n starch.ls which

Physics Letters A 338 (2005) 479–484

www.elsevier.com/locate/pla

Roll waves on flowing cornstarch suspensions

N.J. Balmfortha, J.W.M. Bushb, R.V. Crasterc,∗

a Departments of Mathematics and Earth & Ocean Science, University of British Columbia, Vancouver, Canadab Department of Mathematics, MIT, 77 Mass Av., Cambridge, MA 02139, USA

c Department of Mathematics, Imperial College London, South Kensington Campus, London, SW7 2AZ, UK

Received 1 December 2004; accepted 9 February 2005

Available online 19 March 2005

Communicated by C.R. Doering

Abstract

We report a traveling wave instability in low Reynolds number flows of aqueous concentrated suspensions of corThe experimental observations are difficult to reconcile with theoretical predictions based on simple rheological modeindicate that flows are stable at low Reynolds number. 2005 Elsevier B.V. All rights reserved.

eralionshgy.ing:

ap-ennce,venrn

id; ital-

heax-ids.t ofeen

be-le,

totsost

r-meater,

1. Introduction

Aqueous suspensions of corn starch show sevremarkable features, as many know from observatin the kitchen, or from dining hall trials in Englisschools; these result from its non-Newtonian rheoloOne salient property is an apparent shear thickenit possesses a resistance to flow thatincreases withthe flow rate. For example, a probe, when gentlyplied, slides easily into the material; however, whrapidly inserted, it encounters considerable resistaand causes the surface to almost solidify and efracture. Alternatively, a compressed handful of co

* Corresponding author.E-mail address: r.craster@impererial.ac.uk(R.V. Craster).

0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserveddoi:10.1016/j.physleta.2005.02.071

starch suspension appears dry and temporarily solcan be crumbled and easily breaks. However, whenlowed to relax, it appears to melt and flow away. Ttime over which this change occurs reflects a relation timescale, such as exists for viscoelastic fluNevertheless, the material is not commonly thoughas an archetypal viscoelastic fluid and has never bshown to exhibit classical viscoelastic rheologicalhaviour, such as the rod climbing of a rotating spindopen siphoning, or the elastic recoil of a filament.

Although a commonly encountered material,our knowledge, only two fluid dynamical experimenwith cornstarch suspensions have been reported. Mrecently, Merkt et al.[1] performed the Faraday expeiment with corn starch suspensions, uncovering sonovel flow dynamics. In particular, they found thpersistent “holes” appear in the vibrated fluid lay

.

480 N.J. Balmforth et al. / Physics Letters A 338 (2005) 479–484

teec-n

ithde-ed

ourthis

y-owseti-a-

se

re

ingby

, is(de-hee

l in-ey

dis-ight

arnm-s).te,p-hlyheedof

inrn-

ked

yac-e

-

iand

il-ondndi-ich

urthat

earum-

letpen

s,s atthe

with surrounding elevated collars, which they attributo the shear-thickening property of the material. Sond, Simpson[2] reported the formation of waves oa flowing layer of custard (which is cornstarch wflavouring), meant as a laboratory analogue of abris flow. Simpson’s observations remain unquantifiexperimentally and unexplained theoretically, andpurpose in the present Letter is to record efforts indirection.

Instabilities on flowing films of water are an everday phenomena, being seen on gutters and windon rainy days, and have a well-established theorcal rationalization in terms of the linear instability ofuniform flow[3,4]. This is the so-called Kapitza problem, for which theory predicts that instabilities ariwhen the Reynolds number,Re = UH/ν, based onthe surface flow speed,U , and depth,H , exceeds acritical value of order unity (ν is the kinematic vis-cosity). The instability prompts the growth of what acommonly referred to as “roll waves”[5,6], which re-semble propagating hydraulic jumps. The surprisproperty of the corn-starch suspensions reportedSimpson[2], and examined in greater detail hereinthat similar waves arise, but at Reynolds numbersfined in terms of an effective viscosity) far below tcritical value appropriate for a Newtonian fluid. Wproceed by reporting the results of an experimentavestigation of this phenomenon, indicating how thrun counter to current theoretical explanation, andcussing a number of physical mechanisms that mbe responsible.

