**2.1 Analytical solution of the equation of motion in the case of a Bingham-type fluid**

In this section, we will obtain a mathematical expression that shows how to quantify the velocity profile for a non-Newtonian fluid for the Bingham model.

In the case of a Poiseuille flow, the flow of a Bingham-type fluid in the xdirection is considered, between two plates separated by a distance 2H, taking into account the steady-state conditions, constant cross section, absence of gravitational effects and isothermal flow, and being incompressible, such as the one shown in **Figure 2**.

From the equation of motion, we can obtain the stress profile, as well as the velocity profile:

$$\rho \left[ \frac{\partial \overrightarrow{\mathbf{v}}}{\partial \mathbf{t}} + \overrightarrow{\mathbf{v}} \cdot \nabla \overrightarrow{\mathbf{v}} \right] = \nabla \cdot \overline{\overline{\mathbf{r}}} - \nabla \mathbf{P} + \rho \overrightarrow{\mathbf{g}} \tag{6}$$

Making a separation of variables, in addition to performing the corresponding

The integration constant is zero, when τyx = 0 at y = 0, i.e., at the center of the

<sup>¼</sup> <sup>Δ</sup><sup>P</sup> L

<sup>τ</sup>yx <sup>¼</sup> <sup>Δ</sup><sup>P</sup> L

� �<sup>y</sup> <sup>þ</sup> c1 (9)

� �<sup>y</sup> (10)

� �<sup>y</sup> (11)

y þ c2 (12)

� � (13)

� �y0 (14)

(15)

(16)

<sup>τ</sup>yx <sup>¼</sup> <sup>Δ</sup><sup>P</sup> L

plates, the shear stress is minimal. Therefore, the shear stress profile is

*Bingham Fluid Simulation in Porous Media with Lattice Boltzmann Method*

dvx dy � � <sup>þ</sup> <sup>τ</sup><sup>0</sup> zfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflffl{ Bingham model

Making a second separation of variables, in addition to performing the

μB y2 <sup>2</sup> � <sup>τ</sup><sup>0</sup> μB

In **Figure 2**, it is shown that velocity is zero on the plates; i.e., vx = 0 at y = �H. Using this condition, the value of c2 is obtained. Substituting in Eq. (12) we obtain

> ΔP L � � H2

To know the velocity profile in the region of the plug flow, the condition for the

<sup>y</sup> <sup>¼</sup> y0; <sup>τ</sup><sup>0</sup> <sup>¼</sup> <sup>Δ</sup><sup>P</sup>

Substituting Eq. (14) into Eq. (13), the velocity in the plug flow region is

2μ<sup>B</sup>

Commonly, velocity profiles are usually represented with on the Bingham num-

Bn <sup>¼</sup> <sup>τ</sup><sup>0</sup> μB H v

In Eq. (16), v is a characteristic velocity. Dividing Eq. (13) by this velocity,

ΔP L � � H2

2vμ<sup>B</sup>

y H � �<sup>2</sup>

� 1

� � (17)

v0 <sup>¼</sup> <sup>Δ</sup><sup>P</sup> L � � H<sup>2</sup> 2μ<sup>B</sup>

L

<sup>1</sup> � y0 H � �<sup>2</sup>

y H � �<sup>2</sup>

� 1

Equating Eq. (10) with the Bingham model, we obtain

μB

vx <sup>¼</sup> <sup>Δ</sup><sup>P</sup> L � � 1

H 1 � <sup>y</sup> H � � <sup>þ</sup>

yield stress is proposed according to Eq. (10), when y = y0:

integrals in Eq. (8), we obtain

*DOI: http://dx.doi.org/10.5772/intechopen.90167*

corresponding integrals, we have

obtained:

**47**

ber, which is defined as

vx

<sup>v</sup> <sup>¼</sup> Bn 1 � <sup>y</sup>

H � � <sup>þ</sup>

vx <sup>¼</sup> <sup>τ</sup><sup>0</sup> μB

The pressure gradient effect is considered as a favorable driving force for fluid movement. Usually, the pressure decreases at a constant rate from the initial end to the end in the x direction.

$$\nabla \mathbf{P} \cong \frac{\Delta \mathbf{P}}{\Delta \mathbf{z}} = \frac{(\mathbf{P\_L} - \mathbf{P\_0})}{\mathbf{L}} = \frac{\Delta \mathbf{P}}{\mathbf{L}} \tag{7}$$

The component of the flow density tensor of the amount of movement of Eq. (6) is τyx; therefore, considering the above conditions, we have the following equation to solve:

$$\frac{\mathbf{d}\tau\_{\rm yx}}{\mathbf{d}\mathbf{y}} = \left(\frac{\Delta\mathbf{P}}{\mathcal{L}}\right) \tag{8}$$

**Figure 2.** *Flow of a Bingham fluid between two plates one half view [2].*

*Bingham Fluid Simulation in Porous Media with Lattice Boltzmann Method DOI: http://dx.doi.org/10.5772/intechopen.90167*

Making a separation of variables, in addition to performing the corresponding integrals in Eq. (8), we obtain

$$\mathbf{\tau}\_{\rm yx} = \left(\frac{\Delta \mathbf{P}}{\mathbf{L}}\right) \mathbf{y} + \mathbf{c}\_1 \tag{9}$$

The integration constant is zero, when τyx = 0 at y = 0, i.e., at the center of the plates, the shear stress is minimal. Therefore, the shear stress profile is

$$\mathbf{\tau}\_{\mathbf{yx}} = \left(\frac{\Delta \mathbf{P}}{\mathcal{L}}\right) \mathbf{y} \tag{10}$$

