The Influence of the Proximity of a Solid Wall on the Consistency of Viscous and Plastic Materials

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Schofield, R. K. and Blair, G. W. S. 1930. The Influence of the Proximity

of a Solid Wall on the Consistency of Viscous and Plastic Materials.

Journal of Physical Chemistry (1952). 34 (2), pp. 248-262.

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THE

INFLUENCE

OFTHE

PROXIMITY

OF ASOLID WALL ONTHE CONSISTENCYOFVISCOUS AND PLASTIC MATERIALS*

BY R. K. SCHOFIELD AND G. W. SCOTT BLAIR

Introduction

In attemptingto derivean _expressionfor the rate of flow ofa viscousor

plastic material through a _straight narrow tube of uniform cross-section under a _{pressure gradient, it} is _{usually assumed:}

(1) thateachparticle of thematerialmoves withconstant _velocityin a

straight line parallel to the axis.

(2) thatthereisno _{slip at the wall of the tube.}

(3) thatthevelocitygradient at anypointdependsonlyon the shearing

stressat thatpoint.

Usingtheseassumptions, it isshownbelowthat, no _{matter how complex}

the relationship between _{velocity gradient and shearing} stress (so long as

theformerisfixed when thelatteris_{fixed), the volume extruded in unit time}

will _depend, for a _givenstress at the _{wall of the tube, upon the} cube of the radius. While thisistrue for_{fluids, and} is alsotrue or _nearlytrue for

thick

paste„ of soil and otherminerals, it islound not to begenerally true of such pastes when examined over an extended _range of concentration.

Dis-crepancieswouldoccur if condition (1)were invalidated_{owing toturbulence;}

butreasons are _givenfor_{considering it unlikely that} turbulenceisresponsible

for the effect, It is foundthat the mean _velocity, instead of being

propor-tional to the radius,isdivisible into two parts,one _proportionalto it andthe other independent of it, Thesecond term _{apparentlyrepresents} a _velocity

imparted to the bulk of the materialby an excessive_velocitygradient near

the wall of the tube, suggestingthat the proximityof a solidwall influences the consistency of these materials and causes a breakdown of condition (3). On subtracting the secondterm, the contribution made to themean _velocity

bythe flowing of thebulkofthematerialis_{presumablyleft,}from which

con-sistency constants relating to the material in bulk can be obtained inde-pendent of the dimensions of the tube.

Theoretical.

If condition (1) of the introduction be _granted, so that _{the particles} are

not accelerated,thestress, W, at the wall ofa tube of length L and radius R

to whicha_pressuredifference Pis_appliedis

_PR/aL;

while the stress, S,at a

pointT within the tube anddistantrfrom theaxisis_{Pr/aL. Consequently}

r can _{be expressed}in terms of Sthus:

r =

R/W.

_{S. (1)}

*

Soil Physics Department, RothamstedExperimental Station, Harpenden, England

Downloaded via ROTHAMSTED RESEARCH LTD on August 20, 2019 at 10:15:04 (UTC).

If v bethe_velocityat _T,we _maywrite inaccordance with condition (3)

dv/dr

=

-f(S).

Substitutingthe value ofr _{givenbyequation (i),}andintegrating

v =

R/Wj^f(S)dS,

(ii)

if,in accordancewith (2) v = _o when _S = W. The flow dV, betweenr

andr

₊

dr = 27rdr.v. _{Substituting for}r andv from _{equations (i) and}

(ii) andintegrating,

"

w./"s

/",(s,dsds

»

From thisitisclear_{that, forany given}material,V/ttR3shoulddependonlyon

W if the three conditions are fulfilled.

By making specific assumptions about the form of f (S), V/ttR3 can be evaluated. Thus using the Maxwell assumptionthat

f (S) = MS

where µ (the

fluidity)

is_{the reciprocal of the viscosity, equation (iii)} reduces to Poiseuille’s equation in theform

V

irR3

1

mW. 4

In the same _{way the expression based} on _{the Bingham1 assumption} can

be deduced. Here it is supposed that thematerial does not flow unless a

stress exceeding a critical _{value, SQ,} be _applied to it and _that, at stresses higherthan S0, the velocitygradient equals µ (S —

