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Two-phase models for complex fluid phenomena: some examples
Pergamoo
TWO-PHASE
MODELS
FOR COMPLEX FLUID SOME EXAMPLES
PHENOMENA:
J. R. A. PEARSON Schlumberger Cambridge Research,High Cross, Cambridge, U.K. (Received
for publication 18
August
1993)
Abstract-Three phenomena in suspension continuum mechanics are discussedin this short paper. The firstconcernsthe developmentof gel and ultimatelyyield stress in a stagnant column of initially well-mixed
suspension. The second concerns the steady state that is observed to arise in wall filtration from a confined flowing stream, and the conditions needed to have an interface of discontinuity in solids concentration. The last concerns the dewatering of a dense suspension close to its equilibrium compaction when placed in a long column deviated from the vertical.The need for a multi-phaseapproachis emphasised.
1. INTRODUCTION
Many fluid flows encountered in industrial operations are complex in behaviour; an obvious characteristic usually is that more than one phase is present, as in suspensions, pastes, emulsions or foams. The fluid mechanics of such flows, by analogy with that of simple Newtonian liquids, is however often considered in terms of an equivalent single-phase fluid, albeit with non-Newtonian, thixotropic or elasticoviscous characteristics. The benefits of a continuum representation are thereby gained and formal methods of analysis can be developed. One of the most common assumptions made is that the fluid is rheologically simple: that the response to stress or strain of any fluid “particle” is governed by the history of stress or strain of that particular particle, and not of any of those around it. (This remark often leads to some confusion in that the histories of adjacent particles are themselves subject to the laws of motionenservation of mass and momentum-and hence are not independent; however, the rheological equation of state is.) We do not consider here the role of the temperature, which can be a significant factor in the rheological equation of state. An extremely clear and concise account of such fluid models is to be found in Astarita and Marrucci (1974) and Astarita (1975). There have been many successful applications of such models to industrial and natural flows, particularly where the scale lengths of the flows have been much larger than that of any microstructure of the multi-phase fluid involved and where inertial forces have been important. However, difficulties have arisen when interfacial effects at boundaries have departed from the standard fluid-mechanical concepts of no slip or no shear stress, where boundaries have been porous to one but not all of the phases, or where non-uniformities in the composition of the mixed fluid are evident. In such cases, it sometimes happens that the gradients of field variables become so large that internal length scales of the flow become small enough to approach those of the 727
microstructure; this in turn leads to a weakening of the ‘*simple fluid” approximation. Terms involving higher-order spatial derivatives of field variables become relevant. These high-gradient effects can either be included directly in the rheological equation of state [as in Cosserat materials-see Truesdell and No11 (1965, Section 98)] or as diffusive terms in evolution equations for ancillary structural variables [e.g. Pearson (1968, 1993), Billingham and Ferguson (1993)]. When filtration takes place at porous but undeforming interfaces, the gradient in fluid properties can become so large that no single manageable rheological equation of state is adequate to describe the full range of behaviour. The usual approach is to seek simplified approximations for different regions, and where possible to separate the regions to which these apply by interfaces. Lastly, temporal fluctuations often inevitably accompany spatial fluctuations in fluid properties (chaotic if not turbulent behaviour) and so two- or multiphase models have to be. cast in terms of averaged quantities, thus generating continuum mixture models that are more detailed than single-phase models but not fully descriptive of behaviour on structural length scales. It is thus not always easy to choose the right parametric form for such mixture models. In the rest of this article, a few examples of topical industrial interest are analysed to illustrate some of these points. The aim is not to provide highly detailed, perfectly modelled explanations of precisely measured phenomena, but to point to similarities between apparently separate effects and to argue that a correct choice of class of model (representing the dominant physical and chemical factors) is crucial to any engineering investigation into complex industrial processes involving suspensions. In Section 2, relatively dilute suspensions based on gelling clay “thickeners” (added to increase viscosity and confer a finite yield stress) are considered; phenomena involving relative motion of the two phases, i.e. sedimentation, filtration and migration under shear gradients, are analysed. In Section 3, similar
J. R. A.
728
problems in dense suspensions where the solid phase is the key component, e.g. ceramic pastes and cement slurries, are considered. Here the problem is to retain the necessary amount of liquid to ensure a low enough viscosity and yield stress to allow pumping without suffering severe sedimentation (dewatering) effects. Each problem has to be treated appropriately.
2. CLAY
SUSPENSIONS-LOW
SOLIDS CONTENT
Bentonite suspensions containing no more than a few percent of dispersed clay platelets can exhibit substantial yield stresses. When weighted with particles of barite these suspensions form the basis of most water-based drilling muds used in the oil industry. Conventionally their flow rheology is described in terms of a single-phase model, e.g. a Herschel-Bulkley fluid model, where Z = (Ylyl-’
+ C,lylmp’)y,
y2 = (tr y2)/2 for
Z2 = (tr X2)/2 > Y y=O
forZ
Up to this point a purely single-phase model is sufficient to predict observations with acceptable accuracy. For very small amounts of mud removed an elastic relation I: = Gy.
Z being the extra stress, y the rate of deformation, Y the yield stress, C,,, the consistency and m the power-law index. The parameters Y, C,,, and m are normally obtained from simple-shear viscometry. Recent work [Bay&q et al. (1993) and others] has shown that these parameters, particularly Y, are functions of past shear rate; in simple parlance, these muds are thixotropic. For simple oilfield applications, the Bingham model with m = 1 and C1 = Q,, the plastic viscosity, is used. Quasi-static settling When such a mud is placed in a long vertical cylinder, of length L, and allowed to come to rest, the initial measured (isotropic) pressure P at the bottom is found, as expected, to be p,,,gL, where p,,,is the mean mud density and g is the gravitational acceleration. If a small amount of mud is withdrawn slowly from the bottom of the cylinder, the mud in the column moves downward whilst adhering to the inner wall of the cylinder and it is found that the pressure at the bottom of the hole drops (by much more than would be accounted for by the fall in the mud surface level) to an asymptotic value that is consistent with the full mobilization of a yield stress at the cylinder walls. For a cylinder of inner radius R, this maximum pressure drop (2)
where t, is the maximum wall shear stress. The observed value For rw may be less than the internal yield stress Y used in eq. (1) if the wall is very smooth, but will not exceed it provided the mud is withdrawn very slowly to avoid any viscous forces. For rapid pumping of mud through the cylinder, eq. (1) provides all that is needed to compute the dynamic pressure drop.
