1. Introduction

As an example to the application of Criminale-Ericksen-Filbey (CEF) fluid, we can consider the settling of small particles in a non-Newtonian fluid medium. The geometry of the problem: schematic diagram of a sphere falling through a fluid in a cylinder is shown in (Figure 1). Due to axisymmetry the problem can be considered as 2-D.

cumbersome. On the other hand, the material coefficients used being constant, it is not in good agreement with experimental results in case the shear strain-rate is not very small. The CEF constitutive equation removes this draw-back by taking these coefficients variable and dependent of the shear strain-rate. That is why this model is used in the study of the settling of small

τ ¼ –p I þ η A1 þ ð Þ υ<sup>1</sup> þ υ<sup>2</sup> A<sup>1</sup>

The Rivlin-Ericksen tensors involved in the CEF equation are [5]:

The shear strain-rate in term of the second invariant IId is [6]

1.1.1.1. Material functions in steady-state shear flows

<sup>γ</sup>\_ <sup>¼</sup> <sup>2</sup> ffiffiffiffiffiffi IId <sup>p</sup> <sup>¼</sup>

The merit of CEF constitutive equation is that it stresses the dependence of the viscosity coefficient on shear strain-rate, that is, it takes into account the shear thinning (or shearthickening) effects, and those of normal stresses which are also dependent on the shear strainrate. In steady-state shear flow an extremely wide class of viscoelastic constitutive equations simplifies to CEF equation. The first term of the CEF equation for τ is just ηð Þ γ\_ γ\_; the other two terms, containing υ<sup>1</sup> and υ2, describe the elastic effects associated with the normal stresses [4].

A1 <sup>¼</sup> <sup>2</sup><sup>d</sup> <sup>¼</sup> <sup>∇</sup><sup>V</sup> <sup>þ</sup> <sup>∇</sup>V<sup>T</sup>

A2 <sup>¼</sup> <sup>∇</sup><sup>a</sup> <sup>þ</sup> <sup>∇</sup>aT <sup>þ</sup> <sup>2</sup> <sup>∇</sup><sup>V</sup> <sup>∇</sup>V<sup>T</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 trA<sup>2</sup> 1

¼

<sup>1</sup>, A<sup>2</sup> appearing in Eq. (1) are given in extenso.

r

ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 trA<sup>2</sup>

r

At high shear rates, the viscosity of most polymeric liquids decreases with increasing shear rate (Figure 2). For many engineering applications this is the most important property of

An especially useful form has been described by Carreau [4] for the viscosity coefficient

A1

Particle Settling in a Non-Newtonian Fluid Medium Processed by Using the CEF Model

υ<sup>1</sup> A<sup>2</sup> (1)

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67

<sup>2</sup> <sup>¼</sup> <sup>4</sup>d2 (2)

Id ¼ 0 (3)

(4)

particles in a non-Newtonian fluid medium.

The constitutive equation of the CEF fluid is:

1.1.1. The CEF equation

The first invariant of d is null

The velocity gradients of A1, A<sup>2</sup>

polymeric fluids.

(n = 0.364).

This problem has great importance in a large number of industrial applications. Because these materials are rather polymeric and consequently viscoelastic, it is obligatory to include the elastic behaviour in the analysis. The notations are the usual ones.

The quantities of interest are the stresses: we must consider the effects of the normal stress in addition to those of the shear stresses and to use a constitutive equation which includes normal stress coefficients, υ<sup>1</sup> and υ<sup>2</sup> as well as viscosity coefficient η, while taking into account their shear-thinning variation [1–3].

#### 1.1. The non-Newtonian fluid chosen

Many constitutive equations developed from the continuum mechanics or microstructural viewpoints, are used to describe the behaviour of non-Newtonian fluids. Among them the second-order Rivlin-Ericksen model is generally preferred because it describes the real behaviour of the fluids with sufficient accuracy and also because its application is not very

Figure 1. Schematic diagram of a sphere falling through a fluid in a cylinder.

cumbersome. On the other hand, the material coefficients used being constant, it is not in good agreement with experimental results in case the shear strain-rate is not very small. The CEF constitutive equation removes this draw-back by taking these coefficients variable and dependent of the shear strain-rate. That is why this model is used in the study of the settling of small particles in a non-Newtonian fluid medium.