S„). Againµ is a

con-stant having the dimensions ofa_{reciprocalviscosity, and}is_{usually called the}

mobility. Whensucha materialisforcedthrougha _tube,a central cylinder

of radius RS0/W, within whichthe stressis less than So, moves as a solid plug, andonlythematerialoutsidethis cylinderflows. WhenWislessthan

So, no flow _occurs, andV = _o. In substituting in equation

_(iii)

_to obtain

the value of V whenWexceedsS0,it must berememberedthat, sincef(S) is discontinuous, being zero from o to SQand _µ (S —

S0)fromS0to W, the

in-tegrations must becarriedout intwo stages. Thishasthe effect ofsplitting

V into two _{terms, thus}

y rs. r w z-w z-w

Sv=wd

8 _M(S-So)dS.dS

+^JSa

8

J

M(S-So)dS.dS.

The first is the contribution _{of the plug, the} second is that of _{the flowing}

material between it and the wall. Thisreduces to V

irR3

equation1 deduced by Buckingham,2 and independently by Reiner.3 The graph connectingV/ttR3 andW correspondingto this_equationis_tangential to the W axis atW = _S0. It _is

stronglycurvedforvaluesofWonlyslightly

exceedingSc,butat higher values approximates to a_straightline_{of equation} V

ttR3

This linemakesan _intercepton the Waxisequalto 4/3.S0, and,likethe

cor-responding graph of the Poiseuille equation, hasa _slopeof 1/4. µ. The true

curve is _{steeplyasymptotic}to the

limiting

straight line, the _{discrepancy in}

V beinglessthan1% whenWexceeds 2.2 timesthe_interceptofthe

_limiting

straightline.

If the parabolicrelation

f(S) =

µ Sn

of the Ostwaldtype4 be assumed, equation

(iii)

reduces to

V

3 n

₊

3 . µ .WQ

which is _{equivalent to the equations given by Farrow, Lowe and Neale,5} and PorterandRao.6 The constant µ hasdimensions which depend on the exponent n. The graph in this case starts from the origin, and is curved

throughout its length. The curvature is never _{very strong,} but it decreases

only slowly with increasing W. Experiments.

The modified Bingham plastometer used in this work has already been described.7 The paste to be investigated is made by mixing the soil, clay

or other mineralwith waterinto asmooth_paste,whichisthen forcedthrough

a one hundred mesh-per-inch sieve to remove _any coarse _particles. The

paste is then diluted to the required concentration, and sucked into the plastometer bulbs which, for the high-stress work here described, have a

capacity of 100 c.c. each. The material is forced alternately from one bulb

into the otherthroughone of a seriesof standardised tubes, the level in the bulbs being keptapproximately thesame _by

_tilting

_{the whole system about} a _{pivot. For this}more accurate workwith larger bulbs it becomes necessary to correct the _pressures for the resistance _{offered by the bulbs themselves.} Forthispurpose the bulbsare connected _{directly with} one _{another, and the}

pressurescorrespondingto aseriesof volume-flowsare measured. By

graphi-1

Buckingham’s equation contains an additionalterm whichis negligibly small when

W exceeds S. and whichisreferred to below.

2_E.

Buckingham:J.Am. Chem.Soc.,Test.Mat.,1921, 1154. 3_{Reiner:Kolloid-Z.,39, 80 (1926), etc.}

4_{Wo. Ostwald (and others): Kolloid-Z.,36, 99,}

157, 248 (Zsigmondy Festschrift) 252

(1928);38,261(1926);41,56, 112 _(1927). (Thistypeof equationwas,ofcourse, not origi-nated by Ostwald,buthe hasmademuchuse ofit.)

1_{Farrow, Lowe and Neale: J.Textile Inst., 19,T 18 (1926).} 6_{PorterandRao:Trans. FaradaySoc., 23,}

311 (1927).