(3)
G being the shear modulus of the (incompressible) mud regarded as a gel, has to be added to eq. (I) to explain the rise of Z, from 0 to Y. This leads to a yield strain yr = YjG. However, it has been reliably observed that the pressure falls slowly by an amount less than or equal to the asymptotic limit (2) even when no mud is withdrawn from the bottom-or from anywhere else. How can we explain this? Only by considering the two-phase nature of the mud. Since pm > pw. where p,,, is the water density, the solid phase responsible for the gel structure and additional density in the mud must either sediment or support its own weight. Which of these arises depends on whether (Pm -
(1)
AP = 2r,L/R
PEARSON
p,)gR/Z
z=
or
c Y
(4)
(assuming 5, = Y). If the former then AP = 2 YLJR. If the latter then AP = (pm - p&L. To give some idea of the magnitudes involved, we can typically have p,,, - p,. = 200400 kg/m3, R=
l-5x10-2m,
Y = 10-100 Pa. From eq. (4) we see that (p,,, - p,)gR/2 lies in the range 10-100 Pa which is just that given for Y. For L=lm, AP=20004CM) Pa, an easily measured quantity, being 24% of atmospheric pressure. yr is a less easily measured quantity, and is less frequently quoted than Y. A typical range for )I~ is l-3 x 10e2 Pas, so a shear rate of 30&3OflO s-l is required to give a viscous contribution equal to the long-term yield stress. Consequences for rheometry A Further question can now be asked: What are the consequences of the above observations for rheometry? If we use a concentric cylinder viscometer with its axis vertical, we find that mud placed in the gap between the cylinders will start to sediment and thus to develop a vertical wall shear stress that can approach or even exceed the yield stress. In a very slow start-up experiment (y -+ 0), with the objective of determining G and Y, it is usually assumed that only a circumferential (azimuthal) stress and deformation will arise. Clearly this will not be the case, and in the limit no yield stress will be observed.
Two-phase models for complex fluid phenomena To predict what will happen in any particular case, the time scale of sedimentation and elastic deformation becomes important. In a single-phase model there is only one pressure in an undeforming column, the hydrostatic pressure of a fluid with density p,,,. In a sedimenting system viewed as a two-phase system (with no wall adhesion) the fluid-phase pressure is still the hydrostatic pressure of a fluid of density p,,,, except for a particle-free layer at the top and a particlerich layer at the bottom. The gradient is made up of the hydrostatic pressure gradient of the water phase of density pw together with a Darcy-type gradient that is determined by the permeability of the solid-particle phase falling at a velocity u,,~ relative to water. For well-designed muds this sedimentation velocity is small, although some relative motion of the various components of the solid phase may take place; when walls are present, as explained above, the gel-like nature of the solid phase will reduce or eliminate us.*. In the latter case, the water-phase pressure then becomes the hydrostatic pressure of a liquid of density pw, rather than pm. Once the mud is sheared beyond the yield strain, as in a rotating viscometer, the waterphase pressure then returns to that of a single-phase liquid of density p,,,. Dynamic mud cakes Some of the sedimentary formations into which wells are drilled-for oil or gas-are necessarily permeable. If the circulating mud is not to be lost into the permeable formations, whose own pore fluids tend to be at a pressure close to the hydrostatic pressure of water, the solid particles in the mud have to be chosen to form a filter cake of very low permeability at the surfaces of the permeable formations exposed by the drilled well. If the mud is not circulating, these cakes grow steadily at a rate proportional to t’12 (Philip and Smiles, 1982). If, however, the mud is circulating steadily giving rise to a constant wall shear stress r,, then the mud cakes reach an equilibrium thickness (Fordham et al., 1988). In an earlier paper [Pearson and Sherwood (1988), henceforth PS], this process was analysed using a two-phase fluid model that combined the soil-mechanics approach of Philip and Smiles for the compacted mud cake, an osmotic pressure interpretation leading to particle diffusion against the advective hltration flow, and a rheological equation of state generalising eq. (1) above for the bulk flowing mud. It was concluded that the model led to a filter cake that must grow as xi/J in the flow direction x. It was also pointed out that this did not agree with experiments on muds, although it correctly modelled observations in some other cross-flow-filtration processes. The key to resolving this difficulty is probably to be found in a solid concentration (or equivalently porosity) discontinuity at an effective interface between flowing mud and compacted filter cake. No more will be attempted here than an indication of how this could arise and of how a solution essentially independCES
49:5-K
729
ent of distance x can be obtained. As far as possible nomenclature compatible with PS will be used below. For simplicity we take a nearly unidirectional mean flow velocity WX(Y,ax) = wo(l - =+)f’(y), WJ= rwof(y)/h, s 4 1, f(0) = 0
(5)
between two parallel filtration walls at y = f h. The filtration velocity rwOf(h - 6)/h, where S < h is the thickness of the two equal filter cakes, is thus E times the mean axial flow velocity. Any slight variation of 8 with EXwill be neglected. We seek a steady concentration profile of the form c = c(y)(l
+ o&x).
(6)
For the core of the flow, dominated by advection, we suppose that II, the fluid velocity, and v, the particle velocity, are both equal to w, the mean velocity used in eq. (5). From the mass conservation relation d,(cw,) + d,(cw,) = 0
(7)
we see that u = h- r eliminates terms up to O(E’). We now address the relative motion of particles and fluid that is required to provide the steady-state filtration process at any glven x. The key equation for the region lyl c h - 6 is that which combines the advective particle flux u,c with the diffusive flux Dd~/ay driven by the “equivalent osmotic pressure” ~(c, ~?w,,Gy) to give a nil balance v,. = 0. For this flowing region, the strong colloidal forces [causing the moisture potential $(c)] that dominate in the closely packed filter cake can be neglected; the dominant contribution to the particle stress is the viscous extra stress (8)
z = rt(GY)Y + Y(e)r/r
together with the particle pressure ~1. The balance equation is, therefore, u,c = D(c)&(G
MY
(9)
with the shear rate 7 = aw,/ay given by the momentum conservation equation V(C,Y)Y + Y(c) = r&r-
(10)
If, for simplicity, we suppose that Y(c) + ?f = $-Jc,,
-c)_‘,
x=x0&
B>O
a> 0
TV,
and write (11) (12)
then eq. (9) leads, near the wall where u,. x urill/(1 - c) and y = h, to nn1tc/(l- c) = - D(c)a{xorv(e,,
- e)“cbltloVa~. (13)
Here we suppose that c,. is the value at which a fllter cake would exist, and that in the centre of the flowing layer c is significantly less than c,,. Equation (13) can
730
J. R. A.
be written as
PEARSON
subject to the Coulomb yield criterion [eq. (9) of PS]
UIilt)lolXOrW= (I - c)~(c)~-“(c,*x
-c)“_’
c/W,.. - c) - ac] dc/dy.
