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April 25, 2005
(Written for Powders and Grains 2005. In the seminar, I also mentioned experiments on clustering mediated by hydrodynamic interactions in a fluid, and on vibrated chiral particles.)
ABSTRACT: The structure of dense granular materials can evolve during shear, and this evolution can af-fect the flow properties. We describe high precision shear flow measurements using index-matched imaging in which the internal velocity fields are mapped out quantitatively, while the total volume and shear stress are measured simultaneously. Results include the development of locally ordered structure consisting of sliding hexagonally ordered layers, the response to different shearing protocols and boundary conditions, the internal velocity profiles and rheology, the origin of shear banding, the anomalous mobility due to shear reversal, and the effects of polydispersity.
Slow or quasi-static granular flows are less well studied than rapid “gas” flows. In fact, there is not even a universally agreed definition of such flows. We describe an experimental investigation of quasi-static shear flows in an annular version of a plane Couette geometry. A novel feature of this work is that the internal flow profiles are measured. Since the work has been submitted or published in refereed journals, we give only a summary of the main results here, with references to the original work (Tsai et al. 2003, 2004, and 2005).
The most important result of this study is the ob-servation that slow shearing can lead to internal or-dering over a long period of time, to a state consist-ing of hexagonally close packed planes that slide over each other during shear. The ordering substan-tially changes the rheological behavior of the mate-rial, such as the shear force and the internal velocity profiles. Generally, the effect of ordering on dense flows has been noted in many numerical studies of hard sphere systems, as explained in our papers, but has not been included in rheological models.
We are concerned with flow in the regime of per-sistent contacts, where kinetic energy is far less than the elastic energy of the packing. There are many previous experiments on sheared dense systems, and full references are given in our papers.
Spherical glass beads of diameter d=0.6 mm fill the annular channel formed by two concentric smooth stationary cylinders. They are driven by a floating upper boundary, which rotates at a constant speed while exerting a fixed normal load. The driving sur-face is roughened by a glued layer of beads, and we consider various boundary conditions at the lower surface. The interstitial space is filled with a fluo-rescent fluid whose refractive index is matched to that of the particles. Slices illuminated by a horizon-tal or vertical laser sheet yield images that highlight the particles, which can be tracked using an imaging system. Fluid can enter freely as the volume occu-pied by the particles varies, while the shear force and volume are monitored. Hydrodynamic forces are demonstrated to be negligible, although lubrica-tion effects do occur.
3.1. Ordering transition
Long term shearing of the largely monodisperse par-ticles leads to an ordering transition. We have also found recently that in a binary packing, segregation occurs followed by ordering of the segregated parti-cles.
At the ordering transition, the material self-organizes into a stack of hexagonally packed layers that slide over each other. The transition results in a step change in the granular volume of about 3%, and a 15% reduction of the shear force. The typical par-ticle speed far below the shearing surface declines at the transition, i.e. the ordering causes the lower lay-ers to be influenced less strongly by the shearing.
The timescale of the transition depends on the layer thickness, but is also somewhat stochastic. It occurs after a much longer time for thick layers, and more time is also required when the material is dry. In general, sufficient time is required that the slow-est particles (i.e. those farthest from the shearing surface) translate by several particle diameters.
When the packing is thin (say 6–14 layers) and a monolayer bottom boundary condition is used (far from the shearing surface), the ordering is imperfect for certain filling volumes, but nearly perfect for other volumes. The sample volume as a function of filling mass M is then the sum of a linear rise plus a periodic function of M. We refer to these phenom-ena as “layer quantization effects”, as they are re-lated to the proximity of M to the value required to produce an integral number of complete layers.
The ordering depends on the surface properties of the solid surface that is farthest from the shearing surface, i.e. at the bottom of the sheared layer. If this surface is covered by an ordered array of parti-cles, this induces ordering above after a sufficient interval. On the other hand, for an irregular bottom surface, either an ordered or disordered packing can occur as the final state, depending on the history of the sample and the shearing protocol. We say that the system has multiple final states. We also find that AC shear can nucleate the ordered state even if ordering is otherwise suppressed, provided that the packing is not too dense. The ordered state is al-ways stable once it occurs.
3.2. Velocity profiles and rheology
The velocity profile, normalized by the driving speed, is found to be the same for all driving speeds over a wide range. In general, the velocity profile falls off faster than exponentially with depth. It can be fitted by the exponential of a quadratic function, and it falls off more steeply in the ordered state than in the disordered state. We find that the profile is sensitive to the channel width. It falls off faster than the profile of a Newtonian fluid in the same geome-try. Furthermore, the velocity profile does not de-pend strongly on the particle size, even though one might have expected some scaling with this dimension.
The transverse (i.e. radial) structure of the veloc-ity profile changes dramatically at the ordering tran-sition. It is Poiseuille-like (parabolic) in the disor-dered state, and nearly flat in the ordered state, where entire layers move coherently. In the disor-dered state, sidewalls induce local order in planes parallel to the sidewalls.