Equating Eq. (10) with the Bingham model, we obtain

$$\overbrace{\mu\_{\rm B}\left(\frac{\mathrm{dv}\_{\rm x}}{\mathrm{dy}}\right) + \tau\_{0}}^{\mathrm{Bingham}\ \mathrm{model}} + \tau\_{0} = \left(\frac{\Delta\mathrm{P}}{\mathrm{L}}\right)\mathrm{y} \tag{11}$$

Making a second separation of variables, in addition to performing the corresponding integrals, we have

$$\mathbf{v\_x} = \left(\frac{\Delta \mathbf{P}}{\mathcal{L}}\right) \frac{\mathbf{1}}{\mu\_\mathcal{B}} \frac{\mathbf{y}^2}{2} - \frac{\tau\_0}{\mu\_\mathcal{B}} \mathbf{y} + \mathbf{c\_2} \tag{12}$$

In **Figure 2**, it is shown that velocity is zero on the plates; i.e., vx = 0 at y = �H. Using this condition, the value of c2 is obtained. Substituting in Eq. (12) we obtain

$$\mathbf{v}\_{\mathbf{x}} = \frac{\mathbf{\tau}\_{0}}{\mu\_{\mathrm{B}}} \mathbf{H} \left( \mathbf{1} - \frac{\mathbf{y}}{\mathrm{H}} \right) + \left( \frac{\Delta \mathbf{P}}{\mathrm{L}} \right) \frac{\mathrm{H}^{2}}{2 \mu\_{\mathrm{B}}} \left[ \left( \frac{\mathrm{y}}{\mathrm{H}} \right)^{2} - \mathbf{1} \right] \tag{13}$$

To know the velocity profile in the region of the plug flow, the condition for the yield stress is proposed according to Eq. (10), when y = y0:

$$\mathbf{y} = \mathbf{y}\_0; \qquad \mathbf{\tau}\_0 = \left(\frac{\Delta \mathbf{P}}{\mathcal{L}}\right) \mathbf{y}\_0 \tag{14}$$

Substituting Eq. (14) into Eq. (13), the velocity in the plug flow region is obtained:

$$\mathbf{v}\_0 = \left(\frac{\Delta \mathbf{P}}{\mathbf{L}}\right) \frac{\mathbf{H}^2}{2\mu\_\mathbf{B}} \left(\mathbf{1} - \frac{\mathbf{y}\_0}{\mathbf{H}}\right)^2 \tag{15}$$

Commonly, velocity profiles are usually represented with on the Bingham number, which is defined as

$$\mathbf{Bn} = \frac{\mathbf{\pi}\_0}{\mu\_\mathbf{B}} \frac{\mathbf{H}}{\mathbf{v}} \tag{16}$$

In Eq. (16), v is a characteristic velocity. Dividing Eq. (13) by this velocity,

$$\frac{\mathbf{v\_x}}{\mathbf{v}} = \text{Bn}\left(\mathbf{1} - \frac{\mathbf{y}}{\mathbf{H}}\right) + \left(\frac{\Delta\mathbf{P}}{\mathbf{L}}\right) \frac{\mathbf{H}^2}{2\mathbf{v}\mu\_\text{B}} \left[\left(\frac{\mathbf{y}}{\mathbf{H}}\right)^2 - \mathbf{1}\right] \tag{17}$$

Other examples of Bingham-type fluids in foods are tomato sauce, whipped cream, whipped egg white, margarine, and mustard-type condiments [7, 8].

**2.1 Analytical solution of the equation of motion in the case of a Bingham-type**

In this section, we will obtain a mathematical expression that shows how to quantify the velocity profile for a non-Newtonian fluid for the Bingham model. In the case of a Poiseuille flow, the flow of a Bingham-type fluid in the xdirection is considered, between two plates separated by a distance 2H, taking into account the steady-state conditions, constant cross section, absence of gravitational effects and isothermal flow, and being incompressible, such as the one shown in

From the equation of motion, we can obtain the stress profile, as well as the

The pressure gradient effect is considered as a favorable driving force for fluid movement. Usually, the pressure decreases at a constant rate from the initial end to

<sup>Δ</sup><sup>z</sup> <sup>¼</sup> ð Þ PL � P0

dy <sup>¼</sup> <sup>Δ</sup><sup>P</sup> L � �

The component of the flow density tensor of the amount of movement of Eq. (6) is τyx; therefore, considering the above conditions, we have the following equation

¼ ∇ � τ � ∇P þ ρg

<sup>L</sup> <sup>¼</sup> <sup>Δ</sup><sup>P</sup>

! (6)

<sup>L</sup> (7)

(8)

ρ ∂v ! ∂t þ v ! � ∇v !

" #

∇P ffi

ΔP

dτyx

**fluid**

*Computational Fluid Dynamics Simulations*

**Figure 2**.

to solve:

**Figure 2.**

**46**

*Flow of a Bingham fluid between two plates one half view [2].*

velocity profile:

the end in the x direction.

**Figure 3.** *Velocity profile for a Bingham fluid; Bn = 0.1, 0.2, 0.3, and 0.4 [4].*

velocity of a particle and limited to a discrete group of allowed velocities. During each time step, the movement or jump of particle to nearby lattice sites, along its direction of movement, a collision with another particle can occur when they reach the same site. The result of the collisions is determined by means of the solution of the kinetic equation of Boltzmann for the new function of distribution of the particle to that site, and in this way the function of distribution of the particle is

*Allowed directions for particle movement. (a) Model D2Q9, (b) model D2Q15.*

*Bingham Fluid Simulation in Porous Media with Lattice Boltzmann Method*

*DOI: http://dx.doi.org/10.5772/intechopen.90167*

There are different lattice models, which are given by DmQn; m indicates the dimension and n the permitted velocity, thus, the D2Q9 model (two-dimensional with nine speed directions), of which four sites correspond to nearby neighbors (right, left, up, and down), four other points of the lattice vectors along the

diagonal faces of the following sites, along these diagonals. In this way the particles can travel in eight directions for each lattice site. The circle in the center of the square represents the vector, which has a value of zero and represents particles that have no movement, that is, particles at rest. Then, there are a total of nine real numbers that describe the distribution function of the particle for a lattice site (see

