**Incompressible Non-Newtonian Fluid Flows**

Quoc-Hung Nguyen and Ngoc-Diep Nguyen

*Mechanical Faculty, Ho Chi Minh University of Industry, Vietnam* 

### **1. Introduction**

46 Continuum Mechanics – Progress in Fundamentals and Engineering Applications

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A non-Newtonian fluid is a fluid whose flow properties differ in many ways from those of Newtonian fluids. Most commonly the viscosity of non-Newtonian fluids is not independent of shear rate or shear rate history. In practice, many fluid materials exhibits non-Newtonian fluid behavior such as: salt solutions, molten, ketchup, custard, toothpaste, starch suspensions, paint, blood, and shampoo etc. In a Newtonian fluid, the relation between the shear stress and the shear rate is linear, passing through the origin, the constant of proportionality being the coefficient of viscosity. In a non-Newtonian fluid, the relation between the shear stress and the shear rate is different, and can even be time-dependent. Therefore a constant coefficient of viscosity cannot be defined. In the previous parts of this book, the mechanics of Newtonian fluid have been mentioned. In this chapter, the common rheological models of non-Newtonian fluids are introduced and several approaches concerned with non-Newtonian fluid flows are considered. In addition, the solution of common non-Newtonian fluid flows in a circular pipe, annular and rectangular duct are presented.

## **2. Classification of non-Newtonian fluid**

As above mentioned, a non-Newtonian fluid is one whose flow curve (shear stress versus shear rate) is nonlinear or does not pass through the origin, i.e. where the apparent viscosity, shear stress divided by shear rate, is not constant at a given temperature and pressure but is dependent on flow conditions such as flow geometry, shear rate, etc. and sometimes even on the kinematic history of the fluid element under consideration. Such materials may be conveniently grouped into three general classes:


Among the three groups, the time independent Non-Newtonian fluids are the most popular and easiest to handle in analysis. In this chapter, only this group of Non-Newtonian fluids are considered.

Incompressible Non-Newtonian Fluid Flows 49

models is given here; more complete descriptions of such models are available in many

The relationship between shear stress and shear rate for this type of fluid can be

 

<sup>1</sup> ( )*<sup>n</sup>*

 *K yx*

In these equations, *K* and *n* are two empirical curve-fitting parameters and are known as the fluid consistency coefficient and the flow behaviour index respectively. For a shear thinning fluid, the index may have any value between 0 and 1. The smaller the value of *n,* the greater is the degree of shear-thinning. For a shear-thickening fluid, the index *n* will be greater than unity. When *n*=1, equations (3) becomes the constitutive equation of Newtonian fluid.

Although the power-law model offers the simplest representation of shear-thinning behaviour, it does have a number of limitations. Generally, it applies over only a limited range of shear rates and therefore the fitted values of *K* and *n* will depend on the range of shear rates considered. Furthermore, it does not predict the zero and infinite shear viscosities. Finally, it should be noted that the dimensions of the flow consistency coefficient, *K,* depend on the numerical value of *n* and therefore the *K* values must not be compared when the *n* values differ. On the other hand, the value of *K* can be viewed as the value of apparent viscosity at the shear rate of unity and will therefore depend on the time unit (e.g. second, minute or hour) employed. Despite these limitations, this is perhaps the most widely used model in the literature dealing with process engineering applications. Table 1 provides a compilation of the power-law constants (*K* and *n)* for a variety of substances.

When there are significant deviations from the power-law model at very high and very low shear rates, it is necessary to use a model which takes account of the limiting values of viscosities *μ*0 and *μ* . Based on the molecular network considerations, Carreau (1972) put

> [1 ( ) ] *<sup>n</sup> yx*

where *n* (< 1) and λ are two curve-fitting parameters. This model can describe shear thinning behaviour over wide ranges of shear rates but only at the expense of the added complexity of four parameters. This model predicts Newtonian fluid behaviour *μ* = *μ*0 when

0

 

  2 ( 1)/2

(4)

*yx yx K* (2)

(3)

books (Bird *et al* ., 1987 ; Carreau *et al* ., 1997) and in a review paper (Bird, 1976).

So the apparent viscosity for the so-called power-law fluid is thus given by:

i. The power-law model

mathematically expressed as follows:

ii. The Carreau viscosity equation

forward the following viscosity model.

either *n* = 1 or *λ* =0 or both.

()*<sup>n</sup>*

For *n* < 1, the fluid exhibits shear-thinning properties  *n* = 1, the fluid shows Newtonian behaviour  *n >* 1, the fluid shows shear-thickening behaviour

Fig. 1. Types of time-independent non-Newtonian fluid

In simple shear, the flow behaviour of this class of materials may be described by the following constitutive relation,

$$
\pi\_{yx} = f(\dot{\boldsymbol{\gamma}}\_{yx}) \tag{1}
$$

This equation implies that the value of *yx* at any point within the sheared fluid is determined only by the current value of shear stress at that point or vice versa. Depending upon the form of the function in equation (1), these fluids may be further subdivided into three types: shear-thinning or pseudoplastic, shear-thickening or dilatant and viscoplastic

#### **2.1 Shear-thinning or pseudo-plastic fluids**

The most common type of time-independent non-Newtonian fluid behaviour observed is Pseudo-plasticity or shear-thinning, characterized by an apparent viscosity which decreases with increasing shear rate. Both at very low and at very high shear rates, most shearthinning polymer solutions and melts exhibit Newtonian behaviour, i.e., shear stress–shear rate plots become straight lines and on a linear scale will pass through origin. The resulting values of the apparent viscosity at very low and high shear rates are known as the zero shear viscosity, *μ*0 , and the infinite shear viscosity, *μ*, respectively. Thus, the apparent viscosity of a shear-thinning fluid decreases from *μ*0 to *μ* with increasing shear rate. Many mathematical expressions of varying complexity and form have been proposed in the literature to model shear-thinning characteristics; some of these are straightforward attempts at curve fitting, giving empirical relationships for the shear stress (or apparent viscosity)–shear rate curves for example, while others have some theoretical basis in statistical mechanics – as an extension of the application of the kinetic theory to the liquid state or the theory of rate processes, etc. Only a selection of the more widely used viscosity models is given here; more complete descriptions of such models are available in many books (Bird *et al* ., 1987 ; Carreau *et al* ., 1997) and in a review paper (Bird, 1976).

i. The power-law model

48 Continuum Mechanics – Progress in Fundamentals and Engineering Applications

In simple shear, the flow behaviour of this class of materials may be described by the

( ) *yx yx*

determined only by the current value of shear stress at that point or vice versa. Depending upon the form of the function in equation (1), these fluids may be further subdivided into three types: shear-thinning or pseudoplastic, shear-thickening or dilatant and viscoplastic

The most common type of time-independent non-Newtonian fluid behaviour observed is Pseudo-plasticity or shear-thinning, characterized by an apparent viscosity which decreases with increasing shear rate. Both at very low and at very high shear rates, most shearthinning polymer solutions and melts exhibit Newtonian behaviour, i.e., shear stress–shear rate plots become straight lines and on a linear scale will pass through origin. The resulting values of the apparent viscosity at very low and high shear rates are known as the zero shear viscosity, *μ*0 , and the infinite shear viscosity, *μ*, respectively. Thus, the apparent viscosity of a shear-thinning fluid decreases from *μ*0 to *μ* with increasing shear rate. Many mathematical expressions of varying complexity and form have been proposed in the literature to model shear-thinning characteristics; some of these are straightforward attempts at curve fitting, giving empirical relationships for the shear stress (or apparent viscosity)–shear rate curves for example, while others have some theoretical basis in statistical mechanics – as an extension of the application of the kinetic theory to the liquid state or the theory of rate processes, etc. Only a selection of the more widely used viscosity

Shear rate

fluid Pseudoplastic

Pseudoplastic Bingham

(1)

at any point within the sheared fluid is

Dilatant fluid

Newtonian

plastic fluid

*f*

Fig. 1. Types of time-independent non-Newtonian fluid

fluid

Yield -

This equation implies that the value of *yx*

**2.1 Shear-thinning or pseudo-plastic fluids** 

following constitutive relation,

Shear stress

The relationship between shear stress and shear rate for this type of fluid can be mathematically expressed as follows:

$$
\tau\_{yx} = K(j\dot{\nu}\_{yx})^n \tag{2}
$$

So the apparent viscosity for the so-called power-law fluid is thus given by:

$$
\mu = K(\dot{\nu}\_{yx})^{n-1} \tag{3}
$$

For *n* < 1, the fluid exhibits shear-thinning properties

 *n* = 1, the fluid shows Newtonian behaviour

 *n >* 1, the fluid shows shear-thickening behaviour

In these equations, *K* and *n* are two empirical curve-fitting parameters and are known as the fluid consistency coefficient and the flow behaviour index respectively. For a shear thinning fluid, the index may have any value between 0 and 1. The smaller the value of *n,* the greater is the degree of shear-thinning. For a shear-thickening fluid, the index *n* will be greater than unity. When *n*=1, equations (3) becomes the constitutive equation of Newtonian fluid.

Although the power-law model offers the simplest representation of shear-thinning behaviour, it does have a number of limitations. Generally, it applies over only a limited range of shear rates and therefore the fitted values of *K* and *n* will depend on the range of shear rates considered. Furthermore, it does not predict the zero and infinite shear viscosities. Finally, it should be noted that the dimensions of the flow consistency coefficient, *K,* depend on the numerical value of *n* and therefore the *K* values must not be compared when the *n* values differ. On the other hand, the value of *K* can be viewed as the value of apparent viscosity at the shear rate of unity and will therefore depend on the time unit (e.g. second, minute or hour) employed. Despite these limitations, this is perhaps the most widely used model in the literature dealing with process engineering applications. Table 1 provides a compilation of the power-law constants (*K* and *n)* for a variety of substances.

ii. The Carreau viscosity equation

When there are significant deviations from the power-law model at very high and very low shear rates, it is necessary to use a model which takes account of the limiting values of viscosities *μ*0 and *μ* . Based on the molecular network considerations, Carreau (1972) put forward the following viscosity model.

$$\frac{\mu-\mu\_{\infty}}{\mu\_{0}-\mu\_{\infty}} = \left[1+\left(\lambda\dot{\gamma}\_{yx}\right)^{2}\right]^{(n-1)/2} \tag{4}$$

where *n* (< 1) and λ are two curve-fitting parameters. This model can describe shear thinning behaviour over wide ranges of shear rates but only at the expense of the added complexity of four parameters. This model predicts Newtonian fluid behaviour *μ* = *μ*0 when either *n* = 1 or *λ* =0 or both.

Incompressible Non-Newtonian Fluid Flows 51

Another four parameter model which has gained wide acceptance is due to Cross (1965)

Here *n* (<1) and *k* are two fitting parameters whereas *μ*0 and *μ* are the limiting values of the apparent viscosity at low and high shear rates, respectively. This model reduces to the Newtonian fluid behaviour as *k* → 0. Similarly, when *μ* << *μ*0 and *μ* >>*μ*, it reduces to the familiar power-law model, equation (3). Though initially Cross (1965) suggested that a constant value of *n* =2/3 was adequate to approximate the viscosity data for many systems, it is now thought that treating the index, *n*, as an adjustable parameter offers considerable

When the deviations from the power-law model are significant only at low shear rates, it is more appropriate to use the Ellis model. The three viscosity equations presented so far are examples of the form of equation (1). The three-constant Ellis model is an illustration of the inverse form. In simple shear, the apparent viscosity of an Ellis model fluid is given

0

1/2 1( / ) *yx*

In this equation, *μ*0 is the zero shear viscosity and the remaining two constants *α* (> 1) and *τ*1/2 are adjustable parameters. While the index *α* is a measure of the degree of shear thinning behaviour (the greater the value of *α* , greater is the extent of shear-thinning), *τ*1/2 represents the value of shear stress at which the apparent viscosity has dropped to half its zero shear value. Equation (6) predicts Newtonian fluid behaviour in the limit of *τ* 1/2 → . This form of equation has advantages in permitting easy calculation of velocity profiles from a known stress distribution, but renders the reverse operation tedious and cumbersome. It can easily be seen that in the intermediate range of shear stress (or shear rate), (*τyx* / *τ*1/2)-1>> 1, and

This type of fluid behaviour is characterized by the existence of a yield stress (*τ*0) which must be exceeded before the fluid will deform or flow. Conversely, such a material will deform elastically (or flow *en masse* like a rigid body) when the externally applied stress is smaller than the yield stress. Once the magnitude of the external stress has exceeded the value of the yield stress, the flow curve may be linear or non-linear but will not pass through origin (Figure 1). Hence, in the absence of surface tension effects, such a material