7_{G.W. Scott}Blair_{and E.} M. Crowther:_J.

cal _{intrapolation} the correction, Pb, corresponding to eachvolume-flow can

be estimated. As no _appreciable increase in resistance is caused by

intro-ducing a fewmillimeters of narrow _tubingbetween the bulbs, it _{may safely}

be concluded

that

no kinetic _{energy correction}isnecessarywiththese

meas-urements. The tubes hadbeencarefullyselectedwith a view to uniformity

ofboreand were _{standardised by weighing thequantity} of mercury required

to fill them. A series of constant pressures are _{applied by} means of

com-pressed air, the pressure being measuredon awater or _{mercury manometer}

according to its magnitude. The air displaced by the clay is allowed to escape through an air-capillary of suitable dimensions. The pressure dif-ference (negligible in comparison with the applied pressure) ismeasured on an alcohol manometer at an _angle of one _{in ten (the flow-meter), and} is

directly proportional to the volume of flow ofpaste persecond. The mois-ture content of the _{paste is}determined byheating a sample for one hour in

an oven at a _{temperature of i6o°C. The concentration, K, is expressed} as

the number of grams ofdrymatter per ioogpaste. Volume concentrations

are calculatedon thebasisofa constant _{specificgravity for}the _{dry material} of 2.7.

In_{carrying out} thiswork effortshave been madetouse as widearangeof

radii as _{possible. Although it} is hoped in the future to increase this still

further, difficulties will firsthavetobeovercome. Thus theuse of very wide

capillaries involves large volumes of material and consequently big bulb correction (always difficult to _{determine accurately).} Moreover the

in-creased lengththat must be _{given to the tube necessarily entails}a sacrifice of uniformity in thebore. Thus beyond certainlimits, thelossin accuracy rendersfurtherincrease in radius ofno _advantage. With _very narrow tubes

so coarse a _systemas a_soil-paste behaveserratically.

In Table I are _given, as an _{example, the complete data}for a _sample of Broadbalk Field subsoil similarto that of which the clay-fraction has been used _{in the previous work.}

Table I

Plastometric datafor Broadbalksubsoilpaste (33.5gdrysoilper 100 _gpaste).

Cap. II. R =

0.093 cm. L = 12.

P a V

8.0 _3.0 2.0

9.0 4.2 2.7

10.0 4.4 3.0

II-O _{5-2 3-5}

12.0 5-7 3-7

13.0 6.7 4.6

14.0 7.0 4.8

cm.

Pb S V.

I .I 3-5 75

I -3 3-9 1.05

1 _{4 4-4} 1.10

i-5 49 I -3°

i-5 5-4 1.42

1.8 _5-7 1.68

1.8 6.2 1_-75

252

Cap.III. R = ₀

.073 cm. L = _{12.10 cm.}

P a V Pb S V.

14.0 2.8 1.9 1.0 _{5-2 i-i3}

16.0 3-7 2-5 I -3 5·9 1.49

20.0 4.2 2.9 1_{-4 7·4 i-73}

22.0 _5-i 3-5 1_-4 8.2 2.08

24.0 5-7 3-9 i-5 9.0 2.32

26.0 6.1 4.2 1.6 9-7 2.50

28.0 6.8 4.6 1.8 . 4 2-74

O 7-3 5'° 1.8 I I .2 2.97

Cap. 7 R = 0.059 cm L = 10 60.cm

P a V Pb s V.

15.0 i-5 1.0 _{0.7 5-3 0-93}

20.0 2.4 1.6 I .0 _7· 1.48

25.0 3-2 2.0 I .I 8.g 1.98

0>Ob 4.0 2.6 I -3 I I

.I 2.50

35-o 4-8 3-i i-4 12.5 3 00

Cap. IV. R = ₀

.048 cm. L = _{12.25 cm.}

P a V Pb s V.

16.0 °-5 34 0.6 3-9 47

24.0 1_{-°S .72 0.7} 6.0 99

28.0 1_-3 .88 0 ₇ 7-o 1.22

32.0 1.6 1.09 0.8 8.0 1 _5°

38.0 i-95 i-33 0.8 9-5 1.85

40.0 2.0 1_.36 0.8 10.0 1.88

42.0 2.1 _i-43 0.8 i°-5 1.98

44.0 2.25 i-53 1.0 II .0 2.12

Cap. V. R = _0.040 L = _{12.30 cm.}

P a* V Pb s V.

20.0 _o-3 .06 0.6 4.2 .40

25-o o-5 .10 0.6 _{5 3} .66

3°·° 0.7 T3 0.6 6.4 93

35 ° _0.85 .16 0.6 ₇ 4 1-i3

40.0 1_-°5 .20 0.6 8-5 1_-35

45·0 1.2 _•23 0.6 9-6 1.60

Bulbsalonewithout capillary.