The process of the propagation and collision of particles generally occurs in two stages: the first is to denote the advance of the particles to the next lattice site along the directions of movement; this is for each time step Δt. The second stage is to simulate the collisions of the particles, which causes them to propagate in different directions at each lattice site [11, 12]. These stages can be described through the

To simplify Eq. (18), the BGK approximation is usually used; this approximation

eq � <sup>f</sup> τ

This operator models the effect of the collision as a relaxation of the distribution function towards the Maxwell equilibrium distribution. The parameter τ (relaxation

<sup>Ω</sup> <sup>¼</sup> <sup>f</sup>

!, t ¼ �<sup>Ω</sup> (18)

(19)

updated [9, 10].

**Figure 5.**

**Figure 5**)**.**

replaces the term Ω:

**49**

discretized Boltzmann equation on a lattice.

fi x ! þ e ! i, t þ 1 � fi <sup>x</sup>

**Figure 4.** *Shear stress vs. normalized shear rate with analytical solution.*

is obtained. Eq. (17) represents the velocity profile for the Poiseuille flow between two parallel plates, in the case of a Bingham-type fluid, and its graphic representation is that shown in **Figure 3**, for values of Bn = 0.1, 0.2, 0.3, and 0.4, with dimensionless values [4].

The graph of the shear stress vs. normalized shear rate (rheogram) is shown in **Figure 4**.

### **3. Lattice Boltzmann Bhatnagar-Gross-Krook (BGK)**

The Lattice Boltzmann Method (LBM) generally consists of a discrete lattice; each site (node) is represented by the distribution function, which is defined by the *Bingham Fluid Simulation in Porous Media with Lattice Boltzmann Method DOI: http://dx.doi.org/10.5772/intechopen.90167*

**Figure 5.** *Allowed directions for particle movement. (a) Model D2Q9, (b) model D2Q15.*

velocity of a particle and limited to a discrete group of allowed velocities. During each time step, the movement or jump of particle to nearby lattice sites, along its direction of movement, a collision with another particle can occur when they reach the same site. The result of the collisions is determined by means of the solution of the kinetic equation of Boltzmann for the new function of distribution of the particle to that site, and in this way the function of distribution of the particle is updated [9, 10].

There are different lattice models, which are given by DmQn; m indicates the dimension and n the permitted velocity, thus, the D2Q9 model (two-dimensional with nine speed directions), of which four sites correspond to nearby neighbors (right, left, up, and down), four other points of the lattice vectors along the diagonal faces of the following sites, along these diagonals. In this way the particles can travel in eight directions for each lattice site. The circle in the center of the square represents the vector, which has a value of zero and represents particles that have no movement, that is, particles at rest. Then, there are a total of nine real numbers that describe the distribution function of the particle for a lattice site (see **Figure 5**)**.**

The process of the propagation and collision of particles generally occurs in two stages: the first is to denote the advance of the particles to the next lattice site along the directions of movement; this is for each time step Δt. The second stage is to simulate the collisions of the particles, which causes them to propagate in different directions at each lattice site [11, 12]. These stages can be described through the discretized Boltzmann equation on a lattice.

$$\mathbf{f\_i(\overrightarrow{x} + \overrightarrow{e}\_i, t + 1) - f\_i(\overrightarrow{x}, t) = -\Omega} \tag{18}$$

To simplify Eq. (18), the BGK approximation is usually used; this approximation replaces the term Ω:

$$
\Omega = \frac{\mathbf{f}^{\text{eq}} - \mathbf{f}}{\pi} \tag{19}
$$

This operator models the effect of the collision as a relaxation of the distribution function towards the Maxwell equilibrium distribution. The parameter τ (relaxation

is obtained. Eq. (17) represents the velocity profile for the Poiseuille flow between two parallel plates, in the case of a Bingham-type fluid, and its graphic representation is that shown in **Figure 3**, for values of Bn = 0.1, 0.2, 0.3, and 0.4,

**3. Lattice Boltzmann Bhatnagar-Gross-Krook (BGK)**

*Velocity profile for a Bingham fluid; Bn = 0.1, 0.2, 0.3, and 0.4 [4].*

*Computational Fluid Dynamics Simulations*

The graph of the shear stress vs. normalized shear rate (rheogram) is shown in

The Lattice Boltzmann Method (LBM) generally consists of a discrete lattice; each site (node) is represented by the distribution function, which is defined by the

with dimensionless values [4].

*Shear stress vs. normalized shear rate with analytical solution.*

**Figure 4**.

**48**

**Figure 4.**

**Figure 3.**

time) has dimensions of time and controls the frequency with which the distribution function relaxes to reach equilibrium, that is, this time determines the rate at which the fluctuations in the system lead to this state of equilibrium [13].

$$\mathbf{f}\_{\mathbf{i}}\left(\overrightarrow{\mathbf{x}} + \overrightarrow{\mathbf{e}}\_{\mathbf{i}}, \mathbf{t} + \mathbf{1}\right) - \mathbf{f}\_{\mathbf{i}}\left(\overrightarrow{\mathbf{x}}, \mathbf{t}\right) = -\frac{\mathbf{1}}{\pi} \left[\mathbf{f}\_{\mathbf{i}}\left(\overrightarrow{\mathbf{x}}, \mathbf{t}\right) - \mathbf{f}\_{\mathbf{i}}^{\mathrm{eq}}\left(\overrightarrow{\mathbf{x}}, \mathbf{t}\right)\right] \tag{20}$$

