**Mechanics of Multi-Phase Frictional Visco-Plastic, Non-Newtonian, Depositing Fluid Flow in Pipes, Disks and Channels**

Habib Alehossein

*CSIRO Earth Science and Resource Engineering, University of Queensland Australia* 

#### **1. Introduction**

150 Fluid Dynamics, Computational Modeling and Applications

Vincenti W., and Krouger C., Jr. (1967). *Introduction to physical Gas Dynamics*, Wiley, New

Wagner H.P., Kaeppeler H.J. and Auweter-Kurts M. (1998). *Instabilities in MPD thruster* 

Yalim, J., (2001). *Implementation of An Upwind FVM Solver for Ideal MHD for Space Weather* 

*flows: Investigation of drift and gradient driven instabilities using multi-fluid plasma* 

York, pp 21, 197-244.

*models*, Journal of Physics 31 559-541.

*Applications*, Ph.D. thesis, University of Washington.

*Background***:** Fresh concrete, fly ash and mining slurries are all frictional-visco-plastic fluids. Fresh concrete flow in Tremie pipes is used to control concrete flow rate and minimise bleeding and dilution when concrete is poured into deep submerged excavations for pile foundation construction. Slurries with very fine aggregates are used to backfill underground voids and mines to prevent subsidence and surface structural damage. Backfilling and injection of granular materials into mining induced voids, separated beddings and cracks, as either diluted slurry or concrete paste, is widely used to control subsidence. As a viable environmental solution, mine waste and rejected materials from underground coal seams are used in both backfilling and injection mine operations. For example, during longwall mining the grout slurry is pumped into the separated beds of the fractured rock mass through a pipeline connected to a central vertical borehole, which is drilled deep into the inter-burden rock strata above the coal mine seam. Either as dilute slurry or thick paste or cake, the fill material normally needs to travel a significant longitudinal distance either in a channel, a tremie pipe, a long pipeline, or radially in a disk shaped crack in the rock mass. An undesirable blockage can occur in the central borehole, in the crack or in the transportation channel or pipeline system when the slurry velocity falls below a certain critical threshold velocity, indicating a material phase change from cohesive-viscous to cohesive-frictional. This chapter presents complete analytical solutions of the required pump pressure versus fluid volume rate for such multi-phase fluids, which are categorised as frictional Bingham-Herschel-Bulkley fluids. The theory derived can be applied to flow of such fluids in pipes, disks and open channels. Furthermore, general analytical solutions have been developed for such fluids in terms of velocity and pressure gradients and velocity and pressure, as a function of flow length (e.g. pipe length, disk radial distance, or channel length) from which special and familiar equations for simpler fluids are derivable by mathematical reduction of the general equations. The formulation is distinct in considering many new aspects including: variable shear parameters rather than fixed values; inclusion of total nonlinear behaviour; and, implementation of a friction function to mimic behaviour of the depositing and consolidating stiff slurry or paste, which can cause a significant pressure rise as a result of the increased shear resistance.

*Bingham-Herschel-Bulkley fluids*: Recent laboratory and field experiments on mine-backfill fluids, slurries, cements, pastes and concretes proved their wide range of shear resistance

Mechanics of Multi-Phase Frictional Visco-Plastic,

past for their specific problems of interest [3-4].

u(t,x) Concrete

void backfilling.

h Concrete

y2 y1

L0 L <sup>w</sup> y x

Non-Newtonian, Depositing Fluid Flow in Pipes, Disks and Channels 153

slurry solid particles must deposit in the opened strata bed separation gaps or cracks before crack closure [10-13]. The mechanics of non-Newtonian fluids flowing between parallel disks is a classical fluid mechanics problem that has been studied by a number of researchers in the

D

Ho/C He

Hf

s)

(a) Concrete flow testing (b) Concrete tremie pipe flow (c) Slurry flow in pipe and strata

Hb

Hs

Slurry casing

Slurry level

*Concrete*: Fresh concrete flow through Tremie pipes is used to control concrete flow rate and minimise segregation, bleeding and dilution when poured or placed into deep submerged excavations for pile foundation construction. Slurries with very fine aggregates are used to backfill underground voids and mines to prevent subsidence and surface structural damage. Backfilling and injection of granular materials into the mining induced voids, separated beddings and cracks, as either diluted slurry or concrete paste, is widely used to control coal mine subsidence. As a viable environmental solution, mine waste and rejected materials from underground coal seams in the form of either cementitious or non-cementatious grout, are used in both backfilling and injection mine operations. For example, during longwall mining the grout slurry is pumped into the separated beds of the rock mass through a central vertical borehole, which is drilled deep into the inter-burden rock strata above the coal seam. Either as dilute slurry or thick paste or cake, the fill material normally needs to travel a significant distance in either a long pipeline, or radially in a disk type crack formation of the rock mass. An undesirable blockage can occur in the central borehole, in the disk gap or in the transportation channel or pipeline system, when the slurry velocity falls below a certain critical threshold velocity, indicating a material phase change from cohesiveviscous to cohesive-frictional. Indirect index measure of concrete viscosity and plastic yield is made via an L-box channel measuring workability and flowability of tremie pipe concrete. The L-box test, originally developed for super-workable concrete [6-9] is a relatively newly introduced concrete test to measure the consistency, workability and flowability of a tremie pipe concrete, hence, it is indirectly related to concrete plastic yield and viscosity [6]. Concrete is poured in the rectangular vertical chimney part of the L-box and is allowed to flow in the horizontal channel part, once a sliding gate is opened. The time and profile of the concrete flow in the horizontal channel is measured to compare viscous-plastic behaviour of different concretes. The variation of flow velocity with time and position from the time the sliding gate is opened until the flow reaches static equilibrium has been simulated and formulated by a representative dimensionless partial differential equation (PDE). Mathematically, the resulting

Fig. 1. Various applications of viscous slurry and paste fluids: (a) channel flow for workability and consistency testing of concrete; (b) Concrete termie pipe flow into submerged foundations; (c) multi-phase slurry flow in pipes and fractured rock strata for

c = Cw

Tremie pipe

Pressure balance point o

Lt

equation is of the same form as a non-homogeneous heat-conduction equation [6].

**Extracted underground coal seam** 

Bed separation gap Injection borehole

Total surface ground subsidence

Pump

Plant Pipeline

Batch

Water level

w)

Hw

and complex behaviour in response to shearing necessitating development of a general, nonlinear, cohesive, viscous, frictional, nonlinear, non-Newtonian model of shear stress versus shear strain rate, as an extension to the classical Bingham-Herschel-Bulkley fluid [1- 9]. Viscous plastic behaviour of such fluids are further simplified or idealised as a reduced or special case of the general nonlinear case. In practice, and for various engineering applications, this generic shear stress function is central in all mathematical formulations to describe fluid flow as a function of pressure gradient. Examples of such applications are flow of slurry, paste and concrete through pipes and tremie pipes and channels for fluid transportation and testing purposes, flow through disks, cracks, joints and rock fractures for injection and backfilling purposes. As a first approach, the shear stress vs. strain rate relation may be idealised by a simple linear function, the so called viscous-plastic Bingham line, which may be derived from a simple linear regression analysis of laboratory experimental data [4-5]. The value of the shear strength function at zero shear strain rate, i.e. plastic yield strength (also called cohesion), and slope of the linear curve (viscosity) are two important parameters of the fluid property in the simple linear idealised case.

*Grouts and slurries*: The large cavern created by an underground mine may eventually lead to failure of the overburden rock, propagating layer by layer to the surface, resulting in substantial ground surface subsidence [10-13], as schematically shown in Figure 1(c). As a major potential hazard, mining induced subsidence significantly affects mining costs where major surface structures and natural environment need to be protected, e.g. mining under river systems, gorges, cliffs, power lines, pipelines, communication cables, major roads and bridges, and other significant surface facilities [11]. Remedial measures to manage damage caused by subsidence can often be very costly with potentially damaging impacts and irreversible consequences. Backfilling and injection of granular materials into the mining induced voids, separated beddings and cracks, as either diluted slurry or concrete paste, is widely used to control mine subsidence. Grouts and slurries made of mine and power plant wastes and rejects are viable environmental backfill solutions to both ground stability and mine waste management problems [12]. Like concrete paste, the flowing slurry can be categorised as a generally nonlinear viscous cohesive (Bingham Herschel-Bulkley) fluid [5-11]. However, in mining applications, to reduce ground surface subsidence and control the propagation of the overburden movement to the surface, the solid particles in the injected slurry must deposit in the bed separation gaps of the coal seam over-burden strata, e.g. in longwall mining the grout slurry is pumped into the separated beds of the rock mass from a batching plant source through pipelines connected to a central vertical borehole, which is drilled deep into the over-burden rock above the coal seam (Figure 1(c)). Flow blockage can occur in the injection system, when the slurry velocity falls below a certain critical threshold velocity. The stiffening, consolidating non-flow slurry can generally be categorised as a frictional cohesive soil [14]. In other words, a change of material phase from cohesive-viscous to cohesive-frictional will occur. Using a smaller scale model, this field injection practice has been simulated at the QCAT laboratory of the Commonwealth Scientific & Industrial Research Organisation (CSIRO) in Brisbane, Australia, to study the influence of various grout injection parameters by pumping slurries through various pipes of different sizes and diameters and for different applications (Figures 1-2). As an important industrial application, grout injection into the inter-burden strata is used as a modern technology to control and reduce coal mine subsidence [10-13]. Slurry mixes of coal mine and power plant waste materials, e.g. fly ash or any other coal wash rejects, are injected back into the inter-burden rock strata during longwall mining [4-5]. To reduce subsidence and control inter-burden strata movement, the injected

and complex behaviour in response to shearing necessitating development of a general, nonlinear, cohesive, viscous, frictional, nonlinear, non-Newtonian model of shear stress versus shear strain rate, as an extension to the classical Bingham-Herschel-Bulkley fluid [1- 9]. Viscous plastic behaviour of such fluids are further simplified or idealised as a reduced or special case of the general nonlinear case. In practice, and for various engineering applications, this generic shear stress function is central in all mathematical formulations to describe fluid flow as a function of pressure gradient. Examples of such applications are flow of slurry, paste and concrete through pipes and tremie pipes and channels for fluid transportation and testing purposes, flow through disks, cracks, joints and rock fractures for injection and backfilling purposes. As a first approach, the shear stress vs. strain rate relation may be idealised by a simple linear function, the so called viscous-plastic Bingham line, which may be derived from a simple linear regression analysis of laboratory experimental data [4-5]. The value of the shear strength function at zero shear strain rate, i.e. plastic yield strength (also called cohesion), and slope of the linear curve (viscosity) are two

important parameters of the fluid property in the simple linear idealised case.

*Grouts and slurries*: The large cavern created by an underground mine may eventually lead to failure of the overburden rock, propagating layer by layer to the surface, resulting in substantial ground surface subsidence [10-13], as schematically shown in Figure 1(c). As a major potential hazard, mining induced subsidence significantly affects mining costs where major surface structures and natural environment need to be protected, e.g. mining under river systems, gorges, cliffs, power lines, pipelines, communication cables, major roads and bridges, and other significant surface facilities [11]. Remedial measures to manage damage caused by subsidence can often be very costly with potentially damaging impacts and irreversible consequences. Backfilling and injection of granular materials into the mining induced voids, separated beddings and cracks, as either diluted slurry or concrete paste, is widely used to control mine subsidence. Grouts and slurries made of mine and power plant wastes and rejects are viable environmental backfill solutions to both ground stability and mine waste management problems [12]. Like concrete paste, the flowing slurry can be categorised as a generally nonlinear viscous cohesive (Bingham Herschel-Bulkley) fluid [5-11]. However, in mining applications, to reduce ground surface subsidence and control the propagation of the overburden movement to the surface, the solid particles in the injected slurry must deposit in the bed separation gaps of the coal seam over-burden strata, e.g. in longwall mining the grout slurry is pumped into the separated beds of the rock mass from a batching plant source through pipelines connected to a central vertical borehole, which is drilled deep into the over-burden rock above the coal seam (Figure 1(c)). Flow blockage can occur in the injection system, when the slurry velocity falls below a certain critical threshold velocity. The stiffening, consolidating non-flow slurry can generally be categorised as a frictional cohesive soil [14]. In other words, a change of material phase from cohesive-viscous to cohesive-frictional will occur. Using a smaller scale model, this field injection practice has been simulated at the QCAT laboratory of the Commonwealth Scientific & Industrial Research Organisation (CSIRO) in Brisbane, Australia, to study the influence of various grout injection parameters by pumping slurries through various pipes of different sizes and diameters and for different applications (Figures 1-2). As an important industrial application, grout injection into the inter-burden strata is used as a modern technology to control and reduce coal mine subsidence [10-13]. Slurry mixes of coal mine and power plant waste materials, e.g. fly ash or any other coal wash rejects, are injected back into the inter-burden rock strata during longwall mining [4-5]. To reduce subsidence and control inter-burden strata movement, the injected slurry solid particles must deposit in the opened strata bed separation gaps or cracks before crack closure [10-13]. The mechanics of non-Newtonian fluids flowing between parallel disks is a classical fluid mechanics problem that has been studied by a number of researchers in the past for their specific problems of interest [3-4].

(a) Concrete flow testing (b) Concrete tremie pipe flow (c) Slurry flow in pipe and strata

Fig. 1. Various applications of viscous slurry and paste fluids: (a) channel flow for workability and consistency testing of concrete; (b) Concrete termie pipe flow into submerged foundations; (c) multi-phase slurry flow in pipes and fractured rock strata for void backfilling.

*Concrete*: Fresh concrete flow through Tremie pipes is used to control concrete flow rate and minimise segregation, bleeding and dilution when poured or placed into deep submerged excavations for pile foundation construction. Slurries with very fine aggregates are used to backfill underground voids and mines to prevent subsidence and surface structural damage. Backfilling and injection of granular materials into the mining induced voids, separated beddings and cracks, as either diluted slurry or concrete paste, is widely used to control coal mine subsidence. As a viable environmental solution, mine waste and rejected materials from underground coal seams in the form of either cementitious or non-cementatious grout, are used in both backfilling and injection mine operations. For example, during longwall mining the grout slurry is pumped into the separated beds of the rock mass through a central vertical borehole, which is drilled deep into the inter-burden rock strata above the coal seam. Either as dilute slurry or thick paste or cake, the fill material normally needs to travel a significant distance in either a long pipeline, or radially in a disk type crack formation of the rock mass. An undesirable blockage can occur in the central borehole, in the disk gap or in the transportation channel or pipeline system, when the slurry velocity falls below a certain critical threshold velocity, indicating a material phase change from cohesiveviscous to cohesive-frictional. Indirect index measure of concrete viscosity and plastic yield is made via an L-box channel measuring workability and flowability of tremie pipe concrete. The L-box test, originally developed for super-workable concrete [6-9] is a relatively newly introduced concrete test to measure the consistency, workability and flowability of a tremie pipe concrete, hence, it is indirectly related to concrete plastic yield and viscosity [6]. Concrete is poured in the rectangular vertical chimney part of the L-box and is allowed to flow in the horizontal channel part, once a sliding gate is opened. The time and profile of the concrete flow in the horizontal channel is measured to compare viscous-plastic behaviour of different concretes. The variation of flow velocity with time and position from the time the sliding gate is opened until the flow reaches static equilibrium has been simulated and formulated by a representative dimensionless partial differential equation (PDE). Mathematically, the resulting equation is of the same form as a non-homogeneous heat-conduction equation [6].