(A water manometer was used,but P isgiven convertedintocm. _Mercury).

P 0.5 1.0 _1.5 2.0

a *0.2 0 2.0 6.0 8.0

3' O.4 0 _{1.4 4.I 5-4}

*

(V/a = _.19)·

P is pressureincm. mercury,

a isflowmeter reading (V/a = _0.68).

V isvolume flowincm5/secs.

Pb is pressure due to bulbsincm. mercury.

S isstressin_{dynes/mm.2 (calculated from} P—Pb).

It is well known that the data obtained with such an _{apparatus when}

plottedon a _V/ttR3— Wbasisfall into two

groups. In one, which may

con-venientlybecalled_typeA the pointsfora _singletube lie,

within

thelimitsof

experimental error,on astraightline passingthroughthe origin. Intheother

?o

20

NO RIGIDITY, NOSPREADING

GLYCERINE WATER MIXTURE

V/lTR3 I

/ ·_{tPAOujs O.H5lcm.} 0.093$cm I

a 0766c mfROUG»£N£D

0.0730cm.

---0.0593cm. O.OAÜO cm.

OOAOAcm.

—

/

STRESS(w)

(OYNES/mm*)

PISIOITY NO PEGULAf? 5PP£AD/NG

9POAOPALK P/ELO A/P-OPY

y/7r/%3 SO/L·fN=5?.OX) AO

50

J3,

-20

STRESS (YV)

(OYNES/MM.2)

NO R/G/O/TY,QUTSPREADING

PASTE Of CLAY ERACT/ON _fK=2.36Z)

v/m?3

)l-

-09-5TRE55 _(x/J

DYNESfNXp)

-0 9-0.7.

RlG/OITY ANO SPREADING

KA OUN {K=₃7_2%)

v/mz3

-Bp

STRESS (YV)

OYRES/MN1)

10 14 2 i 10

Fig. 1

Legendsincorporated infourcorners ofeachquarterof drawing.

(type B) thisis not the case. Our own measurements not _only confirmthis fact butshow thateach group must be further subdivided accordingas the

curve obtainedisor is not _{independent of R. There}are thusfour_{possibilities,} an instanceofeachwhichisgiven in Fig. 1.

With the water-glycerinemixture _(type

_Ai)

a _{singlestraightline through}

254

(type A2) Here thebestline thoughpassingthrough theoriginhas a_larger

slopethe smallerthe radius. Thus the disconcertingfactishererevealedthat

a straightUne_throughtheoriginobtained witha _singletubeisnot _{by itself}a

proofthat Poiseuille’s lawisbeingobeyed.

ThethickpasteofBroadbalkFieldsurface soil gives points which,though

more erraticthanthosefor_{water-glycerine,}showno _regulartrendwith_change

of radius. Nevertheless _they cannot be_represented as _falUng on a _straight

linethroughtheorigin (type

Bi).

Thebehaviour of thick soilpastes has al-readybeendescribedindetail in the earlierpaper,1 whereitis shownthatthe curves can be interpreted inthe _{light of the Bingham postulate. New} and

more accurate measurements over a wider concentration range has shown

thata sUghtspreading noticeablein some of theearlier data andattributed

to experimental error cannot be so _explained. In thinner _pastes of _soils,

clays andsimple mineralssuchas_{barytes and gypsum the spreading} is_very marked. A kaolin paste of moderate consistency is given as an _example

(type B2). Here as with_type A2 _{V/ttR3 for}a _{given value of W}increases as

Rdecreases.

An alternative _way is to regard type B2 as the general case: the other three beingspecialand simpler cases. Such a viewraises_{the question} as to

whetherall_{these systems}are _{susceptible to the}same _treatment,andcan

there-fore be _{represented by}a _{single though complex, equation. Already} in the

interpretationof curves oftheBi type,two distinctschoolsofthoughthave developed, one of whichbasesitstreatment on _{Bingham’s postulate and the}

otheron the Ostwald postulate. Thisisnot theplaceto enter intoa _general

discussionoftherelative advantages of the two methods,sufficeitto saythat

allthe soil and mineralpastesinvestigated inthis laboratoryare more

amen-ableto the first method; and that although there undoubtedly are _systems

suchas benzene-rubber and_pastesof atleast some starches that give curves

ofa_shapenot accounted_{for by}the simple Bingham postulate,itis neverthe-less true that much of the data which is represented as _conforming to a

relationship of the Ostwaldtype can aswellbecited in support of the simple linear relationship. (Vide Herschel and Bulkley,2 Porst and Moskowitz,3 Scott Blair,4 Ostwald5).