2. Experiments

Experiments were conducted on the flow downconstant incline of a concentrated solution of costarch in water. We used a number of brands of comercially available cornstarch (e.g., Arco, SafewayThe laboratory set-up involved a rectangular chu10 cm wide and 1.5 or more meters long, fed ustream from a reservoir whose level was kept rougconstant by continually adding fluid. By changing tangle of inclination, we were able to vary flow speand depth. Typical fluid thickness were of the order0.5–1 cm; flow speeds were in the 1–10 cm/s range.We also varied the concentration of the cornstarchsolution; suspensions slightly in excess of 1 part co

starch to 1 part water, by weight, were those marby the most pronounced surface waves.

If the fluid were Newtonian, the force of gravitdown the plane, associated with the gravitationalcelerationg sinθ (whereθ is the angle of the planandg ≈ 9.81 m/s2), would be balanced by that stemming from the viscous shear stress,µuzz (µ being theviscosity, and orientating a two-dimensional Cartescoordinate system so thatz = 0 denotes the inclineplane,x points directly downslope, and(u,w) denotethe velocity field). This demands that

νU

H 2∼ g sinθ;

hence,Re = UH/ν ∼ U2/(gH sinθ). For the rollwave typically seen on water films, the fluid is mlimeters in depth and flows at centimeters per secon a shallow slope of perhaps 10 degrees. This icates a Reynolds number of order 10 or more, whis comfortably above the critical value of(5/4)cotθ[3,4] anticipated for the Kapitza problem. For ocornstarch solutions, on the other hand, assumingthe flow is controlled by an equivalent viscous shstress, we observe waves at effective Reynolds nbers of order 0.1 or less.

Photographs of the roll waves are shown inFig. 1(a)and (b). Naturally arising perturbations at the inseed growing, propagating disturbances that stee

Fig. 1. A typical experimental view of (a) developed roll-wave(b) a detailed view of a pair of waves, and (c) merging wavea downstream location; to give an idea of scale, the width ofchute is 10 cm.

N.J. Balmforth et al. / Physics Letters A 338 (2005) 479–484 481

n,nlyheys;

tionhtseds,thentocess

msrs-wedof

thes),d tore-, itopties

typi-telydis-ith

tionad.

aarcham-ws ins the

nt,

atichgercay.n is

tly,lythe

it ista-

gerive.

tly.

nex-

rim-ly,ner-entm-theer-

omons,ingum-at

m-

rses ofre-

ionrcheri-uror ae-

into an unsteady train of waves. Near initiatiothe waves are remarkably regular, and fairly evespaced, with a wavelength of a few centimeters. Tquickly grow and reach relatively large amplitudethe heights of their crests can be a significant fracof the fluid depth. There is a range of crest heigand as a result, the waves travel with different speovertaking one another in their progression downchute. The collision of two waves leads to merger ia larger wave. Thus the wavetrain undergoes a proof coarsening, as illustrated inFig. 1(c), which is typ-ical of many non-linear wave propagation proble(e.g.,[7]). Both the initial wavelengths and the coaened wavelengths further down the channel sholittle dependence on the chute inclination. In mostthe flows, there was little cross-stream variation ofroll-wave profiles (except adjoining the side walland what remaining structure emerged appearestem from perturbations at the inlet (but see themarks below regarding aged material). Howevercould well be possible that the roll wave develtransverse variations through secondary instabiliin wider channels.

Though a range was observed, wave speedscally exceeded the mean flow speed by approxima33–50 percent. The structure of the waves is alsotinctive: the fluid surface appears to be advected wthe waves, creating a characteristic caterpillar acas the disturbances “roll” over the fluid layer ahe

Fig. 2. Flows with and without unstable roll waves, plotted ongraph of slope angle against concentration (fraction of cornstby weight). The stars show flows in which waves appeared toplify as they propagated downstream, the squares represent flowhich the waves appeared to decay. The dotted line indicatebest-fit border between stable and unstable flows.

This is particularly evident at the front of the currewhich progresses through a series of surges.