{ti(c) + x1 1 - q(c) Iti (14)
Since we expect dc/dy to be positive we are constrained to those values of c which make B(c*,* - c) - ac positive. This means that c < ci = Bcm../(B + co
(15)
in the flowing region. For a = O(p) this means a substantial discontinuity at y = h - 6 where the particle concentration goes from ci to a value 3 c,... The reason that PS rejected a solution of eq. (14) was that it seemed to lead rapidly to c = 0 as y decreased, and this was inconsistent with a c, > 0. However if, in eq. (14), r+u, were replaced by &w,f(y)/h, withf(y) R y near y = 0, to match up with eq. (S), a non-zero value of c(0) would be obtained with c(y) a monotonically increasing function. The matter will not be pursued further analytically here because we know little about the validity of the approximations (11) and (12) in the case of muds, or about D(c). Across the interface y = h - 6, we must satisfy the no-slip boundary conditions w, = u, = v, = 0
(16)
and stress continuity, i.e. S,, and S,
continuous
(17)
where the total stress tensor s=-pI+l-II+Z
(18)
p being the pressure in the liquid phase, n=e+x
(19)
being the sum of the “osmotic pressure” x and the colloidal interparticle pressure I,%,.with X the elastoplastic stress caused by shear strain or strain rate. The S,,, condition in eq. (17) implies p continuous, and (Z,, + l(l)lh--6+ = XL&
(20)
where Zur and $ are taken to be zero in the flowing region and x zero in the solid-cake region. The S, condition in eq. (17) is the constant shear stress condition &h,-6+
=
xxyik-6-
=
e,-
(21)
Our solution in the flowing region specifies the righthand sides of eqs (20) and (21). It merely requires that we investigate whether a solution exists within the compacting cake satisfying eqs (20) and (21). Using I;1 and & to denote the principal extra stresses in the mud cake, with Zi inclined at angle 0 to the y-axis, we write eqs (20) and (21) as Zr ~0s’ 0 + & sin* 0 + Jl(c) = xW (X1 + Z,)sinOcosO
= z,
(22) (23)
+ ~1
G X(c)
(24)
where q(c) is related to the internal angle of friction of the compacted cake and X(c) is the cohesive stress. Note that we appear to have four variables c, 0, Zi and X3 that we can alter to satisfy three conditions, (22)-(24), one of which is an inequality. A unique solution would probably be obtained if we sought the least value of c that would do this. (There is no guarantee that a solution will exist for all suspensions.) Again, to attempt any further analysis would not be justified in this article, because we have little information about the quantities q, I& and X in the just compacted layers of a mud cake formed from a flowing mud. It should be remembered that interfacial behaviour at the edge of the mud cake will be determined by the particulate behaviour of the flowing mud; by analogy with the bed load of a river, there may be a dynamic rather than quasi-static equilibrium, with the variables used in eqs (9)-(24) no more than temporal continuum averages. Recent work by Brady (Nott and Brady, personal communication) has interpreted the motion of particles dispersed and interacting in a flowing suspension in terms of an effective continuum “temperature’--different from a thermodynamic temperature because it is due wholly to the mean shear flow. This provides an alternative to the usual way of deriving diffusion and osmotic pressure from Brownian motion, itself a random process driven by the thermodynamic temperature. Although his simulations of a channel flow considered a suspension of spheres, it seems likely that a similar idea might be relevant for clay platelets and barite particles. Such an approach might provide estimates for some of the unknown functions introduced above.
3. DENSE
SUSPENSIONS-HIGH
SOLIDS
CONTENT
The previous section dealt with relatively dilute suspensions, where the solid phase was chosen to alter the properties of the liquid phase. In this section we consider the opposite situation, where only sufficient liquid is added to a solid powder to make it flow and deform smoothly, and in the case of cement slurries to provide the right amount of water for chemical hydration and reaction. As in the previous section, a general&d Newtonian model, currently the Herschel-Bulkley model (1) proves satisfactory for large-scale flow problems. Furthermore, filtration behaviour broadly similar, but by no means identical, to that described earlier arises. The main difference is that contributions to the particle pressure II from t&see eq. (19jwill cover a much smaller range in c, when the suspension is more sand-like than silt-like. As our illustrative problem, we consider here setting in an inclined cylinder where a process analogous to the “Boycott effect” can accelerate the formation of
Two-phase
free water, and hence, enhance very significantly the sedimentation rate over that predicted by a onedimensional vertical model. For convenience, we represent the cylinder by two flat parallel plates y = 0, H, which are inclined at angle 8 to the vertical with the x-axis defined by the intersection of the plane y = 0 with some vertical plane. We consider only plane flows (and deformations) in the (x, y)-plane. Initially the gap between the planes is filled from x = 0 to x = L by slurry of a uniform porosity cbo.The particle pressure II is related to the porosity in the range of interest by d4/dII = (4, - 4)/K,
4 z- 4, and II = trl’l (25)
where II is the particle stress. The density of the liquid is pv and of the solid p,; the permeability of the slurry is k and the liquid viscosity is Q,,. Both phases are assumed incompressible. We seek to model a phenomenon that is thought to involve two sequential processes: (a) sedimentation and compaction of the slurry in the y-direction, (b) gravity-driven counter-current flow of the compacted-slurry and free-water phases, both of which are treated as liquid-like. For process (a), we seek a solution independent of x and assume (as in much sedimentation and filtration theory) that the matrix stress II is isotropic, defined by TI. We assume that 41 differs by little from 4,,_ II, 4, p, the liquid pressure, ‘“y, the liquid velocity and v,,, the solid velocity, are functions of y and t. The equations governing the process can be shown to be
4 * 40 - 67sinfV40- 4M
- ~o)(P, - PA (33)
x(H-y-_-)/K where h = (P. - PA
(40
-
41) B sin OH*PK
(34)
is the small thickness of the free water layer. The solution has assumed that 4 x 4e at the interface. The time scale of the settling process is given by H%wt40 - 4,)/k(l - 4o)K. We now turn to process (b), the gravity-driven expulsion of water from the thin layer h in the x-direction. Some subtle arguments are needed to explain why sedimentation in the x-direction under the action of the component of gravity g cos 0 B g sin tI is neglected in the following analysis. These depend upon the ratio H/h being very much smaller than sin 8 and imply that advective terms (v, and u,) dominate over sedimentation ones. The same arguments also allow us to approximate the two-phase settled slurry lying between y = 0 and y = H - h as a uniform single-phase liquid, with density Pslr yield stress Y and plastic viscosity tlsr$- Q,,. The least satisfactory assumption that we shall make is that the interface between slurry and free water is stable. Thus, we look, over most of the height L of the cylinder, for a steady solution involving only the velocity components wX(y) in the slurry and u,(y) in the free-water layer, the shear stress x&y), the water layer pressure p(x) and the slurry pressure P(x). We defme the two fluxes H-h %I
=
I
0
H
W,(Y) dy,
qv =
s B-L
a,(Y) dY. (35)
Mass conservation requires
momentum: QH + P)/~Y = CA1 -
9.1+ 4w = 0.