Hatschek6 has_{criticised the practice of}extrapolatingflow-curves bymeans

of straight lines, and considersthat, failing a discontinuous change from a

curvedtoa_{straight portion,}thechoiceof the_portiontoberegardedas_straight

is arbitrary and a matter of scale. _According to the Bingham treatment thesestraightlinesare _{asymptotes to which the true flow-curve approximates} more and more _closelyasthestress increases. It is clearingeneralthat the

error involved in drawingan_asymptotetoan_experimentalcurve _dependson the

1_{G.W.ScottBlairandE.M. Crowther:J.}

Phys. Chem.,33,321 (1929). 8_Herscheland

Bulkley: Ind. Eng. Chem., 16,927 (1924) etc.; Proc. Am. Soc. Teat. Mat., 26, 621 (1926).

3_{Porst andMoskowitz: J.Ind.}

Eng. Chem.,14, 49(1922).

4_{ScottBlair: Kolloid-Z.,}

47,76(1929).

5_{Ostwald:Kolloid-Z.,}

47, 176 (1329).

4_{Hatschek:“The}

steepnessof approach. It wouldcertainlybedifficult to draw_{the asymptote} toarectangular hyperbola givenonlya_portionof the curve;butthe criticism

losesits force where the curve is of the Buckingham-Reiner type. In this

case, already noted, the discrepancy in Vis less than 1% for values of W

greater than2.2 timesthe intercept. Abovethis

limit

the difference between

the true curve and the

limiting

straightline should be outside thelimits of experimentalerror. This isborne out _{in practice} with soil and clay pastes,

so

that

linear_{extrapolationappearsjustified with}thesematerials. Discussion.

These _experiments yield the result that for many viscous suspensions (typeA)as wellas plasticpastes(typeB),V/ttR3doesnot_dependonlyon W. As thevariation oftheformer _quantitysometimes approachestwofold for a

twofold variation of radius, the effectis_{evidently quite}outsidethelimitsof

ordinaryexperimental error. Moreover thefact that the apparatus_gives a

very satisfactory verification of Poiseuille’s lawfortrue fluids indicates

that

it works satisfactorily. The effect has _{every appearance of being} genuine. The next stepthereforeisto seekitscause. Thismust lieina breakdown of

one or more of the three conditions set out in the introduction. These will

beconsideredin turn.

A breakdown of the condition that the particles move with a constant

velocityparallel to theaxiswouldoccur ifthe flowwere _{turbulent. Although} no _{complete theory}of turbulent flowhas yet been advanced,it is _generally consideredthat itspresence ismarkedbya_fallingoffintheslopeofthe

flow-curve as the stress isincreased. Such a _falling off does occur at _veryhigh

stresses,particularly withthemore dilute_suspensions;butthecurves obtained

with a view to _elucidating the effect underdiscussion were not followed far enoughfor this to happen, andas can beseen fromthe_{examplesgiven,}show

no _signs of curvature over the range examined. It mightbe _urged

that

the close_{approximation}to_linearity arisesfrom a chancecancellationof opposite curvations, inamanner similartothatsuggested toexplain the“Laminarast’’

or linear _portion of the flow-curves obtained by Ostwald and Auerbach.1 As_{against this,} it should be_{pointed out} thatmany hundreds of flow-curves

for _{clay and soil pastes} have now been accumulated in thislaboratory, and notone reliablecurve has beenobtained which cannotbe

_fairly

_{represented by}

a _straight line at _{sufficiently high} stresses. It seems inconceivable that an

exact cancellation of two unconnected tendencies should occur in all these

cases. It is more reasonable to _{interpret the straightness} as an indication

that

thesematerialsare _{obeying both condition(1)and the Bingham postulate}

andtoendeavourtodeducean equation of the Buckingham-Reinertypebased

on a modificationof conditions _{(2) and (3).}

Until the results to be expected when flow is turbulent have been more

fully

worked out, it is _impossible to exclude _{it altogether} from the possible

causes _contributingtothe effect. All

that

can besaid_{at present}isthatthere

1_{Oatwald andAuerbach:Kolloid-Z.,}

isno _positiveevidencethatturbulenceis presentin these_experiments. This

statement applieswithequalforceto the‘structure’ typeofturbulence postu-lated by Ostwaldasto themore _{general type.}