The macroscopic variables (mass (ρ) and velocity u!� �) can be calculated directly from the values of the distribution function as

$$\begin{aligned} \rho\left(\overrightarrow{\mathbf{x}},\mathbf{t}\right) &= \sum\_{i=1}^{n} \mathbf{f}\_{i}\left(\overrightarrow{\mathbf{x}},\mathbf{t}\right) \\ \overrightarrow{\mathbf{u}}\left(\overrightarrow{\mathbf{x}},\mathbf{t}\right) &= \frac{\mathbf{1}}{\rho\left(\overrightarrow{\mathbf{x}},\mathbf{t}\right)} \sum\_{i=1}^{n} \mathbf{f}\_{i}\left(\overrightarrow{\mathbf{x}},\mathbf{t}\right) \overrightarrow{\mathbf{e}}\_{i} \end{aligned} \tag{21}$$

Darcy's law; for this the permeability is calculated based on the apparent viscosity

τ0 γ\_ <sup>v</sup>*<sup>=</sup>* dP

In Eq. (23), K is the permeability, v is the velocity, and dP/dx is a pressure force. The discrete macroscopic pressure P is given by the state equation that relates

The relationship is valid for incompressible fluid simulations and is only allowed

The criterion used by Wang and Ho [14] was taken, for which yielding occurs when the magnitude of the extra shear stress tensor exceeded the yield stress, τ0,

**Figure 6** shows the validation of the velocity profiles with the analytical solution and the simulations with LBM. The error between both solutions was less than 2.0%. For the development of the work, three porous media with a deterministic structure and nine random were proposed; in each of them all the simulations were

In **Figures 7**–**15**, the speed patterns for all simulations are shown; for this the values of the yield stress, the pressure forces, and the viscosities were varied.

*Comparison of velocity profiles with different Bingham numbers for the analytical solutions and the proposed*

*LBM. Normalized velocity profiles for (a) Bin = 0.1, (b) Bin = 0.2, (c) Bin = 0.3, (d) Bin = 0.4.*

 ≤τo.

to fluctuate locally around a fixed value [18]. Hidemitsu Hayashi proposed two LBMs for the flow generated by the pressure gradient (FGPG) and the flow driven

by an external force (FDEF), which are consistent with each other [19].

iτ<sup>o</sup> and unyielded if τyx

*=*

dx ð Þ (23)

<sup>s</sup>ρ, where cs is the speed of sound and ρ is

K ¼ μ<sup>B</sup> þ

*Bingham Fluid Simulation in Porous Media with Lattice Boltzmann Method*

according to the following equation:

*DOI: http://dx.doi.org/10.5772/intechopen.90167*

the discrete density to the pressure P <sup>¼</sup> <sup>c</sup><sup>2</sup>

 

performed for three Bingham numbers (0.2, 0.3, and 0.4).

i.e., be yielded when τyx

**Figure 6.**

**51**

the density that is calculated through Eq. (21).

In the case of a Newtonian fluid, the ratio of kinematic viscosity and relaxation time is given by [13] υ = 1/3 (τ-1/2).

## **4. Simulation of Bingham fluids with the lattice Boltzmann method**

For the simulations of the Bingham fluid with the Lattice Boltzmann Method, a modification was made to the LBGK approach presented by Wang and Ho [14] and Tang et al. [15], for a D2Q9 model, which consists in proposing the relaxation parameter τ based on the apparent viscosity:

$$
\pi = \Im[\mu\_{\mathsf{B}} + \pi\_0(\dot{\mathsf{y}})] + \frac{1}{2} \tag{22}
$$

In Eq. (22), μB, τ0, and γ\_ are the Bingham viscosity, yield stress, and shear rate, respectively. Parameter τ was used in the Lattice Boltzmann equation (Eq. (20)). The simulations were carried out on a 64 � 64 lattice, using "bounce back" conditions on the solid walls to ensure that the velocities are zero and periodic boundary conditions at the fluid inlet and outlet, so that the nodes located in the border will have their neighboring nodes on the opposite border. The steady state was reached at 360,000 time steps.

The validation of the proposal in the LBM was performed by comparing the results of the analytical solutions for a Poiseuille flow between two separate plates a distance 2H, shown in **Figure 2** using Eq. (17). The used conditions were pressure force = 2.66E-2, yield stress = 2.00E-5, and Bingham viscosity of 0.4 for a Bin = 0.1; pressure force = 5.83E-3, yield stress = 1.10E-5, and Bingham viscosity of 0.08 for a Bin = 0.2; pressure force = 5.19E-3, yield stress = 1.40E-5, and Bingham viscosity of 0.07 for a Bin = 0.3; and pressure force = 1.88E-3, yield stress = 6.50E-6, and Bingham viscosity of 0.025 for a Bin = 0.4.

Simulations were performed in porous media, applying the LBM in the case of deterministic porous medium and random porous media. A modification of a "Box-Muller method," which is a random number generator, was inserted in random porous media, and blocks were inserted arbitrarily in the lattice for deterministic porous media [16, 17].

An alternative way or method is proposed for obtaining local permeabilities for deterministic and random porous media. This one consists in a modification of

*Bingham Fluid Simulation in Porous Media with Lattice Boltzmann Method DOI: http://dx.doi.org/10.5772/intechopen.90167*

Darcy's law; for this the permeability is calculated based on the apparent viscosity according to the following equation:

$$\mathbf{K} = \left(\mu\_{\rm B} + \frac{\tau\_0}{\dot{\mathbf{y}}}\right) \mathbf{v} / (^{\rm d}\mathbf{\dot{\mathbf{y}}} \angle \mathbf{x}) \tag{23}$$

In Eq. (23), K is the permeability, v is the velocity, and dP/dx is a pressure force.

