \kappa\_2 \sin^2 \theta \sin^2 \rho + \kappa\_3 \cos^2 \theta,\tag{173}$$

$$\kappa\_{12} = \kappa\_1 \cos \theta \sin \theta \cos^2 \rho + \kappa\_2 \cos \theta \sin \theta \sin^2 \rho - \kappa\_3 \cos \theta \sin \theta,\tag{174}$$

$$
\kappa\_{13} = (\kappa\_2 - \kappa\_1) \sin \theta \cos \varphi \sin \varphi,\tag{175}
$$

$$\kappa\_{22} = \kappa\_1 \cos^2 \theta \cos^2 \varphi + \kappa\_2 \cos^2 \theta \sin^2 \varphi + \kappa\_3 \sin^2 \theta,\tag{176}$$

$$\kappa\_{23} = (\kappa\_2 - \kappa\_1) \cos \theta \cos \varphi \sin \varphi,\tag{177}$$

$$
\kappa\_{33} = \kappa\_1 \sin^2 \rho + \kappa\_2 \cos^2 \rho,\tag{178}
$$

$$
\kappa\_{21} = \kappa\_{12} \tag{179}
$$

κ<sup>31</sup> ¼ κ13, (180)

$$
\kappa\_{32} = \kappa\_{23}.\tag{181}
$$

Or, alternatively in the matrix form:

$$\mathbf{k} = \kappa\_1 \mathbf{I} + (\kappa\_2 - \kappa\_1)\mathbf{A} + (\kappa\_3 - \kappa\_1)\mathbf{B},\tag{182}$$

where I is the unit matrix 3 � 3 and A and B are given by:

$$\mathbf{A} = \begin{pmatrix} \sin^2\theta\sin^2\varphi & \cos\theta\sin\theta\sin^2\varphi & \sin\theta\cos\varphi\sin\varphi\\ \cos\theta\sin\theta\sin^2\varphi & \cos^2\theta\sin^2\varphi & \cos\theta\cos\varphi\sin\varphi\\ \sin\theta\cos\varphi\sin\varphi & \cos\theta\cos\varphi\sin\varphi & \cos^2\theta \end{pmatrix},\tag{183}$$

$$\mathbf{B} = \begin{pmatrix} \cos^2 \theta & -\cos \theta \sin \theta & 0 \\ -\cos \theta \sin \theta & \sin^2 \theta & 0 \\ 0 & 0 & 0 \end{pmatrix}. \tag{184}$$

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