Lastly, by varying the inclination, we observe whappears to be a critical threshold in slope below whsmall waves initiated at the top of the chute no longrow or persist as they propagate, but instead deThe dependence of critical angle on concentratiopresented inFig. 2. The critical angle is difficult toestablish for two reasons. First, and less importanthe chute is not sufficiently long to state definitivewhether a small disturbance grows or decays overduration of the experiment. Second, even thoughclear that small amplitude waves decay in the “sble” regime in the experiment, it appears that laramplitude disturbances are able to persist and survIn other words, finite-amplitude effects significancomplicate the identification of the threshold slope

3. Other observations

Jitter In addition to the main wave generatioprocess, the flow of the corn starch suspensionhibited another curious feature resembling a supeposed high-frequency jitter or flutter. More precisethe large-scale flow of the material appeared to geate a high-frequency vibration in a manner reminiscof the generation of acoustic waves by flow in a copressible fluid. The jitter could be seen clearly atcrests of the roll waves, where the caterpillar-like ovturning motions vibrated with periods of order 0.1seconds. However, simply pouring the material frone receptacle to another also excited the vibratito the degree that they could be easily felt on holdone of the containers. This process was also presably responsible for creating substantial agitationthe inlet, which in turn seeded the roll waves theselves.

Ageing After repeating experiments over the couof several days it became clear that the propertiethe corn starch suspensions were slowly evolving, psumably owing to some combination of evaporatof the suspending fluid and the swelling of the stagrains. For example, the material used in the expment illustrated inFig. 1 was fresh (prepared an hobefore the experiment). Leaving the suspension fperiod of a day or two led to partial separation b

482 N.J. Balmforth et al. / Physics Letters A 338 (2005) 479–484

anelthat

ver,uchereere

ed,il-andthe

inat-rela-neodpen-

rn-ist-heaap-e,

ds.icaluchm-stic

an-y,

his

-

gniourd in

ws

is-estanand

areforsedu-ar

ldspen-omhin

re-flu-it: thero-te,

atesglyoththeto

Fig. 3. Roll waves on fluid layers of aged material. The second pshows an experiment in which the inclination was steeper thanshown in the first panel.

tween the cornstarch and suspending fluid; howeremixing produced a suspension that appeared mthe same as before. When the flow experiments wrepeated at this time, the roll waves that appeared wmarkedly different: the critical slope angle increasthe roll-wavelength was visibly smaller, the instabity appeared to saturate at much lower amplitudethere was little sign of coarsening. Snapshots ofre-run experiment are shown inFig. 3. Tranverse vari-ations in the roll-wave profiles were also evidentseveral of the flows of aged material, with wave pterns propagating across the stream even in ourtively narrow channel. We note that solutions of oparticular brand of cornstarch (Challenge-ACH focompanies, inc.) behaved more like the aged sussions of the other brands.

4. Theoretical background

The main theoretical problem posed by the costarch suspension is characterizing its rheology. Exing studies suggest that the suspension can be sthickening (and even thinning over some ranges ofplied shear)[1] and show a definite relaxation tim

r

much as we anticipated on observational grounHowever, there is currently no accepted rheologmodel for this material. Instead, in the absence of sa model, we review known theoretical results for comonly used models of shear thickening or visco-elafluids.

Shear-thickening behaviour is captured in the stdard power-law fluid model, for which the viscositµ, depends on the local deformation rate,γ̇ : ν =Kγ̇ n−1, wheren andK are constants (n > 1 impliesshear thickening). Long-wave stability theory[8] es-tablishes that the critical Reynolds number for tflow, Re = Un−2Hn/K , is given byRe = Rec ≡ (2n+3)/(4 tanθ). (For n = 1 we recover the familiar Newtonian result of Benjamin[3] and Yih[4].) The criticalthreshold increases withn, and so shear thickeninfluids (with n > 1) aremore stable than Newtoniaand shear thinning ones. Thus, the observed behavof the cornstarch suspension cannot be rationalizeterms of shear-thickening rheology alone.