The discrete macroscopic pressure P is given by the state equation that relates the discrete density to the pressure P <sup>¼</sup> <sup>c</sup><sup>2</sup> <sup>s</sup>ρ, where cs is the speed of sound and ρ is the density that is calculated through Eq. (21).

The relationship is valid for incompressible fluid simulations and is only allowed to fluctuate locally around a fixed value [18]. Hidemitsu Hayashi proposed two LBMs for the flow generated by the pressure gradient (FGPG) and the flow driven by an external force (FDEF), which are consistent with each other [19].

The criterion used by Wang and Ho [14] was taken, for which yielding occurs when the magnitude of the extra shear stress tensor exceeded the yield stress, τ0, i.e., be yielded when τyx iτ<sup>o</sup> and unyielded if τyx ≤τo.

**Figure 6** shows the validation of the velocity profiles with the analytical solution and the simulations with LBM. The error between both solutions was less than 2.0%.

For the development of the work, three porous media with a deterministic structure and nine random were proposed; in each of them all the simulations were performed for three Bingham numbers (0.2, 0.3, and 0.4).

In **Figures 7**–**15**, the speed patterns for all simulations are shown; for this the values of the yield stress, the pressure forces, and the viscosities were varied.

**Figure 6.**

*Comparison of velocity profiles with different Bingham numbers for the analytical solutions and the proposed LBM. Normalized velocity profiles for (a) Bin = 0.1, (b) Bin = 0.2, (c) Bin = 0.3, (d) Bin = 0.4.*

time) has dimensions of time and controls the frequency with which the distribution function relaxes to reach equilibrium, that is, this time determines the rate at

> ¼ � <sup>1</sup> τ fi x !, t � �

� f eq <sup>i</sup> x !, t

) can be calculated

<sup>2</sup> (22)

(20)

(21)

h i � �

which the fluctuations in the system lead to this state of equilibrium [13].

<sup>¼</sup> <sup>X</sup><sup>n</sup> i¼1

<sup>¼</sup> <sup>1</sup> ρ x !, t � �X<sup>n</sup>

fi x !, t � �

i¼1

In the case of a Newtonian fluid, the ratio of kinematic viscosity and relaxation

For the simulations of the Bingham fluid with the Lattice Boltzmann Method, a modification was made to the LBGK approach presented by Wang and Ho [14] and Tang et al. [15], for a D2Q9 model, which consists in proposing the relaxation

<sup>τ</sup> <sup>¼</sup> <sup>3</sup>½ �þ <sup>μ</sup><sup>B</sup> <sup>þ</sup> <sup>τ</sup>0ð Þγ\_ <sup>1</sup>

In Eq. (22), μB, τ0, and γ\_ are the Bingham viscosity, yield stress, and shear rate, respectively. Parameter τ was used in the Lattice Boltzmann equation (Eq. (20)). The simulations were carried out on a 64 � 64 lattice, using "bounce back" conditions on the solid walls to ensure that the velocities are zero and periodic boundary conditions at the fluid inlet and outlet, so that the nodes located in the border will have their neighboring nodes on the opposite border. The steady state was reached

The validation of the proposal in the LBM was performed by comparing the results of the analytical solutions for a Poiseuille flow between two separate plates a distance 2H, shown in **Figure 2** using Eq. (17). The used conditions were pressure force = 2.66E-2, yield stress = 2.00E-5, and Bingham viscosity of 0.4 for a Bin = 0.1; pressure force = 5.83E-3, yield stress = 1.10E-5, and Bingham viscosity of 0.08 for a Bin = 0.2; pressure force = 5.19E-3, yield stress = 1.40E-5, and Bingham viscosity of 0.07 for a Bin = 0.3; and pressure force = 1.88E-3, yield stress = 6.50E-6, and

Simulations were performed in porous media, applying the LBM in the case of deterministic porous medium and random porous media. A modification of a "Box-Muller method," which is a random number generator, was inserted in random porous media, and blocks were inserted arbitrarily in the lattice for deterministic

An alternative way or method is proposed for obtaining local permeabilities for deterministic and random porous media. This one consists in a modification of

**4. Simulation of Bingham fluids with the lattice Boltzmann method**

fi x !, t � � e ! i

� fi x !, t � �

The macroscopic variables (mass (ρ) and velocity u!� �

directly from the values of the distribution function as

ρ x !, t � �

u ! x !, t � �

fi x ! þ e ! i, t þ 1 � �

*Computational Fluid Dynamics Simulations*

time is given by [13] υ = 1/3 (τ-1/2).

parameter τ based on the apparent viscosity:

Bingham viscosity of 0.025 for a Bin = 0.4.

at 360,000 time steps.

porous media [16, 17].

**50**

**Figure 7.**

*Bingham = 0.2, yield stress = 1.1E-5, pressure force = 5.83E-3, and viscosity = 0.08. Porosity 81.68%, deterministic porous medium: (a) vectorized flow, (b) velocity patterns. Porosity 81.62%, random porous medium: (c) vectorized flow, (d) velocity patterns.*

**Figure 9.**

**Figure 10.**

**53**

*medium: (c) vectorized flow, (d) velocity patterns.*

*medium: (c) vectorized flow, (d) velocity patterns.*

*Bingham = 0.2, yield stress = 1.1E-5, pressure force = 5.83E-3, and viscosity = 0.08. Porosity 65.82%, deterministic porous medium: (a) vectorized flow, (b) velocity patterns. Porosity 65.75%, random porous*

*Bingham Fluid Simulation in Porous Media with Lattice Boltzmann Method*

*DOI: http://dx.doi.org/10.5772/intechopen.90167*

*Bingham = 0.3, yield stress = 1.4E-5, pressure force = 5.19E-3, and viscosity = 0.07. Porosity 81.68%, deterministic porous medium: (a) vectorized flow, (b) velocity patterns. Porosity 81.62%, random porous*