Mechanics of Multi-Phase Frictional Visco-Plastic,

C = 60 % = 0.0499 = 10.346

yield (Bingham intercept) is 1.6432 Pa.

0

10

20

30

**Shear stress (Pa)**

40

50

**3. Governing equations** 

In Equation (2), **u**

Non-Newtonian, Depositing Fluid Flow in Pipes, Disks and Channels 155

Weight solid concentration (%) = C Viscosity (Pa.s) = Plastic yield (Pa) =

Fig. 3. Shear stress vs shear rate (range 0-700/s) for a sample at solid weight concentration range 30%-60%. Numbers in legend table show Bingham plastic linear fit model to these results in the range of 0-100/s. For instance, for 50% solid concentration in the shear rate range of 0-100/s of the sample, the linear viscosity (Bingham slope) is 0.0106 Pa.s and the

C = 40 % = 0.005 = 0.1967

0 100 200 300 400 500 600 700

**Shear rate (1/s)**

We can measure these parameters by a viscometer-testing device [4-5]. Sometimes for slurries of various solid particle concentrations, the viscometer test results can conveniently fit into one or two linear models (Bingham plastic) for the whole range of shear strain rate. Figure 3 shows examples of multi-linear or bilinear model for slurries of different concentrations for two distinct range of shear strain rate 0-100 /s and 100-700 /s. The Bingham models can be

Governing equations of most fluid mechanics problems normally start with the general basic Reynolds transport theorem of continuum mechanics [15]. This is initially an integral relation stating that the sum of the changes of any intensive fluid property, such as mass, momentum and energy, defined over a control volume CV, denoted here by symbol , must be equal to what is gained or lost through the boundaries of the volume, or control surface (CS), plus what is created or consumed by sources and sinks inside the control volume [15].

> CV CS CV <sup>d</sup> dV . dA QdV 0 dt

Q is the fluid source or sink. Using integration by parts, the second integral can also be

 **u n**

(2)

C = 50 % = 0.0106 = 1.6432

> C = 30 % = 0.0051 = 0.0187

is normal vector to the control surface dA, and

identified by its two main parameters (yield intercept and viscosity slope.

is fluid velocity vector, **<sup>n</sup>**

In this chapter complete analytical solutions of the required pump pressure versus fluid volume rate are discussed for such multi-phase fluids, which are categorised as frictional Bingham-Herschel-Bulkley fluids. The discussed theory can be applied to flow of such fluids in pipes, disks and open channels. Furthermore, general analytical solutions have been developed for complex fluids in terms of velocity and pressure gradients and velocity and pressure, as a function of flow length (e.g. pipe length, disk radial distance, or channel length) from which special and familiar equations for simpler fluids are derivable by mathematical reduction of the general equations. The formulation is distinct in considering many new aspects including: variable shear parameters rather than fixed values; inclusion of total nonlinear behaviour; and, implementation of a friction function to mimic behaviour of the depositing and consolidating stiff slurry or paste, which can cause a significant pressure rise as a result of the increased shear resistance.

#### **2. Frictional Bingham-Herschel-Bulkley fluid**

The general constitutive equation, relating fluid shear stress to shear rate for such general nonlinear, non-Newtonian, viscous, plastic, frictional fluids, which can be applied to fresh concrete, mine backfill slurries and high frictional multiphase fluids, is as follows [4-9]

$$\mathbf{r}(\mathbf{t}, \mathbf{x}) = \mathbf{\hat{n}(t, \mathbf{x})} \left( -\frac{\partial \mathbf{u}(\mathbf{t}, \mathbf{x})}{\partial \hat{\mathbf{x}}} \right) + \mathbf{\hat{n}(t, \mathbf{x})} \left( -\frac{\partial \mathbf{u}(\mathbf{t}, \mathbf{x})}{\partial \hat{\mathbf{x}}} \right)^{n} + \mathbf{\tau}\_{\mathbf{0}}(\mathbf{t}, \mathbf{x}) + \xi(\mathbf{t}, \mathbf{x}) \mathbf{p}(\mathbf{t}, \mathbf{x}) \tag{1}$$

In Equation (1) is shear stress tensor, **u** is velocity vector, and are linear and nonlinear viscosities, **0** is plastic yield, p is pipe pressure and is concrete friction coefficient. The last term, involving the friction and pressure terms (p), is a frictional resistance term which can be applied only when a pipe blockage occurs due to the concrete granular material friction and needs to be reopened by a higher pressure flow, otherwise it can be ignored [4-9]**.** See Figure 2 for a visual definition of the different shear terms and parameters involved in Equation (1).

Fig. 2. Schematic diagrams showing various shear stress components in Equation (1). 0 is the constant uniform plastic yield component, with no viscosity; is the Newtonian linear viscosity coefficient of the linear velocity gradient y with a wall value yh.; is the non-linear viscosity; is the friction coefficient of the fluid pressure p.

In this chapter complete analytical solutions of the required pump pressure versus fluid volume rate are discussed for such multi-phase fluids, which are categorised as frictional Bingham-Herschel-Bulkley fluids. The discussed theory can be applied to flow of such fluids in pipes, disks and open channels. Furthermore, general analytical solutions have been developed for complex fluids in terms of velocity and pressure gradients and velocity and pressure, as a function of flow length (e.g. pipe length, disk radial distance, or channel length) from which special and familiar equations for simpler fluids are derivable by mathematical reduction of the general equations. The formulation is distinct in considering many new aspects including: variable shear parameters rather than fixed values; inclusion of total nonlinear behaviour; and, implementation of a friction function to mimic behaviour of the depositing and consolidating stiff slurry or paste, which can cause a significant

The general constitutive equation, relating fluid shear stress to shear rate for such general nonlinear, non-Newtonian, viscous, plastic, frictional fluids, which can be applied to fresh concrete, mine backfill slurries and high frictional multiphase fluids, is as follows [4-9]

<sup>n</sup> (t, ) (t, ) (t, ) <sup>μ</sup>(t, ) <sup>η</sup>(t, ) (t, ) <sup>ξ</sup>(t, )p(t, ) ˆ ˆ

In Equation (1) is shear stress tensor, **u** is velocity vector, and are linear and nonlinear viscosities, **0** is plastic yield, p is pipe pressure and is concrete friction coefficient. The last term, involving the friction and pressure terms (p), is a frictional resistance term which can be applied only when a pipe blockage occurs due to the concrete granular material friction and needs to be reopened by a higher pressure flow, otherwise it can be ignored [4-9]**.** See Figure 2 for a visual definition of the different shear

Fig. 2. Schematic diagrams showing various shear stress components in Equation (1). 0 is the constant uniform plastic yield component, with no viscosity; is the Newtonian linear viscosity coefficient of the linear velocity gradient y with a wall value yh.; is the non-linear

x

**<sup>τ</sup> xx x <sup>τ</sup> x xx x x** (1)

**p yh yn**

h

 **<sup>0</sup> u x u x**

pressure rise as a result of the increased shear resistance.

**2. Frictional Bingham-Herschel-Bulkley fluid** 

terms and parameters involved in Equation (1).

**Shear resistance components** 

**Pipe flow direction** 

**Q** 

viscosity; is the friction coefficient of the fluid pressure p.

Fig. 3. Shear stress vs shear rate (range 0-700/s) for a sample at solid weight concentration range 30%-60%. Numbers in legend table show Bingham plastic linear fit model to these results in the range of 0-100/s. For instance, for 50% solid concentration in the shear rate range of 0-100/s of the sample, the linear viscosity (Bingham slope) is 0.0106 Pa.s and the yield (Bingham intercept) is 1.6432 Pa.

We can measure these parameters by a viscometer-testing device [4-5]. Sometimes for slurries of various solid particle concentrations, the viscometer test results can conveniently fit into one or two linear models (Bingham plastic) for the whole range of shear strain rate. Figure 3 shows examples of multi-linear or bilinear model for slurries of different concentrations for two distinct range of shear strain rate 0-100 /s and 100-700 /s. The Bingham models can be identified by its two main parameters (yield intercept and viscosity slope.

#### **3. Governing equations**

Governing equations of most fluid mechanics problems normally start with the general basic Reynolds transport theorem of continuum mechanics [15]. This is initially an integral relation stating that the sum of the changes of any intensive fluid property, such as mass, momentum and energy, defined over a control volume CV, denoted here by symbol , must be equal to what is gained or lost through the boundaries of the volume, or control surface (CS), plus what is created or consumed by sources and sinks inside the control volume [15].

$$\frac{\text{d}}{\text{d}t} \int\_{\text{CV}} \varphi \text{dV} + \int\_{\text{CS}} \varphi \text{\vec{u}} \, \text{\vec{n}dA} + \int\_{\text{CV}} \text{QdV} = 0 \tag{2}$$

In Equation (2), **u** is fluid velocity vector, **<sup>n</sup>** is normal vector to the control surface dA, and Q is the fluid source or sink. Using integration by parts, the second integral can also be

Mechanics of Multi-Phase Frictional Visco-Plastic,

circumferential axis direction identical to hoop angle .

**4. Reduced one dimensional equations** 

**x2**

**u(x1, x2) <sup>h</sup>x1**

**2h x2**

**h** 

fluid flow parameters

Slurry x1 x1 x1 x1 n

**Q** 

with time. In other words, we have

and (b) gradient with respect to x2, viz.

Non-Newtonian, Depositing Fluid Flow in Pipes, Disks and Channels 157

We now introduce a new set of independent variables to represent coordinate axes of both radial disk and pipe flow systems, namely: x1 axis is always in the direction of the main flow direction, i.e. either radial disk flow, or longitudinal pipe or channel flow, x2 is the axis normal to the flow direction in the flow cross-section plane, and x3 is the hoop or

Consider now the simple problem of fluid flow through either (i) a uniform circular pipe of inside diameter 2h, as shown in Figure 4 (left), or (ii) a radial disk of thickness 2h, as in Figure 4 (right), or (iii) a channel, as part of an L-Box testing device shown in Figure 1 (a).

Fig. 4. Non-Newtonian viscous-plastic flow in a pipe (left) and in a radial disk (right) with

u1

Q

x1

**x1**

Fluid flow through a pipe of uniform, circular, cross-section is known as the Hagen– Poiseuille flow problem [5]. It is assumed that the circular pipe flow is symmetric around the pipe longitudinal x-axis, the normal stresses are simply the fluid pressure, p, the fluid is incompressible and non-Newtonian in a steady state condition, there is no velocity component in the pipe circular cross-sectional plane, i.e. the plane normal to the pipe length direction. Similarly, it is assumed that the flow in a disk is also non-Newtonian, steady state, incompressible and laminar and the radial disk flow is also cylindrically axi-symmetric [4]. Hence, implementing all these assumptions implies that in all flow cases both normal to flow velocity components (u2, u3) are zero and there is no variation in velocity or pressure

We now define two separate flow gradient functions [4-5], (a) gradient with respect to x1

y

1 2 u

x

u u u(x ,x ) 1 12 (8)

Q, xxxxn

x1

*y*

h

2h h x1 u1(x1, x2) x2

*r*

uuu0 2 3 <sup>θ</sup> (9)

(11a)

u/ t u/ i i i2 2 θ p/ t p/ θ u / x p/ x 0 (10)

transformed to a volume integral by the divergence theorem. Since the whole grouped volume integral must be zero for any arbitrary control volume CV, it implies that the integrand itself must be zero. Therefore, our theory can be started with the following general basic differential equation:

$$\frac{d\mathbf{c}}{dt} + \vec{\nabla} \cdot \mathbf{c} \vec{\mathbf{u}} + \mathbf{Q} = \mathbf{0} \tag{3}$$

Applying the general conservation Equation (3) to mass and momentum, the Navier- Stokes isothermal equations of continuity and momentum [15] are recovered:

$$
\vec{\nabla}.\mathfrak{p}\,\vec{\mathfrak{u}} + \dot{\mathfrak{p}} = 0\tag{4}
$$

$$
\vec{\nabla} \cdot \vec{\mathbf{\bar{o}}} + \rho \vec{\mathbf{g}} = \rho \frac{\mathbf{D} \vec{\mathbf{u}}}{\mathbf{D}t} \tag{5}
$$

where x**i i / <sup>e</sup>** is the gradient vector, is the fluid density, **<sup>σ</sup>** is the stress tensor and **<sup>g</sup>** is the body acceleration or gravity vector. The stress tensor depends on a mean fluid normal stress or pressure ( <sup>1</sup> p δ σ jj ii ) and a deviator stress representing shear stresses ij, which depends on fluid viscosity and velocity gradients.

$$\mathbf{r}\mathbf{o}\_{\text{ij}} = -\mathbf{p}\mathbf{\hat{o}}\_{\text{ij}} + \mathbf{r}\_{\text{ij}} \tag{6}$$

As shown by [3-4], the deviatoric shear stress in (6) for cohesive, frictional, viscous, non-Newtonian slurries depends not only on fluid velocity gradients and yield plastic shear strength, but also on fluid pressure causing frictional resistance to flow, particularly during a blockage (Q →0). As discussed earlier in Equation (1), on the basis of several laboratory experiments on soil like slurries, a general shear stress versus shear strain constitutive material law is proposed for viscous, cohesive, frictional, plastic slurries in which the fluid shear stress is a nonlinear function of shear rate and longitudinal distance. Equation (1) has the following general form when written tensor notation is applied:

$$\mathbf{r}\_{\text{ij}} = \mathbf{\mu} \left( \mathbf{u}\_{i,\text{j}} + \mathbf{u}\_{\text{j},\text{i}} \right) + \mathbf{\eta} \left( \mathbf{u}\_{i,\text{j}} + \mathbf{u}\_{\text{j},\text{i}} \right)^{\text{n}} + \mathbf{r}\_{0\text{ij}} + \xi \mathbf{p} \delta\_{\text{ij}} \tag{7}$$

As discussed earlier and also shown in Figure 2, the first term on the right hand side is the familiar linear Newtonian component, the second term is the nonlinear pseudo-plastic component, the third term is the yield component and the forth term is the pressure component, in which is a coefficient of granular material friction which is the same tangent function of the material friction angle [14]. In the theoretical analysis discussed here, it is assumed that:


We now introduce a new set of independent variables to represent coordinate axes of both radial disk and pipe flow systems, namely: x1 axis is always in the direction of the main flow direction, i.e. either radial disk flow, or longitudinal pipe or channel flow, x2 is the axis normal to the flow direction in the flow cross-section plane, and x3 is the hoop or circumferential axis direction identical to hoop angle .