On _plotting the mean _{velocity V/ttR2 rather than V/ttR3 against W,} a

regularity becomes apparent. A typical set of curves plotted in this way

fromthe data in the tableisshown_{in Fig.} 2. Sinceaccordingto the

Bucking-ham-Reiner equation (see above) the points should lie above the

limiting

straightline bymore than_{1% at values}ofWlessthan2.2 timestheintercept

these have been _omitted, and_onlythose for stresses above 4.4 dynes/mm2 have been used for locating the

limiting

straight lines. As these have a

common _{intercept their slopes} would be proportional to R were _V/ttR3

straightline can in all cases be drawnthrough the points, but for typesA2 and B2 thisdoes_not, when _{extrapolated,} pass through the origin, but gives

a _{positive intercept} on the _{slope axis.} Curves connecting and R which

may convenientlybe spoken of as derivedcurves are _{given in Fig. 3}for the

foursetsofcurves _{of Fig.} 1. Thederivedcurve obtained from Fig.2_together

withones for_pastesof barytes and_gypsumare _{given in Fig.}4. If turbulence

0 .05 .10 .15

Fig. 3 Derivedcurves

were the sole cause of the effect, the disturbance would presumablybemost

marked in the widesttube. In theseconditions themean _{velocity for}a_given

stress on the wall should be more _{nearlyproportional} to the radius with the smaller tubes than with the wider ones. In other words the derived curve

should approximate more _{closely to} a _{straight line through the origin the}

smallerthe radius. It will beseen thatthecurves in Figs.3and4takenas a

whole donot support thisidea.

If,

on the other hand, the derived curves are in _{reality straight} lines (as shown), the factthatsome _givean _{interceptmight}be_interpretedas_indicating

that conditions (1) and ₍₃₎are _{fulfilled, but that} in such cases there isa _slip

258

the radiusandone _{independent of}it. It seems more _{probable however}that

thesecond_{component instead of representing}an _{actual slip}atthewall should be_regardedas a _{velocity imparted to the}bulk of the _{material by} excessive flowing of the material in the immediate

vicinity

of the wall. Provided

that

the thickness of the region in which thisexcessive flow takes _{place is} both

independent of the radiusand also_{small in comparison}with

it,

the effecton

Fio.₄ Derived Curves

themean _velocitywouldbethesame asthatofa_{slip at the wall itself. There}

would however be a difference in tubes so narrow that the radius is of the

same order of magnitudeasthe thickness of the region ofexcessive_flow,as in

thiscase the derivedcurve wouldbend_{round towards the origin.}

Thefirst _{component of the}mean _velocity,sinceit is_proportionalto Ris presumably due to the flow of the material in bulk, and _{therefore equal} to

V/tR2

calculated from _{equation (iii) using the appropriate value} for f(S).

In cases wherethe Bingham postulate is _{applicable to the} material in bulk

thefirst _component will be that given by the Buckingham-Reiner equation andthe slope of the derivedcurve will _{equal \µ.} Thesecond component of

mean _{velocity is equal,according to the constructions in Figs.} 2 and 3, to

thecommon _interceptoftheflowcurves _(Fig.2)on theWaxis _(which accord-ing to the Buckaccord-ingham-Reiner equation should equal 4/3 So)

The existence of a _very

thin

_layer of fluid of the consistency of water, separating the paste fromthe wallhas beenassumedbyBuckingham to

ac-countforthe small movement whichoccurs at stressesso low thatthe bulk

of the material does not flow. Thisview, slightly modified, was _{adopted in}

the earlier paper,where it was _{shown experimentally}that thisflowisrelated to thestressthus:

V/irR2 = _v =

_{(W—A) (vii)}

A beingaconstantstressbelow whichno movement occurs,ethe thickness

of the layer and its mobility. It mightat firstappearthatthesecond

com-ponent of the velocityis thesame as theabove. _{This, however,} can _hardly

bethecase since 0 isfound to havea _magnitudesome 100 times

that

of

_{c .}

Forthis reason thesecond_component, unlikethe_{Buckinghamterm,} cannot

be_{neglected at high}stresseswhenthematerialisflowing. Assumingavalue of equal to the

fluidity

of water in bulk, the thickness e is of the order io~5cm. The corresponding thickness calculatedfrom 0isof the orderio_3cm Asthemobilityofthe modified layer cannotbe_greaterthanthatof water in

bulkandisprobably less,thislatterisa minimumestimate.