#### 1.1.1. The CEF equation

1. Introduction

66 Polymer Rheology

shear-thinning variation [1–3].

1.1. The non-Newtonian fluid chosen

to axisymmetry the problem can be considered as 2-D.

elastic behaviour in the analysis. The notations are the usual ones.

Figure 1. Schematic diagram of a sphere falling through a fluid in a cylinder.

As an example to the application of Criminale-Ericksen-Filbey (CEF) fluid, we can consider the settling of small particles in a non-Newtonian fluid medium. The geometry of the problem: schematic diagram of a sphere falling through a fluid in a cylinder is shown in (Figure 1). Due

This problem has great importance in a large number of industrial applications. Because these materials are rather polymeric and consequently viscoelastic, it is obligatory to include the

The quantities of interest are the stresses: we must consider the effects of the normal stress in addition to those of the shear stresses and to use a constitutive equation which includes normal stress coefficients, υ<sup>1</sup> and υ<sup>2</sup> as well as viscosity coefficient η, while taking into account their

Many constitutive equations developed from the continuum mechanics or microstructural viewpoints, are used to describe the behaviour of non-Newtonian fluids. Among them the second-order Rivlin-Ericksen model is generally preferred because it describes the real behaviour of the fluids with sufficient accuracy and also because its application is not very The constitutive equation of the CEF fluid is:

$$\boldsymbol{\pi} = -\boldsymbol{p} \, \boldsymbol{I} + \boldsymbol{\eta} \, \boldsymbol{A}\_1 + \left(\boldsymbol{\upsilon}\_1 + \boldsymbol{\upsilon}\_2\right) \boldsymbol{A}\_1 \, ^2 - \frac{1}{2} \boldsymbol{\upsilon}\_1 \, \boldsymbol{A}\_2 \tag{1}$$

The merit of CEF constitutive equation is that it stresses the dependence of the viscosity coefficient on shear strain-rate, that is, it takes into account the shear thinning (or shearthickening) effects, and those of normal stresses which are also dependent on the shear strainrate. In steady-state shear flow an extremely wide class of viscoelastic constitutive equations simplifies to CEF equation. The first term of the CEF equation for τ is just ηð Þ γ\_ γ\_; the other two terms, containing υ<sup>1</sup> and υ2, describe the elastic effects associated with the normal stresses [4].

The Rivlin-Ericksen tensors involved in the CEF equation are [5]:

$$A\_1 = 2d = \nabla V \,\,\, + \nabla V^T$$

$$A\_1^2 = 4d^2$$

$$A\_2 = \nabla a + \nabla a^T + 2\nabla V \,\nabla V^T$$

The first invariant of d is null

$$I\_d = \mathbf{0} \tag{3}$$

The shear strain-rate in term of the second invariant IId is [6]

$$\dot{\gamma} = 2\sqrt{\Pi\_\mathrm{d}} = \sqrt{\frac{1}{2}\mathrm{tr}\,\mathrm{A}\_1^2} = \sqrt{\frac{1}{2}\mathrm{tr}\,\mathrm{A}\_2} \tag{4}$$

The velocity gradients of A1, A<sup>2</sup> <sup>1</sup>, A<sup>2</sup> appearing in Eq. (1) are given in extenso.

#### 1.1.1.1. Material functions in steady-state shear flows

At high shear rates, the viscosity of most polymeric liquids decreases with increasing shear rate (Figure 2). For many engineering applications this is the most important property of polymeric fluids.

An especially useful form has been described by Carreau [4] for the viscosity coefficient (n = 0.364).

Figure 2. Shear-thinning in a typical non-Newtonian fluid.

Figure 3. Non-linear results.

$$
\eta = \eta\_0 \left( 1 + 32.32 \cdot tr \cdot A\_1^2 \right)^{-0.318} \tag{5}
$$

The most important points to note about υ<sup>2</sup> are that its magnitude is much smaller than υ1,

<sup>η</sup> ) is plotted against the shear-rate.