#### **4. Reduced one dimensional equations**

156 Fluid Dynamics, Computational Modeling and Applications

transformed to a volume integral by the divergence theorem. Since the whole grouped volume integral must be zero for any arbitrary control volume CV, it implies that the integrand itself must be zero. Therefore, our theory can be started with the following

<sup>d</sup><sup>ς</sup> .<sup>ς</sup> Q 0 dt

Applying the general conservation Equation (3) to mass and momentum, the Navier- Stokes

<sup>D</sup> . ρ ρ Dt **<sup>u</sup> σ g**

is the body acceleration or gravity vector. The stress tensor depends on a mean fluid normal

As shown by [3-4], the deviatoric shear stress in (6) for cohesive, frictional, viscous, non-Newtonian slurries depends not only on fluid velocity gradients and yield plastic shear strength, but also on fluid pressure causing frictional resistance to flow, particularly during a blockage (Q →0). As discussed earlier in Equation (1), on the basis of several laboratory experiments on soil like slurries, a general shear stress versus shear strain constitutive material law is proposed for viscous, cohesive, frictional, plastic slurries in which the fluid shear stress is a nonlinear function of shear rate and longitudinal distance. Equation (1) has

As discussed earlier and also shown in Figure 2, the first term on the right hand side is the familiar linear Newtonian component, the second term is the nonlinear pseudo-plastic component, the third term is the yield component and the forth term is the pressure component, in which is a coefficient of granular material friction which is the same tangent function of the material friction angle [14]. In the theoretical analysis discussed here, it is

1. the flow is laminar and the fluid is incompressible, steady state, stationary, and

2. axially symmetric condition implies that the radial flow component (in a pipe) and

4. the classical term "fluid" has loosely been used interchangeably with "slurry"; to refer to a "slurry", whenever an equivalent "fluid" model can represent the overall, average,

isothermal and axisymmetric with no eddies and no gravity effects;

circumferential flow (in a disk) must vanish. In other words ux2 = ux3 = 0; 3. when there is a full blockage (Q = 0), the friction term p is a dominant term;

isothermal equations of continuity and momentum [15] are recovered:

where x**i i / <sup>e</sup>** is the gradient vector, is the fluid density, **<sup>σ</sup>**

the following general form when written tensor notation is applied:

depends on fluid viscosity and velocity gradients.

mechanical behaviour of the "slurry".

**<sup>u</sup>** (3)

.ρ ρ **<sup>u</sup>** <sup>0</sup> (4)

(5)

σij pδ τ ij ij (6)

) and a deviator stress representing shear stresses ij, which

n ij i,j j,i i,j j,i 0ij ij τ μ u u η u u τ ξpδ (7)

is the stress tensor and **<sup>g</sup>**

general basic differential equation:

stress or pressure ( <sup>1</sup> p δ σ jj ii

assumed that:

Consider now the simple problem of fluid flow through either (i) a uniform circular pipe of inside diameter 2h, as shown in Figure 4 (left), or (ii) a radial disk of thickness 2h, as in Figure 4 (right), or (iii) a channel, as part of an L-Box testing device shown in Figure 1 (a).

Fig. 4. Non-Newtonian viscous-plastic flow in a pipe (left) and in a radial disk (right) with fluid flow parameters

Fluid flow through a pipe of uniform, circular, cross-section is known as the Hagen– Poiseuille flow problem [5]. It is assumed that the circular pipe flow is symmetric around the pipe longitudinal x-axis, the normal stresses are simply the fluid pressure, p, the fluid is incompressible and non-Newtonian in a steady state condition, there is no velocity component in the pipe circular cross-sectional plane, i.e. the plane normal to the pipe length direction. Similarly, it is assumed that the flow in a disk is also non-Newtonian, steady state, incompressible and laminar and the radial disk flow is also cylindrically axi-symmetric [4]. Hence, implementing all these assumptions implies that in all flow cases both normal to flow velocity components (u2, u3) are zero and there is no variation in velocity or pressure with time. In other words, we have

$$\mathbf{u} = \mathbf{u}\_1 = \mathbf{u}(\mathbf{x}\_1 \mathbf{x}\_2) \tag{8}$$

$$\mathbf{u}\_2 = \mathbf{u}\_3 = \mathbf{u}\_\theta = \mathbf{0} \tag{9}$$

$$
\partial \mathbf{\hat{u}}\_i / \partial \mathbf{t} = \partial \mathbf{u}\_i / \partial \mathbf{\hat{o}} = \partial \mathbf{p} / \partial \mathbf{t} = \partial \mathbf{p} / \partial \mathbf{\hat{o}} = \partial \mathbf{u}\_i / \partial \mathbf{x}\_2 = \partial \mathbf{p} / \partial \mathbf{x}\_2 = 0 \tag{10}
$$

We now define two separate flow gradient functions [4-5], (a) gradient with respect to x1 and (b) gradient with respect to x2, viz.

$$\mathbf{y} = -\frac{\partial \mathbf{u}\_1}{\partial \mathbf{x}\_2} \tag{11a}$$

Mechanics of Multi-Phase Frictional Visco-Plastic,

PF: <sup>n</sup>

1

PF: h 2 <sup>2</sup> x h

terms of the integral coefficients A1, B1, C1 [5].

coefficients similar to those produced for radial flow [4], viz.

<sup>u</sup> y y(x h)

words,

PF:

RF:

friction x

Non-Newtonian, Depositing Fluid Flow in Pipes, Disks and Channels 159

The pressure gradient Equations (16) and (17) must be satisfied at all points, including the

h dp (x ) τ μ(x )y <sup>η</sup>(x )y <sup>τ</sup> (x ) <sup>ξ</sup>(x )p(x ) F(y ) 2 dx

dp 1 <sup>1</sup> <sup>h</sup> <sup>μ</sup>(x ) ψ η(x ) <sup>ψ</sup> f(<sup>ψ</sup> /x ) f(y ) dx x <sup>x</sup> *<sup>n</sup>* 

where yh, h and h are the boundary values of y, and , i.e. at the point x2 = h. In other

If yh in (18), or h in (19), are known, the pipe pressure p can be calculated by integrating these equations directly. The result can still be in integral forms depending on the complexity of the coefficient functions such as: viscosity x or x, plasticity, x or

1

x

2 <sup>p</sup> <sup>τ</sup> (x )dx <sup>h</sup> *<sup>x</sup>*

0

1

x

0

For example, the pipe pressure can be written in a general integral form (Equation (22)) in

n 1 p pv Ay By C v 00 1h 1h 1

In Equation (22), v(x1) is an exponential function of x1 and h and A1, B1, C1 are integral

0

0

x

x

x

x

1

1

<sup>2</sup> <sup>ξ</sup>dx

x <sup>1</sup> p f(<sup>ψ</sup> /x )dx <sup>h</sup>

h1 1

h1 1

1 h 1h 1h 01 1 1 h

(18)

, h 2 τ τ (x h) , ψ ψ h 2 (x h) (20)

(21a)

(21b)

(22)

v(x) e <sup>h</sup> (23)

<sup>2</sup> A v(x)μ(x)dx <sup>h</sup> (24a)

<sup>2</sup> B v(x)η(x)dx <sup>h</sup> (24b)

(19)

boundary point, h. Therefore, at the wall boundary point (x2=h) we have

RF: 1 h 1 h h1 h 11 1

2

x

$$
\Psi = -\chi\_1 \mathbf{u}\_1 \tag{11b}
$$

$$\mathbf{u}\boldsymbol{\Psi} = -\mathbf{x}\_1 \frac{\partial \mathbf{u}\_1}{\partial \mathbf{x}\_2} = \mathbf{x}\_1 \mathbf{y} = \boldsymbol{\Psi}' \tag{11c}$$

Hence, the general, basic equations of continuity (4) and momentum (5) reduce to Equations (12) for pipe flow and (13a,b) for radial flow.

$$\text{PF (Pipe flow):}\tag{1} \\ \begin{aligned} \frac{1}{\mathbf{x}\_{2}} \frac{\partial \{\mathbf{x}\_{2} \mathbf{r}\}}{\partial \mathbf{x}\_{2}} = \frac{\partial \mathbf{p}}{\partial \mathbf{x}\_{1}} \end{aligned} \tag{12}$$

$$\text{RF (Radial Flow):}\\\text{i.e., }\begin{aligned} -\frac{1}{\mathbf{x}\_1}\frac{\partial \Psi}{\partial \mathbf{x}\_1} + \frac{\partial \mathbf{u}\_2}{\partial \mathbf{x}\_2} = 0 \end{aligned} \tag{13a}$$

$$\frac{\partial \mathbf{\hat{p}}}{\partial \mathbf{x}\_1} + \frac{\partial \mathbf{\hat{r}}}{\partial \mathbf{x}\_2} = \mathbf{0} \tag{13b}$$

It should be noticed that gravity effect can be incorporated into the pressure gradient term in pipe flow Equation (12), where is the fluid unit weight and is the inclination angle of the pipe with respect to horizontal axis. In order to consider gravity in a pipe flow, <sup>p</sup> x in (12) should be replaced with <sup>p</sup> <sup>γ</sup>sin(β) 

x , as suggested in [2]. The shear stress 1 2 <sup>τ</sup>(x ,x ) in (12) and (13) is a 2D version of the general case shown in (7), which is reproduced in Equation (14) below [4-5].

$$\mathbf{r}(\mathbf{x}\_1, \mathbf{x}\_2) = \mathbf{\mu}(\mathbf{x}\_1)\mathbf{y} + \mathbf{\eta}(\mathbf{r})\mathbf{y}^n + \mathbf{\tau}\_1(\mathbf{x}\_1) + \xi(\mathbf{x}\_1)\mathbf{p}(\mathbf{x}\_1) \tag{14}$$

No slip boundary conditions, i.e. no velocity at the pipe or disk walls, and a full axial or radial symmetry of the flow are assumed. Hence, Equations (11-13) must be solved subject to the following boundary conditions:

$$\mathbf{u} = \mathbf{u}\_1 = \mathbf{u}(\mathbf{x}\_2 = \mathbf{h}) = \mathbf{0} \tag{15a}$$

$$\frac{\partial \mathbf{u}}{\partial \mathbf{x}\_2}(\mathbf{x}\_2 = \mathbf{0}) = \mathbf{0} \tag{15b}$$

Substituting (14) in (12) and (13) and integrating over x2, will give us the following pressuregradient equations

$$\text{PF:}\tag{1}\text{PF:}\tag{2}\text{ }\frac{\text{x}\_2}{\text{2}}\frac{\text{dp}}{\text{dx}\_1}(\text{x}\_1) = \text{\textquotedblleft}\mathfrak{r} = \mathfrak{p}(\text{x}\_1)\mathfrak{p}(\text{x}\_2) + \mathfrak{p}(\text{x}\_1)\mathfrak{p}^n(\text{x}\_2) + \mathfrak{r}\_0(\text{x}\_1) + \mathfrak{f}(\text{x}\_1)\mathfrak{p}(\text{x}\_1) \tag{16}$$

$$\text{RF:}\tag{1}\text{RF:}\tag{1}\tag{2}\text{ }\text{d}\mathbf{x}\_{1}\text{ }^{\text{d}}\mathbf{x}\_{1}=\mathfrak{p}(\mathbf{x}\_{1})\mathbf{y}+\mathfrak{n}(\mathbf{r})\mathbf{y}^{\text{v}}\tag{2}$$

1 1 1 2 u ψ x x' x

Hence, the general, basic equations of continuity (4) and momentum (5) reduce to Equations

22 1 1 p x τ xx x

11 2 1 u <sup>0</sup> xx x 

> 1 2 <sup>p</sup> <sup>τ</sup> <sup>0</sup> x x

It should be noticed that gravity effect can be incorporated into the pressure gradient term in pipe flow Equation (12), where is the fluid unit weight and is the inclination angle of the pipe with respect to horizontal axis. In order to consider gravity in a pipe flow, <sup>p</sup>

(12) and (13) is a 2D version of the general case shown in (7), which is reproduced in

n

No slip boundary conditions, i.e. no velocity at the pipe or disk walls, and a full axial or radial symmetry of the flow are assumed. Hence, Equations (11-13) must be solved subject

> 2 2

Substituting (14) in (12) and (13) and integrating over x2, will give us the following pressure-

x dp (x ) τ μ(x )y(x ) <sup>η</sup>(x )y (x ) <sup>τ</sup> (x ) <sup>ξ</sup>(x )p(x ) 2 dx

dp x (x ) <sup>μ</sup>(x )y <sup>η</sup>(r)y dx

1 1 2 1 2 01 1 1

x

1

<sup>u</sup> (x 0) 0

(12) for pipe flow and (13a,b) for radial flow.

(12) should be replaced with <sup>p</sup> <sup>γ</sup>sin(β)

to the following boundary conditions:

PF: <sup>2</sup> <sup>n</sup>

RF: 211

1

Equation (14) below [4-5].

gradient equations

x 

PF (Pipe flow): <sup>2</sup>

RF (Radial Flow): <sup>2</sup>

*y*

1 1 x u (11b)

(11c)

(12)

(13a)

(13b)

, as suggested in [2]. The shear stress 1 2 <sup>τ</sup>(x ,x ) in

u u u(x h) 0 1 2 (15a)

(16)

(15b)

*<sup>n</sup>* (17)

12 1 11 1 1 τ(x ,x ) μ(x )y η(r)y τ (x ) ξ(x )p(x ) (14)

x in The pressure gradient Equations (16) and (17) must be satisfied at all points, including the boundary point, h. Therefore, at the wall boundary point (x2=h) we have

PF: <sup>n</sup> 1 h 1h 1h 01 1 1 h 1 h dp (x ) τ μ(x )y <sup>η</sup>(x )y <sup>τ</sup> (x ) <sup>ξ</sup>(x )p(x ) F(y ) 2 dx (18)

$$\text{RF:}\\\qquad \mathbf{h} \frac{\text{dp}}{\text{d} \, \text{x}\_1} = \mathfrak{p}(\mathbf{x}\_1) \Big( \frac{1}{\mathbf{x}\_1} \mathfrak{w}\_{\text{h}} \Big) + \mathfrak{q}(\mathbf{x}\_1) \Big( \frac{1}{\mathbf{x}\_1} \mathfrak{w}\_{\text{h}} \Big)^n \\ = \mathfrak{f}(\mathfrak{w}\_{\text{h}}/\mathbf{x}\_1) = \mathfrak{f}(\mathbf{y}\_{\text{h}}) \tag{19}$$

where yh, h and h are the boundary values of y, and , i.e. at the point x2 = h. In other words,

$$\text{PF:}\\\qquad \mathbf{y}\_{\text{h}} = \mathbf{y}(\mathbf{x}\_{2} = \mathbf{h}) = -\frac{\partial \mathbf{u}}{\partial \mathbf{x}\_{2}}\Big|\_{\mathbf{x}\_{2} = \mathbf{h}},\\\mathbf{r}\_{\text{h}} = \mathbf{r}(\mathbf{x}\_{2} = \mathbf{h}),\\\ \mathbf{u}\_{\text{h}} = \mathbf{u}\mathbf{y}(\mathbf{x}\_{2} = \mathbf{h})\tag{20}$$

