Bingham Fluid Simulation in Porous Media with Lattice Boltzmann Method

*José Luis Velázquez Ortega*

## **Abstract**

Generate a lattice Boltzmann model (LBM), which allows to simulate the behavior of a Bingham fluid through a rectangular channel with the D2Q9 model. For this purpose, a relaxation parameter is proposed based on the rheological parameters of the Bingham model. The validation will be carried out with the solution of the movement equation, and velocity profiles will be obtained for three different Bingham numbers (Bn). Other simulations will be made in a rectangular channel in the presence of arbitrarily and randomly generated porous media. The main objective is to propose a method to predict the behavior of non-Newtonian fluids (Bingham fluid) through porous media, which have many applications in the chemical industry avoiding at the same time the expensive experimentation of these systems, with predicting models.

**Keywords:** lattice Boltzmann model, non-Newtonian fluids, Bingham fluid, porous media, velocity profiles

### **1. Introduction**

A continuous medium is characterized by the fact that its atoms or molecules are so close together, in such way that they could be considered macroscopically as a homogeneous mass, whose behavior can be foreseen without considering the movement of each of its elementary particles that compose it. In this sense it is assumed that there are no gaps or separations between the particles.

The movement of fluids can have a wide variety of behaviors for both simple and complex flows (biological and food systems). In addition to this, using Reynolds number, we can know if the flow has turbulent or laminar regime.

Mass conservation is the basic principle of fluid movement, which requires that when the fluid is in motion, it moves so that the mass is preserved. The movement of fluids is governed in general by the continuity equation.

$$\frac{\partial \mathbf{p}}{\partial \mathbf{t}} + \nabla \cdot \left(\rho \vec{\mathbf{v}}\right) = \mathbf{0} \tag{1}$$

On the other hand, the Navier-Stokes equation, which in general terms corresponds to the application of Newton's second law of classical mechanics to fluid movement, is described as follows:

$$
\rho \left[ \frac{\partial \overrightarrow{\mathbf{v}}}{\partial \mathbf{t}} + \overrightarrow{\mathbf{v}} \nabla \cdot \overrightarrow{\mathbf{v}} \right] = -\nabla \mathbf{P} + \mu \nabla^2 \overrightarrow{\mathbf{v}} + \rho \overrightarrow{\mathbf{g}} \tag{2}
$$

In the case of these fluids, the viscosity is no longer constant; therefore, the relationship between the shear stress and the shear rate of the fluid is no longer linear. For this reason, a new term is now introduced and is known as apparent

Some fluids have a yield stress, from which the fluid begins to move. Below this tension the shear rate would be zero. This relationship is not linear or, if it is, it does not pass through the origin [4]. Complex mixtures are considered non-Newtonian fluids: grouts, pastes, gels, polymer solutions, etc. Most non-Newtonian fluids are mixtures with constituents of very different sizes. For example, toothpaste is composed of solid particles suspended in an aqueous solution of several polymers. Solid particles are much larger than water molecules, and polymer molecules are much

Much of the research that is carried out in the field of non-Newtonian fluids has been focused in the measure of its shear stress-shear strain curves and to look for mathematical descriptions of these curves. The study of the behavior of the flow of materials is called rheology (a term that originates from Greek words that give the meaning of "the study of flow"); thus, diagrams such as the one shown in **Figure 1**

In the case of Bingham fluids, sometimes called Bingham plastics, they resist a small shear force indefinitely, but they flow easily under large shear stresses. In other words, at low stresses the plastic viscosity of Bingham is infinite, and at greater stresses the viscosity decreases with the increase in the velocity gradient. Examples are bread dough, toothpaste, apple sauce, some paints, plastics, mayon-

Eugene C. Bingham described the paintings with this model in 1919, published in his book *Fluidity and Plasticity* in 1922. The model was analyzed by Oldroyd (1947),

The main feature of the Bingham model is the yield stress, necessary for the fluid to deform or flow. Above this minimum yield stress*,* the fluid begins to move. If this yield stress is not exceeded, the fluid behaves like a rigid or quasi-rigid body, with