Particle Settling in a Non-Newtonian Fluid Medium Processed by Using the CEF Model

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69

The choice of Carreau formula is justified by the behaviour similarity of inelastic and viscoelastic fluids concerning the viscosity, and that of the formula about the normal stress coefficients by the

As an example to the application of the CEF fluid, we can consider the settling of small particles in a non-Newtonian fluid medium. The simulation of this problem according to the fluid mechanics principles may be realized by the flow of a non-Newtonian fluid around a

The knowledge of the rate of settling of particles in practice is particularly significant in determining the shelf life of materials such as foodstuffs, cleaning materials and many others. Also, in oil and gas drilling it is important to understand the distribution of loose material, removed by the drill bit and carried to the surface by the drilling mud. Thus, this problem has great importance in many natural and physical processes and in a large number of industrial applications such as chemical, genetic and biomedical engineering operations. The cylindrical tube is considered as stationary. The drag coefficient must be calculated. The equations determining

fact that the particle settling problem has characteristics close to dilute suspensions.

υ<sup>2</sup> ¼ –0:15υ<sup>1</sup> (7)

υ<sup>1</sup> þ υ<sup>2</sup> ¼ 0:85υ<sup>1</sup> (8)

usually about 10–20% of υ1, and that it is negative [4].

τγ\_ <sup>¼</sup> <sup>υ</sup><sup>1</sup>

2. An example to CEF fluid application

sphere falling along the centerline of a cylindrical tube [9–14].

Hence, we shall take

Figure 4. The relaxation time (defined as <sup>N</sup><sup>1</sup>

Consequently

The normal stress coefficients may be handled as below:

If η = η<sup>o</sup> and υ<sup>1</sup> = υ<sup>10</sup> for γ\_ = 0, according to λγ\_ Weissenberg number, η/η<sup>o</sup> and <sup>N</sup><sup>1</sup> <sup>γ</sup>\_ <sup>2</sup>υ<sup>10</sup> <sup>¼</sup> <sup>υ</sup><sup>1</sup> <sup>υ</sup><sup>10</sup> curves may be displayed as in the Figure 3 [7]. Elastic effects are observable in a steady simple-shear flow through normal stress effects. This is demonstrated in Figure 4 [8].

Treating the curve in the Figure 4 by the least square method, the formula below can be found for the first normal stress coefficient υ1:

$$\frac{\mathbf{U}\_1}{\eta} \approx \mathbf{10} \left[ {}^{-0.169 \left( \log\_{10} \dot{\mathbf{y}} \right)} {}^{2} - 0.76 \log\_{10} \dot{\mathbf{y}} - 0.821 \right] \tag{6}$$

Particle Settling in a Non-Newtonian Fluid Medium Processed by Using the CEF Model http://dx.doi.org/10.5772/intechopen.75977 69

Figure 4. The relaxation time (defined as <sup>N</sup><sup>1</sup> τγ\_ <sup>¼</sup> <sup>υ</sup><sup>1</sup> <sup>η</sup> ) is plotted against the shear-rate.

The most important points to note about υ<sup>2</sup> are that its magnitude is much smaller than υ1, usually about 10–20% of υ1, and that it is negative [4].

Hence, we shall take

$$\nu\_2 = -0.15\nu\_1\tag{7}$$

Consequently

<sup>η</sup> <sup>¼</sup> <sup>η</sup><sup>0</sup> <sup>1</sup> <sup>þ</sup> <sup>32</sup>:<sup>32</sup> � tr � <sup>A</sup><sup>2</sup>

may be displayed as in the Figure 3 [7]. Elastic effects are observable in a steady simple-shear

Treating the curve in the Figure 4 by the least square method, the formula below can be found

If η = η<sup>o</sup> and υ<sup>1</sup> = υ<sup>10</sup> for γ\_ = 0, according to λγ\_ Weissenberg number, η/η<sup>o</sup> and <sup>N</sup><sup>1</sup>

<sup>η</sup> <sup>≈</sup> <sup>10</sup> �0:169 log10 ð Þ<sup>γ</sup>\_ <sup>2</sup>

flow through normal stress effects. This is demonstrated in Figure 4 [8].