4) f ~~41Bsin0 (24)
(36)
Stress equilibrium yields
mass conservation: a4/at = - a(4u,)/aY
(27)
4u, + (1 - 4)v, = 0
(28)
dpfdx + p,,.gcos 0 = dz,/dy = q,,.d2u,/dY2 (37) dP/dx + pJg sin tI = dr,/dy
(38)
dp/dx = dP/dx.
(39)
where
Darcy’s law: (u, - upI4 = WpPy
+ P,+.Bsin Wrtw
(29)
with boundary conditions u,(O, t) = Yy(O,t) = 0
(30)
and initial condition
zXg in eq. (38) will be given by the Bingham fluid version of eq. (1) with y = dw,/dy. The boundary and interfacial conditions are u,(H)
4(y,O)=40-
(31)
Note that all inertia and acceleration terms are neglected, as is usual in such analyses. From eqs (25)-(29) we obtain the compaction equation ~(40
731
models for complex fluid phenomena
- 4,)a4lat
= k(1 - 4o)K a24/8y2. (32)
For sufficiently long times an approximately linear steady-state solution for 4(y) is obtained:
= w,(O) = 0;
u,(H - h) = w,(H
rx,, continuous at H -
h.
-
hk
W)
Here, again for simplicity, a no-slip condition at the wall has been used for the slurry, but a more realistic condition might be cL,Iw,I = IL1 - y
lw,l=
0
for
for 174
lzwl s- y -c
for some frictional slip coefficient Pi.
Y
(41)
J. R. A.
732
Solution of eqs (35)-(41) is tedious but straightforward. The details are omitted. For the case Y = 0, pv + a~, we find that 4w = h”(p,, - P&7 cos ~(ttalh + Mf)/3~74 x(4rlJJ f %vH).
(42)
The time to expel the free layer of water is thus Lb/q,. A typical case is given by L = 100 m, H = 0.02 m, Ap = 1500 kg/m3, rlw= 10-j Pa s, h=4x 10V4m, g = 10 m/s2, giving an expulsion time of about 10 min. It is much more difficult to estimate values for Qo, &, k and K, which are needed to estimate the time scale of process (a); however for the cases studied, a value of about 1 min was observed to give substantial development of a free water layer on a shallow sedimenting column of slurry. The separation in time of processes (a) and (b) is therefore not unreasonable. If we now revert to the question of the stability of the interface between settled slurry and free water, we find that this has been studied in connection with the Boycott effect for initially much less dense suspensions, and is by no means a simple matter. Any instability will however change the distribution of weight and velocity in the inclined column and so affect any estimate of the expulsion time for free water. It is likely to increase it, possibly by one or two orders of magnitude. Even so, the settling time with which our figure of 10 min (or possibly several hours) has to be compared is that for vertical settling involving process (a) alone. Noting that the settling time would then be (L/H)’ times as great as the 1 min estimate given for crosschannel settling, we see that many days would be required. Clearly, only small departures from the vertical would accelerate the formation of free water at the top of a tall column of slurry.
4. DISCUSSION Readers may be disappointed that the analysis presented above is inconclusive and sketchy. They will no doubt have realised, and may even have expected, that detailed numerical (numerical because of the non-linear nature of the systems studied) solutions could have been obtained for wide ranges of all the dimensionless variables governing already very approximate model equations. To have done so would have been contrary to the philosophy underlying the work reported. The aim has been to provide conjectures, approximate models and elementary arguments to draw general conclusions that can be further investigated in
PEARSON
particular cases when necessary. Experience has shown that real industrial or natural processes are very complex and that use can seldom be made of detailed numerical calculations except to indicate trends. What I hope has been shown is that (1) even gelling suspensions of small particles can sediment sufficiently when the bulk is at rest to develop shear stresses of significance; (2) such behaviour can make low-shear-rate rheometry, particularly investigation of yield stress and strain, very ambiguous if parameters in singlephase models are being sought; (3) dynamic filtration is likely to be extremely sensitive to the nature of the particulate phase: a delicate balance between colloidal, drag and shear-induced forces, none of which is at all easy to estimate either theoretically or experimentally, will characterise the narrow interfacial region between the highly mobile flowing phase and any compacted filter c&e; (4) dewatering of dense suspensions in very tall coIumns with inclined walls is likely to be dominated by a version of the Boycott effect.
Acknowledgements-It is a pleasure to acknowledge the interesting and helpful conversations I have had with my colleagues at this laboratory, particularly John Sherwood, on the matters discussed above. REFERENCES Astarita, G., 1975, An Introduction to Non-linear Continuum Thermodynamics. Societa Editrice di Chin&a, Milan. Astarita, G. and Marrucci, G., 1974, Principles of NonNewtonian Fluid Mechanics. McGraw-Hill, Maidenhead. Baylock, P., Cartalos, U. and Piau, J.-M., 1993, Steady state and transient phenomena in a simple shear flow of model drilling fluids. Paper D19 presented at 64th Annual Mecting, Society of Rheology, Santa Barbara, U.S.A. Billingham, J. and Fetguson, J. W. J., 1993, Laminar unidirectional flow of a thixotropic fluid in a circular pipe, J. Non-Newtonian Fluid Mech. 47, 21-56. Fordham, E. J., Ladva, H. K. J., Hall, C., Baret, J.-F. and Sherwood, I. D., 1988, Dynamic filtration of bsntonite muds under different flow conditions. SPE Paper 18038. Annual Technical Conference, Houston, U.S.A. Pearson, J. R. A., 1968, Mass transfer in rheological behaviour, in Proceedings of 3rd All Union Heat and Mass Tramfer Conference, Minsk, Vol. 3, p. 167. Pearson, J. R. A., 1993, Flow curves with a maximum. .I. Rheol. (in press). Pearson, I. R. A. and Sherwood, J. D., 1988, Continuum modelling of cross-flow filtration. PhysicoChem. Hydrodyn. 10, 64761. Philip, J. R. and Smiles, D. E., 1982, Macroscopic analysis of the behaviour of colloidal suspensions. Adu. Colloid Interface Sci. 17, 83. Truesdell, C. and NOB, W.. 1965, The Non-linear Field Theories of Mechanics. Encyclopedia of Physics (Edited by S. Fliigge), Vol. III/3. Springer, Berlin.