Experimentswithtubes, the walls of which hadbeenetchedwithfluoride alsorevealan essential difference between_{the two phenomena.} It was found

(loe. cit.) thatetchinggreatlyinterfereswiththe motion at very lowstresses and renders equation (vii) inapplicable. No corresponding influence on 0 isobserved. Thusit willbe seen from Figs. 1 and_3,thatthe points obtained

withthe etchedtubefallinto finewiththosefor the smooth tubes. Afurther distinctionis _{apparent in the behaviour of} pastes madefromsoils

that

have previously beenair-dried. Movement at low stress isinhibited whereas no

similarinterferenceis_{found at high}stresses. It would appear thereforethat,

wherethe derived curve doesnot _{passthroughthe origin, there exsits}in the immediate neighbourhood of the wall of the tubea_{region in which the viscous} or _{plastic properties}ofamaterial flowing through itare modified. Whena

lami-natedmaterialsuchas a_claypasteflowsthroughatube particlesnear thewall

willtend to align themselves in the direction of flow, andbeunableto rotate under the influence of the_{viscous couple acting}inthem. In thebulkof the

material the particles will rotate _{sufficiently to prevent any alignment.} If

indeed there be _any such an orientation near thewall _{it might} give rise to

an increase in

mobility

as the wall is approached and thus to a deviation

from equation(iii)suchasis_actuallyobserved. Onthe otherhandattemptsto eliminate the 0 term _bytheuse of materials whichare believed tobedevoid of

laminarstructure hasso far_provedunsuccessful. Onlytrue fluidsshowboth

no term and no _rigidity. An alternative_{explanation would}involvean in-creased concentration of the _suspended material towards the centre of the tuberelative_{to the region}near _{the wall, the relatively}more dilute material

having a _{greater mobility.} In order to test thisidea a _sample of clay

the sideofwhicha hole had beendrilled. This hole was _{very small} so that

the exuding of material through it would scarcelyinterfere with the flow in the tube. Althougha _large variation in concentration would have to be

as-sumed to account for the value of <r0 observed, no appreciable difference

n concentration was found between the exudedmaterialandtherest. Weare _{not, therefore,}ina_{position to offer}a _{detailed physical explanation}

of the effect observed. Yet theview that_{the properties of the} materialare

modified inthe neighbourhood of thewallis_supportedbyanotherfact which has been_{repeatedly observed,} but which has hitherto received no

explana-tion. It will be seen that _{in Fig.} 2 ofthe previous paper _{(p. 326) the ratio}

of the extrapolated intercept on the _{pressure axis} in the V-P curve, to the

pressure at which flow at thewall juststarts, is _{somewhat greater}than the 4/3 necessitated by the Buckingham-Reiner equation. This _discrepancy

may wellbe dueto a decreasedvalue of S0in the neighbourhood of the wall.

The _{Determination ofConsistency Constants.}

Methods for _{determining absolute consistency constants} have hitherto

been basedon _{the supposition that, provided the motion} is not _turbulent,

V/ttR3dependsonlyon Wand the nature andconcentration of thematerial. Wherethisisnot _{true (Types}A2 and B2) the methods requiremodification

whether the constants are definedwith referenceto the curves obtained by

plottingV/7rR3 against W (Buckingham-Reiner equation)or _byplotting the logarithms of these _{quantities (Farrow, Lowe and Neale equation).} In such cases,itisimpossible to defineaviscous constant forthe materialfrom

meas-urements made with a _{single capillary,} since there would be a _progressive

changein its value withradiuswere _{the usual method adopted. An instance}

is given in Fig. 5 wherethe viscosity as given(1) by the Poiseuille-Bingham

method for wide and narrow _{capillaries respectively and} (2) fromtheslope of the derived curve is _{plotted against the concentration.} The latter

construction has the advantage of giving a _{single constant independent of}

radius, which,ifthe considerations advancedaboveare correctisameasure of

theviscous_{properties of the}bulkofthematerial.