υ1

The normal stress coefficients may be handled as below:

Figure 2. Shear-thinning in a typical non-Newtonian fluid.

for the first normal stress coefficient υ1:

Figure 3. Non-linear results.

68 Polymer Rheology

1

�0:76log10γ\_�0:821

�0:<sup>318</sup> (5)

<sup>γ</sup>\_ <sup>2</sup>υ<sup>10</sup> <sup>¼</sup> <sup>υ</sup><sup>1</sup>

<sup>υ</sup><sup>10</sup> curves

(6)

$$
\nu\_1 + \nu\_2 = 0.85\nu\_1\tag{8}
$$

The choice of Carreau formula is justified by the behaviour similarity of inelastic and viscoelastic fluids concerning the viscosity, and that of the formula about the normal stress coefficients by the fact that the particle settling problem has characteristics close to dilute suspensions.

#### 2. An example to CEF fluid application

As an example to the application of the CEF fluid, we can consider the settling of small particles in a non-Newtonian fluid medium. The simulation of this problem according to the fluid mechanics principles may be realized by the flow of a non-Newtonian fluid around a sphere falling along the centerline of a cylindrical tube [9–14].

The knowledge of the rate of settling of particles in practice is particularly significant in determining the shelf life of materials such as foodstuffs, cleaning materials and many others. Also, in oil and gas drilling it is important to understand the distribution of loose material, removed by the drill bit and carried to the surface by the drilling mud. Thus, this problem has great importance in many natural and physical processes and in a large number of industrial applications such as chemical, genetic and biomedical engineering operations. The cylindrical tube is considered as stationary. The drag coefficient must be calculated. The equations determining the motion of a sphere in an incompressible fluid under isothermal conditions will be given for a non-Newtonian viscous fluid exhibiting shear-thinning and using cylindrical coordinates [15]. As this study is the simulation of the slow motion of small particles, we can suppose the flow irrotational. Furthermore, due to the data of the considered problem, the motion may be assumed steady, axisymmetric and the fluid incompressible. The global cylindrical coordinate system (r, θ, z) is shown in Figure 1, cited firstly in the introduction section showing schematic diagram of the problem studied. The local area coordinates are L1, L2, and L3.

Due to the lack of analytical solutions, one will have to resort to numerical methods. The finite element method (FEM) will be used for this purpose.

#### 2.1. Governing equations

Continuity equation

$$
\nabla \cdot \mathbf{V} = \frac{\partial v\_r}{\partial r} + \frac{\partial v\_z}{\partial z} + \frac{v\_r}{r} = 0 \tag{9}
$$

Motion equations

$$
\rho \frac{DV}{Dt} = \nabla.\pi \tag{10}
$$

2.3. Explicit expressions of the dimensionless governing equations

ð Þ A<sup>1</sup> zr þ η

<sup>∂</sup><sup>z</sup> ð Þ <sup>A</sup><sup>2</sup> zr <sup>þ</sup> <sup>υ</sup><sup>1</sup>

ð Þ A<sup>1</sup> zz þ η

<sup>∂</sup><sup>z</sup> ð Þ <sup>A</sup><sup>2</sup> zz <sup>þ</sup> <sup>υ</sup><sup>1</sup>

a/R = 0.2, <sup>a</sup> = 0.05 m, and Vs <sup>=</sup> 0.016 m/s, and because for <sup>γ</sup>\_ ! 0, we have <sup>υ</sup><sup>1</sup>

notice that for K = 0 we go back to the generalized Newtonian fluid.