 

1

0 1/2 *m* ( ) 

 

(6)

1 1()*<sup>n</sup> yx k*

(5)

0

 

improvement over the use of the constant value of *n* (Barnes *et al.* , 1989).

equation (6) reduces to equation (3) with *n* =(1/) and 1 1/

 

 

iii. The Cross viscosity equation

iv. The Ellis fluid model

**2.2 Viscoplastic fluid behaviour** 

by:

which, in simple shear, is written as:


*Source*: Modified after Johnson (1999)

Table 1. Typical values of power-law constants for a few systems

#### iii. The Cross viscosity equation

50 Continuum Mechanics – Progress in Fundamentals and Engineering Applications

System Temperature (*K*) *n M* (Pa sn )

High density polyethylene (HDPE) 453–493 0.6 3.75–6.2 x 103 High impact polystyrene 443–483 0.20 3.5–7.5 x 104 Polystyrene 463–498 0.25 1.5–4.5 x 104 Polypropylene 453–473 0.40 4.5–7 x 103 Low density polyethylene (LDPE) 433–473 0.45 4.3–9.4 x 103 Nylon 493–508 0.65 1.8–2.6 x 103 Polymethylmethyacrylate (PMMA) 493–533 0.25 2.5–9 x 104 Polycarbonate 553–593 0.65–0.8 1–8.5 x 103

Nail polish 298 0.86 750 Mascara 298 0.24 200 Toothpaste 298 0.28 120 Sunscreen lotions 298 0.28 75 Ponds cold cream 298 0.45 25 Oil of Olay 298 0.22 25

Ammonium alginate solution (3.37%) 297 0.5 13 Apple butter – 0.15 200 Apple sauce 300 0.3–0.45 12–22 Apricot puree 300 0.3–0.4 5–20 Banana puree 293–315 0.33–0.5 4–10 Carrot puree 298 0.25 25 Chicken (minced) 296 0.10 900 Chocolate 303 0.5 0.7 Guava puree 296.5 0.5 40 Human blood 300 0.9 0.004 Mango pulp 300–340 0.3 3–10 Marshmallow cream – 0.4 560 Mayonnaise 298 0.6 5–100 Papaya puree 300 0.5 10 Peach puree 300 0.38 1–5 Peanut butter – 0.07 500 Pear puree 300 0.4–0.5 1–5 Plum puree 287 0.35 30–80 Tomato concentrate (5.8% solid) 305 0.6 0.22 Tomato ketch up 295 0.24 33 Tomato paste – 0.5 15 Whipped desert toppings – 0.12 400 Yoghurt 293 0.5–0.6 25

**Agro- and food-related products**

**Polymer melts**

Table 1. Typical values of power-law constants for a few systems

**Personal care products** 

*Source*: Modified after Johnson (1999)

Another four parameter model which has gained wide acceptance is due to Cross (1965) which, in simple shear, is written as:

$$\frac{\mu - \mu\_{\infty}}{\mu\_0 - \mu\_{\infty}} = \frac{1}{1 + k(\dot{\boldsymbol{\gamma}}\_{\text{yx}})^n} \tag{5}$$

Here *n* (<1) and *k* are two fitting parameters whereas *μ*0 and *μ* are the limiting values of the apparent viscosity at low and high shear rates, respectively. This model reduces to the Newtonian fluid behaviour as *k* → 0. Similarly, when *μ* << *μ*0 and *μ* >>*μ*, it reduces to the familiar power-law model, equation (3). Though initially Cross (1965) suggested that a constant value of *n* =2/3 was adequate to approximate the viscosity data for many systems, it is now thought that treating the index, *n*, as an adjustable parameter offers considerable improvement over the use of the constant value of *n* (Barnes *et al.* , 1989).

#### iv. The Ellis fluid model

When the deviations from the power-law model are significant only at low shear rates, it is more appropriate to use the Ellis model. The three viscosity equations presented so far are examples of the form of equation (1). The three-constant Ellis model is an illustration of the inverse form. In simple shear, the apparent viscosity of an Ellis model fluid is given by:

$$\mu = \frac{\mu\_0}{1 + (\tau\_{yx} / \tau\_{1/2})^{\alpha - 1}} \tag{6}$$

In this equation, *μ*0 is the zero shear viscosity and the remaining two constants *α* (> 1) and *τ*1/2 are adjustable parameters. While the index *α* is a measure of the degree of shear thinning behaviour (the greater the value of *α* , greater is the extent of shear-thinning), *τ*1/2 represents the value of shear stress at which the apparent viscosity has dropped to half its zero shear value. Equation (6) predicts Newtonian fluid behaviour in the limit of *τ* 1/2 → . This form of equation has advantages in permitting easy calculation of velocity profiles from a known stress distribution, but renders the reverse operation tedious and cumbersome. It can easily be seen that in the intermediate range of shear stress (or shear rate), (*τyx* / *τ*1/2)-1>> 1, and equation (6) reduces to equation (3) with *n* =(1/) and 1 1/ 0 1/2 *m* ( ) 

#### **2.2 Viscoplastic fluid behaviour**

This type of fluid behaviour is characterized by the existence of a yield stress (*τ*0) which must be exceeded before the fluid will deform or flow. Conversely, such a material will deform elastically (or flow *en masse* like a rigid body) when the externally applied stress is smaller than the yield stress. Once the magnitude of the external stress has exceeded the value of the yield stress, the flow curve may be linear or non-linear but will not pass through origin (Figure 1). Hence, in the absence of surface tension effects, such a material

Incompressible Non-Newtonian Fluid Flows 53

A simple generalization of the Bingham plastic model to embrace the non-linear flow curve (for *τyx* > *τ*0) is the three constant Herschel–Bulkley fluid model. In one dimensional steady

> for *yx* <sup>0</sup>

It is again noted that, the dimensions of *K* depend upon the value of *n.* With the use of the third parameter, this model provides a somewhat better fit to some experimental data.

Many foodstuffs and biological materials, especially blood, are well described by this two

 for *yx* <sup>0</sup> 

This model has often been used for describing the steady shear stress–shear rate behaviour of blood, yoghurt, tomato purée, molten chocolate, etc. The flow behaviour of some particulate suspensions also closely approximates to this type of behaviour. The comparative performance of these three as well as several other models for viscoplastic behaviour has been thoroughly evaluated in an extensive review paper by Bird *et al* . (1983) and a through discussion on the existence, measurement and implications of yield stress has

Dilatant fluids are similar to pseudoplastic systems in that they show no yield stress but their apparent viscosity increases with increasing shear rate; thus these fluids are also called shear-thickening. This type of fluid behaviour was originally observed in concentrated suspensions and a possible explanation for their dilatant behaviour is as follows: At rest, the voidage is minimum and the liquid present is sufficient to fill the void space. At low shear rates, the liquid lubricates the motion of each particle past others and the resulting stresses are consequently small. At high shear rates, on the other hand, the material expands or dilates slightly (as also observed in the transport of sand dunes) so that there is no longer sufficient liquid to fill the increased void space and prevent direct solid–solid contacts which result in increased friction and higher shear stresses. This mechanism causes the apparent viscosity to rise rapidly with increasing rate of shear. The term dilatant has also been used for all other fluids which exhibit increasing apparent viscosity with increasing rate of shear. Many of these, such as starch pastes, are not true suspensions and show no dilation on shearing. The above explanation therefore is not applicable but nevertheless such materials

 

> 

(8)

(9)

<sup>0</sup> ( )*<sup>n</sup> yx K yx*

> 0 *yx*

1/2 1/2 1/2 <sup>0</sup> (/ ) *yx yx*

 for *yx* <sup>0</sup>

 

 for *yx* <sup>0</sup>

ii. The Herschel-Bulkley fluid model

shearing motion, the model is written as:

**2.3 Shear-thickening or dilatant fluid behaviour** 

are still commonly referred to as dilatant fluids.

0 *yx*

been provided by Barnes (1999).

iii. The Casson fluid model

constant model as:

will not level out under gravity to form an absolutely flat free surface. One can, however, explain this kind of fluid behaviour by postulating that the substance at rest consists of three-dimensional structures of sufficient rigidity to resist any external stress less than *τ*0. For stress levels greater than *τ*0, however, the structure breaks down and the substance behaves like a viscous material. In some cases, the build-up and breakdown of structure has been found to be reversible, i.e., the substance may regain its initial value of the yield stress.

A fluid with a linear flow curve for |*τyx* | > | *τ*0 | is called a Bingham plastic fluid and is characterized by a constant plastic viscosity (the slope of the shear stress versus shear rate curve) and a yield stress. On the other hand, a substance possessing a yield stress as well as a non-linear flow curve on linear coordinates (for |*τyx*| > |*τ*0|), is called a yield pseudoplastic material. It is interesting to note that a viscoplastic material also displays an apparent viscosity which decreases with increasing shear rate. At very low shear rates, the apparent viscosity is effectively infinite at the instant immediately before the substance yields and begins to flow. It is thus possible to regard these materials as possessing a particular class of shear-thinning behaviour.

Strictly speaking, it is virtually impossible to ascertain whether any real material has a true yield stress or not, but nevertheless the concept of a yield stress has proved to be convenient in practice because some materials closely approximate to this type of flow behaviour, e.g. see Barnes and Walters (1985) , Astarita (1990) , Schurz (1990) and Evans (1992) . Many workers in this field view the yield stress in terms of the transition from a solid-like (high viscosity) to a liquid-like (low viscosity) state which occurs abruptly over an extremely narrow range of shear rates or shear stress (Uhlherr *et al* ., 2005). It is not uncommon for the two values of viscosity to differ from each other by several orders of magnitude. The answer to the question whether a fluid has a yield stress or not seems to be related to the choice of a time scale of observation. Common examples of viscoplastic fluid behaviour include particulate suspensions, emulsions, foodstuffs, blood and drilling mud, etc. (Barnes, 1999).

Over the years, many empirical expressions have been proposed as a result of straightforward curve-fitting exercises. A model based on sound theory is yet to emerge. Three commonly used models for viscoplastic fluids are: Bingham plastic model, Herschel-Bulkley model and Casson model.

i. The Bingham plastic model

This is the simplest equation describing the flow behaviour of a fluid with a yield stress and, in steady one-dimensional shear, it is described by

$$
\tau\_{yx} = \tau\_0 + \mu(j\tau\_{yx}) \text{ for } \left| \tau\_{yx} \right| > \left| \tau\_0 \right|
$$

$$
j\tau\_{yx} = 0 \text{ for } \left| \tau\_{yx} \right| < \left| \tau\_0 \right| \tag{7}
$$

Often, the two model parameters *τ*0 and *μ* are treated as curve-fitting constants irrespective of whether or not the fluid possesses a true yield stress.

will not level out under gravity to form an absolutely flat free surface. One can, however, explain this kind of fluid behaviour by postulating that the substance at rest consists of three-dimensional structures of sufficient rigidity to resist any external stress less than *τ*0. For stress levels greater than *τ*0, however, the structure breaks down and the substance behaves like a viscous material. In some cases, the build-up and breakdown of structure has been found to be reversible, i.e., the substance may regain its initial value of the yield stress. A fluid with a linear flow curve for |*τyx* | > | *τ*0 | is called a Bingham plastic fluid and is characterized by a constant plastic viscosity (the slope of the shear stress versus shear rate curve) and a yield stress. On the other hand, a substance possessing a yield stress as well as a non-linear flow curve on linear coordinates (for |*τyx*| > |*τ*0|), is called a yield pseudoplastic material. It is interesting to note that a viscoplastic material also displays an apparent viscosity which decreases with increasing shear rate. At very low shear rates, the apparent viscosity is effectively infinite at the instant immediately before the substance yields and begins to flow. It is thus possible to regard these materials as possessing a

Strictly speaking, it is virtually impossible to ascertain whether any real material has a true yield stress or not, but nevertheless the concept of a yield stress has proved to be convenient in practice because some materials closely approximate to this type of flow behaviour, e.g. see Barnes and Walters (1985) , Astarita (1990) , Schurz (1990) and Evans (1992) . Many workers in this field view the yield stress in terms of the transition from a solid-like (high viscosity) to a liquid-like (low viscosity) state which occurs abruptly over an extremely narrow range of shear rates or shear stress (Uhlherr *et al* ., 2005). It is not uncommon for the two values of viscosity to differ from each other by several orders of magnitude. The answer to the question whether a fluid has a yield stress or not seems to be related to the choice of a time scale of observation. Common examples of viscoplastic fluid behaviour include particulate suspensions, emulsions, foodstuffs, blood and drilling

Over the years, many empirical expressions have been proposed as a result of straightforward curve-fitting exercises. A model based on sound theory is yet to emerge. Three commonly used models for viscoplastic fluids are: Bingham plastic model, Herschel-

This is the simplest equation describing the flow behaviour of a fluid with a yield stress and,

 for *yx* <sup>0</sup> 

Often, the two model parameters *τ*0 and *μ* are treated as curve-fitting constants irrespective

 for *yx* <sup>0</sup> 

(7)

<sup>0</sup> ( ) *yx yx*

 

particular class of shear-thinning behaviour.

mud, etc. (Barnes, 1999).

