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

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

slurry solid particles must deposit in the opened strata bed separation gaps or cracks before crack closure [10][11][12][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]. 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][7][8][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]. 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.

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][5][6][7][8][9] 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][5][6][7][8][9]. See Figure 2 for a visual definition of the different shear terms and parameters involved in Equation (1 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.

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].
In Equation (2), u  is fluid velocity vector, n  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 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: Applying the general conservation Equation (3) to mass and momentum, the Navier-Stokes isothermal equations of continuity and momentum [15] are recovered: Dt  is the gradient vector,  is the fluid density, σ   is the stress tensor and g  is the body acceleration or gravity vector. The stress tensor depends on a mean fluid normal stress or pressure ( ) and a deviator stress representing shear stresses  ij , which depends on fluid viscosity and velocity gradients.
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: 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: 1. the flow is laminar and the fluid is incompressible, steady state, stationary, and isothermal and axisymmetric with no eddies and no gravity effects; 2. axially symmetric condition implies that the radial flow component (in a pipe) and circumferential flow (in a disk) must vanish. In other words u x2 = u x3 = 0; 3. when there is a full blockage (Q = 0), the friction term p is a dominant term; 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, mechanical behaviour of the "slurry".
We now introduce a new set of independent variables to represent coordinate axes of both radial disk and pipe flow systems, namely: x 1 axis is always in the direction of the main flow direction, i.e. either radial disk flow, or longitudinal pipe or channel flow, x 2 is the axis normal to the flow direction in the flow cross-section plane, and x 3 is the hoop or circumferential axis direction identical to hoop angle .

Reduced one dimensional equations
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 (u 2 , u 3 ) are zero and there is no variation in velocity or pressure with time. In other words, we have 11 2 uu u ( x , x ) We now define two separate flow gradient functions [4][5], (a) gradient with respect to x 1 and (b) gradient with respect to x 2 , viz.
Hence, the general, basic equations of continuity (4) and momentum (5) reduce to Equations (12) for pipe flow and (13a,b) for radial flow.
PF (Pipe flow): RF (Radial Flow): 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, p x , as suggested in [2]. The shear stress 12 (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].
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: Substituting (14) in (12) and (13) and integrating over x 2 , will give us the following pressuregradient equations PF: RF: The pressure gradient Equations (16) and (17) must be satisfied at all points, including the boundary point, h. Therefore, at the wall boundary point (x 2 =h) we have PF: where y h ,  h and  h are the boundary values of y,  and , i.e. at the point x 2 = h. In other words, If y h 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    PF: For example, the pipe pressure can be written in a general integral form (Equation (22)) in terms of the integral coefficients A 1 , B 1 , C 1 [5].
In Equation (22), v(x 1 ) is an exponential function of x 1 and h and A 1 , B 1 , C 1 are integral coefficients similar to those produced for radial flow [4], viz.
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 in which  s is the normalised slope and n  is the general power factor for any nonlinear behaviour. In other words, Therefore, the integrals (24) have the following general non-dimensional form which can be integrated numerically [4][5]: However, for constant properties of slurry, the A 1 , B 1 , C 1 parameters reduce to either a simple exponential function of x=x 1 , in the presence of a friction coefficient, i.e.  ≠0, or a simple linear function in terms of pipe length (L = x-x 0 ), 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. 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].

Solutions for pipe and radial flow
The pressure gradient dependency in all Equations (16) In Equation (31) where u(0) is the maximum velocity at the flow centre line given by 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 y h and g(y h ) 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 y h . 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].

Tremie pipe concrete flow
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 x 2 is the radial distance from the tremie pipe's axis of symmetry (Figure 1, middle) [6][7][8][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)  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), x 2 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 4 th order polynomial function of the tremie pipe diameter, D. Furthermore, the flow rate is inversely proportional to the viscosity,  0 and (partially) proportional to the differential pressure at the two ends of the tremie pipe [6], p, of length L t . In other words we have: 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,  c and inclination angle .

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 x 2 and x 3 compared to the longitudinal main flow direction x 1 . In other words, which gives a solution in terms of Fourier coefficients [6]   In the solution (43)  is an arbitrary constant satisfying both the differential equation and the boundary conditions, while A n and B n 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.

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.
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 0h 1 g( y ) 2  confirming the classical result, max rr 1 u( r ) u 2  . Furthermore, we have: 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. 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].   . Required pipe distance (from x 0 to x) is nonlinearly proportional to pressure (p 0 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).

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.  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.

Notation
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/r 0 represents the normalised form of the radial distance variable r with respect to a reference distance r 0 , i.e. the radius of the central vertical pipe.
Italic symbols indicate normalized, or dimensionless quantities, e.g. the fluid velocity, u = u/U 0 is the dimensionless form of the dimensional counterpart quantity, u. Radius of central vertical pipe connected to disk x 1 , x 2 , x 3  Subscripts indicating longitudinal (radial in disk flow), normal to flow cross-section and circumferential coordinates, respectively x Longitudinal distance or coordinate axis along pipe length x 0 Initial reference point in a pipeline section along x 1 direction www.intechopen.com u = u 1 Longitudinal velocity (u 1 ) in both pipe and disk radial flow u Velocity vector with velocity components (u 1 , u 2 , u 3 =u  ) u t Terminal velocity or free fall, submerged solid particle limit speed u D Deposition velocity or particle speed at minimum pressure gradient  Viscosity coefficient (linear term)  Volume concentration of solids in slurry mix =  ux 1 Radial velocity times radial distance function ' = yx 1 Derivative of  with respect to