ψ ¼ constant �

<sup>K</sup> <sup>¼</sup> <sup>υ</sup><sup>10</sup> ηo Vs

The coefficient K is the special case for γ\_ = 0 of the Weissenberg number. It is worthwhile to

ðr 0

rvzdr ≈ π �

ðr 0

Projection of the motion equation on the r axis

∂η ∂z

∂υ<sup>1</sup>

∂η ∂z

∂υ<sup>1</sup>

ð Þ A<sup>1</sup> rr þ

<sup>∂</sup><sup>r</sup> ð Þ <sup>A</sup><sup>2</sup> rr <sup>þ</sup>

Projection of the motion equation on the z axis

∂r A2 1 � � rz þ ∂υ<sup>1</sup> ∂z A2 1 � �

ð Þ A<sup>1</sup> rz þ

<sup>∂</sup><sup>r</sup> ð Þ <sup>A</sup><sup>2</sup> rz <sup>þ</sup>

∂r A2 1 � � rr þ ∂υ<sup>1</sup> ∂z A2 1 � �

∂vr ∂r þ ∂vz ∂z þ vr

> ∂ð Þ A<sup>1</sup> rr ∂r

zr þ υ<sup>1</sup>

þ

∂ A<sup>2</sup> 1 � � rr ∂r

∂ð Þ A<sup>2</sup> rr ∂r

� � � �

∂ð Þ A<sup>1</sup> rz ∂r

( ) " #

zz þ υ<sup>1</sup>

� � � �

( ) " #

∂ð Þ A<sup>1</sup> zr ∂z

> þ ∂ A<sup>2</sup> 1 � � zr ∂z

∂ð Þ A<sup>2</sup> zr ∂z

> ∂ð Þ A<sup>1</sup> zz ∂z

> > þ ∂ A<sup>2</sup> 1 � � zz ∂z

∂ð Þ A<sup>2</sup> zz ∂z

þ

� �

þ

þ

∂ A<sup>2</sup> 1 � � rz ∂r

∂ð Þ A<sup>2</sup> rz ∂r

� �

Particle Settling in a Non-Newtonian Fluid Medium Processed by Using the CEF Model

<sup>r</sup> <sup>¼</sup> <sup>0</sup> (14)

þ

http://dx.doi.org/10.5772/intechopen.75977

�

¼ 0

�

¼ 0

<sup>η</sup> ! <sup>υ</sup><sup>10</sup> η0 ¼ 1

rvzdr (18)

(15)

71

(16)

rr � <sup>A</sup><sup>2</sup> 1 � � θθ

r

þ

þ A2 1 � � rz r

<sup>þ</sup> ð Þ <sup>A</sup><sup>2</sup> rz r

<sup>a</sup> <sup>¼</sup> <sup>0</sup>:<sup>32</sup> (17)

<sup>þ</sup> ð Þ <sup>A</sup><sup>2</sup> rr � ð Þ <sup>A</sup><sup>2</sup> θθ r

<sup>þ</sup> ð Þ <sup>A</sup><sup>1</sup> rz r

<sup>þ</sup> ð Þ <sup>A</sup><sup>1</sup> rr � ð Þ <sup>A</sup><sup>1</sup> θθ r

> þ A2 1 � �

Continuity equation

� ∂p ∂r þ ∂η ∂r

> � 1 2

<sup>þ</sup>0:85<sup>K</sup> <sup>∂</sup>υ<sup>1</sup>

<sup>K</sup> <sup>∂</sup>υ<sup>1</sup>

� ∂p ∂z þ ∂η ∂r

� 1 2

Taking as numerical example

which ensues

2.3.1. Stream function

<sup>þ</sup>0:85<sup>K</sup> <sup>∂</sup>υ<sup>1</sup>

<sup>K</sup> <sup>∂</sup>υ<sup>1</sup>

where K is the normalization coefficient.

2.3.2. Boundary conditions (dimensionless)

Along the cylindrical tube (r = R): vr = 0 vz = 1.

At the inlet of the flow: vr = 0 vz = 1.