If yh in (18), or h in (19), are known, the pipe pressure p can be calculated by integrating these equations directly. The result can still be in integral forms depending on the complexity of the coefficient functions such as: viscosity x or x, plasticity, x or friction x

$$\text{PF:}\tag{2}\tag{2}\tag{3}\tag{4}\tag{4}\tag{4}\tag{4}\tag{10}\tag{10}\tag{11a}\tag{11}\tag{11}\tag{11}\tag{12}\tag{12}\tag{12}\tag{12}\tag{12}\tag{11}\tag{13}\tag{13}\tag{13}\tag{13}\tag{13}\tag{14}\tag{14}\tag{14}\tag{14}\tag{15}\tag{15}\tag{15}\tag{15}\tag{15}\tag{16}\tag{16}\tag{16}\tag{16}\tag{17}\tag{17}\tag{17}\tag{17}\tag{17}\tag{18}\tag{18}\tag{18}\tag{18}\tag{19}\tag{19}\tag{19}\tag{19}\tag{11}\tag{11}\tag{11}\tag{11}\tag{11}\tag{11}\tag{11}\tag{11}\tag{11}\tag{12}\tag{12}\tag{12}\tag{12}\tag{12}\tag{13}\tag{13}\tag{13}\tag{13}\tag{13}\tag{14}\tag{14}\tag{14}\tag{14}\tag{15}\tag{15}\tag{15}\tag{15}\tag{15}\tag{15}\tag{16}\tag{16}\tag{16}\tag{17}\tag{17}\tag{17}\tag{17}\tag{17}\tag{17}\tag{18}\tag{18}\tag{18}\tag{18}\tag{19}\tag{19}\tag{19}\tag{19}\tag{19}\tag{11}\tag{11}\tag{11}\tag{11}\tag{11}\tag{11}\tag{11}\tag{11}\tag{11}\tag{11}\tag{12}\tag{12}\tag{12}\tag{12}\tag{12}\tag{13}\tag{13}\tag{13}\tag{13}\tag{13}\tag{14}\tag{14}\tag{14}\tag{14}\tag{15}\tag{15}\tag{15}\tag{15}\tag{15}\tag$$

$$\text{RF:}\tag{2\,\text{RF}}\tag{2\,\text{F}}\tag{2\,\text{M}}\underbrace{\text{g}}\_{\text{x}\_{0}}\mathbf{f}\{\text{\"(}\mathbf{\bar{y}}\_{\text{h}}/\mathbf{x}\_{1}\text{)}\mathbf{d}\mathbf{x}\_{1}\tag{21\,\text{b}}$$

For example, the pipe pressure can be written in a general integral form (Equation (22)) in terms of the integral coefficients A1, B1, C1 [5].

$$\mathbf{p} = \left(\mathbf{p}\_0 \mathbf{v}\_0 + \left(\mathbf{A}\_1 \mathbf{y}\_h + \mathbf{B}\_1 \mathbf{y}\_h^n + \mathbf{C}\_1\right)\right) \mathbf{v}^{-1} \tag{22}$$

In Equation (22), v(x1) is an exponential function of x1 and h and A1, B1, C1 are integral coefficients similar to those produced for radial flow [4], viz.

$$\mathbf{v}(\mathbf{x}) = \mathbf{e}^{-\frac{2}{h} \int \xi dx} \tag{23}$$

$$\mathbf{A}\_{1} = \frac{2}{\hbar} \int\_{\mathbf{x}\_{0}}^{\mathbf{x}} \mathbf{v}(\mathbf{x}) \mu(\mathbf{x}) d\mathbf{x} \tag{24a}$$

$$\mathbf{B}\_1 = \frac{2}{\hbar} \int\_{\mathbf{x}\_0}^{\mathbf{x}} \mathbf{v}(\mathbf{x}) \mathbf{\eta}(\mathbf{x}) d\mathbf{x} \tag{24b}$$

Mechanics of Multi-Phase Frictional Visco-Plastic,

Fig. 5. Values of integral function *I*

0

5

10

*Normalised integral I*

15

**(***r***,** *n, n* **,** *s* **)**

20

25

Fig. 6. Values of integral function *I*

10

100

*I***(***r***,** *n* **,** *n*

**,** *s***)**

1000

10000

100000

1000000

*<sup>s</sup>*, as indicated in Equation (27b)

*<sup>s</sup>*, as indicated in Equation (27b)

*n* and 

*n* and  0 20 40 60 80 100 **Normalised radial distance** *r*

0 20 40 60 80 100 **Normalised radial distance** *r*

with radial distance *r* for some values of parameters *n*,

*I***(***r***,** *n* **,** *n*

 **,** *<sup>s</sup>* **)**

I(r,0,0,0) I(r,0,0,0.5) I(r,0.5,0,0) I(r,0.5,0,0.5) I(r,1,0,0) I(r,1,0,0.5) I(r,1.5,0,0) I(r,2,0,0) I(r,1.5,0,0.5) I(r,2,0,0.5)

I(r ,0.0 ,2.0 ,1.0) I(r ,0.0 ,2.0 ,0.5) I(r ,0.5 ,2.0 ,1.0) I(r ,0.5 ,2.0 ,0.5) I(r ,0.0 ,1.0 ,1.0) I(r ,0.5 ,1.5 ,0.5)

with radial distance *r* for some values of parameters *n*,

Non-Newtonian, Depositing Fluid Flow in Pipes, Disks and Channels 161

$$\mathbf{C}\_{1} = \frac{2}{\hbar} \int\_{\mathbf{x}\_{0}}^{\mathbf{x}} \mathbf{v}(\mathbf{x}) \mathbf{r}\_{0}(\mathbf{x}) d\mathbf{x} \tag{24c}$$

As shown in [4-5], evaluations of these integrals become straight forward and generic, if we first find the normalised forms of the dimensional functions, (x), (x), (x) and (x) in the shear stress Equations (16) and (17). Using the same symbols, but in *Italic* fonts, let the *italic* symbols , , and be the normalised counterparts of (x), (x), (x) and (x), respectively. As discussed in detail in [4-5], all the four functions , , , , can be represented by one symbolic generic function, , i.e. (0, s, r, n). The normalised form of is simply

$$\alpha(r) = 1 - \alpha\_s(r-1)^{\eta\_a} \tag{25}$$

in which s is the normalised slope and *n* is the general power factor for any nonlinear behaviour. In other words,

$$\alpha = \frac{\mathbf{a}}{\mathbf{a}\_0}, \alpha\_s = \left(1 - \frac{\mathbf{a}\_\alpha}{\mathbf{a}\_0}\right) \left(\frac{\mathbf{r}\_\alpha}{\mathbf{r}\_0} - 1\right)^{-\eta\_a} \tag{26}$$

Therefore, the integrals (24) have the following general non-dimensional form which can be integrated numerically [4-5]:

$$\text{PF:}\tag{2}\text{PF:}\tag{1}\tag{3}\text{ V}=\underset{1}{\overset{x}{\text{I}}}{\text{I}}v(\mathbf{x})\mathbf{a}(\mathbf{x})\text{dx}=\underset{1}{\overset{k}{\underset{1}{\text{I}}}}{\text{I}}e^{\frac{1}{\mathbf{x}}\mathbf{\hat{d}}\mathbf{x}}\mathbf{a}(\mathbf{x})\mathbf{d}\mathbf{x}\tag{27a}$$

$$\text{(RF: } I(\mathbf{x}) = \left\{ \mathbf{x}^{-n} [1 - \alpha\_s(\mathbf{x} - \mathbf{1})^{n\_a}] \mathbf{dx} = \frac{\mathbf{x}^{1-n} - \mathbf{1}}{\mathbf{1} - n} - \frac{\alpha\_s}{\mathbf{1} - n} \left\{ \mathbf{x}^{1-n} \left(\mathbf{x} - \mathbf{1}\right)^{n\_a} - n\_a \int\_{\mathbf{1}}^{\mathbf{x}} \mathbf{x}^{1-n} \left(\mathbf{x} - \mathbf{1}\right)^{n\_a - 1} d\mathbf{x} \right\} \tag{27b}$$

However, for constant properties of slurry, the A1, B1, C1 parameters reduce to either a simple exponential function of x=x1, in the presence of a friction coefficient, i.e. ≠0, or a simple linear function in terms of pipe length (L = x-x0), in the absence of friction coefficient, i.e. =0. In other words, the values of these coefficients, as described by the integrals (24), can be calculated from relations (28) for the case where = = 0 = constant, but =0, and from relations (29) for the case where = = 0 = constant, but ≠0.

$$\mathbf{A}\_{1} = \mathbf{2h}^{-1} \boldsymbol{\mu}(\mathbf{x} - \mathbf{x}\_{0}) = \mathbf{2h}^{-1} \boldsymbol{\mu} \mathbf{L} \,, \; \mathbf{B}\_{1} = \mathbf{2h}^{-1} \boldsymbol{\eta} \mathbf{L} \,, \; \mathbf{C}\_{1} = \mathbf{2h}^{-1} \mathbf{r}\_{0} \mathbf{L} \tag{28}$$

$$\mathbf{A}\_{1} = -\frac{\mu}{\xi} \mathbf{v}\_{\perp} \mathbf{v}\_{\perp} \ \mathbf{B}\_{1} = -\frac{\Pi}{\xi} \mathbf{v}\_{\perp} \ \mathbf{v}\_{\perp} \ \mathbf{C}\_{1} = -\frac{\mathbf{r}\_{0}}{\xi} \mathbf{v}\_{\perp} \ \mathbf{v}\_{\perp} \ \mathbf{v}\_{\perp} = e^{-\frac{2}{\mathbf{h}} \xi \mathbf{l}} \tag{29}$$

Figures 5 and 6 show some typical values of the function *I* for a range of viscosity and plasticity parameters in a radial flow, namely *n* from (0) to (2), *n* from (0) to (2), s from (0) to (2) and normalised *r=x* from (1) to (100) [4].

<sup>2</sup> C v(x)<sup>τ</sup> (x)dx <sup>h</sup> (24c)

(25)

is the general power factor for any nonlinear

(26)

 

(29)

s from (0)

be the normalised counterparts of (x), (x), (x) and (x),

*n*

(27a)

1 1 1

<sup>2</sup> <sup>ξ</sup><sup>L</sup>

<sup>h</sup> vL *e* 

from (0) to (2), *n* from (0) to (2),

0 0 <sup>α</sup> <sup>r</sup> 1 1 α r

 

1 1 A 2h 1 0 <sup>μ</sup>(x-x ) 2h <sup>μ</sup><sup>L</sup> , <sup>1</sup> B 2h <sup>1</sup> <sup>η</sup><sup>L</sup> , <sup>1</sup> C 2h 1 0 <sup>τ</sup> <sup>L</sup> (28)

1 L <sup>τ</sup> C v <sup>ξ</sup> ,

 

0

symbols

is simply

in which

PF:

, , and 

behaviour. In other words,

integrated numerically [4-5]:

*s*

As shown in [4-5], evaluations of these integrals become straight forward and generic, if we first find the normalised forms of the dimensional functions, (x), (x), (x) and (x) in the shear stress Equations (16) and (17). Using the same symbols, but in *Italic* fonts, let the *italic*

respectively. As discussed in detail in [4-5], all the four functions , , , , can be represented by one symbolic generic function, , i.e. (0, s, r, n). The normalised form of

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

Therefore, the integrals (24) have the following general non-dimensional form which can be

*x x k dx I x v(x) (x)dx e (x)dx*

*n n s n n n n*

(27b)

However, for constant properties of slurry, the A1, B1, C1 parameters reduce to either a simple exponential function of x=x1, in the presence of a friction coefficient, i.e. ≠0, or a simple linear function in terms of pipe length (L = x-x0), in the absence of friction coefficient, i.e. =0. In other words, the values of these coefficients, as described by the integrals (24), can be calculated from relations (28) for the case where = = 0 = constant, but =0, and

<sup>ξ</sup> , <sup>0</sup>

Figures 5 and 6 show some typical values of the function *I* for a range of viscosity and

*<sup>x</sup> I x x x dx x x n x x dx n n*

1 1

RF: 1

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

from relations (29) for the case where = = 0 = constant, but ≠0.

<sup>ξ</sup> , 1 L <sup>η</sup> B v

 

0 α α ,

( )

1 L <sup>μ</sup> A v

plasticity parameters in a radial flow, namely *n*

to (2) and normalised *r=x* from (1) to (100) [4].

*s*

s is the normalised slope and *n*

x 1 0 x

Fig. 5. Values of integral function *I* with radial distance *r* for some values of parameters *n*, *n* and *<sup>s</sup>*, as indicated in Equation (27b)

Fig. 6. Values of integral function *I* with radial distance *r* for some values of parameters *n*, *n* and *<sup>s</sup>*, as indicated in Equation (27b)

Mechanics of Multi-Phase Frictional Visco-Plastic,

integration of the velocity gradient.