Bulkley model and Casson model. i. The Bingham plastic model

in steady one-dimensional shear, it is described by

of whether or not the fluid possesses a true yield stress.

0 *yx*

#### ii. The Herschel-Bulkley fluid model

A simple generalization of the Bingham plastic model to embrace the non-linear flow curve (for *τyx* > *τ*0) is the three constant Herschel–Bulkley fluid model. In one dimensional steady shearing motion, the model is written as:

$$
\tau\_{yx} = \tau\_0 + K(\dot{\nu}\_{yx})^n \text{ for } \left| \tau\_{yx} \right| > \left| \tau\_0 \right|
$$

$$
\dot{\nu}\_{yx} = 0 \text{ for } \left| \tau\_{yx} \right| < \left| \tau\_0 \right| \tag{8}
$$

It is again noted that, the dimensions of *K* depend upon the value of *n.* With the use of the third parameter, this model provides a somewhat better fit to some experimental data.

iii. The Casson fluid model

Many foodstuffs and biological materials, especially blood, are well described by this two constant model as:

$$\left|\tau\_{yx}\right|^{1/2} = \left|\tau\_0\right|^{1/2} + (\mu \ne \dot{\tau}\_{yx})^{1/2} \text{ for } \left|\tau\_{yx}\right| > \left|\tau\_0\right|$$

$$\dot{\gamma}\_{yx} = 0 \text{ for } \left|\tau\_{yx}\right| < \left|\tau\_0\right| \tag{9}$$

This model has often been used for describing the steady shear stress–shear rate behaviour of blood, yoghurt, tomato purée, molten chocolate, etc. The flow behaviour of some particulate suspensions also closely approximates to this type of behaviour. The comparative performance of these three as well as several other models for viscoplastic behaviour has been thoroughly evaluated in an extensive review paper by Bird *et al* . (1983) and a through discussion on the existence, measurement and implications of yield stress has been provided by Barnes (1999).

#### **2.3 Shear-thickening or dilatant fluid behaviour**

Dilatant fluids are similar to pseudoplastic systems in that they show no yield stress but their apparent viscosity increases with increasing shear rate; thus these fluids are also called shear-thickening. This type of fluid behaviour was originally observed in concentrated suspensions and a possible explanation for their dilatant behaviour is as follows: At rest, the voidage is minimum and the liquid present is sufficient to fill the void space. At low shear rates, the liquid lubricates the motion of each particle past others and the resulting stresses are consequently small. At high shear rates, on the other hand, the material expands or dilates slightly (as also observed in the transport of sand dunes) so that there is no longer sufficient liquid to fill the increased void space and prevent direct solid–solid contacts which result in increased friction and higher shear stresses. This mechanism causes the apparent viscosity to rise rapidly with increasing rate of shear. The term dilatant has also been used for all other fluids which exhibit increasing apparent viscosity with increasing rate of shear. Many of these, such as starch pastes, are not true suspensions and show no dilation on shearing. The above explanation therefore is not applicable but nevertheless such materials are still commonly referred to as dilatant fluids.

Of the time-independent fluids, this sub-class has received very little attention; consequently very few reliable data are available. Until recently, dilatant fluid behaviour was considered to be much less widespread in the chemical and processing industries. However, with the recent growing interest in the handling and processing of systems with high solids loadings, it is no longer so, as is evidenced by the number of recent review articles on this subject (Barnes *et al*., 1987; Barnes, 1989; Goddard and Bashir, 1990). Typical examples of materials exhibiting dilatant behaviour include concentrated suspensions of china clay, titanium dioxide (Metzner and Whitlock, 1958) and of corn fl our in water (Griskey *et al*., 1985). The limited information reported so far suggests that the apparent viscosity–shear rate data often result in linear plots on double logarithmic coordinates over a limited shear rate range and the flow behaviour may be represented by the power-law model, with the flow behaviour index, *n*, greater than unity, i.e.,

$$
\mu = K(\dot{\nu}\_{xy})^{n-1} \tag{10}
$$

Incompressible Non-Newtonian Fluid Flows 55

Consider a one-directional flow of fluid through a circular tube with radius R, Figure 2. The volumetric flow rate through an annular element of area perpendicular to the flow and of

2. *Q r rvx*

 

0

2 2

*dr <sup>Q</sup>*

*<sup>i</sup> <sup>i</sup> <sup>r</sup> <sup>r</sup> x x r v <sup>r</sup> dv dr*

0 0 + 2 2

Provided there is no slip at the tube wall, the first term in equation (13) vanishes. Equation

0 ( ) *R*

If the fluid is time-independent and homogeneous, the shear stress is a function of shear rate

Changing variables in equation (14), using equation (15), and dropping the subscripts *rx*,

2 3 0 0 () () *w w i w w w R R <sup>Q</sup> d d*

instead of *r*.

*<sup>R</sup> <sup>r</sup> <sup>v</sup> dr <sup>x</sup>*

with *r* is known from the following well-known equation:

 *r dr* 

> *r R*

2 2 . . 3

*x*

 

> 

*Q* 

*rx w*

<sup>2</sup> *<sup>R</sup> <sup>x</sup> Q* 

(11)

*rv dr* (12)

(14)

(15)

only and the

(13)

, is a function of shear stress *rx*

2

(16)

 

 

and, consequently, the flow rate through the whole tube is

2

Fig. 2. Geometric presentation of MR fluid in o circular tube

 

. <sup>2</sup>

only. The inverse is that the shear rate

*<sup>w</sup>* is the wall shear stress.

is interpreted as a function of

equation (14) can be written as

**3. Rabinowitsch-Mooney equation** 

width *r* is given by

Integrating by parts gives

(13) then can be written as

variation of *rx*

where 

where 

One can readily see that for *n* > 1, equation (10) predicts increasing viscosity with increasing shear rate. The dilatant behaviour may be observed in moderately concentrated suspensions at high shear rates, and yet, the same suspension may exhibit pseudoplastic behaviour at lower shear rates.

This section is concluded by Table 2 providing a list of materials displaying a spectrum of non-Newtonian flow characteristics in diverse applications to reinforce idea yet again of the ubiquitous nature of such flow behaviour.



Of the time-independent fluids, this sub-class has received very little attention; consequently very few reliable data are available. Until recently, dilatant fluid behaviour was considered to be much less widespread in the chemical and processing industries. However, with the recent growing interest in the handling and processing of systems with high solids loadings, it is no longer so, as is evidenced by the number of recent review articles on this subject (Barnes *et al*., 1987; Barnes, 1989; Goddard and Bashir, 1990). Typical examples of materials exhibiting dilatant behaviour include concentrated suspensions of china clay, titanium dioxide (Metzner and Whitlock, 1958) and of corn fl our in water (Griskey *et al*., 1985). The limited information reported so far suggests that the apparent viscosity–shear rate data often result in linear plots on double logarithmic coordinates over a limited shear rate range and the flow behaviour may be represented by the power-law model, with the flow

<sup>1</sup> ( )*<sup>n</sup>*

(10)

Good spreadability and adhesive properties

Permit tamping operations in which small impulses produce almost complete settlement

Flows readily in a pipe, but difficult to restart

not run excessively at low speeds

 *K xy*

One can readily see that for *n* > 1, equation (10) predicts increasing viscosity with increasing shear rate. The dilatant behaviour may be observed in moderately concentrated suspensions at high shear rates, and yet, the same suspension may exhibit pseudoplastic behaviour at

This section is concluded by Table 2 providing a list of materials displaying a spectrum of non-Newtonian flow characteristics in diverse applications to reinforce idea yet again of the

**Practical fluid Characteristics Consequence of non-Newtonian behaviour**  Toothpaste Bingham Plastic Stays on brush and behaves more liquid-like

Drilling muds Bingham Plastic Good lubrication properties and ability to

' Bouncing Putty ' Visco-elastic Will fl ow if stretched slowly, but will bounce

Printing inks Pseudoplastic Spread easily in high speed machines yet do

the flow

Non-drip paints Thixotropic Thick in the tin, thin on the brush

Egg white Visco-elastic Easy air dispersion (whipping) Molten polymers Visco-elastic Thread-forming properties

while brushing

convey debris

(or shatter) if hit sharply

behaviour index, *n*, greater than unity, i.e.,

ubiquitous nature of such flow behaviour.

Wallpaper paste Pseudoplastic and

Waxy crude oils Viscoplastic and

Viscoelastic

Dilatant and thixotropic

Thixotropic

Table 2. Non-Newtonian characteristics of some common materials

lower shear rates.

Wet cement aggregates

#### **3. Rabinowitsch-Mooney equation**

Consider a one-directional flow of fluid through a circular tube with radius R, Figure 2. The volumetric flow rate through an annular element of area perpendicular to the flow and of width *r* is given by

$$
\delta \mathcal{Q}\_{\!\!\!\/} = \ 2\pi r \delta r. v\_{\!\!\/} \tag{11}
$$

and, consequently, the flow rate through the whole tube is

$$Q = 2\pi \int\_0 rv\_x dr\tag{12}$$

Integrating by parts gives

$$Q = 2\pi \left| \left[ \frac{r^2 \upsilon\_x}{2} \right]\_0^{r\_i} + \left[ \frac{r^2}{2} \left( -\frac{dv\_x}{dr} \right) dr \right] \right| \tag{13}$$

Provided there is no slip at the tube wall, the first term in equation (13) vanishes. Equation (13) then can be written as

$$Q = \underset{0}{\pi} \underset{0}{\dot{\pi}} r^2(-\dot{\gamma}) dr \tag{14}$$

If the fluid is time-independent and homogeneous, the shear stress is a function of shear rate only. The inverse is that the shear rate , is a function of shear stress *rx* only and the variation of *rx* with *r* is known from the following well-known equation:

$$\frac{\tau\_{\rm rx}}{\tau\_w} = \frac{r}{R} \tag{15}$$

where *<sup>w</sup>* is the wall shear stress.

Changing variables in equation (14), using equation (15), and dropping the subscripts *rx*, equation (14) can be written as

$$\hat{\mathbf{Q}} = \underset{\mathbf{0}}{\pi} \int \frac{\tau^2 R^2}{\tau\_w^2} (-\dot{\boldsymbol{\gamma}}) \frac{\tau\_{\dot{\boldsymbol{\gamma}}}}{\tau\_w} d\boldsymbol{\tau} = \left. \frac{\tau}{\tau\_w^3} \right|\_0^{\tau\_w} \hat{\mathbf{r}}^2(-\dot{\boldsymbol{\gamma}}) d\boldsymbol{\tau} \tag{16}$$

where is interpreted as a function of instead of *r*.

Fig. 2. Geometric presentation of MR fluid in o circular tube

Writing equation (16) in terms of the flow characteristic gives

$$\frac{8\mu}{D} = \frac{4Q}{\pi R^3} = \frac{4}{\pi\_w^3} \int\_0^\epsilon \tau^2 (-\dot{\gamma}) d\tau \tag{17}$$

Incompressible Non-Newtonian Fluid Flows 57

4. Calculate the true wall shear rate from equation (20) with the derivative determined in

The flow rate-pressure drop measurements shown in Table 3 were made in a horizontal tube having an internal diameter *D* = 6 mm, the pressure drop being measured between two

0.0864 0.463 1.37 2.76 4.13 5.20 6.78 8.15

8 / *du D* gradient *n' (3n'+1)/4n'* -1 .

0.157 0.232 0.375 0.439 0.475 0.475 0.475 0.475

**4. Calculation of flow rate-pressure drop relationship for laminar flow using** 

Flow rate-pressure drop calculations for laminar non-Newtonian flow in pipes may be made in various ways depending on the type of flow information available. When the flow data are in the form of flow rate and pressure gradient measured in a tubular viscometer or in a

stress-flow characteristic curve and the shear stress-true shear rate curve for this material.