#### **Figure 8.**

*Bingham = 0.2, yield stress = 1.1E-5, pressure force = 5.83E-3, and viscosity = 0.08. Porosity 73.75%, deterministic porous medium: (a) vectorized flow, (b) velocity patterns. Porosity 73.68%, random porous medium: (c) vectorized flow, (d) velocity patterns.*

*Bingham Fluid Simulation in Porous Media with Lattice Boltzmann Method DOI: http://dx.doi.org/10.5772/intechopen.90167*

**Figure 9.**

**Figure 7.**

**Figure 8.**

**52**

*medium: (c) vectorized flow, (d) velocity patterns.*

*Computational Fluid Dynamics Simulations*

*medium: (c) vectorized flow, (d) velocity patterns.*

*Bingham = 0.2, yield stress = 1.1E-5, pressure force = 5.83E-3, and viscosity = 0.08. Porosity 81.68%, deterministic porous medium: (a) vectorized flow, (b) velocity patterns. Porosity 81.62%, random porous*

*Bingham = 0.2, yield stress = 1.1E-5, pressure force = 5.83E-3, and viscosity = 0.08. Porosity 73.75%, deterministic porous medium: (a) vectorized flow, (b) velocity patterns. Porosity 73.68%, random porous*

*Bingham = 0.2, yield stress = 1.1E-5, pressure force = 5.83E-3, and viscosity = 0.08. Porosity 65.82%, deterministic porous medium: (a) vectorized flow, (b) velocity patterns. Porosity 65.75%, random porous medium: (c) vectorized flow, (d) velocity patterns.*

#### **Figure 10.**

*Bingham = 0.3, yield stress = 1.4E-5, pressure force = 5.19E-3, and viscosity = 0.07. Porosity 81.68%, deterministic porous medium: (a) vectorized flow, (b) velocity patterns. Porosity 81.62%, random porous medium: (c) vectorized flow, (d) velocity patterns.*

**Figure 11.**

*Bingham = 0.3, yield stress = 1.4E-5, pressure force = 5.19E-3, and viscosity = 0.07. Porosity 73.75%, deterministic porous medium: (a) vectorized flow, (b) velocity patterns. Porosity 73.68%, random porous medium: (c) vectorized flow, (d) velocity patterns.*

**Figure 13.**

**Figure 14.**

**55**

*medium: (c) vectorized flow, (d) velocity patterns.*

*medium: (c) vectorized flow, (d) velocity patterns.*

*Bingham = 0.4, yield stress = 6.5E-6, pressure force = 1.88E-3, and viscosity = 0.025. Porosity 81.68%, deterministic porous medium: (a) vectorized flow, (b) velocity patterns. Porosity 81.62%, random porous*

*Bingham Fluid Simulation in Porous Media with Lattice Boltzmann Method*

*DOI: http://dx.doi.org/10.5772/intechopen.90167*

*Bingham = 0.4, yield stress = 6.5E-6, pressure force = 1.88E-3, and viscosity = 0.025. Porosity 73.75%, deterministic porous medium: (a) vectorized flow, (b) velocity patterns. Porosity 73.68%, random porous*

*Bingham = 0.3, yield stress = 1.4E-5, pressure force = 5.19E-3, and viscosity = 0.07. Porosity 65.82%, deterministic porous medium: (a) vectorized flow, (b) velocity patterns. Porosity 65.75%, random porous medium: (c) vectorized flow, (d) velocity patterns.*

*Bingham Fluid Simulation in Porous Media with Lattice Boltzmann Method DOI: http://dx.doi.org/10.5772/intechopen.90167*

**Figure 13.**

**Figure 11.**

**Figure 12.**

**54**

*medium: (c) vectorized flow, (d) velocity patterns.*

*Computational Fluid Dynamics Simulations*

*medium: (c) vectorized flow, (d) velocity patterns.*

*Bingham = 0.3, yield stress = 1.4E-5, pressure force = 5.19E-3, and viscosity = 0.07. Porosity 73.75%, deterministic porous medium: (a) vectorized flow, (b) velocity patterns. Porosity 73.68%, random porous*

*Bingham = 0.3, yield stress = 1.4E-5, pressure force = 5.19E-3, and viscosity = 0.07. Porosity 65.82%, deterministic porous medium: (a) vectorized flow, (b) velocity patterns. Porosity 65.75%, random porous*

*Bingham = 0.4, yield stress = 6.5E-6, pressure force = 1.88E-3, and viscosity = 0.025. Porosity 81.68%, deterministic porous medium: (a) vectorized flow, (b) velocity patterns. Porosity 81.62%, random porous medium: (c) vectorized flow, (d) velocity patterns.*

#### **Figure 14.**

*Bingham = 0.4, yield stress = 6.5E-6, pressure force = 1.88E-3, and viscosity = 0.025. Porosity 73.75%, deterministic porous medium: (a) vectorized flow, (b) velocity patterns. Porosity 73.68%, random porous medium: (c) vectorized flow, (d) velocity patterns.*

**Figure 15.**

*Bingham = 0.4, yield stress = 6.5E-6, pressure force = 1.88E-3, and viscosity = 0.025. Porosity 65.82%, deterministic porous medium: (a) vectorized flow, (b) velocity patterns. Porosity 65.75%, random porous medium: (c) vectorized flow, (d) velocity patterns.*

**Figure 16.**

**Figure 17.**

*porous media.*

**57**

*(d) pattern porous media.*

*Local permeabilities for deterministic porous media with porosity of 81.68%, Bingham = 0.2, yield stress = 1.1E-5, pressure force = 5.83E-3 and viscosity = 0.08. (a–c) The rough numerical date of permeability,*