Roll waves have been observed on mud flo(kaolin suspensions) in laboratory flumes[8,9], andmud is usually thought to be a shear-thinning vcoplastic fluid. The experimental observations suggcritical Reynolds numbers that are much higher ththose observed for our cornstarch suspensions,the observed wavelengths of muddy roll wavesmuch longer. Unlike cornstarch, the observationsmud are consistent with theoretical predictions baon standard rheological models for viscoplastic flids [10]. Interestingly, the theory applied to shethickening, viscoplastic fluids[10] suggests that yieldstresses could substantially lower the critical Reynonumber. However we observed that cornstarch sussions show negligible yield stresses, draining frinclined surfaces over long timescales to very tfilms.

Long-wavelength stability theories have been psented for a number of archetypal viscoelasticids [11,12]. In particular, for the Oldroyd-B model,has been shown that elasticity can be destabilizingcritical Reynolds number is lowered by an amount pportional to the dimensionless polymer relaxation rathe Deborah number. If the relaxation and shear rare comparable, the dynamics is likely to be stroninfluenced by the elasticity. In our experiments, bthe observed relaxation rate (inferred roughly fromtime taken for compressed, rigid, material to return

N.J. Balmforth et al. / Physics Letters A 338 (2005) 479–484 483

hearex-l isdsrn-measis-

ingeels

ng.es,ser-ve

ourose

ap-in

log-orero-rizeret-ible

uldfor-k-thatin-andto

orn-

sm-en-

ti-atesg

exts,ug-sity,n-pts-

hearti-ns

all,rhe-iatrixthes, aesss-

to

leare-

reowvesfer

ithpres-lede-

re-

he-er

ars,his

a fluid state after compression is released) and srate are of order one Hertz. The possibility thusists that the viscoelastic behaviour of the materiaresponsible for the diminution of the critical Reynolnumber. However, as previously mentioned, the costarch suspension is not usually thought of in the savein as polymeric, viscoelastic fluids. Moreover, it hbeen questioned previously whether this type of vcoelastic instability could ever be observed[12].

Roll waves have also been reported on flowgranular layers[13]. The theory used to rationalizthis phenomenon is based on shallow-fluid modakin to the St. Venant model of hydraulic engineeriSimple friction laws model the granular fluid stressand yield some agreement between theory and obvation. However, the flow speeds required for waformation are well above those encountered incornstarch suspensions, and more in line with thencountered for Newtonian viscous films.

5. Discussion

In summary, we are unable to rationalize thepearance of roll waves at low Reynolds numbercornstarch suspensions by adopting a simple rheoical model. To provide an answer to this puzzle, mstudies are required on this material, both at the micscopic and macroscopic levels, in order to characteits rheology. Because we have no convincing theoical explanation, we can only speculate as to possmechanisms.

Most obviously, the cornstarch suspension cobe visco-elastic, and the known stability resultsOldroyd-B fluids[12] might hold the key to the puzzle. However, Oldroyd-B fluids are not shear thicening. Nevertheless, more general versions ofconstitutive law (such as the Oldroyd-8 family) cancorporate shear thickening in simple steady shearextensional flow[14]. The question then arises aswhether one can fit rheological measurements of cstarch suspensions to such a model.

Alternatively, wave excitation at low Reynoldnumbers could be attributed to a jamming phenoenon as may arise in highly concentrated suspsions[15,16]: a localized perturbation may jam parcles together into a coherent structure that accelerto collect and jam further particles into a growin

mass. Related notions have appeared in other contsuch as in thixotropic fluids where it has been sgested that particle interaction can lead to a viscobifurcation and thence instability[17,18]. In a sensethis behaviour results from what might be called “costitutive instability”, which is also a common concein non-Newtonian fluid mechanics, primarily for vicoelastic fluids[19,20].

Another possibility is related to the idea that tsuspension develops inhomogeneity: suspended pcles are known to migrate in solute, leaving regioof high shear or boundary layers[21]. Effective slip inthe particle-depleted layers, both internal and at a wcan result from this (a phenomenon that plaguesologists). A similar dynamic sorting arises in medlike wet sand, where shear perturbs the solid mafrom its state of optimal packing, thus increasingfluid fraction at the base of the layer. In both casemore viscous, particle-rich layer develops over a lviscous layer. A stick-slip instability is then a posible consequence, as is interfacial instability duethe abrupt change in effective viscosity[22]. Althoughsuch explanations are appealing, it remains uncwhether the form of the resulting instability could rsemble roll waves.