Alternatively, calculate the gradient from the differences between the successive values

and measure the gradient at various points on the curve.

will not be a straight line and the

(s ) 

, was 870 kg/m3. Determine the wall shear

Mass flow rate x 103 (kg/s)

> 2.34 1.83 1.42 1.32 1.28 1.28 1.28 1.28

3. Plot ln(8 / ) *du D* against ln *<sup>w</sup>*

tapings 2.0*m* apart. The density of the fluid,

(bar)

0.384 0.519 0.716 0.965 1.16 1.29 1.46 1.60

Pressure drop

of these quantities.

**Example 1** 

Table 3.

28.8 38.9 53.7 72.4 87.0 96.8 110 120

Table 4

 **data** 

The results are shown in Table 4

 (Pa) 

step 3. In general, the plot of ln(8 / ) *du D* against ln *<sup>w</sup>*

gradient must be evaluated at the appropriate points on the curve.

where *u* is the average velocity of the fluid flow and *D* is the diameter of the tube. For flow in a pipe or tube the shear rate is negative so the integral in equation (17) is positive. For a given relationship between and , the value of the integral depends only on the value of *<sup>w</sup>* . Thus, for a non-Newtonian fluid, as well as for a Newtonian fluid, the flow characteristic *8u/D* is a unique function of the wall shear stress *<sup>w</sup>* .

The shear rate can be extracted from equation (17) by differentiating with respect to . Moreover, if a definite integral is differentiated w.r.t. the upper limit ( *<sup>w</sup>* ), the result is the integrand evaluated at the upper limit. It is convenient first to multiply equation (17) by <sup>3</sup> *w* throughout, then differentiating w.r.t. *<sup>w</sup>* gives

$$43\tau\_w^2 \frac{8u}{D} + \tau\_w^3 \frac{d(8u/D)}{d\tau\_w} = 4\tau\_w^2 (-\dot{\gamma})\_w \tag{18}$$

Rearranging equation (18) gives the wall shear rate *w*as

$$-\dot{\gamma}\_w = \frac{8u}{D} \left[ \frac{3}{4} + \frac{1}{4} \frac{\tau\_w}{(8u/D)} \frac{d(8u/D)}{d\tau\_w} \right] \tag{19}$$

Making use of the relationship *dx/x = dlnx*, equation (19) can be written as

$$-\dot{\gamma}\_w = \frac{8u}{D} \left[ \frac{3}{4} + \frac{1}{4} \frac{d \ln(8u/D)}{d \ln \tau\_w} \right] \tag{20}$$

As the wall shear rate *wN* for a Newtonian fluid in laminar flow is equal to (*-8u/D*), equation (20) can be expressed as

$$\dot{\gamma}\_w = \dot{\gamma}\_{wN} \left\lfloor \frac{3}{4} + \frac{1}{4} \frac{d \ln(8u/D)}{d \ln \tau\_w} \right\rfloor \tag{21}$$

Equations (20) and (21) are forms of the Rabinowitsch-Mooney equation. It shows that the wall shear rate for a non-Newtonian fluid can be calculated from the value for a Newtonian fluid having the same flow rate in the same pipe, the correction factor being the quantity in the square brackets. The derivative can be estimated by plotting *ln(8u/D*) against ln *<sup>w</sup>* and measuring the gradient. Alternatively the gradient may be calculated from the (finite) differences between values of *ln(8du/D*) and ln *<sup>w</sup>* . Thus the flow curve *<sup>w</sup>* against *w* can be determined. The measurements required and the calculation procedure are as follows.


#### **Example 1**

56 Continuum Mechanics – Progress in Fundamentals and Engineering Applications

3 3

 

where *u* is the average velocity of the fluid flow and *D* is the diameter of the tube. For flow in a pipe or tube the shear rate is negative so the integral in equation (17) is positive. For a

*<sup>w</sup>* . Thus, for a non-Newtonian fluid, as well as for a Newtonian fluid, the flow

integrand evaluated at the upper limit. It is convenient first to multiply equation (17) by <sup>3</sup>

. 2 3 8 (8 / ) <sup>2</sup> 3 + = 4 ( ) *w w w w w*

*D R*

844 = ( )

. <sup>2</sup>

 

(17)

, the value of the integral depends only on the value of

*<sup>w</sup>* .

(18)

.

*w*

*<sup>w</sup>* ), the result is the

 against *w*can

and

0

*w u Q <sup>d</sup>*

*w*

can be extracted from equation (17) by differentiating with respect to

 

*w*

(19)

(20)

(21)

. Thus the flow curve *<sup>w</sup>*

from the flow rate measurements.

*w*

*w*

from the pressure drop measurements and the corresponding values of

*wN* for a Newtonian fluid in laminar flow is equal to (*-8u/D*),

as

8 3 1 (8 / ) 4 4 (8 / )

8 3 1 ln(8 / )

*u d uD*

3 1 ln(8 / )

 

*d uD d*

4 4 ln *<sup>w</sup>*

4 4 ln *w wN*

Equations (20) and (21) are forms of the Rabinowitsch-Mooney equation. It shows that the wall shear rate for a non-Newtonian fluid can be calculated from the value for a Newtonian fluid having the same flow rate in the same pipe, the correction factor being the quantity in the square brackets. The derivative can be estimated by plotting *ln(8u/D*) against ln *<sup>w</sup>*

measuring the gradient. Alternatively the gradient may be calculated from the (finite)

be determined. The measurements required and the calculation procedure are as follows.

1. Measure Q at various values of / *P L <sup>f</sup>* , preferably eliminating end effects.

*D d*

*u d u D D uD d* 

Writing equation (16) in terms of the flow characteristic gives

 and 

characteristic *8u/D* is a unique function of the wall shear stress

Rearranging equation (18) gives the wall shear rate *w*

differences between values of *ln(8du/D*) and ln *<sup>w</sup>*

the flow characteristic <sup>3</sup> (8 / 4 / ) *du D Q R*

Moreover, if a definite integral is differentiated w.r.t. the upper limit (

*<sup>w</sup>* gives

 

*u duD D d*

*<sup>w</sup> <sup>w</sup>*

Making use of the relationship *dx/x = dlnx*, equation (19) can be written as

 

given relationship between

As the wall shear rate

2. Calculate

equation (20) can be expressed as

throughout, then differentiating w.r.t.

The shear rate

The flow rate-pressure drop measurements shown in Table 3 were made in a horizontal tube having an internal diameter *D* = 6 mm, the pressure drop being measured between two tapings 2.0*m* apart. The density of the fluid, , was 870 kg/m3. Determine the wall shear stress-flow characteristic curve and the shear stress-true shear rate curve for this material.


Table 3.

The results are shown in Table 4


Table 4

#### **4. Calculation of flow rate-pressure drop relationship for laminar flow using data**

Flow rate-pressure drop calculations for laminar non-Newtonian flow in pipes may be made in various ways depending on the type of flow information available. When the flow data are in the form of flow rate and pressure gradient measured in a tubular viscometer or in a

Incompressible Non-Newtonian Fluid Flows 59

0.00 0.0 0.00 (Centerline) 1.70 3.91 11.24 3.39 12.41 142.8 5.09 24.38 631.0 6.78 39.39 1812 8.48 57.14 4108 10.18 77.43 8016 (pipe wall)

10.18 . <sup>2</sup> 3 -1


The above is the general method but in this case the viscometric data can be well

 

10.18 . <sup>2</sup> 3.667 3 1

*d dP* = 1.62 17510 *a s*

 

(Table 6) illustrates the fact that flow in the centre makes a small contribution

 

show that the fluid is strongly shear thinning. If the data were available as values of

is negative for flow in a pipe. In the above calculations, the absolute values of

This agrees with the value found by numerical integration and would give the same value

As mentioned previously, the minus sign in equation (17) reflects the fact that the shear rate

**5. Wall shear stress-flow characteristic curves and scale-up for laminar flow**  When data are available in the form of the flow rate-pressure gradient relationship obtained in a small diameter tube, direct scale-up for flow in larger pipes can be done. It is not

curve with the true value of

1.62 *s-*1. This allows the integral in equation

should be calculated as their product. The table of

0 were not used; they were provided to

0 at

, and

calculated from the

10.18 /6[0+8016+4(11.24+613+4108)+2(142.8+1812)]= 17490 Pa s 3

. <sup>2</sup>

 

(*Pa*)

Table 6.

for *u*.

values of <sup>2</sup> 

By Simpson's rule

 *<sup>d</sup>*

From equation (17)

represented by 0.60 

corresponding values of

necessary to determine the

(17) to be evaluated analytically.

*u*

 

Note that the values of the apparent viscosity

, then

> 

0.749 *Pa*, thus 1.667

0 0

to the total flow: flow in the outer parts of the pipe is most significant.

have been used and the minus sign has therefore been dropped.

 -  0

pilot scale pipeline, direct scale-up can be done as described in Section 5. When the data are in the form of shear stress-shear rate values (tabular or graphical), the flow rate can be calculated directly using equation (17), where *D* is the diameter of the pipe to be used and *w* is the wall shear stress corresponding to the specified pressure gradient. Whether obtained with a rotational instrument or with a tubular viscometer, the data provide the relationship between and . Numerical evaluation of the integral in equation (17) can be done using selected pairs of values of and ranging from 0 to *w*.

If the , relationship can be accurately represented by a simple algebraic expression, such as the power law, over the required range then this may be used to substitute for , in equation (17), allowing the integral to be evaluated analytically. Both these methods are illustrated in the following example.

#### **Example 2**


Using the viscometric data given in Table 5 calculate the average velocity for the material flowing through a pipe of diameter 37m*m* when the pressure gradient is 1.1*kPa/m*.

Table 5.

*Calculations* 

The wall shear stress is given by

$$\tau\_w = \frac{D \Delta P}{4L} = \frac{(37 \times 10^{-3} \text{m})(1100 \text{ Pa/m})}{4} = 10.18 \text{ Pa}$$

the flow characteristic

$$\frac{8u}{D} = \frac{4}{\pi\_w^3} \int\_0^{\tau\_w} \tau^2 (-\dot{\gamma}) d\tau$$

It is necessary to evaluate the integral from = 0 to = 10.18*Pa*. This can be done by calculating . <sup>2</sup> for each of the values given in the table and plotting . <sup>2</sup> against . The area under the curve between = 0 and = 10.18*Pa* can then be measured. An alternative, which will be used here, is to use a numerical method such as Simpson's rule. This requires values at equal intervals of . Dividing the range of integration into six strips and interpolating the data allows Table 6 to be constructed.


Table 6.

58 Continuum Mechanics – Progress in Fundamentals and Engineering Applications

pilot scale pipeline, direct scale-up can be done as described in Section 5. When the data are in the form of shear stress-shear rate values (tabular or graphical), the flow rate can be calculated directly using equation (17), where *D* is the diameter of the pipe to be used and

 is the wall shear stress corresponding to the specified pressure gradient. Whether obtained with a rotational instrument or with a tubular viscometer, the data provide the

> and

such as the power law, over the required range then this may be used to substitute for

flowing through a pipe of diameter 37m*m* when the pressure gradient is 1.1*kPa/m*.

equation (17), allowing the integral to be evaluated analytically. Both these methods are

Using the viscometric data given in Table 5 calculate the average velocity for the material


. <sup>2</sup>

 

= 0 to

. Dividing the range of integration into six strips and

= 10.18*Pa*. This can be done by

against

. The

 

= 10.18*Pa* can then be measured. An alternative,

3 0 8 4 () *w*

for each of the values given in the table and plotting . <sup>2</sup>

which will be used here, is to use a numerical method such as Simpson's rule. This requires

*D*

= 0 and

interpolating the data allows Table 6 to be constructed.

*w <sup>u</sup> <sup>d</sup>*

0.0417 0.175 0.708 2.82 11.22 12.03

. Numerical evaluation of the integral in equation (17) can be

.

> , in

ranging from 0 to *w*

( ) *Pa* ( s)

4.58 1.95 0.777 0.310 0.123 0.118

*<sup>a</sup> Pa*

, relationship can be accurately represented by a simple algebraic expression,

*w*

If the 

**Example 2** 

Table 5.

*Calculations* 

relationship between

 and 

<sup>1</sup> . ( ) *s* 

done using selected pairs of values of

illustrated in the following example.