PF:

RF:

PF:

RF:

Non-Newtonian, Depositing Fluid Flow in Pipes, Disks and Channels 163

3 2 3 4 23 2 3

3 3 3 2 2 2 3 3 3n 2 2n 1

32 2 2 3 3n-1 2 2n

2 2n-1 2 n1 n 2 n-1 G''(y) 3A y 6A Cy 3AC 3nB y 3(2n 1)AB y 6nB Cy 3(n 2)A By 6(n 1)ABCy 3nBC y

Determination of other parameters is rather straight forward. Once, yh or h are determined, the wall shear stress, radial pressure gradient and pressure functions can be determined directly from Equations (16)-(19). For instance, we can calculate the fluid velocity by direct

<sup>u</sup> (n 1)Ay 2nBy u(x ) u(0) dx u(0) h

<sup>u</sup> (n 1)a<sup>ψ</sup> 2nb<sup>ψ</sup> u(x ) u(0) dx u(0)

(n 1)Ay 2nBy u(0) u u h 2(n 1)F

(n 1)a<sup>ψ</sup> 2nb<sup>ψ</sup> u(0) u u h 2(n 1)x f

<sup>2</sup> x 2 n1

x 2(n 1)F(y ) 

<sup>2</sup> x 2 n1

x 2(n 1)x f(ψ ) 

0 2 h

0 2 2 h

3n 1 2n 2 2n 1

<sup>A</sup> 3AC G(y) Fdy y A Cy y <sup>C</sup> <sup>y</sup> 4 2

3 22

B 3AB 3B C yyy 3n 1 2n 2 2n 1

 

G'(y) F A y 3A Cy 3AC y C B y 3AB y

2 2 n 3 n 2 n 1

3A B 6ABC 3BC y yy n3 n2 n1

2 2n 2 n 2 n1 2 n

+3B Cy 3A By 6ABCy 3BC y

2 2

2 2

where u(0) is the maximum velocity at the flow centre line given by

0 max

0 max

 

<sup>n</sup> f ψ aψ bψ (33d)

n 1 f' ψ a nbx (33e)

<sup>n</sup> F(y) τ Ay By C (34d)

n 1 F'(y) A nBy (34e)

(35a)

(35b)

2 n1 h h

(36a)

h

2 n1 h h

(36b)

2 h

(34a)

(34b)

(34c)

#### **5. Solutions for pipe and radial flow**

The pressure gradient dependency in all Equations (16)-(17) can be removed by dividing general functions or Equations (16)-(17) to their corresponding boundary values, i.e. Equations (18)-(19). Thus, we have a ratio of two polynomial functions, with a numerator that is a function of y, and a denominator that is a function of yh, as shown in Equation (30a, b), in which a, b, A, B, C are functions of flow line distance x1 only.

$$\text{PF:}\\\text{r:}\\\qquad \mathbf{x}\_2 = \mathbf{h} \frac{\mathbf{r}}{\mathbf{r}\_h} = \mathbf{h} \frac{\mathbf{A} \mathbf{y} + \mathbf{B} \mathbf{y}^\mathbf{n} + \mathbf{C}}{\mathbf{A} \mathbf{y}\_h + \mathbf{B} \mathbf{y}\_h^\mathbf{n} + \mathbf{C}} = \mathbf{h} \frac{\mathbf{F}(\mathbf{y})}{\mathbf{F}(\mathbf{y}\_h)} = \mathbf{h} \frac{\mathbf{F}}{\mathbf{F}\_h} \tag{30a}$$

$$\text{RF:}\\\text{r:}\\\qquad \mathbf{x}\_2 = \mathbf{h} \frac{\mathbf{r}}{\mathbf{r}\_h} = \mathbf{h} \frac{\mathbf{a}\mathbf{y} + \mathbf{b}\mathbf{y}^\mathbf{n}}{\mathbf{a}\mathbf{y}\_h + \mathbf{b}\mathbf{y}^\mathbf{n}\_h} = \mathbf{h} \frac{\mathbf{f}(\mathbf{y})}{\mathbf{f}(\mathbf{y}\_h)} = \mathbf{h} \frac{\mathbf{f}}{\mathbf{f}\_h} \tag{30b}$$

To solve (30a) or (30b) for our primary unknowns, either y or , we need another equation in terms of the flow rate, Q, which must be conserved at any section normal to x1 direction. The results are integral equations relating velocity gradient y, or yh (or , or h in the case of radial flow) to the flow rate Q [4-5].

$$\text{PF:}\\
\text{d} = \int\_{\text{A}} \vec{\text{u}}.\\\vec{\text{d}} = 2\pi \int\_{0}^{\text{h}} \mathbf{x}\_{2} \mathbf{u}\_{1} \text{d} \mathbf{x}\_{2} = \frac{\Pi}{3} \mathbf{h}^{3} \left(\mathbf{y}\_{\text{h}} - \mathbf{F}\_{\text{h}}^{-3} \mathbf{G}\_{\text{h}}\right) \tag{31a}$$

$$\text{RF:}\\
\text{a. }\qquad \mathbf{Q} = \int\_{\mathbf{A}} \vec{\mathbf{u}}.\\\mathbf{d}\vec{\mathbf{A}} = 2\mathbf{n} \oint\_{0} \mathbf{x}\_{2} \mathbf{u}\_{1} \mathbf{d}\mathbf{x}\_{2} = 2\mathbf{n} \mathbf{h}^{2} \left(\mathbf{\upmu}\_{\mathbf{h}} - \mathbf{f}\_{\mathbf{h}}^{-2} \mathbf{g}\_{\mathbf{h}}\right) \tag{31b}$$

Values of velocity gradient at the wall boundary, yh, or the function h= x1 yh, needs be calculated generally by the Newton-Raphson iteration [16]. Hence solutions to (31) take the following general forms:

$$\text{PF:}\\\qquad\qquad\mathbf{y}\_{\text{h}\_{i+1}}\Longleftrightarrow\mathbf{y}\_{\text{h}\_{i}} - \frac{(\mathbf{Q}\_{\text{h}} - \mathbf{y}\_{\text{h}})\mathbf{G}\prime(\mathbf{y}\_{\text{h}}) + \mathbf{G}\prime(\mathbf{y}\_{\text{h}\_{i}})}{(\mathbf{Q}\_{\text{h}} - \mathbf{y}\_{\text{h}})\mathbf{G}\prime(\mathbf{y}\_{\text{h}})},\\\ \mathbf{Q}\_{\text{h}} = \frac{\mathbf{3}}{\mathbf{n}\text{h}^{3}}\mathbf{Q} \tag{32a}$$

$$\text{RF:}\\\qquad\qquad\qquad\boldsymbol{\Psi}\_{\mathbf{h}\_{\mathrm{i}+1}}\Leftrightarrow\\\boldsymbol{\Psi}\_{\mathbf{h}\_{\mathrm{i}}}-\frac{(\mathbf{Q}\_{\mathrm{h}}-\boldsymbol{\Psi}\_{\mathrm{h}})\mathbf{g}'(\boldsymbol{\upmu}\_{\mathrm{h}})+\mathbf{g}(\boldsymbol{\upmu}\_{\mathrm{h}})}{(\mathbf{Q}\_{\mathrm{h}}-\boldsymbol{\upmu}\_{\mathrm{h}})\mathbf{g}''(\boldsymbol{\upmu}\_{\mathrm{h}})},\ \mathbf{Q}\_{\mathrm{h}}=\frac{1}{2\pi\mathbf{h}^{2}}\mathbf{Q}\tag{32b}$$

In Equation (31)-(32) f, F, g and G are polynomial functions of the unknown variable y or .

$$\log\left(\boldsymbol{\upmu}\right) = \int \mathbf{f}^2 \mathbf{d}\boldsymbol{\upmu} = \frac{\mathbf{a}^2}{3}\boldsymbol{\upmu}^3 + \frac{\mathbf{b}^2}{2\mathbf{n}+1}\boldsymbol{\upmu}^{2\mathbf{n}+1} + \frac{2\mathbf{a}\mathbf{b}}{\mathbf{n}+2}\boldsymbol{\upmu}^{\mathbf{n}+2}\tag{33a}$$

$$\mathbf{g'(\mu) = f^2\left(\mu\right) = a^2 \mu^2 + b^2 \mu^{2n} + 2ab\mu\nu^{n+1} \tag{33b}$$

$$\mathbf{g''(\mu) = 2a^2 \mu + 2nb^2 \mu^{2n-1} + 2(n+1)ab\mu^n} \tag{33c}$$

The pressure gradient dependency in all Equations (16)-(17) can be removed by dividing general functions or Equations (16)-(17) to their corresponding boundary values, i.e. Equations (18)-(19). Thus, we have a ratio of two polynomial functions, with a numerator that is a function of y, and a denominator that is a function of yh, as shown in Equation (30a,

h

h

To solve (30a) or (30b) for our primary unknowns, either y or , we need another equation in terms of the flow rate, Q, which must be conserved at any section normal to x1 direction. The results are integral equations relating velocity gradient y, or yh (or , or h in the case of

<sup>π</sup> Q u.dA 2<sup>π</sup> x u dx h <sup>y</sup> F G

Values of velocity gradient at the wall boundary, yh, or the function h= x1 yh, needs be calculated generally by the Newton-Raphson iteration [16]. Hence solutions to (31) take the

hh h h

 , h <sup>3</sup>

hh h (Q y )G'(y ) G(y )

hh h h

 , h <sup>2</sup>

hh h (Q ψ )g'(ψ ) g(ψ )

In Equation (31)-(32) f, F, g and G are polynomial functions of the unknown variable y or .

2 2 <sup>2</sup> a b <sup>3</sup> 2n 1 2ab n 2 <sup>g</sup> <sup>ψ</sup> f dψψ ψ ψ 3 2n 1 n 2 

(30b)

3 3

(31a)

2 2

<sup>3</sup> Q Q

<sup>1</sup> Q Q 2 h

(33a)

2 2 2 2 2n n 1 g' ψ f ψ a ψ b ψ 2abψ (33b)

2 2 2n 1 <sup>n</sup> g'' ψ 2a ψ 2nb ψ 2(n 1)abψ (33c)

<sup>π</sup><sup>h</sup> (32a)

(32b)

Q u.dA 2<sup>π</sup> x u dx 2π<sup>h</sup> <sup>ψ</sup> <sup>f</sup> <sup>g</sup> (31b)

21 2 h hh

21 2 h hh

3

n <sup>2</sup> <sup>n</sup> <sup>h</sup> <sup>h</sup> h h <sup>τ</sup> ay by f(y) <sup>f</sup> xh h h h τ ay by f(y ) f

(30a)

n <sup>2</sup> <sup>n</sup> <sup>h</sup> <sup>h</sup> h h <sup>τ</sup> Ay By C F(y) <sup>F</sup> xh h h h τ Ay By C F(y ) F 

**5. Solutions for pipe and radial flow** 

radial flow) to the flow rate Q [4-5].

following general forms:

PF:

RF:

b), in which a, b, A, B, C are functions of flow line distance x1 only.

PF:

RF:

y y (Q y )G''(y )

ψ ψ (Q <sup>ψ</sup> )g''(<sup>ψ</sup> )

A 0

PF: i i1 i

RF: i i 1 i

h h

h h

A 0

h

h

$$\mathbf{f}(\mathsf{u}\boldsymbol{\upmu}) = \mathbf{a}\boldsymbol{\upmu} + \mathsf{b}\boldsymbol{\upmu}^{\mathsf{n}} \tag{33d}$$

$$\mathbf{f}^{\circ}(\mathsf{u}\mathsf{y}) = \mathsf{a} + \mathsf{n}\mathsf{b}\mathsf{x}^{\mathrm{n}-1} \tag{33e}$$

$$\begin{aligned} \mathbf{G(y)} &= \int \mathbf{F}^3 \mathbf{d} \mathbf{y} = \frac{\mathbf{A}^3}{4} \mathbf{y}^4 + \mathbf{A}^2 \mathbf{C} \mathbf{y}^3 + \frac{3 \mathbf{A} \mathbf{C}^2}{2} \mathbf{y}^2 + \mathbf{C}^3 \mathbf{y} + \\ &+ \frac{\mathbf{B}^3}{3 \mathbf{n} + 1} \mathbf{y}^{3 \mathbf{n} + 1} + \frac{3 \mathbf{A} \mathbf{B}^2}{2 \mathbf{n} + 2} \mathbf{y}^{2 \mathbf{n} + 2} + \frac{3 \mathbf{B}^2 \mathbf{C}}{2 \mathbf{n} + 1} \mathbf{y}^{2 \mathbf{n} + 1} + \\ &+ \frac{3 \mathbf{A}^2 \mathbf{B}}{\mathbf{n} + 3} \mathbf{y}^{n + 3} + \frac{6 \mathbf{A} \mathbf{B} \mathbf{C}}{\mathbf{n} + 2} \mathbf{y}^{n + 2} + \frac{3 \mathbf{B} \mathbf{C}^2}{\mathbf{n} + 1} \mathbf{y}^{n + 1} \end{aligned} \tag{34a}$$

$$\begin{aligned} \mathbf{G}'(\mathbf{y}) &= \mathbf{F}^3 = \mathbf{A}^3 \mathbf{y}^3 + 3\mathbf{A}^2 \mathbf{C} \mathbf{y}^2 + 3\mathbf{A} \mathbf{C}^2 \mathbf{y} + \mathbf{C}^3 + \mathbf{B}^3 \mathbf{y}^{3n} + 3\mathbf{A} \mathbf{B}^2 \mathbf{y}^{2n+1} + \\ &+ 3\mathbf{B}^2 \mathbf{C} \mathbf{y}^{2n} + 3\mathbf{A}^2 \mathbf{B} \mathbf{y}^{n+2} + 6\mathbf{A} \mathbf{B} \mathbf{C} \mathbf{y}^{n+1} + 3\mathbf{B} \mathbf{C}^2 \mathbf{y}^n \end{aligned} \tag{34b}$$

$$\begin{aligned} \mathbf{G}"\text{(y)} &= 3\mathbf{A}^3 \mathbf{y}^2 + 6\mathbf{A}^2 \mathbf{C} \mathbf{y} + 3\mathbf{A} \mathbf{C}^2 + 3\mathbf{n} \mathbf{B}^3 \mathbf{y}^{3n+1} + 3(2\mathbf{n} + 1) \mathbf{A} \mathbf{B}^2 \mathbf{y}^{2n} + \\ &+ 6\mathbf{n} \mathbf{B}^2 \mathbf{C} \mathbf{y}^{2n+1} + 3(\mathbf{n} + 2) \mathbf{A}^2 \mathbf{B} \mathbf{y}^{n+1} + 6(\mathbf{n} + 1) \mathbf{A} \mathbf{B} \mathbf{C} \mathbf{y}^n + 3\mathbf{n} \mathbf{B} \mathbf{C}^2 \mathbf{y}^{n+1} \end{aligned} \tag{34c}$$

$$\mathbf{F(y) = \mathbf{r} = A\mathbf{y} + B\mathbf{y}^n + C} \tag{34d}$$

$$\mathbf{F}'(\mathbf{y}) = \mathbf{A} + \mathbf{n} \mathbf{B} \mathbf{y}^{n-1} \tag{34e}$$

Determination of other parameters is rather straight forward. Once, yh or h are determined, the wall shear stress, radial pressure gradient and pressure functions can be determined directly from Equations (16)-(19). For instance, we can calculate the fluid velocity by direct integration of the velocity gradient.

$$\text{PF:}\\\qquad\qquad\mathbf{u}(\mathbf{x}\_{2})=\mathbf{u}(\mathbf{0})+\int\limits\_{0}^{\mathbf{x}\_{2}}\frac{\partial\mathbf{u}}{\partial\mathbf{x}\_{2}}d\mathbf{x}\_{2}=\mathbf{u}(\mathbf{0})-\mathbf{h}\frac{(\mathbf{n}+1)\mathbf{A}\mathbf{y}^{2}+2\mathbf{n}\mathbf{B}\mathbf{y}^{n+1}}{2(\mathbf{n}+1)\mathbf{F}(\mathbf{y}\_{h})}\tag{35a}$$

$$\text{RF:}\\
\qquad\qquad\mathbf{u}(\mathbf{x}\_{2})=\mathbf{u}(\mathbf{0})+\int\_{0}^{\frac{\mathbf{x}\_{2}}{2}}\frac{\partial\mathbf{u}}{\partial\mathbf{x}\_{2}}d\mathbf{x}\_{2}=\mathbf{u}(\mathbf{0})-\frac{(\mathbf{n}+1)\mathbf{a}\mathbf{u}^{2}+2\mathbf{n}\mathbf{b}\mathbf{u}^{n+1}}{2(\mathbf{n}+1)\mathbf{x}\_{2}\mathbf{f}(\mathbf{u}\_{h})}\tag{35b}$$

where u(0) is the maximum velocity at the flow centre line given by

$$\text{PF:}\\\text{\textbullet (0)} = \mathbf{u}\_0\\= \mathbf{u}\_{\text{max}} = \mathbf{h} \frac{(\mathbf{n} + 1) \mathbf{A} \mathbf{y}\_{\text{h}}^2 + 2 \mathbf{n} \mathbf{B} \mathbf{y}\_{\text{h}}^{n+1}}{2(\mathbf{n} + 1) \mathbf{F}\_{\text{h}}} \tag{36a}$$

$$\text{RF:}\tag{1\text{RF}:}\tag{1\text{\textquotedbl{}}}=\mathbf{u}\_{0}=\mathbf{u}\_{\text{max}}=\mathbf{h}\frac{(\mathbf{n}+1)\mathbf{a}\mathbf{p}\_{\text{h}}^{2}+2\mathbf{n}\mathbf{b}\mathbf{p}\_{\text{h}}^{n+1}}{2(\mathbf{n}+1)\mathbf{x}\_{2}\mathbf{f}\_{\text{h}}}\tag{36b}$$