TWO-PHASE
MODELS
FOR COMPLEX FLUID SOME EXAMPLES
PHENOMENA:
J. R. A. PEARSON Schlumberger Cambridge Research,High Cross, Cambridge, U.K. (Received
for publication 18
August
1993)
Abstract-Three phenomena in suspension continuum mechanics are discussedin this short paper. The firstconcernsthe developmentof gel and ultimatelyyield stress in a stagnant column of initially well-mixed
suspension. The second concerns the steady state that is observed to arise in wall filtration from a confined flowing stream, and the conditions needed to have an interface of discontinuity in solids concentration. The last concerns the dewatering of a dense suspension close to its equilibrium compaction when placed in a long column deviated from the vertical.The need for a multi-phaseapproachis emphasised.
1. INTRODUCTION
Many fluid flows encountered in industrial operations are complex in behaviour; an obvious characteristic usually is that more than one phase is present, as in suspensions, pastes, emulsions or foams. The fluid mechanics of such flows, by analogy with that of simple Newtonian liquids, is however often considered in terms of an equivalent single-phase fluid, albeit with non-Newtonian, thixotropic or elasticoviscous characteristics. The benefits of a continuum representation are thereby gained and formal methods of analysis can be developed. One of the most common assumptions made is that the fluid is rheologically simple: that the response to stress or strain of any fluid “particle” is governed by the history of stress or strain of that particular particle, and not of any of those around it. (This remark often leads to some confusion in that the histories of adjacent particles are themselves subject to the laws of motionenservation of mass and momentum-and hence are not independent; however, the rheological equation of state is.) We do not consider here the role of the temperature, which can be a significant factor in the rheological equation of state. An extremely clear and concise account of such fluid models is to be found in Astarita and Marrucci (1974) and Astarita (1975). There have been many successful applications of such models to industrial and natural flows, particularly where the scale lengths of the flows have been much larger than that of any microstructure of the multi-phase fluid involved and where inertial forces have been important. However, difficulties have arisen when interfacial effects at boundaries have departed from the standard fluid-mechanical concepts of no slip or no shear stress, where boundaries have been porous to one but not all of the phases, or where non-uniformities in the composition of the mixed fluid are evident. In such cases, it sometimes happens that the gradients of field variables become so large that internal length scales of the flow become small enough to approach those of the 727
microstructure; this in turn leads to a weakening of the ‘*simple fluid” approximation. Terms involving higher-order spatial derivatives of field variables become relevant. These high-gradient effects can either be included directly in the rheological equation of state [as in Cosserat materials-see Truesdell and No11 (1965, Section 98)] or as diffusive terms in evolution equations for ancillary structural variables [e.g. Pearson (1968, 1993), Billingham and Ferguson (1993)]. When filtration takes place at porous but undeforming interfaces, the gradient in fluid properties can become so large that no single manageable rheological equation of state is adequate to describe the full range of behaviour. The usual approach is to seek simplified approximations for different regions, and where possible to separate the regions to which these apply by interfaces. Lastly, temporal fluctuations often inevitably accompany spatial fluctuations in fluid properties (chaotic if not turbulent behaviour) and so two- or multiphase models have to be. cast in terms of averaged quantities, thus generating continuum mixture models that are more detailed than single-phase models but not fully descriptive of behaviour on structural length scales. It is thus not always easy to choose the right parametric form for such mixture models. In the rest of this article, a few examples of topical industrial interest are analysed to illustrate some of these points. The aim is not to provide highly detailed, perfectly modelled explanations of precisely measured phenomena, but to point to similarities between apparently separate effects and to argue that a correct choice of class of model (representing the dominant physical and chemical factors) is crucial to any engineering investigation into complex industrial processes involving suspensions. In Section 2, relatively dilute suspensions based on gelling clay “thickeners” (added to increase viscosity and confer a finite yield stress) are considered; phenomena involving relative motion of the two phases, i.e. sedimentation, filtration and migration under shear gradients, are analysed. In Section 3, similar
J. R. A.
728
problems in dense suspensions where the solid phase is the key component, e.g. ceramic pastes and cement slurries, are considered. Here the problem is to retain the necessary amount of liquid to ensure a low enough viscosity and yield stress to allow pumping without suffering severe sedimentation (dewatering) effects. Each problem has to be treated appropriately.
2. CLAY
SUSPENSIONS-LOW
SOLIDS CONTENT
Bentonite suspensions containing no more than a few percent of dispersed clay platelets can exhibit substantial yield stresses. When weighted with particles of barite these suspensions form the basis of most water-based drilling muds used in the oil industry. Conventionally their flow rheology is described in terms of a single-phase model, e.g. a Herschel-Bulkley fluid model, where Z = (Ylyl-’
+ C,lylmp’)y,
y2 = (tr y2)/2 for
Z2 = (tr X2)/2 > Y y=O
forZ
Up to this point a purely single-phase model is sufficient to predict observations with acceptable accuracy. For very small amounts of mud removed an elastic relation I: = Gy.
Z being the extra stress, y the rate of deformation, Y the yield stress, C,,, the consistency and m the power-law index. The parameters Y, C,,, and m are normally obtained from simple-shear viscometry. Recent work [Bay&q et al. (1993) and others] has shown that these parameters, particularly Y, are functions of past shear rate; in simple parlance, these muds are thixotropic. For simple oilfield applications, the Bingham model with m = 1 and C1 = Q,, the plastic viscosity, is used. Quasi-static settling When such a mud is placed in a long vertical cylinder, of length L, and allowed to come to rest, the initial measured (isotropic) pressure P at the bottom is found, as expected, to be p,,,gL, where p,,,is the mean mud density and g is the gravitational acceleration. If a small amount of mud is withdrawn slowly from the bottom of the cylinder, the mud in the column moves downward whilst adhering to the inner wall of the cylinder and it is found that the pressure at the bottom of the hole drops (by much more than would be accounted for by the fall in the mud surface level) to an asymptotic value that is consistent with the full mobilization of a yield stress at the cylinder walls. For a cylinder of inner radius R, this maximum pressure drop (2)
where t, is the maximum wall shear stress. The observed value For rw may be less than the internal yield stress Y used in eq. (1) if the wall is very smooth, but will not exceed it provided the mud is withdrawn very slowly to avoid any viscous forces. For rapid pumping of mud through the cylinder, eq. (1) provides all that is needed to compute the dynamic pressure drop.