0.00911 0.0911 0.911 9.111 91.11 102.3

The wall shear stress is given by

the flow characteristic

calculating . <sup>2</sup>

 

area under the curve between

values at equal intervals of

4 *<sup>w</sup>*

*D P L*

It is necessary to evaluate the integral from

By Simpson's rule

$$\int\_0^{10.18} \tau^2 \gamma d\tau \approx \frac{10.18 \,/\, 6}{3} [0 + 8016 + 4(11.24 + 613 + 4108) + 2(142.8 + 1812)] = -17490 \,\text{Pa}^3\text{s}^{-1}$$

From equation (17)

$$\mu = \frac{(37 \times 10^{-3} \text{m})(17490 \text{ Pa}^{\text{3}} \text{s}^{-1})}{2(10.18 \text{ Pa})^{3}} = 0.307 \text{ m/s}^{-1}$$

The above is the general method but in this case the viscometric data can be well represented by 0.60 0.749 *Pa*, thus 1.667 1.62 *s-*1. This allows the integral in equation (17) to be evaluated analytically.

$$\int\_0^{\tau\_w} \tau^2 \dot{\gamma} d\tau = 1.62 \int\_0^{10.18} \tau^{3.667} d\tau = 17510 \text{ Pa}^3 \text{s}^{-1}$$

This agrees with the value found by numerical integration and would give the same value for *u*.

Note that the values of the apparent viscosity 0 were not used; they were provided to show that the fluid is strongly shear thinning. If the data were available as values of 0 at corresponding values of , then should be calculated as their product. The table of values of <sup>2</sup> (Table 6) illustrates the fact that flow in the centre makes a small contribution to the total flow: flow in the outer parts of the pipe is most significant.

As mentioned previously, the minus sign in equation (17) reflects the fact that the shear rate is negative for flow in a pipe. In the above calculations, the absolute values of , and have been used and the minus sign has therefore been dropped.

#### **5. Wall shear stress-flow characteristic curves and scale-up for laminar flow**

When data are available in the form of the flow rate-pressure gradient relationship obtained in a small diameter tube, direct scale-up for flow in larger pipes can be done. It is not necessary to determine the - curve with the true value of calculated from the Rabinowitsch-Mooney equation (Equation (20)). Equation (17) shows that the flow characteristic is a unique function of the wall shear stress for a particular fluid:

$$\frac{8\mu}{D} = \frac{4}{\pi\_w^3} \int\_0^{\tau\_w} \tau^2 (-\dot{\gamma}) d\tau$$

In the case of a Newtonian fluid, substituting / into the above equation and evaluating the integral gives

$$\frac{8\mu}{D} = \frac{\tau\_w}{\mu} \tag{22}$$

Incompressible Non-Newtonian Fluid Flows 61

Decreasing diameter

*D0 D1 D3* 

Fig. 3. Shear stress at the pipe wall against flow characteristic for a non-Newtonian fluid

*8u/D* 

Turbulent

It is found useful to define two quantifies *K'* and *n'* in order to describe the *w*

ln '

*n*

*w*

values should be found at the point corresponding to the required value of

characteristic curve. If the laminar flow data are plotted on logarithmic axes as in Figure 4,

ln(8 / ) *<sup>w</sup> d*

'

*n*

*d uD* 

8 '

In general, both *K'* and *n'* have different values at different points along the curve. The

cases, the curve in Figure 4 will be virtually straight over the range required and a single value may be used for each of *K'* and *n'*. Although equation (27) is similar to the equation of

The reason for defining *n'* in this way can be seen from equation (21) where the inverse of the derivative occurs in the correction factor. Equation (20) can be written in terms of *n'* as

*<sup>u</sup> <sup>K</sup> D*

Given a wall shear stress-flow characteristic curve such as that in Figure 3, the flow ratepressure drop relationship can be found for any diameter of pipe provided the flow remains laminar and is within the range of the graph. For example, if it is required to calculate the pressure drop for flow in a pipe of given diameter at a specified volumetric flow rate, the

is calculated and the corresponding

*<sup>w</sup>*. In some


read from the graph. The pressure gradient, and hence the

Laminar

(26)

(27)

flowing in a pipe

value of the flow characteristic <sup>3</sup> (8 / 4 / ) *uD Q R*

then the gradient of the curve defines the value of *n'* :

The equation of the tangent can be written as

a power law fluid, the two must not be confused.

pressure drop for a given pipe length, can then be calculated.

value of the wall shear stress *w*

Recall that the wall shear rate for a Newtonian fluid in laminar flow in a tube is equal to 8 /*u D* . In the case of a non-Newtonian fluid in laminar flow, the flow characteristic is no longer equal to the magnitude of the wall shear rate. However, the flow characteristic is still related uniquely to *w* because the value of the integral, and hence the right hand side of equation (17), is determined by the value of *w*.

If the fluid flows in two pipes having internal diameters *D1* and *D2* with the same value of the wall shear stress in both pipes, then from equation (17) the values of the flow characteristic are equal in both pipes:

$$\frac{8u\_1}{D\_1} = \frac{8u\_2}{D\_2} \tag{23}$$

So the average velocities are related by

$$\frac{\mu\_1}{\mu\_2} = \frac{D\_1}{D\_2} \tag{24}$$

By substituting for *u* or by writing the flow characteristic as <sup>3</sup> 4 / *Q R* , the volumetric flow rates are related by

$$\frac{Q\_1}{Q\_2} = \left(\frac{D\_1}{D\_2}\right)^3\tag{25}$$

It is important to appreciate that the same value of *w* requires different values of the pressure gradient in the two pipes. It is convenient to represent the flow behaviour as a graph of *w* plotted against 8 /*u D* , as shown in Figure 3. In accordance with the above discussion, all data fit a single line for laminar flow. The graph is steeper for turbulent flow and different lines are found for different pipe diameters. It is noteworthy that the same would be found for Newtonian flow if the data were plotted in this way and the laminar flow line would be a straight line of gradient *µ* passing through the origin. The plot in Figure 3 is not a true flow curve because the flow characteristic is equal to the magnitude of the wall shear rate only in the case of Newtonian laminar flow.

Rabinowitsch-Mooney equation (Equation (20)). Equation (17) shows that the flow

8*u <sup>w</sup> D*

Recall that the wall shear rate for a Newtonian fluid in laminar flow in a tube is equal to 8 /*u D* . In the case of a non-Newtonian fluid in laminar flow, the flow characteristic is no longer equal to the magnitude of the wall shear rate. However, the flow characteristic is still

 . If the fluid flows in two pipes having internal diameters *D1* and *D2* with the same value of the wall shear stress in both pipes, then from equation (17) the values of the flow

> 1 2 1 2 8 8 *u u*

1 1 2 2 *u D*

1 1 2 2 *Q D Q D* 

pressure gradient in the two pipes. It is convenient to represent the flow behaviour as a

discussion, all data fit a single line for laminar flow. The graph is steeper for turbulent flow and different lines are found for different pipe diameters. It is noteworthy that the same would be found for Newtonian flow if the data were plotted in this way and the laminar flow line would be a straight line of gradient *µ* passing through the origin. The plot in Figure 3 is not a true flow curve because the flow characteristic is equal to the magnitude of

3

plotted against 8 /*u D* , as shown in Figure 3. In accordance with the above

By substituting for *u* or by writing the flow characteristic as <sup>3</sup> 4 / *Q R*

It is important to appreciate that the same value of *w*

the wall shear rate only in the case of Newtonian laminar flow.

because the value of the integral, and hence the right hand side of

3 0 8 4 () *w*

*w <sup>u</sup> <sup>d</sup>*

. <sup>2</sup>

 

 

/ into the above equation and

(22)

*D D* (23)

*u D* (24)

(25)

requires different values of the

, the volumetric flow

characteristic is a unique function of the wall shear stress for a particular fluid:

*D*

In the case of a Newtonian fluid, substituting

characteristic are equal in both pipes:

So the average velocities are related by

equation (17), is determined by the value of *w*

evaluating the integral gives

related uniquely to *w*

rates are related by

graph of *w*

Fig. 3. Shear stress at the pipe wall against flow characteristic for a non-Newtonian fluid flowing in a pipe

Given a wall shear stress-flow characteristic curve such as that in Figure 3, the flow ratepressure drop relationship can be found for any diameter of pipe provided the flow remains laminar and is within the range of the graph. For example, if it is required to calculate the pressure drop for flow in a pipe of given diameter at a specified volumetric flow rate, the value of the flow characteristic <sup>3</sup> (8 / 4 / ) *uD Q R* is calculated and the corresponding value of the wall shear stress *w* read from the graph. The pressure gradient, and hence the pressure drop for a given pipe length, can then be calculated.

It is found useful to define two quantifies *K'* and *n'* in order to describe the *w* -flow characteristic curve. If the laminar flow data are plotted on logarithmic axes as in Figure 4, then the gradient of the curve defines the value of *n'* :

$$m^{\prime} = \frac{d\ln\pi\_w}{d\ln(8u/D)}\tag{26}$$

The equation of the tangent can be written as

$$\left(\tau\_w\right) = K \left(\frac{8\mu}{D}\right)^{n'} \tag{27}$$

In general, both *K'* and *n'* have different values at different points along the curve. The values should be found at the point corresponding to the required value of *<sup>w</sup>*. In some cases, the curve in Figure 4 will be virtually straight over the range required and a single value may be used for each of *K'* and *n'*. Although equation (27) is similar to the equation of a power law fluid, the two must not be confused.

The reason for defining *n'* in this way can be seen from equation (21) where the inverse of the derivative occurs in the correction factor. Equation (20) can be written in terms of *n'* as

Incompressible Non-Newtonian Fluid Flows 63

shear stress at wall flow characteristic (8 / )

*ap u d*

For laminar flow, *µap* has the property that it is the viscosity of a Newtonian fluid having the same flow characteristic as the non-Newtonian fluid when subjected to the same value of wall shear stress. In particular, this corresponds to the same volumetric flow rate for the same pressure gradient in the same pipe. This suggests that *µap* might be a useful quantity for correlating flow rate-pressure gradient data for non-Newtonian flow in pipes. This is

Re' *<sup>i</sup>*

8 '

8 ' 8 /

> ' 1 ' 8 '

Use of this generalized Reynolds number was suggested by Metzner and Reed (1955). For Newtonian behaviour, *K'* = *µ* and *n'* = 1 so that the generalized Reynolds number reduces to

Turbulent flow of Newtonian fluids is described in terms of the Fanning friction factor, which is correlated against the Reynolds number with the relative roughness of the pipe wall as a parameter. The same approach is adopted for non-Newtonian flow but the

> 1 2 2 *<sup>f</sup> <sup>u</sup>*

It is straightforward to show that the Fanning friction factor for laminar non-Newtonian

**7. Turbulent flow of Inelastic non-Newtonian fluids in pipes and circular** 

generalized Reynolds number is used. The Fanning friction factor is defined by

*n u d Re*

*i <sup>u</sup> <sup>K</sup> d*

> *i i <sup>u</sup> <sup>K</sup>*

> > 2''

*n n i*

*K*

*ud d* 

*ap ud* 

'

' 1

*n*

*n*

found to be the case and it is on *µap* that a generalized Reynolds number *Re'* is based

Representing the fluid's laminar flow behaviour in terms of *K'* and *n'*

The pipe flow apparent viscosity, defined by equation 31, is given by

*ap*

Using *µap* in Equation (34), the generalized Reynolds number takes the form

the normal Reynolds number.

**ducts** 

flow becomes

=

*i*

(32)

(35)

(36)

*f* 16 /Re' (37)

(33)

(34)

(31)

Fig. 4. Logarithmic plot of wall shear stress against flow characteristic: the gradient at a point defines n'

$$\dot{\mathcal{Y}}\_{o0} = \dot{\mathcal{Y}}\_{o0\text{N}} \left(\frac{3\text{n}'+1}{4\text{n}'}\right) \tag{28}$$

Equation (28) is helpful in showing how the value of the correction factor in the Rabinowitsch-Mooney equation corresponds to different types of flow behaviour. For a Newtonian fluid, *n'* = 1 and therefore the correction factor has the value unity. Shear thinning behaviour corresponds to *n'* < 1 and consequently the correction factor has values greater than unity, showing that the wall shear rate *w* is of greater magnitude than the value for Newtonian flow. Similarly, for shear thickening behaviour, *w* is of a smaller magnitude than the Newtonian value *wN* . The value correction factor varies from 2.0 for *n'* = 0.2 to 0.94 for *n'* = 1.3.