Mechanics of Multi-Phase Frictional Visco-Plastic,

be ignored. Therefore, equation (12) becomes

[6], p, of length Lt. In other words we have:

**7. Concrete flow in a rectangular channel** 

which gives a solution in terms of Fourier coefficients [6]

0

 

*n*

inclination angle .

direction x1. In other words,

**6. Tremie pipe concrete flow** 

Non-Newtonian, Depositing Fluid Flow in Pipes, Disks and Channels 165

The flow theory developed for general viscous-plastic-frictional fluids can be applied to fresh wet concrete pastes and slurries as well. Again, an important question would be the relation between the flow rate, Q, and the pressure, p. Similar to the pipe flow discussed above, we may assume a fully developed laminar, one dimensional flow, where x2 is the radial distance from the tremie pipe's axis of symmetry (Figure 1, middle) [6-9]. Gravity plays a major role as the main driving force for concrete flow in tremie pipes, thus, it cannot

2 c

 4 3

c 0

2 1 ,x 1 ,x 1 ,t 1 τ (t,x ) p (t,x ) ρu (t,x ) 0 (42)

 

(43)

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

*n nnn*

(, ) cos( ) sin( ) *<sup>n</sup> a t*

*u t x e A x B x mx*

(40)

(41)

(x <sup>τ</sup>) p <sup>x</sup> <sup>γ</sup> sin(θ)

It is assumed that the tremie pipe generally makes an angle with the horizontal x axis, where = 90 indicates a vertical tremie. In Equation (40), x2 is the coordinate radius or radial distance from the pipe's cross-sectional centre, p is pressure and c is the effective concrete unit weight. As a special case, a specific analytical solution from the general solution (32a) can be derived for a linear Bingham-plastic model [5]. In this particular solution, the tremie pipe flow rate, Q, becomes a 4th order polynomial function of the tremie pipe diameter, D. Furthermore, the flow rate is inversely proportional to the viscosity, 0and (partially) proportional to the differential pressure at the two ends of the tremie pipe

 

2 1

t

<sup>π</sup><sup>D</sup> <sup>Δ</sup><sup>p</sup> <sup>π</sup><sup>D</sup> <sup>Q</sup> <sup>γ</sup> sin θ τ 128μ L 24μ 

The pressure differential between the two ends of the tremie, p, can in theory accept any arbitrary value; from negative to zero and positive numbers. In the case of a zero p, the driving pressure is simply the gravity term containing the concrete unit weight, cand

Concrete flow during pouring and flowing in channels, chutes and testing equipment for testing purposes are normally not at a steady state situation [6]. General time-dependent 2D and 3D differential equations governing flow of concrete in rectangular channels and chutes can be developed and solved numerically, as shown in [6]. However, for the sake of understanding, it is also possible to reduce these equations to a simple 1D form, based on an assumption that there is no significant independent variation in any variable or function in the normal directions x2 and x3 compared to the longitudinal main flow

x x

2

Hence, the velocity profile across the flow cross-section is given by

$$\text{PF:}\\\text{ }\mathbf{u}=\mathbf{u}\_{\text{max}}\left(\mathbf{1}-\frac{(\mathbf{n}+1)\mathbf{A}\mathbf{y}^2+2\mathbf{n}\mathbf{B}\mathbf{y}^{n+1}}{(\mathbf{n}+1)\mathbf{A}\mathbf{y}\_{\text{h}}^2+2\mathbf{n}\mathbf{B}\mathbf{y}\_{\text{h}}^{n+1}}\right)\tag{37a}$$

$$\text{RF:}\tag{1} \\ \text{RF:}\tag{2} \\ \text{u} = \text{u}\_{\text{max}} \left( 1 - \frac{(\text{n} + 1) \text{a} \upmu^2 + 2 \text{nb} \upmu^{n+1}}{(\text{n} + 1) \text{a} \upmu^2\_{\text{h}} + 2 \text{nb} \upmu^{n+1}\_{\text{h}}} \right) \tag{37b}$$

The average flow velocity can also be determined in the usual manner by integrating (36) directly, namely

$$\overline{\mathbf{u}} = \frac{1}{\int\_0^h \mathbf{x}\_2 d\mathbf{x}\_2} \int\_0 \mathbf{u}(\mathbf{x}\_2) \mathbf{x}\_2 d\mathbf{x}\_2 = \mathbf{u}\_{\text{max}} \left( 1 - \lambda\_0 \right) \tag{38}$$

Where 0 is a function of yh, or h (in the case of radial flow [4]).

$$\text{PF:}\\\qquad\qquad\lambda\_0 = \mathbf{G}\_0\\\text{(y}\_{\text{h}}) = 1 - \frac{\mathbf{y}\_{\text{h}}\mathbf{F}\_{\text{h}} - \mathbf{G}\_{\text{h}}\mathbf{F}\_{\text{h}}^{-2}}{\frac{3}{2}\mathbf{A}\mathbf{y}\_{\text{h}}^2 + \frac{3\mathbf{n}}{\mathbf{n}+1}\mathbf{B}\mathbf{y}\_{\text{h}}^{n+1}}\tag{39a}$$

$$\mathbf{RF}: \qquad \lambda\_0 = \mathbf{g}\_0(\mathbf{x}\_h) = \frac{\mathbf{1} + \mathbf{n}\_1 \mathbf{a}^{-1} \mathbf{b} \mathbf{w}\_h^{n-1} + \mathbf{n}\_2 \mathbf{a}^{-2} \mathbf{b}^2 \mathbf{w}\_h^{2(n-1)}}{\mathbf{3} + \mathbf{n}\_3 \mathbf{a}^{-1} \mathbf{b} \mathbf{w}\_h^{n-1} + \mathbf{n}\_4 \mathbf{a}^{-2} \mathbf{b}^2 \mathbf{w}\_h^{2(n-1)}} \tag{39b}$$

In Equation (39b) ni is a constant depending only on the power factor n, given by

$$\mathbf{n}\_1 = \frac{\mathbf{3n(n+3)}}{(\mathbf{n}+1)(\mathbf{n}+2)}, \quad \mathbf{n}\_2 = \frac{6\mathbf{n}^2}{(\mathbf{n}+1)(2\mathbf{n}+1)}, \quad \mathbf{n}\_3 = 3\frac{(3\mathbf{n}+1)}{(\mathbf{n}+1)}, \; \mathbf{n}\_4 = \frac{6\mathbf{n}}{(\mathbf{n}+1)}\tag{39c}$$

All the above solutions (e.g. Equations (32)) are also reducible to the classical solutions. For example, the average flow velocity becomes half of the maximum flow velocity for pipe flow, and 2/3 of the maximum flow velocity in the case of radial flow, for the case of pure Newtonian fluid [4-5].

Slurry flow may be assumed to stop in the case of a blockage (Q 0), which means the values of yh and g(yh) are identically zero. This is due to the effects of the cohesive frictional terms ( and 0) introduced in the shear stress Equations (16-17), which now become dominant in blocking the slurry flow. In the slurry industry, a critical question always arises on what the minimum pump pressure is required for a given slurry flow rate either to transport it to a given distance, or be able to reopen a blockage in a specified pipe length. The minimum required pump pressure can be calculated from Equation (21), which depends on the wall shear resistance in the pipeline or the disk. The wall shear stress is a function of the longitudinal distance x and velocity gradient yh. Practically, during the field slurry injection, the minimum required pump pressure to transport the slurry to a given distance is one of the most important questions that needs to be addressed [4-5].

#### **6. Tremie pipe concrete flow**

164 Fluid Dynamics, Computational Modeling and Applications

max 2 n1

max 2 n1

h 2 2 2 max 0

u u(x )x dx u 1 λ

yF GF <sup>λ</sup> G (y ) 1 3 3n Ay By

0 0h 1 n1 2 2 2(n 1)

1 na b<sup>ψ</sup> na b <sup>ψ</sup> <sup>λ</sup> g (x ) 3 na b<sup>ψ</sup> na b <sup>ψ</sup>

6n

(n 1)(2n 1) , 3

All the above solutions (e.g. Equations (32)) are also reducible to the classical solutions. For example, the average flow velocity becomes half of the maximum flow velocity for pipe flow, and 2/3 of the maximum flow velocity in the case of radial flow, for the case of pure

Slurry flow may be assumed to stop in the case of a blockage (Q 0), which means the values of yh and g(yh) are identically zero. This is due to the effects of the cohesive frictional terms ( and 0) introduced in the shear stress Equations (16-17), which now become dominant in blocking the slurry flow. In the slurry industry, a critical question always arises on what the minimum pump pressure is required for a given slurry flow rate either to transport it to a given distance, or be able to reopen a blockage in a specified pipe length. The minimum required pump pressure can be calculated from Equation (21), which depends on the wall shear resistance in the pipeline or the disk. The wall shear stress is a function of the longitudinal distance x and velocity gradient yh. Practically, during the field slurry injection, the minimum required pump pressure to transport the slurry to a given distance is one of the most important questions that needs to be

In Equation (39b) ni is a constant depending only on the power factor n, given by

The average flow velocity can also be determined in the usual manner by integrating (36)

(n 1)Ay 2nBy

(n 1)aψ 2nbψ

(n 1)Ay 2nBy uu 1

(n 1)a<sup>ψ</sup> 2nb<sup>ψ</sup> uu 1

h

<sup>0</sup> 2 2 <sup>0</sup>

x dx

Where 0 is a function of yh, or h (in the case of radial flow [4]).

 , <sup>2</sup> 2

n

0 0h

1

2 n1

 

> 

(38)

2

(39a)

6n

(39c)

(n 1)

n

(3n 1) n 3 (n 1) , 4

hh hh

2 n1

 

3 h4 h

2 n 1 h h

1 n1 2 2 2(n 1) 1 h2 h

 (39b)

(37a)

(37b)

h h

2 n1

h h

Hence, the velocity profile across the flow cross-section is given by

PF:

RF:

PF:

RF:

1

Newtonian fluid [4-5].

addressed [4-5].

3n(n 3) <sup>n</sup> (n 1)(n 2)

directly, namely

The flow theory developed for general viscous-plastic-frictional fluids can be applied to fresh wet concrete pastes and slurries as well. Again, an important question would be the relation between the flow rate, Q, and the pressure, p. Similar to the pipe flow discussed above, we may assume a fully developed laminar, one dimensional flow, where x2 is the radial distance from the tremie pipe's axis of symmetry (Figure 1, middle) [6-9]. Gravity plays a major role as the main driving force for concrete flow in tremie pipes, thus, it cannot be ignored. Therefore, equation (12) becomes

$$\frac{\partial \langle \mathbf{x}\_2 \mathbf{r} \rangle}{\partial \mathbf{x}\_2} = \mathbf{x}\_2 \left( \frac{\partial \mathbf{p}}{\partial \mathbf{x}\_1} + \mathbf{y}\_c \sin(\theta) \right) \tag{40}$$

It is assumed that the tremie pipe generally makes an angle with the horizontal x axis, where = 90 indicates a vertical tremie. In Equation (40), x2 is the coordinate radius or radial distance from the pipe's cross-sectional centre, p is pressure and c is the effective concrete unit weight. As a special case, a specific analytical solution from the general solution (32a) can be derived for a linear Bingham-plastic model [5]. In this particular solution, the tremie pipe flow rate, Q, becomes a 4th order polynomial function of the tremie pipe diameter, D. Furthermore, the flow rate is inversely proportional to the viscosity, 0and (partially) proportional to the differential pressure at the two ends of the tremie pipe [6], p, of length Lt. In other words we have:

$$\mathbf{Q} = \frac{\mathbf{n}\mathbf{D}^4}{128\mu} \left(\frac{\Delta\mathbf{p}}{\mathcal{L}\_t} + \mathbf{y}\_c \sin(\theta)\right) - \frac{\mathbf{n}\mathbf{D}^3}{24\mu}\mathbf{r}\_0 \tag{41}$$

The pressure differential between the two ends of the tremie, p, can in theory accept any arbitrary value; from negative to zero and positive numbers. In the case of a zero p, the driving pressure is simply the gravity term containing the concrete unit weight, cand inclination angle .

#### **7. Concrete flow in a rectangular channel**

Concrete flow during pouring and flowing in channels, chutes and testing equipment for testing purposes are normally not at a steady state situation [6]. General time-dependent 2D and 3D differential equations governing flow of concrete in rectangular channels and chutes can be developed and solved numerically, as shown in [6]. However, for the sake of understanding, it is also possible to reduce these equations to a simple 1D form, based on an assumption that there is no significant independent variation in any variable or function in the normal directions x2 and x3 compared to the longitudinal main flow direction x1. In other words,

$$
\sigma\_{\mathbf{x}\_2}(\mathbf{t}, \mathbf{x}\_1) - \mathbf{p}\_{,\mathbf{x}\_1}(\mathbf{t}, \mathbf{x}\_1) - \rho \mathbf{u}\_{,t}(\mathbf{t}, \mathbf{x}\_1) = 0 \tag{42}
$$

which gives a solution in terms of Fourier coefficients [6]

$$\ln(t,\mathbf{x}) = \sum\_{n=0}^{\infty} e^{-a\kappa\_n^2 t} \left( A\_n \cos(\kappa\_n \mathbf{x}) + B\_n \sin(\kappa\_n \mathbf{x}) \right) + m\mathbf{x}^2 \tag{43}$$

Mechanics of Multi-Phase Frictional Visco-Plastic,

u(r,x) <sup>τ</sup> (r,x) <sup>μ</sup>(x) <sup>μ</sup><sup>y</sup>

r

rx

confirming the classical result, max r r

separately demonstrated.