The relationship between the shear stress and the velocity gradient is linear, but it does not go through the origin for a Bingham plastic (**Figure 1**); its mathematical

> dvx dy z}|{ γ\_

> > � � � �<τ<sup>0</sup>

dy <sup>¼</sup> 0; if <sup>τ</sup>yx

τ0

γ\_ ¼ 0; for τyx ≤τ<sup>0</sup>

where τ<sup>0</sup> is the yield stress, μ<sup>B</sup> is the plastic viscosity of Bingham, and μð Þγ\_ is the apparent viscosity, which decreases with the increase in themagnitude of the shear rate γ\_.

if τyx � � �

�> τ<sup>0</sup> (4)

<sup>γ</sup>\_ for <sup>τ</sup>yx <sup>&</sup>gt;τ<sup>0</sup> (5)

τyx ¼ τ<sup>0</sup> þ μ<sup>B</sup>

dvx

μð Þ¼ γ\_ μ<sup>B</sup> þ

viscosity or also known as shear rate-dependent viscosity [3].

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

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

larger than water molecules.

are often called rheograms.

**2. Bingham model**

zero shear rate.

model is given by

**45**

Reiner (1958), and Prager (1961).

naise, ketchup, aleas, and some grouts [5, 6].

for a fluid with velocity v!, density ρ, pressure p, and kinematic shear viscosity μ. The Navier–Stokes equation is a second-order partial differential equation, which can have an analytical solution only for a small number of cases.

There are basically three types of fluids from the point of view of fluid dynamics, which are called Newtonians, non-Newtonians, and viscoelastic.

#### **1.1 Newtonian fluids**

In this type of fluid, the shear stress or shear force per unit area is proportional to the viscosity gradient:

$$\tau\_{\rm yx} = \mu \left( \frac{\rm dv\_x}{\rm dy} \right) \tag{3}$$

In Eq. (3), τyx is the shear stress, (dvx/dy) is the shear rate, and μ is the viscosity, and its graphic representation can be seen in **Figure 1**.

All gases, liquid water, and liquids of single molecules (ammonia, alcohol, benzene, petroleum, chloroform, butane, etc.) are Newtonian. Many food materials such as milk, apple juice, orange juice, wine, and beer have a Newtonian behavior [1].

#### **1.2 Non-Newtonian fluids**

When the relationship between shear stress and shear rate is not linear, the fluid is called non-Newtonian. There are many these types of fluids, and their behaviors are shown in **Figure 1**.

**Figure 1.** *Types of times-independent flow behavior [2].*

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

In the case of these fluids, the viscosity is no longer constant; therefore, the relationship between the shear stress and the shear rate of the fluid is no longer linear. For this reason, a new term is now introduced and is known as apparent viscosity or also known as shear rate-dependent viscosity [3].

Some fluids have a yield stress, from which the fluid begins to move. Below this tension the shear rate would be zero. This relationship is not linear or, if it is, it does not pass through the origin [4]. Complex mixtures are considered non-Newtonian fluids: grouts, pastes, gels, polymer solutions, etc. Most non-Newtonian fluids are mixtures with constituents of very different sizes. For example, toothpaste is composed of solid particles suspended in an aqueous solution of several polymers. Solid particles are much larger than water molecules, and polymer molecules are much larger than water molecules.

Much of the research that is carried out in the field of non-Newtonian fluids has been focused in the measure of its shear stress-shear strain curves and to look for mathematical descriptions of these curves. The study of the behavior of the flow of materials is called rheology (a term that originates from Greek words that give the meaning of "the study of flow"); thus, diagrams such as the one shown in **Figure 1** are often called rheograms.

In the case of Bingham fluids, sometimes called Bingham plastics, they resist a small shear force indefinitely, but they flow easily under large shear stresses. In other words, at low stresses the plastic viscosity of Bingham is infinite, and at greater stresses the viscosity decreases with the increase in the velocity gradient. Examples are bread dough, toothpaste, apple sauce, some paints, plastics, mayonnaise, ketchup, aleas, and some grouts [5, 6].