For the observed high frequency jitter, could thebe a link to three other phenomena? First, the flof compressible fluids can generate acoustic wa(e.g., [23]). Second, elasto-plastic materials sufwhat is referred to as “flutter” instability[24], whichtakes the form of unstable propagating waves wspeeds near those expected for shear and comsional waves. Third, the avalanching of so-calsinging or booming sand and fluidized granular mdia may produce acoustic signals[25,26]. In all threeinstances, a relatively slow flow generates high fquency wave-like disturbances.

Whatever the physical origin of the observed pnomenon, we hope that our study will motivate furthinvestigations of this curious material.

Acknowledgements

We thank Anthony Pierce, Bud Homsy, OmMatar and Gareth McKinley for helpful discussionand John Simpson for drawing our attention to tproblem.

484 N.J. Balmforth et al. / Physics Letters A 338 (2005) 479–484

thedge,

05.

o-

ewt.

ly-

97)

on-les,

46

47

on-

..

References

[1] F.S. Merkt, et al., Phys. Rev. Lett. 92 (2004) 184501.[2] J.E. Simpson, Gravity Currents in the Environment and

Laboratory, second ed., Cambridge Univ. Press, Cambri1999.

[3] T.B. Benjamin, J. Fluid Mech. 2 (1957) 554.[4] C.S. Yih, Phys. Fluids 6 (1963) 321.[5] P.L. Kapitza, S.P. Kapitza, Zh. Eksp. Teor. Fiz. 19 (1949) 1[6] R.R. Brock, J. Hydraul. Div. 95 (1969) 1401.[7] N.J. Balmforth, S. Mandre, J. Fluid Mech. 514 (2004) 1.[8] C.O. Ng, C. Mei, J. Fluid Mech. 263 (1994) 151.[9] P. Coussot, Mudflow Rheology and Dynamics, IAHR Mon

graph Series, Balkema, Rotterdam, 1997.[10] N.J. Balmforth, J.J. Liu, J. Fluid Mech. 519 (2004) 33.[11] A.S. Gupta, J. Fluid Mech. 28 (1967) 17.[12] E.S.G. Shaqfeh, R.G. Larson, G.H. Fredrickson, J. Non-N

Fluid Mech. 31 (1989) 87.[13] Y. Forterre, O. Pouliquen, J. Fluid Mech. 486 (2003) 21.[14] R.B. Bird, R.C. Armstrong, O. Hassager, Dynamics of Po

meric Liquids, vol. 1, Wiley, New York, 1977.

[15] R.S. Farr, J.R. Melrose, R.C. Ball, Phys. Rev. E 55 (197203.

[16] A.J. Liu, S.R. Nagel (Eds.), Jamming and Rheology. Cstrained Dynamics on Microscopic and Macroscopic ScaTaylor and Francis, London, 2001.

[17] P. Coussot, Q.D. Nguyen, H.T. Huynh, D. Bonn, J. Rheol.(2002) 573.

[18] J. Billingham, J.W.J. Ferguson, J. Non-Newt. Fluid Mech.(1993) 21.

[19] Y. Renardy, Theor. Comput. Fluid Dyn. 7 (1995) 463.[20] J. Hunter, M. Slemrod, Phys. Fluids 26 (1983) 2345.[21] D. Leighton, A. Acrivos, J. Fluid Mech. 177 (1987) 109.[22] C.S. Yih, J. Fluid Mech. 27 (1967) 337.[23] E.G. Broadbent, D.W. Moore, Philos. Trans. R. Soc. L

don 290A (1979) 353.[24] L. An, D.G. Schaeffer, J. Mech. Phys. Solids 40 (1992) 683[25] P. Sholtz, M. Bretz, F. Nori, Contemp. Phys. 38 (1997) 329[26] A.J. Patitsas, J. Fluids Struct. 17 (2003) 287.