(3)
G being the shear modulus of the (incompressible) mud regarded as a gel, has to be added to eq. (I) to explain the rise of Z, from 0 to Y. This leads to a yield strain yr = YjG. However, it has been reliably observed that the pressure falls slowly by an amount less than or equal to the asymptotic limit (2) even when no mud is withdrawn from the bottom-or from anywhere else. How can we explain this? Only by considering the two-phase nature of the mud. Since pm > pw. where p,,, is the water density, the solid phase responsible for the gel structure and additional density in the mud must either sediment or support its own weight. Which of these arises depends on whether (Pm -
(1)
AP = 2r,L/R
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p,)gR/Z
z=
or
c Y
(4)
(assuming 5, = Y). If the former then AP = 2 YLJR. If the latter then AP = (pm - p&L. To give some idea of the magnitudes involved, we can typically have p,,, - p,. = 200400 kg/m3, R=
l-5x10-2m,
Y = 10-100 Pa. From eq. (4) we see that (p,,, - p,)gR/2 lies in the range 10-100 Pa which is just that given for Y. For L=lm, AP=20004CM) Pa, an easily measured quantity, being 24% of atmospheric pressure. yr is a less easily measured quantity, and is less frequently quoted than Y. A typical range for )I~ is l-3 x 10e2 Pas, so a shear rate of 30&3OflO s-l is required to give a viscous contribution equal to the long-term yield stress. Consequences for rheometry A Further question can now be asked: What are the consequences of the above observations for rheometry? If we use a concentric cylinder viscometer with its axis vertical, we find that mud placed in the gap between the cylinders will start to sediment and thus to develop a vertical wall shear stress that can approach or even exceed the yield stress. In a very slow start-up experiment (y -+ 0), with the objective of determining G and Y, it is usually assumed that only a circumferential (azimuthal) stress and deformation will arise. Clearly this will not be the case, and in the limit no yield stress will be observed.
Two-phase models for complex fluid phenomena To predict what will happen in any particular case, the time scale of sedimentation and elastic deformation becomes important. In a single-phase model there is only one pressure in an undeforming column, the hydrostatic pressure of a fluid with density p,,,. In a sedimenting system viewed as a two-phase system (with no wall adhesion) the fluid-phase pressure is still the hydrostatic pressure of a fluid of density p,,,, except for a particle-free layer at the top and a particlerich layer at the bottom. The gradient is made up of the hydrostatic pressure gradient of the water phase of density pw together with a Darcy-type gradient that is determined by the permeability of the solid-particle phase falling at a velocity u,,~ relative to water. For well-designed muds this sedimentation velocity is small, although some relative motion of the various components of the solid phase may take place; when walls are present, as explained above, the gel-like nature of the solid phase will reduce or eliminate us.*. In the latter case, the water-phase pressure then becomes the hydrostatic pressure of a liquid of density pw, rather than pm. Once the mud is sheared beyond the yield strain, as in a rotating viscometer, the waterphase pressure then returns to that of a single-phase liquid of density p,,,. Dynamic mud cakes Some of the sedimentary formations into which wells are drilled-for oil or gas-are necessarily permeable. If the circulating mud is not to be lost into the permeable formations, whose own pore fluids tend to be at a pressure close to the hydrostatic pressure of water, the solid particles in the mud have to be chosen to form a filter cake of very low permeability at the surfaces of the permeable formations exposed by the drilled well. If the mud is not circulating, these cakes grow steadily at a rate proportional to t’12 (Philip and Smiles, 1982). If, however, the mud is circulating steadily giving rise to a constant wall shear stress r,, then the mud cakes reach an equilibrium thickness (Fordham et al., 1988). In an earlier paper [Pearson and Sherwood (1988), henceforth PS], this process was analysed using a two-phase fluid model that combined the soil-mechanics approach of Philip and Smiles for the compacted mud cake, an osmotic pressure interpretation leading to particle diffusion against the advective hltration flow, and a rheological equation of state generalising eq. (1) above for the bulk flowing mud. It was concluded that the model led to a filter cake that must grow as xi/J in the flow direction x. It was also pointed out that this did not agree with experiments on muds, although it correctly modelled observations in some other cross-flow-filtration processes. The key to resolving this difficulty is probably to be found in a solid concentration (or equivalently porosity) discontinuity at an effective interface between flowing mud and compacted filter cake. No more will be attempted here than an indication of how this could arise and of how a solution essentially independCES
49:5-K
729
ent of distance x can be obtained. As far as possible nomenclature compatible with PS will be used below. For simplicity we take a nearly unidirectional mean flow velocity WX(Y,ax) = wo(l - =+)f’(y), WJ= rwof(y)/h, s 4 1, f(0) = 0
(5)
between two parallel filtration walls at y = f h. The filtration velocity rwOf(h - 6)/h, where S < h is the thickness of the two equal filter cakes, is thus E times the mean axial flow velocity. Any slight variation of 8 with EXwill be neglected. We seek a steady concentration profile of the form c = c(y)(l
+ o&x).
(6)
For the core of the flow, dominated by advection, we suppose that II, the fluid velocity, and v, the particle velocity, are both equal to w, the mean velocity used in eq. (5). From the mass conservation relation d,(cw,) + d,(cw,) = 0
(7)
we see that u = h- r eliminates terms up to O(E’). We now address the relative motion of particles and fluid that is required to provide the steady-state filtration process at any glven x. The key equation for the region lyl c h - 6 is that which combines the advective particle flux u,c with the diffusive flux Dd~/ay driven by the “equivalent osmotic pressure” ~(c, ~?w,,Gy) to give a nil balance v,. = 0. For this flowing region, the strong colloidal forces [causing the moisture potential $(c)] that dominate in the closely packed filter cake can be neglected; the dominant contribution to the particle stress is the viscous extra stress (8)
z = rt(GY)Y + Y(e)r/r
together with the particle pressure ~1. The balance equation is, therefore, u,c = D(c)&(G
MY
(9)
with the shear rate 7 = aw,/ay given by the momentum conservation equation V(C,Y)Y + Y(c) = r&r-
(10)
If, for simplicity, we suppose that Y(c) + ?f = $-Jc,,
-c)_‘,
x=x0&
B>O
a> 0
TV,
and write (11) (12)
then eq. (9) leads, near the wall where u,. x urill/(1 - c) and y = h, to nn1tc/(l- c) = - D(c)a{xorv(e,,
- e)“cbltloVa~. (13)
Here we suppose that c,. is the value at which a fllter cake would exist, and that in the centre of the flowing layer c is significantly less than c,,. Equation (13) can
730
J. R. A.
be written as
PEARSON
subject to the Coulomb yield criterion [eq. (9) of PS]
UIilt)lolXOrW= (I - c)~(c)~-“(c,*x
-c)“_’
c/W,.. - c) - ac] dc/dy.