Moreover, theshapeof the viscosity-concentration curve as _givenbythe second construction is such as would beexpected if the deviation from

lin-earity were due simply to a decrease in the extent of hydrationof the par-ticles. This shape is not reproduced in the curves derived from the single capillaries. Thecurve fromthe smallest capillary hasan _{upward curvature}

of the typefrequentlyobtainedforconcentration curves for hydrophylic

ma-terials with a _{simple Ostwald viscometer.}

It is at least apparent that measurement of the viscosity of suspensions cannot be considered to be reliable unless made with at least two tubes of reasonably differingradii.

The method at present used in _{this laboratory}for determining the

con-sistency constantsforsoiland claypastes isasfollows:

The data for a series of four or five different capillaries having a total

the values of P _{having previously} been corrected for the resistance of the bulbs. The beststraightlinesare then drawn throughthe pointsin sucha

waythat theyconvergeon thestressaxis. The pointat whichthese extra-polationsconverge is takenas the

rigidity,

C.1 Theslope

( )

of each curve

is then measured (i.e. the rise in V/ttR2 per unit increase in stress) and a

V/

PtGtcjn

(ornes/

VtíATJQNL

nr

MM*)

y~PLASTK

rare

7CONSTAt

ACLAY'A

VTSr/iTh ftact/on

CONC£N TftAT/ON

J

0

0

6 1 1 , 1

1 \XiTOL.COMCM

(/VT£(PCi_ier/i5j

i X

1

¡

\

Tl

1

ZrOL.co>

PS£t/0O

(cenn

-r/scosJT

0O/S£) * Co/cuAafoa'froms·

ctfrtc&y /rc'o/oecr/

Cur

m_{Arrftco/7'#ry} - O,0933cm.) i+o*cm.)

O

*

0

* Q

G

«

A * *

1 <

7rot. concn. 1*/ooxr/(£7o-/-7K) ¿961

Fig. 5

is _{plotted separately against R.} The slope of the derived curve

_{(dR/d )}

divided by 4 is taken as the viscous constant

_'

_{(pseudo-viscosity} =

/µ)

and theintercept ofthe extrapolatedcurve on the <raxisas _0, ameasure of thewall effect.

Fig. 5 shows an interesting relationship for a clay fraction between 0

and concentration, the value of cr0 passing through a maximum at a low

1_{Thisis, ofcourse,the}“limit _of

concentration and disappearing for pure water andalso_{at high concentrations} wherethe high

rigidity

limitsa furtherextensionof the concentration range. Accurate measurements of at low concentrations are not easy, and the values are liable to a

_fairly

_{large error,} buttherecan beno doubtas to the generalshape of the curve. It is of interest

that

the maximum of thecurve

occurs atabout thesame concentration as

that

at which

_rigidity

firstmakes

its appearance.

Since the dimensions of 0 are _{somewhat inconvenient,} an alternative

method is to extrapolate the derived curves

still

further onto the negative

radiusaxis. In thiswaya _{hyperthetical length (R„)}isobtained which must beaddedtoeachradius before the equationstatingtheproportionalityofthe slope of the V/ttR2:

PR/2L

curves _{to (viscosity} X radius) can be _applied.

In other words R3(R

+

R0) takes the place of the R4 in the equations of Poiseuille or _{Buckingham-Reiner when} written so as to give V _{directly in}

terms ofP.

Our thanks are due to Dr. B. A. Keen for his interest and criticisms

throughout theprogress ofthis work.

Summary

If,

in_{considering the flow of}a _plasticmaterial through anarrow _tube, it be assumed that the _velocity gradient at any point depends only on the stressat

that

point, it necessarily followsthatthemean _{velocity for}a _given

stressat the wall of the tube shouldbe directly proportional to the radius

of the tube. Although

thick

soilpastes conform closely to thisrequirement,

thinner pasteswhethertheyshow

_rigidity

or _{not give marked}discrepancies. Thesediscrepanciescan beaccountedfor byassumingthat inthe immediate

proximityof the walla modificationof the plastic properties occurs, which

impartsan additional velocity to the bulk of the material. By first