They have two projections in the axisymmetric problem considered. Hence, there are overall three equations. Unknowns are vr, vz, p. Supposing a creeping flow, the left-hand side may be taken null, and the motion equations reduce to

$$\nabla.\pi = 0\tag{11}$$

According to the assumptions of steady, axisymmetric, incompressible and irrotational flow properties above mentioned, we can write

$$\frac{\partial}{\partial t} = 0 \,\, \frac{\partial}{\partial \theta} = 0 \,\, \nabla \,\, \mathbf{V} = 0 \,\, \mathbf{v}\_{\theta} = 0 \,\, \tag{12}$$

#### 2.2. Nondimensionalization

Nondimensional variables and scales are introduced as

$$r^\* = \frac{r}{a} \quad z^\* = \frac{z}{a} \quad \upsilon\_r^\* = \frac{\upsilon\_r}{V\_s} \quad \upsilon\_\theta^\* = \frac{\upsilon\_\theta}{V\_s} \quad \upsilon\_z^\* = \frac{\upsilon\_z}{V\_s}$$

$$p^\* = p\frac{a}{V\_s\eta\_o} \\ t^\* = = t\frac{\eta\_o}{a^2\rho} \quad \eta^\* = \frac{\eta}{\eta\_o} \quad \upsilon\_1^\* = \frac{\upsilon\_1}{\upsilon\_{10}} \\ \upsilon\_2^\* = \frac{\upsilon\_2}{\upsilon\_{20}} \quad D^\* = \frac{D}{\eta\_0 a V\_s}$$

$$(A\_1)^\* = (A\_1)\frac{a}{V\_s} \qquad \left(A\_1^2\right)^\* = \left(A\_1^2\right)\left(\frac{a}{V\_s}\right)^2 \qquad (A\_2)^\* = (A\_2)\left(\frac{a}{V\_s}\right)^2 \qquad \psi^\* = \frac{\psi}{a\_2 V\_s} \tag{13}$$

For brevity \* notation will be elected by implication.

#### 2.3. Explicit expressions of the dimensionless governing equations

Continuity equation

the motion of a sphere in an incompressible fluid under isothermal conditions will be given for a non-Newtonian viscous fluid exhibiting shear-thinning and using cylindrical coordinates [15]. As this study is the simulation of the slow motion of small particles, we can suppose the flow irrotational. Furthermore, due to the data of the considered problem, the motion may be assumed steady, axisymmetric and the fluid incompressible. The global cylindrical coordinate system (r, θ, z) is shown in Figure 1, cited firstly in the introduction section showing schematic diagram of the

Due to the lack of analytical solutions, one will have to resort to numerical methods. The finite

<sup>r</sup> <sup>¼</sup> <sup>0</sup> (9)

Dt <sup>¼</sup> <sup>∇</sup>:<sup>τ</sup> (10)

∇:τ ¼ 0 (11)

<sup>∂</sup><sup>θ</sup> <sup>¼</sup> <sup>0</sup> <sup>∇</sup>. <sup>V</sup> <sup>¼</sup> <sup>0</sup> νθ <sup>¼</sup> <sup>0</sup> (12)

vz <sup>∗</sup> <sup>¼</sup> vz Vs

> a Vs <sup>2</sup>

<sup>D</sup><sup>∗</sup> <sup>¼</sup> <sup>D</sup> η0aVs

> <sup>ψ</sup><sup>∗</sup> <sup>¼</sup> <sup>ψ</sup> a2Vs

(13)

<sup>∇</sup>:<sup>V</sup> <sup>¼</sup> <sup>∂</sup>vr ∂r þ ∂vz ∂z þ vr

> r DV

They have two projections in the axisymmetric problem considered. Hence, there are overall three equations. Unknowns are vr, vz, p. Supposing a creeping flow, the left-hand side may be

According to the assumptions of steady, axisymmetric, incompressible and irrotational flow

vθ <sup>∗</sup> <sup>¼</sup> <sup>v</sup><sup>θ</sup> Vs

ð Þ A2 <sup>∗</sup> <sup>¼</sup> ð Þ A2

ηo υ1 <sup>∗</sup> <sup>¼</sup> <sup>υ</sup><sup>1</sup> υ<sup>10</sup> υ2 <sup>∗</sup> <sup>¼</sup> <sup>υ</sup><sup>2</sup> υ<sup>20</sup>

problem studied. The local area coordinates are L1, L2, and L3.

element method (FEM) will be used for this purpose.