#### **6. Generalized Reynolds number for flow in pipes**

It is recalled that for Newtonian flow in a pipe, the Reynolds number is defined by

$$Re = \frac{\rho u D}{\mu} \tag{29}$$

In the case of non-Newtonian flow, it is necessary to use an appropriate apparent viscosity. Although the apparent viscosity *µa* is defined in the same way as for a Newtonian fluid, it no longer has the same fundamental significance and other, equally valid, definitions of apparent viscosities may be made. In flow in a pipe, where the shear stress varies with radial location, the value of *µa* varies. It is shown that the conditions near the pipe wall that are most important. The value of *µa* evaluated at the wall is given by

$$\mu\_a = -\frac{\text{shear stress at wall}}{\text{shear rate at wall}} = \frac{\tau\_{ao}}{(-dv\_x \,/\, dr)\_{o}}\tag{30}$$

Another definition is based, not on the true shear rate at the wall, but on the flow characteristic. This quantity, which may be called the apparent viscosity for pipe flow, is given by

Fig. 4. Logarithmic plot of wall shear stress against flow characteristic: the gradient at a

 

greater than unity, showing that the wall shear rate *w*

*ln*()

**6. Generalized Reynolds number for flow in pipes** 

are most important. The value of *µa* evaluated at the wall is given by

magnitude than the Newtonian value

= 0.2 to 0.94 for *n'* = 1.3.

given by

value for Newtonian flow. Similarly, for shear thickening behaviour, *w*

It is recalled that for Newtonian flow in a pipe, the Reynolds number is defined by

 *uD Re* 

In the case of non-Newtonian flow, it is necessary to use an appropriate apparent viscosity. Although the apparent viscosity *µa* is defined in the same way as for a Newtonian fluid, it no longer has the same fundamental significance and other, equally valid, definitions of apparent viscosities may be made. In flow in a pipe, where the shear stress varies with radial location, the value of *µa* varies. It is shown that the conditions near the pipe wall that

shear stress at wall shear rate at wall ( / )

Another definition is based, not on the true shear rate at the wall, but on the flow characteristic. This quantity, which may be called the apparent viscosity for pipe flow, is

*<sup>a</sup> dv dr*

=

*x*

(30)

. . 3n'+1 4n'

*ln*(*8u/D*)

Equation (28) is helpful in showing how the value of the correction factor in the Rabinowitsch-Mooney equation corresponds to different types of flow behaviour. For a Newtonian fluid, *n'* = 1 and therefore the correction factor has the value unity. Shear thinning behaviour corresponds to *n'* < 1 and consequently the correction factor has values

(28)

is of greater magnitude than the

*wN* . The value correction factor varies from 2.0 for *n'*

(29)

is of a smaller

 *N* 

point defines n'

$$
\mu\_{ap} = \frac{\text{shear stress at wall}}{\text{flow characteristic}} = \frac{\tau\_{ap}}{(8u/d\_i)}\tag{31}
$$

For laminar flow, *µap* has the property that it is the viscosity of a Newtonian fluid having the same flow characteristic as the non-Newtonian fluid when subjected to the same value of wall shear stress. In particular, this corresponds to the same volumetric flow rate for the same pressure gradient in the same pipe. This suggests that *µap* might be a useful quantity for correlating flow rate-pressure gradient data for non-Newtonian flow in pipes. This is found to be the case and it is on *µap* that a generalized Reynolds number *Re'* is based

$$\text{Re}' = \frac{\rho u d\_i}{\mu\_{ap}} \tag{32}$$

Representing the fluid's laminar flow behaviour in terms of *K'* and *n'*

$$\left(\tau\_{\alpha}\right) = \left. K' \right| \left(\frac{8\mu}{d\_i}\right)^{n'} \tag{33}$$

The pipe flow apparent viscosity, defined by equation 31, is given by

$$
\mu\_{\text{ap}} = \frac{\tau\_{\text{ao}}}{8\mu \;/\; d\_i} = K \left(\frac{8\mu}{d\_i}\right)^{n-1} \tag{34}
$$

Using *µap* in Equation (34), the generalized Reynolds number takes the form

$$Re^{\prime} = \frac{\rho u^{2-n^{\prime}} d\_i^{n^{\prime}}}{8^{n^{\prime}-1} K^{\prime}} \tag{35}$$

Use of this generalized Reynolds number was suggested by Metzner and Reed (1955). For Newtonian behaviour, *K'* = *µ* and *n'* = 1 so that the generalized Reynolds number reduces to the normal Reynolds number.

#### **7. Turbulent flow of Inelastic non-Newtonian fluids in pipes and circular ducts**

Turbulent flow of Newtonian fluids is described in terms of the Fanning friction factor, which is correlated against the Reynolds number with the relative roughness of the pipe wall as a parameter. The same approach is adopted for non-Newtonian flow but the generalized Reynolds number is used. The Fanning friction factor is defined by

$$f = \frac{\tau\_{\alpha}}{\frac{1}{2}\rho u^{2}}\tag{36}$$

It is straightforward to show that the Fanning friction factor for laminar non-Newtonian flow becomes

$$f \, = 16 \, / \, \text{Re}'\tag{37}$$

Incompressible Non-Newtonian Fluid Flows 65

and *n'* a value of 0.3. Calculate the values of the apparent viscosity for pipe flow *µap*, the generalized Reynolds number *Re'* and the pressure drop across the tube, neglecting end

> 8 8(2.0 / ) <sup>1</sup> = 210 0.0762 *u ms*

' 1 <sup>8</sup> (0.3 1.0) 0.7 210 0.0237

' 1 <sup>8</sup> 0.3 0.7 ' (1.48 Pa s )(0.0237 ) 0.0351 Pa s

<sup>3</sup> (0.0762 m)(2.0 m)(961 kg/m ) ' = 4178

*D m*

*<sup>u</sup> K s*

(0.0351 Pa s) *ap*

*s*

*s*

Source: D. W. Dodge and A. B. Metzner, *AIChE Journal* **5** (1959) 189-204 Fig. 6. Friction factor chart for purely viscous non-Newtonian fluids.

*<sup>n</sup> u*

*D*

*n*

*D*

*uD Re* 

effects.

*Calculations* 

and

Hence

and

The flow characteristic is given by

*ap*

This is of the same form as equation for Newtonian flow and is one reason or using this form of generalized Reynolds number. Equation (37) provides another way of calculating the pressure gradient for a given flow rate for laminar non-Newtonian flow.

### **7.1 Laminar-turbulent transition**

A stability analysis made by Ryan and Johnson (1959) suggests that the transition from laminar to turbulent flow for inelastic non-Newtonian fluids occurs at a critical value of the generalized Reynolds number that depends on the value of *n'*. The results of this analysis are shown in Figure 5. This relationship has been tested for shear thinning and for Bingham plastic fluids and has been found to be accurate. Over the range of shear thinning behaviour encountered in practice, 0.2 ≤ *n'* ≤ 1, the critical value of *Re'* is in the range 2100 ≤ *Re'* ≤ 2400.

Fig. 5. Variation of the critical value of the Reynolds number with n'

#### **7.2 Friction factors for turbulent flow in smooth pipes**

Experimental results for the Fanning friction factor for turbulent flow of shear thinning fluids in smooth pipes have been correlated by Dodge and Metzner (1959) as a generalized form of the yon Kármán equation:

$$\frac{1}{f^{1/2}} = \frac{4.0}{\left(n!\right)^{0.75}} \log[f^{1-n'/2} \text{Re}'] - \frac{0.40}{\left(n!\right)^{1.2}}\tag{38}$$

This correlation is shown in Figure 6. The broken lines represent extrapolation of equation (38) for values of *n'* and *Re'* beyond those of the measurements made by Dodge and Metzner. More recent studies tend to confirm the findings of Dodge and Metzner but do not significantly extend the range of applicability. Having determined the value of the friction factor *f* for a specified flow rate and hence *Re'*, the pressure gradient can be calculated in the normal way.

#### **Example 3**

A general time-independent non-Newtonian liquid of density 961 kg/m3 flows steadily with an average velocity of 2.0*m/s* through a tube 3.048 m long with an inside diameter of 0.0762 m. For these conditions, the pipe flow consistency coefficient *K'* has a value of 1.48 Pa s0.3

This is of the same form as equation for Newtonian flow and is one reason or using this form of generalized Reynolds number. Equation (37) provides another way of calculating

A stability analysis made by Ryan and Johnson (1959) suggests that the transition from laminar to turbulent flow for inelastic non-Newtonian fluids occurs at a critical value of the generalized Reynolds number that depends on the value of *n'*. The results of this analysis are shown in Figure 5. This relationship has been tested for shear thinning and for Bingham plastic fluids and has been found to be accurate. Over the range of shear thinning behaviour encountered in practice, 0.2 ≤ *n'* ≤ 1, the critical value of *Re'* is in the range 2100 ≤ *Re'* ≤ 2400.

Experimental results for the Fanning friction factor for turbulent flow of shear thinning fluids in smooth pipes have been correlated by Dodge and Metzner (1959) as a generalized

This correlation is shown in Figure 6. The broken lines represent extrapolation of equation (38) for values of *n'* and *Re'* beyond those of the measurements made by Dodge and Metzner. More recent studies tend to confirm the findings of Dodge and Metzner but do not significantly extend the range of applicability. Having determined the value of the friction factor *f* for a specified flow rate and hence *Re'*, the pressure gradient can be calculated in the

A general time-independent non-Newtonian liquid of density 961 kg/m3 flows steadily with an average velocity of 2.0*m/s* through a tube 3.048 m long with an inside diameter of 0.0762 m. For these conditions, the pipe flow consistency coefficient *K'* has a value of 1.48 Pa s0.3

1 '/2 1/2 0.75 1.2 1 4.0 0.40 = log[ '] ( ') ( ') *<sup>n</sup> f Re f n <sup>n</sup>*

(38)

the pressure gradient for a given flow rate for laminar non-Newtonian flow.

Fig. 5. Variation of the critical value of the Reynolds number with n'

**7.2 Friction factors for turbulent flow in smooth pipes** 

form of the yon Kármán equation:

normal way. **Example 3** 

**7.1 Laminar-turbulent transition** 

and *n'* a value of 0.3. Calculate the values of the apparent viscosity for pipe flow *µap*, the generalized Reynolds number *Re'* and the pressure drop across the tube, neglecting end effects.

Source: D. W. Dodge and A. B. Metzner, *AIChE Journal* **5** (1959) 189-204

Fig. 6. Friction factor chart for purely viscous non-Newtonian fluids.

#### *Calculations*

The flow characteristic is given by

$$\frac{8\mu}{D} = \frac{8(2.0m/s)}{0.0762m} = 210s^{-1}$$

and

$$\left(\frac{8\mu}{D}\right)^{n'-1} = \text{ } \text{21} \\ 0^{(0.3-1.0)} = 0.0237 \text{s}^{0.7}$$

Hence

$$
\mu\_{\rm ap} = K \left( \frac{8\mu}{D} \right)^{n^\*-1} = \text{(1.48 Pa s}^{0.3} \text{)} \text{(0.0237 s}^{0.7} \text{)} = 0.0351 \text{ Pa s}^{-1}
$$

and

$$Re' = \frac{\rho u D}{\mu\_{ap}} = \frac{(0.0762 \text{ m})(2.0 \text{ m})(961 \text{ kg/m}^3)}{(0.0351 \text{ Pa s})} = 4178 \text{ s}$$

Incompressible Non-Newtonian Fluid Flows 67

(8 / ) <sup>8</sup> ( ) +b( ) *<sup>h</sup> w ww*

where *a* and *b* are two geometric parameters characterizing the cross-section of the duct

4 *<sup>h</sup> <sup>w</sup>*

() ()

, *p*(*x*) =(*b* /*a <sup>w</sup>*

*<sup>u</sup> <sup>f</sup> <sup>d</sup>*

where *ξ* is a dummy variable of integration. The constant *C*0 has been evaluated by using the

 (46)

*n*

/*ρu2*

as:

*<sup>f</sup> Re* (48)

/ (/) 1 0 8 1( ) ( ) *w b a b a*

<sup>8</sup> ( )

*g*

*n n h*

2

8( )

1

 

*<sup>g</sup> n n u D <sup>a</sup> K b n*

*n*

16

*u a K b D n*

For constant values of *a* and *b* , equation (42) is an ordinary differential equation of the form

*D P L*

(*a=*1/4 and *b=* 3/4 for a circular tube, and *a* =1/2 and *b* =1 for the slit) and *<sup>w</sup>*

*p*

value of shear stress at the wall, and is related to the pressure gradient as:

(d *y*/d *x*) + *p*(*x*) *y* = *q*(*x*) which can be integrated to obtain the solution as:

*h*

which can be rewritten in terms of the friction factor, *f* = 2 *<sup>w</sup>*

*D a*

*w*

= 0 and therefore, *C*<sup>0</sup> =0.