Non-Newtonian, Depositing Fluid Flow in Pipes, Disks and Channels 167

Substituting these values in equations (16-32), we recover the well-known Newtonian

<sup>1</sup> u (r) u2 . Furthermore, we have:

3 0 4 h h

> 0 1

y

τ

h 3 4Q

0 hx 3 4μ Q

> 0 4

0 0 4 <sup>8</sup><sup>μ</sup> <sup>Q</sup> pp L

max 2

max 2 2 max h <sup>y</sup> <sup>r</sup> u(r,x) u 1 u 1

2 2

y h

dp 8μ Q

2Q u(0,x) u

Figure 8 shows velocity profiles in normalised form for the "three special cases" discussed above. For the general Bingham fluid with constant non-zero values of 0 and 0, the pipe velocity profile follows a parabolic curve close to the special pure viscous case (i), when fluid viscosity is dominant or the ratio 0/0 is small, and moves towards the uniform profile of the special case (ii), when plastic yield or cohesion is dominant or the ratio 0/0 is large, as shown in the figure. Figure 9 demonstrates an example of a Bingham plastic solution for radial disk flow, where the contribution of each of the two shear parameters is

solutions [15], as expected. It can be seen that for this case the function 0 h

, <sup>μ</sup> <sup>0</sup> (r) <sup>μ</sup> , ητξ n 0 (44a)

<sup>μ</sup> g y <sup>4</sup> (44b)

<sup>g</sup> <sup>2</sup> (44c)

<sup>π</sup><sup>h</sup> (44d)

<sup>π</sup><sup>h</sup> (44e)

dx <sup>π</sup><sup>h</sup> (44f)

<sup>π</sup><sup>h</sup> (44g)

<sup>π</sup><sup>h</sup> (44h)

max 0 max <sup>1</sup> u u 1g u2 (44j)

(44i)

<sup>1</sup> g (y ) <sup>2</sup>

In the solution (43) is an arbitrary constant satisfying both the differential equation and the boundary conditions, while *An* and *Bn* are Fourier coefficients to be determined from the boundary conditions [6]. Figure 7 shows a typical result for various values of n truncating the number of Fourier terms. It shows results of the Fourier analysis for the two cases of *u*(0, *x*) and *u*(0.5, *x*), and the increasing effects of the number of Fourier terms, namely from *n* = 5, 10 to 120. The second line in the figure corresponds to velocity at time *t* = 0.5 for different profile points along the x line using *n* = 120. Notice that since continuity and differentiability is not a requirement at the end points of a Fourier series analysis, it doesn't converge to the numerical solution at point *x* = 1, as expected.

Fig. 7. The function *u*(0, *x*) represented by a Fourier series with different number of Fourier coefficients (*n* = 5, 10, 120).

#### **8. Discussion**

The above general theory is certainly reducible to simpler classical Newtonian and Bingham models with appropriate parameter substitutions [4-5]. Classical special cases can be derived, e.g. (i) pure, uniform, viscous, Newtonian slurry; (ii) Pure, uniform, cohesive (plastic), non-Newtonian slurry; (iii) Linear Bingham viscous plastic slurry. In case (i), (ii) and (iii) the shear stress function (1) reduces to either (i) the simplest, classical, linear function of the shear strain multiplied by a constant viscosity number, i.e. 12 0 τ(x ,x ) μ y ; or (ii) just a pure plastic material with no viscosity, i.e. 12 0 τ(x ,x ) τ ; or a linear Bingham viscoplastic model, i.e. 12 0 0 τ(x ,x ) μ y τ . For instance, for a pipe flow in a pure viscous Newtonian fluid, we have

boundary conditions, while *An* and *Bn* are Fourier coefficients to be determined from the boundary conditions [6]. Figure 7 shows a typical result for various values of n truncating the number of Fourier terms. It shows results of the Fourier analysis for the two cases of *u*(0, *x*) and *u*(0.5, *x*), and the increasing effects of the number of Fourier terms, namely from *n* = 5, 10 to 120. The second line in the figure corresponds to velocity at time *t* = 0.5 for different profile points along the x line using *n* = 120. Notice that since continuity and differentiability is not a requirement at the end points of a Fourier series analysis, it doesn't converge to the

*u* (0, *x* ), *n* = 5

Fig. 7. The function *u*(0, *x*) represented by a Fourier series with different number of Fourier

0 0.2 0.4 0.6 0.8 1 **Distance Ratio** *x* **= (x/X0)**

*u* (0, *x* ), *n* = 10

The above general theory is certainly reducible to simpler classical Newtonian and Bingham models with appropriate parameter substitutions [4-5]. Classical special cases can be derived, e.g. (i) pure, uniform, viscous, Newtonian slurry; (ii) Pure, uniform, cohesive (plastic), non-Newtonian slurry; (iii) Linear Bingham viscous plastic slurry. In case (i), (ii) and (iii) the shear stress function (1) reduces to either (i) the simplest, classical, linear function of the shear strain multiplied by a constant viscosity number, i.e. 12 0 τ(x ,x ) μ y ; or (ii) just a pure plastic material with no viscosity, i.e. 12 0 τ(x ,x ) τ ; or a linear Bingham viscoplastic model, i.e. 12 0 0 τ(x ,x ) μ y τ . For instance, for a pipe flow in a pure viscous

is an arbitrary constant satisfying both the differential equation and the

n = 5 n = 10 n = 120

*u* (0, *x* ), *n* = 120

In the solution (43)

coefficients (*n* = 5, 10, 120).

Newtonian fluid, we have

**8. Discussion** 

0

0.2

0.4

0.6

**Velocity Ratio** *u* **= (u/U0)**

0.8

1

numerical solution at point *x* = 1, as expected.

*u* (0.5, *x* )

$$\mathbf{u}\mathbf{r}\_{\mathbf{x}\mathbf{x}}(\mathbf{r},\mathbf{x}) = \boldsymbol{\mathfrak{u}}(\mathbf{x}) \left( -\frac{\partial \mathbf{u}(\mathbf{r},\mathbf{x})}{\partial \mathbf{r}} \right) = \mathbf{\upmu}\mathbf{y} \,, \ \boldsymbol{\upmu}(\mathbf{r}) = \boldsymbol{\upmu}\_0 \,, \ \boldsymbol{\mathfrak{u}} = \boldsymbol{\mathfrak{r}} = \boldsymbol{\xi} = \mathbf{\upmu} = \mathbf{0} \tag{44a}$$

Substituting these values in equations (16-32), we recover the well-known Newtonian solutions [15], as expected. It can be seen that for this case the function 0 h <sup>1</sup> g (y ) <sup>2</sup> confirming the classical result, max r r <sup>1</sup> u (r) u2 . Furthermore, we have:

$$\mathbf{g}\_{\mathbf{h}} = \frac{\mathbf{h}\_0^3}{4} \mathbf{y}\_{\mathbf{h}}^4 \tag{44b}$$

$$\mathbf{g}\_0 = \frac{1}{2} \tag{44c}$$

$$\mathbf{y}\_h = \frac{4\mathbf{Q}}{\|\mathbf{h}\|^3} \tag{44d}$$

$$\mathbf{r}\_{\text{hx}} = \frac{4\mu\_0 \mathbf{Q}}{\text{m}\text{h}^3} \tag{44e}$$

$$\left| \frac{\text{dp}}{\text{dx}} \right| = \frac{8\mu\_0 \text{Q}}{\text{m} \text{h}^4} \tag{44f}$$

$$\mathbf{p}\_0 - \mathbf{p} = \frac{8\mu\_0 \mathbf{Q}}{\mathbf{n} \cdot \mathbf{h}^4} \mathbf{L} \tag{44g}$$

$$\mathbf{u}(0,\mathbf{x}) = \mathbf{u}\_{\text{max}} = \frac{\mathbf{2Q}}{\|\mathbf{n}\|^2} \tag{44h}$$

$$\mathbf{u}(\mathbf{r},\mathbf{x}) = \mathbf{u}\_{\text{max}} \left( \mathbf{1} - \frac{\mathbf{y}^2}{\mathbf{y}\_h^2} \right) = \mathbf{u}\_{\text{max}} \left( \mathbf{1} - \frac{\mathbf{r}^2}{\mathbf{h}^2} \right) \tag{44i}$$

$$\mathbf{u} = \mathbf{u}\_{\text{max}} \left( \mathbf{1} - \mathbf{g}\_0 \right) = \frac{1}{2} \mathbf{u}\_{\text{max}} \tag{44j}$$

Figure 8 shows velocity profiles in normalised form for the "three special cases" discussed above. For the general Bingham fluid with constant non-zero values of 0 and 0, the pipe velocity profile follows a parabolic curve close to the special pure viscous case (i), when fluid viscosity is dominant or the ratio 0/0 is small, and moves towards the uniform profile of the special case (ii), when plastic yield or cohesion is dominant or the ratio 0/0 is large, as shown in the figure. Figure 9 demonstrates an example of a Bingham plastic solution for radial disk flow, where the contribution of each of the two shear parameters is separately demonstrated.

Mechanics of Multi-Phase Frictional Visco-Plastic,

also consistent with the empirical equations.

experimental measurements reported in the literature [5].

0

5

10

**Log (-dp/dx)**

15

20

25

Fig. 10. Effect of frictional coefficient on pressure gradient for a given slurry

0 0.2 0.4 0.6 0.8 1 **Friction coefficient**

Non-Newtonian, Depositing Fluid Flow in Pipes, Disks and Channels 169

As a numerical pipe flow example, consider a slurry modelled by a linear Bingham plastic model, where 0 = 0.1Pa.s and 0 = 0.1Pa. As shown in Table 1, the maximum and average velocities in the pipe are 3.54 m/s and 1.76 m/s, respectively. The table also shows a list of

Slurry behaviour is controlled by its two distinct material components, i.e. the solid particles and the water. Depending on the velocity of the fluid and the terminal velocity and physical characteristics of the suspending solid particles, the slurry behaviour can evolve by two distinct characteristics, a uniform viscous fluid or a fluid with separated, submerged, sedimentation deposit; where the latter is the favourite mechanism in mining grout injection. The solid particle concentration or viscosity is constant in the former case and (increasingly) variable in the latter. The more the concentration of the particles, the greater is the effect of the frictional viscosity, as observed in our direct viscosity measurements and

yh (1/s) f(yh) g''(yh) g'(yh) g(yh) -dp/dx (Pa/m) umax (m/s) g0(yh) uave (m/s) hx (Pa) pmin (Pa) 141.804 14.280 61.179 2912.220 103969.296 571.217 3.504 0.498 1.760 14.280 571217.381

When working with slurries made of particulate and granular materials for injection operations in the field, it is quite possible to encounter pipe blockage. This is when the last term in Equation (7) or (14) becomes non-zero and hence dominates the process due to high frictional shear resistance against the slurry flow. Several laboratory blockage tests have been carried out to confirm the role and effects of this frictional term in Equation (14). In these experiments, initially the pump pressure was reduced gradually during an injection process to reduce flow velocity causing settlement and sedimentation of the grains until full blockage has occurred. An attempt to reopen the same blockage was made by increasing the pump pressure. However, a much higher than the initial pump pressure was required to reopen the blockage, confirming the effect of the frictional term in Equation (14). Figure 10 demonstrates how the pressure can increase rapidly before or behind a blockage, resulting in a substantial head loss. This theoretical exponential trend agrees with similar

values for several other variables and parameters used in the present theory.

Table 1. Numerical example for cohesive-viscous slurry ( = 0.1 Pa.s, = 0.1)

Fig. 8. Comparisons of normalised velocity profiles for different slurries of various viscosity () and plasticity () in a pipe flow

Fig. 9. Contribution of viscosity and cohesion to pressure drop for a Bingham plastic slurry in a radial disk flow

Case (i), pure viscous Case (ii), pure plastic

(vi),

(iii), (iv), (v),

Fig. 8. Comparisons of normalised velocity profiles for different slurries of various viscosity

0 0.2 0.4 0.6 0.8 1 **Pipe radial distance (r/h)**

> = 0.008 Pa.s 0 = 0.3 Pa

p p

<sup>3</sup> <sup>0</sup> r

πh 6 μQ

0 = 0.3 Pa

 

0

<sup>r</sup> ln

Fig. 9. Contribution of viscosity and cohesion to pressure drop for a Bingham plastic slurry

= 0.008 Pa.s

0 100 200 300 400 500 **Disc radial flow distance (radius), r (mm)**

() and plasticity () in a pipe flow

100

h=4mm Q = 20 lit/hr r0 = 2mm

0

20

40

**Pressure, p**

**0-p (Pa)**

60

80

0

0.2

0.4

0.6

**Pipe velocity (u/umax)**

0.8

1

in a radial disk flow

As a numerical pipe flow example, consider a slurry modelled by a linear Bingham plastic model, where 0 = 0.1Pa.s and 0 = 0.1Pa. As shown in Table 1, the maximum and average velocities in the pipe are 3.54 m/s and 1.76 m/s, respectively. The table also shows a list of values for several other variables and parameters used in the present theory.


Table 1. Numerical example for cohesive-viscous slurry ( = 0.1 Pa.s, = 0.1)

Slurry behaviour is controlled by its two distinct material components, i.e. the solid particles and the water. Depending on the velocity of the fluid and the terminal velocity and physical characteristics of the suspending solid particles, the slurry behaviour can evolve by two distinct characteristics, a uniform viscous fluid or a fluid with separated, submerged, sedimentation deposit; where the latter is the favourite mechanism in mining grout injection. The solid particle concentration or viscosity is constant in the former case and (increasingly) variable in the latter. The more the concentration of the particles, the greater is the effect of the frictional viscosity, as observed in our direct viscosity measurements and also consistent with the empirical equations.

When working with slurries made of particulate and granular materials for injection operations in the field, it is quite possible to encounter pipe blockage. This is when the last term in Equation (7) or (14) becomes non-zero and hence dominates the process due to high frictional shear resistance against the slurry flow. Several laboratory blockage tests have been carried out to confirm the role and effects of this frictional term in Equation (14). In these experiments, initially the pump pressure was reduced gradually during an injection process to reduce flow velocity causing settlement and sedimentation of the grains until full blockage has occurred. An attempt to reopen the same blockage was made by increasing the pump pressure. However, a much higher than the initial pump pressure was required to reopen the blockage, confirming the effect of the frictional term in Equation (14). Figure 10 demonstrates how the pressure can increase rapidly before or behind a blockage, resulting in a substantial head loss. This theoretical exponential trend agrees with similar experimental measurements reported in the literature [5].