{ti(c) + x1 1 - q(c) Iti (14)
Since we expect dc/dy to be positive we are constrained to those values of c which make B(c*,* - c) - ac positive. This means that c < ci = Bcm../(B + co
(15)
in the flowing region. For a = O(p) this means a substantial discontinuity at y = h - 6 where the particle concentration goes from ci to a value 3 c,... The reason that PS rejected a solution of eq. (14) was that it seemed to lead rapidly to c = 0 as y decreased, and this was inconsistent with a c, > 0. However if, in eq. (14), r+u, were replaced by &w,f(y)/h, withf(y) R y near y = 0, to match up with eq. (S), a non-zero value of c(0) would be obtained with c(y) a monotonically increasing function. The matter will not be pursued further analytically here because we know little about the validity of the approximations (11) and (12) in the case of muds, or about D(c). Across the interface y = h - 6, we must satisfy the no-slip boundary conditions w, = u, = v, = 0
(16)
and stress continuity, i.e. S,, and S,
continuous
(17)
where the total stress tensor s=-pI+l-II+Z
(18)
p being the pressure in the liquid phase, n=e+x
(19)
being the sum of the “osmotic pressure” x and the colloidal interparticle pressure I,%,.with X the elastoplastic stress caused by shear strain or strain rate. The S,,, condition in eq. (17) implies p continuous, and (Z,, + l(l)lh--6+ = XL&
(20)
where Zur and $ are taken to be zero in the flowing region and x zero in the solid-cake region. The S, condition in eq. (17) is the constant shear stress condition &h,-6+
=
xxyik-6-
=
e,-
(21)
Our solution in the flowing region specifies the righthand sides of eqs (20) and (21). It merely requires that we investigate whether a solution exists within the compacting cake satisfying eqs (20) and (21). Using I;1 and & to denote the principal extra stresses in the mud cake, with Zi inclined at angle 0 to the y-axis, we write eqs (20) and (21) as Zr ~0s’ 0 + & sin* 0 + Jl(c) = xW (X1 + Z,)sinOcosO
= z,
(22) (23)
+ ~1
G X(c)
(24)
where q(c) is related to the internal angle of friction of the compacted cake and X(c) is the cohesive stress. Note that we appear to have four variables c, 0, Zi and X3 that we can alter to satisfy three conditions, (22)-(24), one of which is an inequality. A unique solution would probably be obtained if we sought the least value of c that would do this. (There is no guarantee that a solution will exist for all suspensions.) Again, to attempt any further analysis would not be justified in this article, because we have little information about the quantities q, I& and X in the just compacted layers of a mud cake formed from a flowing mud. It should be remembered that interfacial behaviour at the edge of the mud cake will be determined by the particulate behaviour of the flowing mud; by analogy with the bed load of a river, there may be a dynamic rather than quasi-static equilibrium, with the variables used in eqs (9)-(24) no more than temporal continuum averages. Recent work by Brady (Nott and Brady, personal communication) has interpreted the motion of particles dispersed and interacting in a flowing suspension in terms of an effective continuum “temperature’--different from a thermodynamic temperature because it is due wholly to the mean shear flow. This provides an alternative to the usual way of deriving diffusion and osmotic pressure from Brownian motion, itself a random process driven by the thermodynamic temperature. Although his simulations of a channel flow considered a suspension of spheres, it seems likely that a similar idea might be relevant for clay platelets and barite particles. Such an approach might provide estimates for some of the unknown functions introduced above.
3. DENSE
SUSPENSIONS-HIGH
SOLIDS
CONTENT
The previous section dealt with relatively dilute suspensions, where the solid phase was chosen to alter the properties of the liquid phase. In this section we consider the opposite situation, where only sufficient liquid is added to a solid powder to make it flow and deform smoothly, and in the case of cement slurries to provide the right amount of water for chemical hydration and reaction. As in the previous section, a general&d Newtonian model, currently the Herschel-Bulkley model (1) proves satisfactory for large-scale flow problems. Furthermore, filtration behaviour broadly similar, but by no means identical, to that described earlier arises. The main difference is that contributions to the particle pressure II from t&see eq. (19jwill cover a much smaller range in c, when the suspension is more sand-like than silt-like. As our illustrative problem, we consider here setting in an inclined cylinder where a process analogous to the “Boycott effect” can accelerate the formation of
Two-phase
free water, and hence, enhance very significantly the sedimentation rate over that predicted by a onedimensional vertical model. For convenience, we represent the cylinder by two flat parallel plates y = 0, H, which are inclined at angle 8 to the vertical with the x-axis defined by the intersection of the plane y = 0 with some vertical plane. We consider only plane flows (and deformations) in the (x, y)-plane. Initially the gap between the planes is filled from x = 0 to x = L by slurry of a uniform porosity cbo.The particle pressure II is related to the porosity in the range of interest by d4/dII = (4, - 4)/K,
4 z- 4, and II = trl’l (25)
where II is the particle stress. The density of the liquid is pv and of the solid p,; the permeability of the slurry is k and the liquid viscosity is Q,,. Both phases are assumed incompressible. We seek to model a phenomenon that is thought to involve two sequential processes: (a) sedimentation and compaction of the slurry in the y-direction, (b) gravity-driven counter-current flow of the compacted-slurry and free-water phases, both of which are treated as liquid-like. For process (a), we seek a solution independent of x and assume (as in much sedimentation and filtration theory) that the matrix stress II is isotropic, defined by TI. We assume that 41 differs by little from 4,,_ II, 4, p, the liquid pressure, ‘“y, the liquid velocity and v,,, the solid velocity, are functions of y and t. The equations governing the process can be shown to be
4 * 40 - 67sinfV40- 4M
- ~o)(P, - PA (33)
x(H-y-_-)/K where h = (P. - PA
(40
-
41) B sin OH*PK
(34)
is the small thickness of the free water layer. The solution has assumed that 4 x 4e at the interface. The time scale of the settling process is given by H%wt40 - 4,)/k(l - 4o)K. We now turn to process (b), the gravity-driven expulsion of water from the thin layer h in the x-direction. Some subtle arguments are needed to explain why sedimentation in the x-direction under the action of the component of gravity g cos 0 B g sin tI is neglected in the following analysis. These depend upon the ratio H/h being very much smaller than sin 8 and imply that advective terms (v, and u,) dominate over sedimentation ones. The same arguments also allow us to approximate the two-phase settled slurry lying between y = 0 and y = H - h as a uniform single-phase liquid, with density Pslr yield stress Y and plastic viscosity tlsr$- Q,,. The least satisfactory assumption that we shall make is that the interface between slurry and free water is stable. Thus, we look, over most of the height L of the cylinder, for a steady solution involving only the velocity components wX(y) in the slurry and u,(y) in the free-water layer, the shear stress x&y), the water layer pressure p(x) and the slurry pressure P(x). We defme the two fluxes H-h %I
=
I
0
H
W,(Y) dy,
qv =
s B-L
a,(Y) dY. (35)
Mass conservation requires
momentum: QH + P)/~Y = CA1 -
9.1+ 4w = 0.