*w*

Re

For the flow of a power-law fluid, *f*(*τ*) = (*τ*/*K)*1/ *<sup>n</sup>* and integration of equation (46) yields:

*y*

*x*

Fig. 7. Laminar flow between parallel plates

Now identifying *y=* (8*V*/*Dh*) and *x* = *<sup>w</sup>*

where the generalized Reynolds number,

to equation (43) is given as:

condition that when *V* =0, *w*

 

*h* 

*z*

*duD <sup>u</sup> f a d D*

*L*

*yz*

*w h*

(43)

*p+p* 

(44)

<sup>0</sup> ( ) *p x dx p x dx <sup>y</sup> e e <sup>q</sup> x dx C* (45)

) and *q*(*x*)= (*f*( *<sup>w</sup>*

 )/*a <sup>w</sup>*

(47)

(49)

is the mean

), the solution

From Figure 6, the Fanning friction factor *f* has a value 0.0047. Therefore the pressure drop is given by

$$\Delta P\_f = 4f \left(\frac{L}{d\_i}\right) \frac{\rho \text{u}^2}{2} = \frac{2(0.0047)(3.048 \text{ m})(961 \text{ kg/m}^3)(2.0 \text{ m/s})^2}{(0.0762 \text{ m})} = 1445 \text{ Pa}$$

### **8. Laminar flow of inelastic fluids in non-circular ducts**

Analytical solutions for the laminar flow of time-independent fluids in non-axisymmetric conduits are not possible. Numerous workers have obtained approximate and/or complete numerical solutions for specific flow geometries including square, rectangular and triangular pipes (Schechter, 1961 ; Wheeler and Wissler, 1965 ; Miller, 1972 ; Mitsuishi and Aoyagi, 1969, 1973). On the other hand, semi-empirical attempts have also been made to develop methods for predicting pressure drop for time-independent fluids in ducts of noncircular cross-section. Perhaps the most systematic and successful friction factor analysis is that provided by Kozicki *et al* . (1966, 1967) . It is useful to recall here that the equation (19) is a generalized equation for the laminar flow of time-independent fluids in a tube and it can be slightly rearranged as:

$$-\dot{\gamma}\_w = f(\tau\_w) = \frac{1}{4}\tau\_w \frac{d(8u \;/\ D)}{d\tau\_w} + \frac{3}{4}(\frac{8u}{D})\tag{39}$$

Similarly, one can parallel this approach for the fully developed laminar flow of time independent fluids in a thin slit (Figure 7) to derive the following relationship:

$$-\dot{\gamma}\_w = \left(-\frac{dV\_z}{dr}\right)\_w = f(\tau\_w) = \tau\_w \frac{d(\mu \,/\, h)}{d\tau\_w} + 2(\frac{\mu}{h})\tag{40}$$

In order to develop a unified treatment for ducts of various cross-sections, it is convenient to introduce the usual hydraulic diameter *Dh* (defined as four times the area for flow/wetted perimeter) into equations (39) and 40).

For a circular pipe, *D*<sup>h</sup> = *D* and hence equation (38) becomes:

$$-\dot{\boldsymbol{\gamma}}\_{oo} = f(\boldsymbol{\tau}\_{oo}) = \frac{1}{4} \boldsymbol{\tau}\_{oo} \frac{d(8\boldsymbol{\mu} \,/\, D\_h)}{d\boldsymbol{\tau}\_{oo}} + \frac{3}{4} (\frac{8\boldsymbol{\mu}}{D\_h}) \tag{41}$$

For the slit shown in Figure 7, the hydraulic diameter *D*h= 4*h* , and thus equation (39) is rewritten as:

$$-\dot{\gamma}\_w = \frac{1}{2}\tau\_w \frac{d(8u/D\_h)}{d\tau\_w} + \frac{8u}{D\_h} \tag{42}$$

By noting the similarity between the form of the Rabinowitsch–Mooney equations for the flow of time-independent fluids in circular pipes (equation (41)) and that in between two plates (equation (42)), they suggested that it could be extended to the ducts having a constant cross-section of arbitrary shape as follows:

From Figure 6, the Fanning friction factor *f* has a value 0.0047. Therefore the pressure drop is

Analytical solutions for the laminar flow of time-independent fluids in non-axisymmetric conduits are not possible. Numerous workers have obtained approximate and/or complete numerical solutions for specific flow geometries including square, rectangular and triangular pipes (Schechter, 1961 ; Wheeler and Wissler, 1965 ; Miller, 1972 ; Mitsuishi and Aoyagi, 1969, 1973). On the other hand, semi-empirical attempts have also been made to develop methods for predicting pressure drop for time-independent fluids in ducts of noncircular cross-section. Perhaps the most systematic and successful friction factor analysis is that provided by Kozicki *et al* . (1966, 1967) . It is useful to recall here that the equation (19) is a generalized equation for the laminar flow of time-independent fluids in a tube and it can

1 3 ( ) +( ) 4 4

*<sup>f</sup> duD <sup>u</sup>*

Similarly, one can parallel this approach for the fully developed laminar flow of time

 

In order to develop a unified treatment for ducts of various cross-sections, it is convenient to introduce the usual hydraulic diameter *Dh* (defined as four times the area for flow/wetted

*w*

( ) +2( ) (/) *<sup>z</sup> w w*

1 3 ( ) +( ) 4 4

1 8 (8 / )+( ) <sup>2</sup>

By noting the similarity between the form of the Rabinowitsch–Mooney equations for the flow of time-independent fluids in circular pipes (equation (41)) and that in between two plates (equation (42)), they suggested that it could be extended to the ducts having a

*duD u d D*

*w h*

*<sup>f</sup> duD <sup>u</sup>*

 

*<sup>h</sup> w w*

 

For the slit shown in Figure 7, the hydraulic diameter *D*h= 4*h* , and thus equation (39) is

(8 / ) *<sup>h</sup>* 8

*d D*

*w w w*

independent fluids in a thin slit (Figure 7) to derive the following relationship:

*w*

*dV <sup>f</sup> dr*

constant cross-section of arbitrary shape as follows:

*w*

For a circular pipe, *D*<sup>h</sup> = *D* and hence equation (38) becomes:

perimeter) into equations (39) and 40).

rewritten as:

(8 / ) 8

*d D*

*w*

*du h u d h*

(40)

*h*

(42)

(41)

(39)

<sup>2</sup> (0.0762 m) *<sup>f</sup>*<sup>4</sup>

**8. Laminar flow of inelastic fluids in non-circular ducts** 

*i L*

 

*<sup>d</sup> P f*

be slightly rearranged as:

<sup>2</sup> 3 2 <sup>u</sup> 2(0.0047)(3.048 m)(961 kg/m )(2.0 m/s) = 1445 Pa

given by

$$-\dot{\gamma}\_w = f(\overline{\tau}\_w) = a\overline{\tau}\_w \frac{d(8u \,/\, D\_h)}{d\overline{\tau}\_w} + b(\frac{8u}{D\_h}) \tag{43}$$

Fig. 7. Laminar flow between parallel plates

where *a* and *b* are two geometric parameters characterizing the cross-section of the duct (*a=*1/4 and *b=* 3/4 for a circular tube, and *a* =1/2 and *b* =1 for the slit) and *<sup>w</sup>* is the mean value of shear stress at the wall, and is related to the pressure gradient as:

$$
\overline{\tau}\_w = \frac{D\_h}{4} \left( \frac{-\Delta P}{L} \right) \tag{44}
$$

For constant values of *a* and *b* , equation (42) is an ordinary differential equation of the form (d *y*/d *x*) + *p*(*x*) *y* = *q*(*x*) which can be integrated to obtain the solution as:

$$y = e^{-\int p(\mathbf{x})d\mathbf{x}} \int e^{\int p(\mathbf{x})d\mathbf{x}} q(\mathbf{x})d\mathbf{x} + \mathbf{C}\_0 \tag{45}$$

Now identifying *y=* (8*V*/*Dh*) and *x* = *<sup>w</sup>* , *p*(*x*) =(*b* /*a <sup>w</sup>* ) and *q*(*x*)= (*f*( *<sup>w</sup>* )/*a <sup>w</sup>* ), the solution to equation (43) is given as:

$$\frac{8u}{D\_h} = \frac{1}{a} (\overline{\tau}\_w)^{-b/a} \int\_0^{\overline{\tau}\_w} \xi^{(b/a)-1} f(\tau) d\xi \tag{46}$$

where *ξ* is a dummy variable of integration. The constant *C*0 has been evaluated by using the condition that when *V* =0, *w*= 0 and therefore, *C*<sup>0</sup> =0.

For the flow of a power-law fluid, *f*(*τ*) = (*τ*/*K)*1/ *<sup>n</sup>* and integration of equation (46) yields:

$$\overline{\tau}\_{w} = K \left\{ \frac{8u}{D\_n} (b + \frac{a}{n}) \right\}^n \tag{47}$$

which can be rewritten in terms of the friction factor, *f* = 2 *<sup>w</sup>* /*ρu<sup>2</sup>* as:

$$f = \frac{16}{Re\_{\chi}}\tag{48}$$

where the generalized Reynolds number,

$$\mathrm{Re}\_g = \frac{\rho u^{2-n} D\_h^n}{8^{n-1} K (b + \frac{a}{n})^n} \tag{49}$$

Incompressible Non-Newtonian Fluid Flows 69

The main virtue of this approach lies in its simplicity and the fact that the geometric parameters *a* and *b* can be deduced from the behaviour of Newtonian fluids in the same flow geometry. Table 7 lists values of *a* and *b* for a range of flow geometries commonly

Kozicki *et al.*(1966) argued that the friction factor of the turbulent flow in non-circular ducts

1 4 0.4 ( ) log (Re ) 4 log[ ] 3 1 *n*

*a a bn f n f n <sup>n</sup> <sup>n</sup>* 

Note that since for a circular tube, *a=* 1/4 and *b* =3/4, equation (50) is consistent with that for circular pipes. The limited data available on turbulent flow in triangular (Irvine Jr, 1988), rectangular (Kostic and Hartnett, 1984) and square ducts (Escudier and Smith, 2001) conforms to equation (48). In the absence of any definite information, Kozicki and Tiu (1988) suggested that the Dodge–Metzner criterion, *Re*<sup>g</sup> 2100, can be used for predicting the limit

Some further attempts have been made to simplify and/or improve upon the two geometric parameter method of Kozicki *et al* . (1966, 1967). Delplace and Leuliet (1995) revisited the definition of the generalized Reynolds number (equation (49)) and argued that while the use of *a* and *b* accounts for the non-circular cross-sections of the ducts, but the factor 8*<sup>n</sup>*-<sup>1</sup> appearing in the denominator is strictly applicable for the flow in circular ducts only. Their reasoning hinges on the fact that for the laminar flow of a Newtonian fluid, the product

48 (. ) *<sup>f</sup> Re*

where both the (Fanning) friction factor and the Reynolds number are based on the use of the hydraulic diameter, *Dh* and the mean velocity of the flow, *u*. Furthermore, they were able

(51)

(52)

to link the geometric parameters *a* and *b* with the new parameter *β* as follows:

1 ; 1 1 *a b*

and finally, the factor of 8*n-*1 in the denominator in equation (49) is replaced (24/ *β)*

these modifications, one can use the relationship *f* =(16/*Re*g) to estimate the pressure gradient for the laminar flow of a power-law fluid in a non-circular duct for which the value of *β* is known either from experiments or from numerical results. Therefore, this approach necessitates the knowledge of only one parameter (*β)* as opposed to the two geometric parameters, namely, *a* and *b* in the method of Kozicki *et al* . (1966) and Kozicki and Tiu (1967), albeit a similar suggestion was also made by Miller (1972) and Liu (1983). Finally, for the limiting case of a circular pipe, evidently *β=*3 thereby leading to *a=*(1/4) and *b=*(3/4) and the two definitions of the Reynolds number coincide, as expected. The values of *β* for a few standard duct shapes are summarized in Table 5. While in laminar flow, these two methods give almost identical predictions, the applicability of the modified method of Delplace and Leuliet (1995) has not been checked in the transitional and turbulent flow regions. Scant

0.75 10 1.2

*g*

(*f*.*Re)* is a function of the conduit shape only. Thus, they wrote

(2 )/2 0.25

(50)

*n-*1. With

encountered in process applications.

of laminar flow in non-circular ducts.

can be calculated from the following equation


Table 6. Values of a and b depending on geometry of the ducts

Table 6. Values of a and b depending on geometry of the ducts

The main virtue of this approach lies in its simplicity and the fact that the geometric parameters *a* and *b* can be deduced from the behaviour of Newtonian fluids in the same flow geometry. Table 7 lists values of *a* and *b* for a range of flow geometries commonly encountered in process applications.