Fig. 10. Effect of frictional coefficient on pressure gradient for a given slurry

Mechanics of Multi-Phase Frictional Visco-Plastic,

through the entire process of concrete pour or discharge.

a, b, c Shear stress function coefficients A, B, C Integral function coefficients

<sup>π</sup><sup>h</sup> A flow rate related constant

**n** Normal vector p Fluid pressure Q Flow rate

h 3 <sup>3</sup> Q Q

Ai, Bi, Ci Constants of Bingham plastic solution

**Appendix**  *Notation* 

Non-Newtonian, Depositing Fluid Flow in Pipes, Disks and Channels 171

Therefore, it must be self compacting, self levelling and maintain its original quality, homogeneity and integrity all the way from the tremie pipe to the discharge point and then through the narrow paths between heavy reinforcements. Similar to a viscous-plastic slurry or paste, shear behaviour of a fresh tremie pipe concrete was explained by a linear Bingham plastic model. Traditional slump and spread tests together with the L-box tests are used as indirect index tests to measure physical visco-plastic properties of concrete. However, the concrete industry needs also to develop a large scale viscometer testing method to measure viscosity and plastic yield of tremie pipe concrete directly. Based on the Bingham parameters, the governing relation between the steady state concrete flow rate and the required pressure gradient was presented. To maintain a successful, uniform, steady state flow in the tremie pipe, a balance pressure height must be determined and controlled

In the following derivations, italic symbols are used for normalised parameters, or quantities representing their counterparts denoted by the same non-italic symbols. For instance, *r* = r/r0 represents the normalised form of the radial distance variable r with

Italic symbols indicate normalized, or dimensionless quantities, e.g. the fluid velocity, *u* =

C Solid concentration by weight or mass = Cweight = Cvolume (solid/mix) f A polynomial function of fluid velocity gradient ( <sup>n</sup> f(y) ay by c )

( )h Index "h" denoting function value at either pipe or disk boundary walls

r Radial distance from pipe centre; polar r coordinate axis, disk radial

x1, x2, x3 Subscripts indicating longitudinal (radial in disk flow), normal to flow

g A polynomial function of fluid velocity gradient ( <sup>3</sup> g(y) fd <sup>y</sup> )

( )0 Index "0" denoting initial or constant value of a variable L Pipe length (L= x –x0) from reference section x0 to any section x.

n Shear strain power factor, Function power factor

 distance from borehole or disk centre in a radial flow r0 Radius of central vertical pipe connected to disk

 cross-section and circumferential coordinates, respectively x Longitudinal distance or coordinate axis along pipe length x0 Initial reference point in a pipeline section along x1 direction

respect to a reference distance r0, i.e. the radius of the central vertical pipe.

u/U0 is the dimensionless form of the dimensional counterpart quantity, u.

Fig. 11. Required pipe distance (from x0 to x) is nonlinearly proportional to pressure (p0 to p) for a linear uniform pipe. The slope of the relation depends on frictional-cohesive properties of the granular materials of the slurry. The higher the friction or cohesion is, the smaller the required distance at a given pressure differential is.

In practice, a blockage is usually reopened by pumping a less viscous fluid (e.g. water) at a very high pump pressure and minimum viscous shear resistance. The pump pressure required is a function of not only frictional properties of the deposited sediment, but also the size distribution of the aggregates (Figure 11).

#### **9. Conclusions**

On the basis of continuum equations of fluid and soil mechanics, a comprehensive, versatile, slurry shear model has been developed for transportation of grout, paste and fill materials used in the civil and mining industries, covering a wide range of material characteristics and behaviour, namely from the flowing fluid slurries to consolidated solid deposits in underground coal mining induced rock fractures. The theory has been specifically tailor made for grout flows through uniform pipes, discs and tremies, in order to transport material to designated injection or backfill targets. The theory can mimic both flow and blockage behaviour of the fill material. The tool can be used to predict variations of pressure and velocity and their gradients, as a function of flow rate, in the entire backfill-placement system from batching plant to the borehole cracks and foundation excavations.

The shear theory can mimic shear resistance of both: (i) a cohesive, viscous flow and (ii) a stationary, cohesive, pressure-dependent, frictional, plastic soil. The pressure dependent frictional term in the shear stress model determines the frictional resistance of the deposited fill material during a blockage. Consistent with laboratory and field experiments, the theoretical pump pressure required to open a blockage is orders of magnitude greater than the amount needed for pumping the same material when it is under a steady state flow. This explains why very high pump pressures are often needed to clean blockages compared with much lower pressures required during steady state slurry flows.

Concrete flow and placement into deep foundations is normally performed under several harsh environmental conditions of tightness, inaccessibility and deep submergence. Therefore, it must be self compacting, self levelling and maintain its original quality, homogeneity and integrity all the way from the tremie pipe to the discharge point and then through the narrow paths between heavy reinforcements. Similar to a viscous-plastic slurry or paste, shear behaviour of a fresh tremie pipe concrete was explained by a linear Bingham plastic model. Traditional slump and spread tests together with the L-box tests are used as indirect index tests to measure physical visco-plastic properties of concrete. However, the concrete industry needs also to develop a large scale viscometer testing method to measure viscosity and plastic yield of tremie pipe concrete directly. Based on the Bingham parameters, the governing relation between the steady state concrete flow rate and the required pressure gradient was presented. To maintain a successful, uniform, steady state flow in the tremie pipe, a balance pressure height must be determined and controlled through the entire process of concrete pour or discharge.

#### **Appendix**

#### *Notation*

170 Fluid Dynamics, Computational Modeling and Applications

Fig. 11. Required pipe distance (from x0 to x) is nonlinearly proportional to pressure (p0 to p) for a linear uniform pipe. The slope of the relation depends on frictional-cohesive properties of the granular materials of the slurry. The higher the friction or cohesion is, the smaller the

= 10 deg , = 0.01 Pa

0.0 0.2 0.4 0.6 0.8 1.0 *p* **0 -** *p* **(MPa)**

= 1 deg , = 0.01 Pa

= 1 deg , = 10 Pa

In practice, a blockage is usually reopened by pumping a less viscous fluid (e.g. water) at a very high pump pressure and minimum viscous shear resistance. The pump pressure required is a function of not only frictional properties of the deposited sediment, but also the

On the basis of continuum equations of fluid and soil mechanics, a comprehensive, versatile, slurry shear model has been developed for transportation of grout, paste and fill materials used in the civil and mining industries, covering a wide range of material characteristics and behaviour, namely from the flowing fluid slurries to consolidated solid deposits in underground coal mining induced rock fractures. The theory has been specifically tailor made for grout flows through uniform pipes, discs and tremies, in order to transport material to designated injection or backfill targets. The theory can mimic both flow and blockage behaviour of the fill material. The tool can be used to predict variations of pressure and velocity and their gradients, as a function of flow rate, in the entire backfill-placement

The shear theory can mimic shear resistance of both: (i) a cohesive, viscous flow and (ii) a stationary, cohesive, pressure-dependent, frictional, plastic soil. The pressure dependent frictional term in the shear stress model determines the frictional resistance of the deposited fill material during a blockage. Consistent with laboratory and field experiments, the theoretical pump pressure required to open a blockage is orders of magnitude greater than the amount needed for pumping the same material when it is under a steady state flow. This explains why very high pump pressures are often needed to clean blockages compared with

Concrete flow and placement into deep foundations is normally performed under several harsh environmental conditions of tightness, inaccessibility and deep submergence.

system from batching plant to the borehole cracks and foundation excavations.

much lower pressures required during steady state slurry flows.

required distance at a given pressure differential is.

Friction angle (deg) = = tan-1 Plastic yiled (Pa) =

size distribution of the aggregates (Figure 11).

0

2

4

*x-x*

**0 (m)**

6

8

10

**9. Conclusions** 

In the following derivations, italic symbols are used for normalised parameters, or quantities representing their counterparts denoted by the same non-italic symbols. For instance, *r* = r/r0 represents the normalised form of the radial distance variable r with respect to a reference distance r0, i.e. the radius of the central vertical pipe.

Italic symbols indicate normalized, or dimensionless quantities, e.g. the fluid velocity, *u* = u/U0 is the dimensionless form of the dimensional counterpart quantity, u.


Mechanics of Multi-Phase Frictional Visco-Plastic,

= fluid acceleration or velocity rate

,t

u

*v* 

W 

x 

**x** 

**x**ˆ 

y 

YD 

X0

y1 , y2

*x* or \* *x*

manuscript.

**11. References** 

direction

**10. Acknowledgements** 

Co, New York.

Ltd., Great Britain.

u

t

Non-Newtonian, Depositing Fluid Flow in Pipes, Disks and Channels 173

U0 = fresh concrete velocity at L-box entrance (reference velocity)

*<sup>k</sup> ui* = finite difference velocity function *u* at time *k* and position *i* 

= width of L-box (in out of plane z direction)

= flow direction and distance in L-box test

= position vector with components x, y, z

ξ(t, ) **x** = friction coefficient function when concrete blockage occurs

= a dimensionless function representing fluid velocity

= x axis and flow direction in tremie pipe and L-box test

= two end coordinates of L-box horizontal open channel

This work was conducted within the Subsidence Control Research Program of CSIRO. Industry support from BHP Billiton, ACARP (C16023; C12019) and CIA of Australia is gratefully acknowledged. The Author would like to thank CSIRO colleagues Dr Jane Hodgkinson and Dr Cameron Huddlestone-Holmes for their review of the initial

[1] Govier, G.W., Aziz, K. 1972. *The flow of complex mixtures in pipes*, Van Nostrand Reinhold

[3] T. Yen Na and A.G. Hansen, 1967, "Radial flow of viscous non-Newotonian Fluids

[4] Alehossein H. 2009, "Viscous, cohesive, non-Newtonian, depositing, radial slurry flow", *International Journal of Mineral Processing*, V. 93 No. 1, 2009, pp. 11-19. [5] Alehossein H., Shen, B., Qin, Z. and Huddlestone-Holmes, C.R. 2012, "Flow analysis,

and mining engineering", *ASCE, Journal of Materials in Civil Engineering*. [6] Alehossein, H., Beckhaus, K. and Larisch, M. 2012, "Analysis of L-Box test for tremie

[7] Beckhaus, K., Larisch, M., Alehossein, H., Ney, P., Northey, S., Lucas, G., Dux, P.,

Recommended Practice published by *the Concrete Institute of Australia (CIA)*.

between disks", *Int. J. Non-linear Mechanics.* Vol. 2, pp. 261-273. Pergamon Press

transportation and deposition of frictional, viscoplastic slurries and pastes in civil

Buttling, S., Lucas, G., Vanderstaay, L. 2011. *Tremie Concrete for Deep Foundations*".

[2] Middleman, S., 1977*. Fundamentals of Polymer Processing, Mcgraw*-*Hill*, NY.

concrete, *ACI, Journal of American Concrete Institute*.

= height drop along L-box horizontal open channel (y1y2)

= vector normal to pisition vector for velocity gradient calculations

= maximum concrete flow distance in L-box test (reference length)

= y-axis coordinate; vertical position in L-box test; pipe flow radial

*u* or \* *u* = dimensionless flow velocity function in L-box test



#### **10. Acknowledgements**

This work was conducted within the Subsidence Control Research Program of CSIRO. Industry support from BHP Billiton, ACARP (C16023; C12019) and CIA of Australia is gratefully acknowledged. The Author would like to thank CSIRO colleagues Dr Jane Hodgkinson and Dr Cameron Huddlestone-Holmes for their review of the initial manuscript.

#### **11. References**

172 Fluid Dynamics, Computational Modeling and Applications

u = u1 Longitudinal velocity (u1) in both pipe and disk radial flow **u** Velocity vector with velocity components (u1, u2, u3=u)

Viscosity coefficient (linear term)

' = yx1 Derivative of with respect to x2

df/dt f Local time derivative of a function f

u1/x2 Shear strain rate (velocity gradient) Viscosity coefficient (non-linear) Viscosity coefficient (linear) Density, slurry density

tan Friction coefficient, friction angle

 Final far field value of a property Value corresponding to property

μ/μ0 Viscosity coefficient (linear)

τ/τ Shear stress, cohesion

 ξ/ξ Friction coefficient p = fluid pressure function

η/η0 Viscosity coefficient (nonlinear)

D /Dt Total time derivative

Gradient vector

 

*Subscripts* 

Q 

0 τ 

u 

uy 

u

,y

0

0

u

y

 Volume concentration of solids in slurry mix = ux1 Radial velocity times radial distance function

d /dj Derivative with respect to a coordinate axis j /j Partial derivative with respect to a coordinate axis j

w Weight concentration of solids in slurry mix

 Volume concentration of solids in slurry mix Circumferential (hoop) coordinate axis

Shear stress, cohesion (yield stress), stress tensor

0 Initial value, reference value for normalisation

= volume rate or fluid flow rate

= plastic yield or cohesion intercept in linearised Bingham plastic model

= fresh concrete or fluid velocity, x-component of velocity in 1D model

= shear stress function of vsicosity, plastic yield and shear strain gradient

= y component of fluid velocity in 2D model

= y gradient of velocity in 1D model, shear strain rate

 Value corresponding to properties , respectively <sup>0</sup> *x* x/ x Gradient velocity divided by radius, 0 0 r0 0 x r u /h X /h

 -Ratio of disk radial velocity gradient and x2, dY/dx2 Y -Radial velocity times distance x1, integral of

Generic symbol representing either one of functions: , , ,

ut Terminal velocity or free fall, submerged solid particle limit speed uD Deposition velocity or particle speed at minimum pressure gradient


**8** 

*USA* 

Timothy J. Madden

*Kirtland Air Force Base, New Mexico* 

**Three Dimensional Simulation of** 

*US Air Force Research Laboratory, Directed Energy Directorate* 

**Gas-Radiation Interactions in Gas Lasers** 

The spectroscopically measured lineshape of an atomic transition provides a wealth of useful information relative to diagnosing the state of a gas. The center of the lineshape is specific to a particular transition of a specific atom. The width of the lineshape indicates the amount of broadening of the transition, due to the effects of both collisions with other particles in the gas and Doppler shift due to the movement of the atom. Since the Doppler shift is proportional to velocity, the width of the transition can be used to estimate the degree of random molecular motion in the gas, expressed macroscopically as temperature. A Doppler shift to the frequencies in the transition can also occur through the bulk motion of the gas, and this can be used to examine the velocity field of the gas. The astronomy community was the first to recognize the utility of these concepts in practical applications, stemming to the early 1930s.1,2 In 1934 Stuve and Elvey3 showed that by including a bulk gas velocity Doppler broadening term in the Voigt equation for the transition lineshape in addition to the random thermal motion term, it was possible to estimate whether a stellar atmosphere was 'turbulent' or not based upon the fit of the Voigt equation to the measured transition lineshapes for that atmosphere. Taking the theory a step further, using estimated optical paths or length scales for the stellar atmospheres that they were measuring, they were able to estimate median gas velocities and correlate increasing velocity magnitude

More recently, continuously tunable diode lasers have been applied to lineshape measurement of transitions within species in the chemical oxygen-iodine laser (COIL) flowfield as a means to determine number density and laser gain on the I 2P1/2→2P3/2 transition. Davis and Allen et al4,5,6,7 applied lineshape measurement and Voigt fitting methods to various COIL species as a mechanism to determine concentration and translational temperature in the COIL flowfield as an experiment diagnostic. While the effect of the bulk gas velocity was not taken into account in these investigations, the diagnostics developed by these investigations provided the means to do so. Nikolaev et al,8 applied the same type of continuously tunable diode laser to the I 2P1/2→2P3/2 transition to investigate the influence of mean gas velocity through a COIL device upon the laser gain. They showed that by varying the angle at which the diode laser beam passes through the COIL flowfield, the line center gain can be varied through bulk velocity induced Doppler broadening. Using this mechanism, they were able to determine the mean flow velocity for the COIL and convert the laser gain

**1. Introduction** 

with increasing temperature of the stellar atmosphere.