4) f ~~41Bsin0 (24)
(36)
Stress equilibrium yields
mass conservation: a4/at = - a(4u,)/aY
(27)
4u, + (1 - 4)v, = 0
(28)
dpfdx + p,,.gcos 0 = dz,/dy = q,,.d2u,/dY2 (37) dP/dx + pJg sin tI = dr,/dy
(38)
dp/dx = dP/dx.
(39)
where
Darcy’s law: (u, - upI4 = WpPy
+ P,+.Bsin Wrtw
(29)
with boundary conditions u,(O, t) = Yy(O,t) = 0
(30)
and initial condition
zXg in eq. (38) will be given by the Bingham fluid version of eq. (1) with y = dw,/dy. The boundary and interfacial conditions are u,(H)
4(y,O)=40-
(31)
Note that all inertia and acceleration terms are neglected, as is usual in such analyses. From eqs (25)-(29) we obtain the compaction equation ~(40
731
models for complex fluid phenomena
- 4,)a4lat
= k(1 - 4o)K a24/8y2. (32)
For sufficiently long times an approximately linear steady-state solution for 4(y) is obtained:
= w,(O) = 0;
u,(H - h) = w,(H
rx,, continuous at H -
h.
-
hk
W)
Here, again for simplicity, a no-slip condition at the wall has been used for the slurry, but a more realistic condition might be cL,Iw,I = IL1 - y
lw,l=
0
for
for 174
lzwl s- y -c
for some frictional slip coefficient Pi.
Y
(41)
J. R. A.
732
Solution of eqs (35)-(41) is tedious but straightforward. The details are omitted. For the case Y = 0, pv + a~, we find that 4w = h”(p,, - P&7 cos ~(ttalh + Mf)/3~74 x(4rlJJ f %vH).
(42)
The time to expel the free layer of water is thus Lb/q,. A typical case is given by L = 100 m, H = 0.02 m, Ap = 1500 kg/m3, rlw= 10-j Pa s, h=4x 10V4m, g = 10 m/s2, giving an expulsion time of about 10 min. It is much more difficult to estimate values for Qo, &, k and K, which are needed to estimate the time scale of process (a); however for the cases studied, a value of about 1 min was observed to give substantial development of a free water layer on a shallow sedimenting column of slurry. The separation in time of processes (a) and (b) is therefore not unreasonable. If we now revert to the question of the stability of the interface between settled slurry and free water, we find that this has been studied in connection with the Boycott effect for initially much less dense suspensions, and is by no means a simple matter. Any instability will however change the distribution of weight and velocity in the inclined column and so affect any estimate of the expulsion time for free water. It is likely to increase it, possibly by one or two orders of magnitude. Even so, the settling time with which our figure of 10 min (or possibly several hours) has to be compared is that for vertical settling involving process (a) alone. Noting that the settling time would then be (L/H)’ times as great as the 1 min estimate given for crosschannel settling, we see that many days would be required. Clearly, only small departures from the vertical would accelerate the formation of free water at the top of a tall column of slurry.
4. DISCUSSION Readers may be disappointed that the analysis presented above is inconclusive and sketchy. They will no doubt have realised, and may even have expected, that detailed numerical (numerical because of the non-linear nature of the systems studied) solutions could have been obtained for wide ranges of all the dimensionless variables governing already very approximate model equations. To have done so would have been contrary to the philosophy underlying the work reported. The aim has been to provide conjectures, approximate models and elementary arguments to draw general conclusions that can be further investigated in
PEARSON
particular cases when necessary. Experience has shown that real industrial or natural processes are very complex and that use can seldom be made of detailed numerical calculations except to indicate trends. What I hope has been shown is that (1) even gelling suspensions of small particles can sediment sufficiently when the bulk is at rest to develop shear stresses of significance; (2) such behaviour can make low-shear-rate rheometry, particularly investigation of yield stress and strain, very ambiguous if parameters in singlephase models are being sought; (3) dynamic filtration is likely to be extremely sensitive to the nature of the particulate phase: a delicate balance between colloidal, drag and shear-induced forces, none of which is at all easy to estimate either theoretically or experimentally, will characterise the narrow interfacial region between the highly mobile flowing phase and any compacted filter c&e; (4) dewatering of dense suspensions in very tall coIumns with inclined walls is likely to be dominated by a version of the Boycott effect.
Acknowledgements-It is a pleasure to acknowledge the interesting and helpful conversations I have had with my colleagues at this laboratory, particularly John Sherwood, on the matters discussed above. REFERENCES Astarita, G., 1975, An Introduction to Non-linear Continuum Thermodynamics. Societa Editrice di Chin&a, Milan. Astarita, G. and Marrucci, G., 1974, Principles of NonNewtonian Fluid Mechanics. McGraw-Hill, Maidenhead. Baylock, P., Cartalos, U. and Piau, J.-M., 1993, Steady state and transient phenomena in a simple shear flow of model drilling fluids. Paper D19 presented at 64th Annual Mecting, Society of Rheology, Santa Barbara, U.S.A. Billingham, J. and Fetguson, J. W. J., 1993, Laminar unidirectional flow of a thixotropic fluid in a circular pipe, J. Non-Newtonian Fluid Mech. 47, 21-56. Fordham, E. J., Ladva, H. K. J., Hall, C., Baret, J.-F. and Sherwood, I. D., 1988, Dynamic filtration of bsntonite muds under different flow conditions. SPE Paper 18038. Annual Technical Conference, Houston, U.S.A. Pearson, J. R. A., 1968, Mass transfer in rheological behaviour, in Proceedings of 3rd All Union Heat and Mass Tramfer Conference, Minsk, Vol. 3, p. 167. Pearson, J. R. A., 1993, Flow curves with a maximum. .I. Rheol. (in press). Pearson, I. R. A. and Sherwood, J. D., 1988, Continuum modelling of cross-flow filtration. PhysicoChem. Hydrodyn. 10, 64761. Philip, J. R. and Smiles, D. E., 1982, Macroscopic analysis of the behaviour of colloidal suspensions. Adu. Colloid Interface Sci. 17, 83. Truesdell, C. and NOB, W.. 1965, The Non-linear Field Theories of Mechanics. Encyclopedia of Physics (Edited by S. Fliigge), Vol. III/3. Springer, Berlin.