Kozicki *et al.*(1966) argued that the friction factor of the turbulent flow in non-circular ducts can be calculated from the following equation

$$\frac{1}{\sqrt{f}} = \frac{4}{n^{0.75}} \log\_{10}(\text{Re}\_g f^{(2-n)/2}) - \frac{0.4}{n^{1.2}} + 4n^{0.25} \log[\frac{a(a+bn)}{3n+1}] \tag{50}$$

Note that since for a circular tube, *a=* 1/4 and *b* =3/4, equation (50) is consistent with that for circular pipes. The limited data available on turbulent flow in triangular (Irvine Jr, 1988), rectangular (Kostic and Hartnett, 1984) and square ducts (Escudier and Smith, 2001) conforms to equation (48). In the absence of any definite information, Kozicki and Tiu (1988) suggested that the Dodge–Metzner criterion, *Re*<sup>g</sup> 2100, can be used for predicting the limit of laminar flow in non-circular ducts.

Some further attempts have been made to simplify and/or improve upon the two geometric parameter method of Kozicki *et al* . (1966, 1967). Delplace and Leuliet (1995) revisited the definition of the generalized Reynolds number (equation (49)) and argued that while the use of *a* and *b* accounts for the non-circular cross-sections of the ducts, but the factor 8*<sup>n</sup>*-1 appearing in the denominator is strictly applicable for the flow in circular ducts only. Their reasoning hinges on the fact that for the laminar flow of a Newtonian fluid, the product (*f*.*Re)* is a function of the conduit shape only. Thus, they wrote

$$
\beta = \frac{48}{\left(f.Re\right)}\tag{51}
$$

where both the (Fanning) friction factor and the Reynolds number are based on the use of the hydraulic diameter, *Dh* and the mean velocity of the flow, *u*. Furthermore, they were able to link the geometric parameters *a* and *b* with the new parameter *β* as follows:

$$a = \frac{1}{1+\beta}; \quad b = \frac{\beta}{1+\beta} \tag{52}$$

and finally, the factor of 8*n-*1 in the denominator in equation (49) is replaced (24/ *β) n-*1. With these modifications, one can use the relationship *f* =(16/*Re*g) to estimate the pressure gradient for the laminar flow of a power-law fluid in a non-circular duct for which the value of *β* is known either from experiments or from numerical results. Therefore, this approach necessitates the knowledge of only one parameter (*β)* as opposed to the two geometric parameters, namely, *a* and *b* in the method of Kozicki *et al* . (1966) and Kozicki and Tiu (1967), albeit a similar suggestion was also made by Miller (1972) and Liu (1983). Finally, for the limiting case of a circular pipe, evidently *β=*3 thereby leading to *a=*(1/4) and *b=*(3/4) and the two definitions of the Reynolds number coincide, as expected. The values of *β* for a few standard duct shapes are summarized in Table 5. While in laminar flow, these two methods give almost identical predictions, the applicability of the modified method of Delplace and Leuliet (1995) has not been checked in the transitional and turbulent flow regions. Scant

Incompressible Non-Newtonian Fluid Flows 71

<sup>2</sup> <sup>2</sup> 919 /

*h <sup>P</sup> f u Pa m*

Astarita, G., The Engineering Reality of the Yield Stress, *J. Rheol.* 34 (1990) 275 Barnes , H.A. and Walters , K. , The yield stress myth, *Rheol. Acta* 24 (1985) 323 .

to dense suspension rheology, *Chem. Eng. Sci.* 42 (1987) 591 .

Barnes , H.A., The yield stress - a review, *J. Non-Newt. Fluid Mech.* 81 (1999) 133. Bird , R.B. , Useful non-Newtonian models, *Annu. Rev. Fluid Mech.* 8 (1976) 13.

dynamics, 2nd edition, Wiley , New York (1987) .

Evans , I. D. , On the nature of the yield stress, *J. Rheol.* 36 (1992) 1313 .

rectangular duct, *Adv. Heat Transf.* 19 (1989) 247.

Barnes , H.A. , Edwards , M.F. and Woodcock , L.V. , Applications of computer simulations

Barnes , H.A. , Hutton , J.F. and Walters , K. , An Introduction to Rheology , *Elsevier*,

Bird , R.B. , Dai , G.C. and Yarusso , B.J. , The rheology and flow of viscoplastic materials,

Bird , R.B. , Armstrong , R.C. and Hassager , O. , Dynamics of polymeric liquids. Fluid

Cross , M.M. , Rheology of non-Newtonian fluids: A new flow equation for pseudoplastic

Carreau , P.J. , Rheological equations from molecular network theories, *Trans. Soc. Rheol.* 16

Carreau , P.J. , Dekee , D. and Chhabra , R.P., Rheology of polymeric systems: Principles and

Delplace , F. and Leuliet , J. C. , Generalized Reynolds number for the flow of power law

Dodge, D. W. and Metzner, A. B., Turbulent flow of non-Newtonian systems, *AIChE Journal.*

Escudier , M.P. and Smith , S. , Non-Newtonian liquids through a square duct, *Proc. Royal* 

Goddard , J.D. and Bashir , Y. , Recent developments in structured continua II (Chapter 2) ,

Griskey, R.G., Nechrebecki, D.G., Notheis , P.J. and Balmer , R.T., Rheological and pipeline

Hartnett , J.P. and Kostic , M. , Heat transfer to Newtonian and non- Newtonian fluids in

flow dehavior of corn starch dispersions, *J. Rheol.* 29(1985) 349

fluids in cylindrical ducts of arbitrary cross-section, Chem. Eng. Journal, 56, 33-37.,

*L D*


iv. elliptical, *b'/a'=* 0.5.


**9. References** 




Amsterdam (1989) .

(1972) 99.

applications , (1997)

5 (1959) 189-204.

*Soc.* 457 (2001) 911 .

*Chem. Eng. J.* 56 (1995) 33.

Longman , London (1990) .

*Rev. Chem. Eng.* 1 (1983) 1 .

systems*, J. Colloid Sci.* 20 (1965) 417.

*L* 

analytical and experimental results suggest that visco-elasticity in a fluid may induce secondary motion in non-circular conduits, even under laminar conditions. However, measurements reported to date indicate that the friction factor–Reynolds number behaviour is little influenced by such secondary flows (Hartnett and Kostic, 1989).

#### **Example 4**

A power-law fluid (*K* = 0.3 Pa.s*<sup>n</sup>* and *n* = 0.72) of density 1000 kg/m3 is flowing in a series of ducts of the same flow area but different cross-sections as listed below:


Estimate the pressure gradient required to maintain an average velocity of 1.25*m/s* in each of these channels. Use the geometric parameter method. Also, calculate the value of the generalized Reynolds number as a guide to the nature of the flow.

#### **Solution**


$$-\frac{\Delta P}{L} = \frac{2f\,\rho u^2}{D\_h} = 1963Pa/m$$


$$-\frac{\Delta P}{L} = \frac{2f\rho u^2}{D\_{\text{fr}}} = 689 Pa \,/\, m^2$$

iii. For a rectangular duct with *H* /*W* = 0.5, *H* =0.0425 m and *W=* 0.085 m (for the same area of flow), and from Table 6: *a*=0.244, *b=* 0.728

$$-\quad \mathbf{D}\_{\mathbf{h}} = 4\mathbf{H}\mathbf{W}/2(\mathbf{H} + \mathbf{W}) = 0.0567\mathbf{m}$$

$$\text{1. } \quad \text{Reynolds number, } \operatorname{Re}\_{\mathcal{S}} = \frac{\rho u^{2-n} D\_h^n}{8^{n-1} K (b + \frac{a}{n})^n} = 960^\circ$$

$$- \quad \text{f=1/960=0.0167 and } - \frac{\Delta P}{L} = \frac{2f \,\rho u^2}{D\_h} = 919 \, Pa/m^2$$

iv. elliptical, *b'/a'=* 0.5.


70 Continuum Mechanics – Progress in Fundamentals and Engineering Applications

analytical and experimental results suggest that visco-elasticity in a fluid may induce secondary motion in non-circular conduits, even under laminar conditions. However, measurements reported to date indicate that the friction factor–Reynolds number behaviour

A power-law fluid (*K* = 0.3 Pa.s*<sup>n</sup>* and *n* = 0.72) of density 1000 kg/m3 is flowing in a series of

Estimate the pressure gradient required to maintain an average velocity of 1.25*m/s* in each of these channels. Use the geometric parameter method. Also, calculate the value of the

is little influenced by such secondary flows (Hartnett and Kostic, 1989).

ducts of the same flow area but different cross-sections as listed below:

generalized Reynolds number as a guide to the nature of the flow.

2

*n n h*

*L D*

*n n h*


*h*


*L D*

*h <sup>P</sup> f u Pa m*

iii. For a rectangular duct with *H* /*W* = 0.5, *H* =0.0425 m and *W=* 0.085 m (for the same area

<sup>2</sup> <sup>2</sup> 1963 /

<sup>2</sup> <sup>2</sup> 689 /

*<sup>P</sup> f u Pa m*

1 Re 579 8( )

 

*<sup>g</sup> n n u D <sup>a</sup> K b n*

2

2

*n n h*

1 Re 960 8( )

 

*<sup>g</sup> n n u D <sup>a</sup> K b n*

1 Re 1070 8( )

 

*<sup>g</sup> n n u D <sup>a</sup> K b n*


of flow), and from Table 6: *a*=0.244, *b=* 0.728

i. concentric annulus with *R=* 37mm and *σ=(R/Ri)=* 0.40

**Example 4** 

**Solution** 

ii. circular pipe of radius *R* iii. rectangular, (*H* /*W)* =0.5 iv. elliptical, *b'/a'=* 0.5

i. For a concentric annulus, *σ* = 0.4

ii. For a circular tube, the area of flow






$$- \qquad f = 0.0168 \text{ and } -\frac{\Delta P}{L} = 864 \, Pa \, / \, m$$

#### **9. References**

Astarita, G., The Engineering Reality of the Yield Stress, *J. Rheol.* 34 (1990) 275


**4** 

*USA* 

**Microstructures** 

Sushrut Vaidya and Jeong-Ho Kim**\***

**Continuum Mechanics of Solid Oxide Fuel Cells** 

A solid oxide fuel cell (SOFC) is a device that converts the chemical energy of fuels into electrical energy (Singhal & Kendall, 2003). SOFCs have received much attention from researchers due to their promise of delivering relatively clean energy at high efficiencies (Singhal & Kendall, 2003). An SOFC consists of a few basic parts: an anode, a cathode, an electrolyte, and interconnect wires (Singhal & Kendall, 2003). The electrolyte in an SOFC is a solid oxide such as Yttria-Stabilized Zirconia (YSZ). The porous anode is usually a ceramicmetal composite (so called cermet) such as the nickel-zirconia cermet (Ni-YSZ). The porous cathode is usually a composite of strontium-doped lanthanum manganite (LSM) and Yttria-Stabilized Zirconia (LSM-YSZ) (Singhal & Kendall, 2003) or a composite such as gadolinium-doped ceria-lanthanum strontium cobalt ferrite (GDC-LSCF) (Anandakumar et al., 2010). Oxygen atoms undergo reduction on the porous cathode surface, and the resulting oxide ions are transported through the electrolyte to the porous anode. Here, the oxide ions react with the fuel (such as hydrogen). Hydrogen is oxidized, and the electrons of the oxide ions are liberated. The free electrons give rise to electric current (Singhal & Kendall, 2003).

Research on SOFCs has concentrated on many different aspects, including anode, cathode, and electrolyte materials; investigating the behavior of different SOFC configurations; modeling and simulating electrochemical, thermal, and flow phenomena; and performing thermal stress and probability of failure analyses. Researchers have employed experimental, analytical, and computational approaches in their investigations. For example, Selcuk and Atkinson (1997, 2000) conducted a number of experimental studies to estimate various mechanical properties of SOFC ceramic materials such as YSZ and NiO-YSZ. They determined the biaxial flexural strength and fracture toughness of YSZ both at room temperature and an operating temperature of 900⁰C (Selcuk & Atkinson, 2000). They also experimentally studied the dependence of the Young's modulus, shear modulus, and Poisson's ratio of YSZ and NiO-YSZ (amongst other ceramic materials) on porosity (Selcuk & Atkinson, 1997). The results of these studies were summarized by the authors (Atkinson & Selcuk, 2000) where they also suggested techniques for improving the mechanical

**1. Introduction** 

 \*

Corresponding Author

**Using Three-Dimensional Reconstructed** 

*Department of Civil and Environmental Engineering, University of Connecticut,* 