taken null, and the motion equations reduce to

properties above mentioned, we can write

2.2. Nondimensionalization

ð Þ A1 <sup>∗</sup> <sup>¼</sup> ð Þ A1

∂ ∂t

<sup>z</sup><sup>∗</sup> <sup>¼</sup> <sup>z</sup> a vr <sup>∗</sup> <sup>¼</sup> vr Vs

> ηo <sup>a</sup><sup>2</sup><sup>r</sup> <sup>η</sup><sup>∗</sup> <sup>¼</sup> <sup>η</sup>

1 a Vs <sup>2</sup>

Nondimensional variables and scales are introduced as

A2 1 <sup>∗</sup> <sup>¼</sup> <sup>A</sup><sup>2</sup>

For brevity \* notation will be elected by implication.

r <sup>∗</sup> <sup>¼</sup> <sup>r</sup> a

a Vsη<sup>o</sup> t <sup>∗</sup> ¼¼ <sup>t</sup>

<sup>p</sup><sup>∗</sup> <sup>¼</sup> <sup>p</sup>

a Vs <sup>¼</sup> <sup>0</sup> <sup>∂</sup>

2.1. Governing equations

Continuity equation

70 Polymer Rheology

Motion equations

$$\frac{\partial v\_r}{\partial r} + \frac{\partial v\_z}{\partial z} + \frac{v\_r}{r} = 0 \tag{14}$$

Projection of the motion equation on the r axis

$$\begin{split} & -\frac{\partial p}{\partial r} + \frac{\partial \eta}{\partial r} (A\_1)\_{rr} + \frac{\partial \eta}{\partial z} (A\_1)\_{zr} + \eta \left[ \frac{\partial (A\_1)\_{rr}}{\partial r} + \frac{\partial (A\_1)\_{zr}}{\partial z} + \frac{(A\_1)\_{rr} - (A\_1)\_{\theta\theta}}{r} \right] + \\ & + 0.85K \left\{ \frac{\partial \upsilon\_1}{\partial r} (A\_1^2)\_{rr} + \frac{\partial \upsilon\_1}{\partial z} (A\_1^2)\_{zr} + \upsilon\_1 \left[ \frac{\partial (A\_1^2)\_{rr}}{\partial r} + \frac{\partial (A\_1^2)\_{zr}}{\partial z} + \frac{(A\_1^2)\_{rr} - (A\_1^2)\_{\theta\theta}}{r} \right] \right\} - \\ & - \frac{1}{2}K \left\{ \frac{\partial \upsilon\_1}{\partial r} (A\_2)\_{rr} + \frac{\partial \upsilon\_1}{\partial z} (A\_2)\_{zr} + \upsilon\_1 \left[ \frac{\partial (A\_2)\_{rr}}{\partial r} + \frac{\partial (A\_2)\_{zr}}{\partial z} + \frac{(A\_2)\_{rr} - (A\_2)\_{\theta\theta}}{r} \right] \right\} = 0 \end{split} \tag{15}$$

Projection of the motion equation on the z axis

$$\begin{split} & -\frac{\partial p}{\partial z} + \frac{\partial \eta}{\partial r} (A\_1)\_{rz} + \frac{\partial \eta}{\partial z} (A\_1)\_{zz} + \eta \left[ \frac{\partial (A\_1)\_{rz}}{\partial r} + \frac{\partial (A\_1)\_{zz}}{\partial z} + \frac{(A\_1)\_{rz}}{r} \right] + \\ & + 0.85K \left\{ \frac{\partial \upsilon\_1}{\partial r} (A\_1^2)\_{rz} + \frac{\partial \upsilon\_1}{\partial z} (A\_1^2)\_{zz} + \upsilon\_1 \left[ \frac{\partial (A\_1^2)\_{rz}}{\partial r} + \frac{\partial (A\_1^2)\_{zz}}{\partial z} + \frac{(A\_1^2)\_{rz}}{r} \right] \right\} - \\ & - \frac{1}{2}K \left\{ \frac{\partial \upsilon\_1}{\partial r} (A\_2)\_{rz} + \frac{\partial \upsilon\_1}{\partial z} (A\_2)\_{zz} + \upsilon\_1 \left[ \frac{\partial (A\_2)\_{rz}}{\partial r} + \frac{\partial (A\_2)\_{zz}}{\partial z} + \frac{(A\_2)\_{rz}}{r} \right] \right\} = 0 \end{split} \tag{16}$$

where K is the normalization coefficient.