*Bingham Fluid Simulation in Porous Media with Lattice Boltzmann Method*

*DOI: http://dx.doi.org/10.5772/intechopen.90167*

*Local permeabilities for random porous media with porosity of 81.62%, Bingham = 0.2, yield stress = 1.1E-5, pressure force = 5.83E-3 and viscosity = 0.08; (a–c) The rough numerical date of permeability, (d) pattern*

In **Figures 7**–**15** the decrease in porosity corresponds to a decrease in velocity. It can be noted that for a Bingham number of 0.2 (**Figures 7**–**9**), higher velocities are shown in deterministic porous media than random media. But in the latter, there are more places to pass the fluid. These behaviors are presented in the simulations for the three Bingham numbers used (0.2, 0.2, and 0.3). Comparing the Bingham number of 0.2 with that of 0.4, a decrease in the velocity in the deterministic porous media is observed as well as the random ones according to the conditions handled; this is due to the decrease of the initial effort, the pressure force, and the viscosity.

Local permeabilities were simulated based on apparent viscosities for all deterministic and random porous media. In **Figures 16**–**19**, only some of the results of local permeabilities for deterministic porous media 81.68 and 65.82% with Bingham numbers of 0.2 and 0.4, respectively, are shown. Likewise, only the result of the simulations for two random porous media 81.62 and 65.75% for Bingham numbers of 0.2 and 0.4, respectively, are shown.

**Figures 16**–**19** show the zones in blue of the local permeabilities. It is remarkable that the blue areas predominate in random porous media. By comparing the Bingham number of 0.2 for the two deterministic and random porous media according to **Figures 16** and **17** with that of 0.4 of **Figures 18** and **19**, an increase in permeabilities can be seen.

Finally, pressures for all porous media were simulated. **Figures 20**–**25** show some of the results obtained in the case of deterministic porous media with porosities of 81.68, 73.75, and 65.82% for Bingham numbers 0.2, 0.3, and 0.4, respectively. Similarly, the results are presented for random porous media with porosities of 81.62, 73.68, and 65.75% for Bingham numbers 0.2, 0.3, and 0.4, respectively.

*Bingham Fluid Simulation in Porous Media with Lattice Boltzmann Method DOI: http://dx.doi.org/10.5772/intechopen.90167*

#### **Figure 16.**

In **Figures 7**–**15** the decrease in porosity corresponds to a decrease in velocity. It can be noted that for a Bingham number of 0.2 (**Figures 7**–**9**), higher velocities are shown in deterministic porous media than random media. But in the latter, there are more places to pass the fluid. These behaviors are presented in the simulations for the three Bingham numbers used (0.2, 0.2, and 0.3). Comparing the Bingham number of 0.2 with that of 0.4, a decrease in the velocity in the deterministic porous media is observed as well as the random ones according to the conditions handled; this is due to the decrease of the initial effort, the pressure force, and the viscosity. Local permeabilities were simulated based on apparent viscosities for all deterministic and random porous media. In **Figures 16**–**19**, only some of the results of local permeabilities for deterministic porous media 81.68 and 65.82% with Bingham numbers of 0.2 and 0.4, respectively, are shown. Likewise, only the result of the simulations for two random porous media 81.62 and 65.75% for Bingham numbers

*Bingham = 0.4, yield stress = 6.5E-6, pressure force = 1.88E-3, and viscosity = 0.025. Porosity 65.82%, deterministic porous medium: (a) vectorized flow, (b) velocity patterns. Porosity 65.75%, random porous*

**Figures 16**–**19** show the zones in blue of the local permeabilities. It is remarkable that the blue areas predominate in random porous media. By comparing the Bingham number of 0.2 for the two deterministic and random porous media according to **Figures 16** and **17** with that of 0.4 of **Figures 18** and **19**, an increase in perme-

Finally, pressures for all porous media were simulated. **Figures 20**–**25** show some of the results obtained in the case of deterministic porous media with porosities of 81.68, 73.75, and 65.82% for Bingham numbers 0.2, 0.3, and 0.4, respectively. Similarly, the results are presented for random porous media with porosities of 81.62, 73.68, and 65.75% for Bingham numbers 0.2, 0.3, and 0.4, respectively.

of 0.2 and 0.4, respectively, are shown.

*medium: (c) vectorized flow, (d) velocity patterns.*

*Computational Fluid Dynamics Simulations*

abilities can be seen.

**56**

**Figure 15.**

*Local permeabilities for deterministic porous media with porosity of 81.68%, Bingham = 0.2, yield stress = 1.1E-5, pressure force = 5.83E-3 and viscosity = 0.08. (a–c) The rough numerical date of permeability, (d) pattern porous media.*

#### **Figure 17.**

*Local permeabilities for random porous media with porosity of 81.62%, Bingham = 0.2, yield stress = 1.1E-5, pressure force = 5.83E-3 and viscosity = 0.08; (a–c) The rough numerical date of permeability, (d) pattern porous media.*

**Figure 20.**

**Figure 21.**

**Figure 22.**

*pressure.*

**59**

*Pressures for deterministic porous medium with porosity of 81.68%, Bingham = 0.2, yield stress = 1.1E-5, pressure force = 5.83E-3 and Viscosity = 0.08. (a) Pressure distribution, (b) the rough numerical date of pressure.*

*Bingham Fluid Simulation in Porous Media with Lattice Boltzmann Method*

*DOI: http://dx.doi.org/10.5772/intechopen.90167*

*Pressures for random porous medium with porosity of 81.62%, Bingham = 0.2, yield stress = 1.1E-5, pressure force = 5.83E-3 and Viscosity = 0.08. (a) Pressure distribution, (b) the rough numerical date of pressure.*

