**2. Bingham model**

ρ ∂v ! ∂t þ v !∇ � v !

*Computational Fluid Dynamics Simulations*

**1.1 Newtonian fluids**

the viscosity gradient:

**1.2 Non-Newtonian fluids**

are shown in **Figure 1**.

**Figure 1.**

**44**

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

" #

can have an analytical solution only for a small number of cases.

and its graphic representation can be seen in **Figure 1**.

ics, which are called Newtonians, non-Newtonians, and viscoelastic.

¼ �∇<sup>P</sup> <sup>þ</sup> <sup>μ</sup>∇2v

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

There are basically three types of fluids from the point of view of fluid dynam-

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

In Eq. (3), τyx is the shear stress, (dvx/dy) is the shear rate, and μ is the viscosity,

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

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].

dvx dy � �

τyx ¼ μ

! þ ρg

! (2)

(3)

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), Reiner (1958), and Prager (1961).

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 zero shear rate.

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 model is given by

$$\begin{aligned} \boldsymbol{\tau}\_{\rm yx} &= \boldsymbol{\tau}\_0 + \mu\_\text{B} \, \mathop{\rm div}\_\text{x} \quad \text{if} \quad |\boldsymbol{\tau}\_{\rm yx}| > \boldsymbol{\tau}\_0 \\\\ \frac{\text{dv}\_\mathbf{x}}{\text{d}\mathbf{y}} &= \mathbf{0}; \quad \text{if} \quad |\boldsymbol{\tau}\_{\rm yx}| < \boldsymbol{\tau}\_0 \end{aligned} \tag{4}$$

$$\mu(\dot{\boldsymbol{\gamma}}) = \mu\_\text{B} + \frac{\boldsymbol{\tau}\_0}{\dot{\boldsymbol{\gamma}}} \quad \text{for} \quad \boldsymbol{\tau}\_{\rm yx} > \boldsymbol{\tau}\_0 \tag{5}$$

$$\dot{\boldsymbol{\gamma}} = \mathbf{0}; \quad \text{for} \quad \boldsymbol{\tau}\_{\rm yx} \le \boldsymbol{\tau}\_0$$

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\_.

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