Taking as numerical example

a/R = 0.2, <sup>a</sup> = 0.05 m, and Vs <sup>=</sup> 0.016 m/s, and because for <sup>γ</sup>\_ ! 0, we have <sup>υ</sup><sup>1</sup> <sup>η</sup> ! <sup>υ</sup><sup>10</sup> η0 ¼ 1 which ensues

$$K = \frac{\nu\_{10}}{\eta\_o} \frac{V\_s}{a} = 0.32\tag{17}$$

The coefficient K is the special case for γ\_ = 0 of the Weissenberg number. It is worthwhile to notice that for K = 0 we go back to the generalized Newtonian fluid.

#### 2.3.1. Stream function

$$
\psi = \text{constant} - \int\_0^r r v\_z dr \approx \pi - \int\_0^r r v\_z dr \tag{18}
$$

2.3.2. Boundary conditions (dimensionless)

At the inlet of the flow: vr = 0 vz = 1.

Along the cylindrical tube (r = R): vr = 0 vz = 1.

Along the centerline of the cylindrical tube (r = 0): vr = 0 <sup>∂</sup>vz <sup>∂</sup><sup>r</sup> ¼ 0

On the surface of the sphere: vr = vz = 0.

At the outlet of the flow: vr = 0 <sup>∂</sup>vz <sup>∂</sup><sup>r</sup> ¼ 0.

$$p = 0 \text{ (atmospheric pressure)}\tag{19}$$

L1ð Þ 2 L1–<sup>1</sup> ; L2 ð Þ 2 L2–<sup>1</sup> ; L3ð Þ 2 L3–<sup>1</sup> ; 4L1L2; 4L2L3; 4L3L1 (21)

Particle Settling in a Non-Newtonian Fluid Medium Processed by Using the CEF Model

http://dx.doi.org/10.5772/intechopen.75977

73

and those concerning the pressure for nodes 1, 2, 3 are L1, L2, L3, (Figure 6).

Figure 5. (a) A linear triangular element; and (b) a quadratic triangular element.

tions for velocity and pressure fields [16].

Figure 6. Area coordinates.

The element shown satisfies the LBB condition and thus gives reliable and convergent solu-

#### 2.4. Dimensionless stress components

Introducing <sup>K</sup> <sup>¼</sup> <sup>υ</sup><sup>10</sup> η0 Vs <sup>a</sup> in the CEF constitutive equation Eq. (1), using Eq.(13) nondimensional formulas and Eq. (7), <sup>τ</sup><sup>∗</sup> <sup>¼</sup> <sup>τ</sup> <sup>a</sup> Vsη<sup>o</sup> stress tensor components can be written electing the \* notation by implication as follows:

$$\begin{aligned} \tau\_{rr} &= -p + \eta (A\_1)\_{rr} + K \Big[ 0.85 \nu\_1 (A\_1^2)\_{rr} - \frac{1}{2} \nu\_1 (A\_2)\_{rr} \Big] \\ \tau\_{rz} &= \eta (A\_1)\_{rz} + K \Big[ 0.85 \nu\_1 (A\_1^2)\_{rz} - \frac{1}{2} \nu\_1 (A\_2)\_{rz} \Big] \\ \tau\_{\theta\theta} &= -p + \eta (A\_1)\_{\theta\theta} + K \Big[ 0.85 \nu\_1 (A\_1^2)\_{\theta\theta} - \frac{1}{2} \nu\_1 (A\_2)\_{\theta\theta} \Big] \\ \tau\_{zz} &= -p + \eta (A\_1)\_{zz} + K \Big[ 0.85 \nu\_1 (A\_1^2)\_{zz} - \frac{1}{2} \nu\_1 (A\_2)\_{zz} \Big] \end{aligned} \tag{20}$$