*Pressures for deterministic porous medium with porosity of 73.75%, Bingham = 0.3, yield stress = 1.4E-5, pressure force = 5.19E-3 and Viscosity = 0.07. (a) Pressure distribution, (b) the rough numerical date of*

#### **Figure 18.**

*Local permeabilities for deterministic porous media with porosity of 65.82%, Bingham = 0.4, yield stress = 6.5E-6, pressure force = 1.88-3 and viscosity = 0.025; (a–c) the rough numerical date of permeability, (d) pattern porous media.*

#### **Figure 19.**

*Local permeabilities for random porous media with porosity of 65.75%, Bingham = 0.4, yield stress = 6.5E-6, pressure force = 1.88-3 and viscosity = 0.025; (a–c) The rough numerical date of permeability, (d) pattern porous media.*

**Figures 20**–**25**, you can see in all cases the difference in pressures, higher pressure in the red colors, and less pressure in the green colors, in addition to observing the pressures in the different zones.

*Bingham Fluid Simulation in Porous Media with Lattice Boltzmann Method DOI: http://dx.doi.org/10.5772/intechopen.90167*

**Figure 20.**

*Pressures for deterministic porous medium with porosity of 81.68%, Bingham = 0.2, yield stress = 1.1E-5, pressure force = 5.83E-3 and Viscosity = 0.08. (a) Pressure distribution, (b) the rough numerical date of pressure.*

**Figure 21.**

*Pressures for random porous medium with porosity of 81.62%, Bingham = 0.2, yield stress = 1.1E-5, pressure force = 5.83E-3 and Viscosity = 0.08. (a) Pressure distribution, (b) the rough numerical date of pressure.*

*Pressures for deterministic porous medium with porosity of 73.75%, Bingham = 0.3, yield stress = 1.4E-5, pressure force = 5.19E-3 and Viscosity = 0.07. (a) Pressure distribution, (b) the rough numerical date of pressure.*

**Figures 20**–**25**, you can see in all cases the difference in pressures, higher pressure in the red colors, and less pressure in the green colors, in addition to

*Local permeabilities for random porous media with porosity of 65.75%, Bingham = 0.4, yield stress = 6.5E-6, pressure force = 1.88-3 and viscosity = 0.025; (a–c) The rough numerical date of permeability, (d) pattern*

*Local permeabilities for deterministic porous media with porosity of 65.82%, Bingham = 0.4, yield*

*stress = 6.5E-6, pressure force = 1.88-3 and viscosity = 0.025; (a–c) the rough numerical date of permeability,*

observing the pressures in the different zones.

**Figure 18.**

**Figure 19.**

*porous media.*

**58**

*(d) pattern porous media.*

*Computational Fluid Dynamics Simulations*

**5. Conclusions**

a Poiseuille flow.

regime.

work.

costs and time.

more complex.

**Acknowledgements**

**Conflict of interest**

paper.

**61**

many engineering problems.

*DOI: http://dx.doi.org/10.5772/intechopen.90167*

In the present work, the Lattice Boltzmann Method was applied to a problem of the flow of a non-Newtonian Bingham-type fluid between two plates, in the case of

This method is an alternative to the conventional ones used in computational fluid mechanics, its programming is not complicated, and today it is applied to

Validations were carried out with the analytical solution of the velocity profiles for the case of a Poiseuille flow and the simulations with Lattice Boltzmann, for the case of Bingham-type fluids, for values of the Bingham number (Bin) of 0.1, 0.2, 0.3, and 0.4. The results of all the simulations were quite acceptable, since the

The LBM proves to be kind for simulations with small lattices, such as the one used in the present work 64 64. All simulations were performed in a laminar

Three deterministic porous media with porosities of 81.68, 75.75, and 65.82% and nine randomized ones with porosities of 81.62, 73.68, and 65.75% were proposed for three Bingham numbers (0.2, 0.3, and 0.4), to perform all simulations. In them the

Profiles of velocities, permeabilities, and local pressures were obtained, in all cases the results and behaviors were acceptable for all porous media, and the three Bingham numbers, although only some of the results obtained, were presented at

The LBM with the necessary restrictions allows to perfectly simulate the behav-

Finally, it would be convenient to perform simulations with turbulent flows to verify the goodness of the method with this type of fluid, in which its description is

The present work was developed under the sponsorship of the Facultad de Estudios Superiores Cuautitlán-Universidad Nacional Autónoma de México.

The author declares no conflicts of interest regarding the publication of this

ior of fluids, as is the case of the Bingham type; the importance of this is the application of multiple industrial processes, in the displacement of fluids reducing

percentage of error between both results did not exceed 2.0%.

*Bingham Fluid Simulation in Porous Media with Lattice Boltzmann Method*

pressure forces, yield stress, and viscosities were varied.

**Figure 23.**

*Pressures for random porous medium with porosity of 73.68%, Bingham = 0.3, yield stress = 1.4E-5, pressure force = 5.19E-3 and Viscosity = 0.07. (a) Pressure distribution, (b) the rough numerical date of pressure.*

**Figure 24.**

*Pressures for deterministic porous medium with porosity of 65.82%, Bingham = 0.4, yield stress = 6.5E-6, pressure force = 1.88E-3 and Viscosity = 0.025. (a) Pressure distribution, (b) the rough numerical date of pressure.*

**Figure 25.**

*Pressures for random porous medium with porosity of 65.75%, Bingham = 0.4, yield stress = 6.5E-6, pressure force = 1.88E-3 and Viscosity = 0.025. (a) Pressure distribution, (b) the rough numerical date of pressure.*

*Bingham Fluid Simulation in Porous Media with Lattice Boltzmann Method DOI: http://dx.doi.org/10.5772/intechopen.90167*