3.3. Origin of shear banding
What is the origin of shear-banding, i.e. the sharp decline of the velocity profile from the sheared sur-face? In our experiments, the localization of shear can be understood by considerations of torque bal-ance on a layer of granular material extending from the top down to a depth z. The torque on the top and bottom of this layer differ somewhat, because of friction at the channel walls. The difference is then proportional to the rate of change of shear stress with height, which may in turn be decomposed into the following product:
The stress is expected to be only weakly dependent on shear rate for a granular packing. Therefore, a rapid rate of decline of shear rate with height can be expected, to generate the torque necessary to balance wall friction. The shear confinement can therefore be interpreted as a consequence of the weak depend-ence of stress on shear rate in conjunction with sidewall friction.
3.4. Particle trajectories
Particle motion in the ordered state is highly coher-ent, and particles do not diffuse vertically to an ap-preciable extent. On the other hand, in the disor-dered state, there is significant vertical diffusion, with a root-mean-square displacement which varies roughly as the square root of the elapsed time.
Studies of particle trajectories during AC shear-ing show that particles move more rapidly during a short transient following the instant of reversal, be-cause internal stresses vanish at that point. We refer to this phenomenon as anomalous mobility.
This work leads to many interesting questions. Why is the crystalline state always stable once reached? What is the role of the nonlinearity of the velocity gradient? How does the response to shear for a polydisperse packing differ from that of monodis-perse packings. (For binary mixtures, we have noted an interesting combination of segregation fol-lowed by local ordering.) How can one incorporate local order into rheological theories of dense granu-lar flows? What is the role of the grain size?
These experiments suggest that the internal struc-ture of natural packings probably evolves over long times, and that the internal velocity profiles and rheology must slowly change as a result. The assumption of steady state shear is unlikely to be valid because of this evolution.
We appreciate the support of the U.S. National Sci-ence Foundation, Division of Materials Research, under Grant DMR-0405187 to Haverford College and DMR-0079909 to the University of Pennsyl-vania.
Full references to related work by other authors are given in these papers:
Tsai, J.-C., Voth, G.A., and Gollub, J.P. 2003. Internal granu-lar dynamics, shear-induced crystallization, and compaction steps. Phys. Rev. Lett. 91:064301.
Tsai, J.-C., & Gollub, J.P. 2004. Slowly sheared dense granu-lar flows: Crystallization and non-unique final states. Phys. Rev. E. 70: 031303.
Tsai, J.-C. & Gollub, J.P. 2005. Shear banding and transient response of granular packings in an annular channel. Submitted to Phys. Rev. E.
April 19, 2005
Vortex ripples and wind ripples
Expts in annular confinement, sand + water driven by oscillatory rotation lead to sharp ripple patterns around their circumferential direction. Ingenious conical mirror for imaging.
Time series indicate increasing wavelength and amplitude, but apparently reach steady state. Not thought to coarsen indefinitely.
Changing drive amplitude:
Selected wavelength \propto drive amplitude; \lambda/L =4/3 approx;
raising amplitude drives it up but lowering does not.
There is a limited range of drive frequency which drives rippling.
Basic viortex ripple picture known since Ayrtou 1910. Separation vortex on lee side of ripple leads to upflow in the lee and pile building. Hydrodynamic simulations (e.g. KH Andersen PhD) reproduce this. The coupling between successive ripples depends non-trivially on ripple height compared to spacing.
Mass transfer model
- models evolution of fully developed ripples;
- mass transfer mainly governed by size of recipient
dmi/dt = 2f(mi)-f(mi+1)-f(m~i-1)
and relabel when mi -> 0.
Uniform pattern stable when f'(m)<0 , so a whole band of stable wavelenghts exits.
f(m) assumed to exhibit a maximum; this is validated by measurements (Arbus sp??). Fitting with different functions for the three f's appearing above does give near agreement as expected.
PRL 88 234302 (2002)
Wavelength selection for vortex ripples
Model by simulation does predicted a selected wavelength with insensitivity to initial conditions. Thus there exsists a mechanism of selection within the stable band.
Model can be viewed as a downhill dynamics, dxi/dt = -dV/dxi .
This suggests selection to minimise under fixed total 'lemgth'. This would amount to a Maxwell construction but (in the approximations to f tested) this does not compare well with the data.
Selection mechanism remains a puzzle.
Stochastic coarsenning of wind ripples
Wind ripples propagate (e.g. rightwards) at speed v(\lamda) \propto 1/ \lamda. This maps into a particle dynamics with extinction.
One sided: d\lambdai/dt=f(\lambdai)-f(\lambdai-1)
f(l) \propto 1/l.
This can to some extent be solved exactly, \lamda \propto ln(t) emerging due to stochastic merging (to some extent confirmed by data).
Some analogy with compartmented models for granular gases (D. van der Meer et al, JSTAT (2004) P04004), but in these cases the fluxes are governed by the source boxes rather than predominantly by the destination.
Form of tail for f leads to sublogarithmic coarsenning.
All fits into wider context of condensation in zero range processes.
More complete sand-transport models.
Symmetries and desiderata:
- conservation of mass;
- invariance with resepect to global height
- x <
> -x but not h <> -h.
- uphill current
- slope selection
ht + hxx + hxxxx + (hx2)xx - (hx3)x = 0
but this gives unbounded coarsenning. \lambda \propto ln(t).
Some support for logarithmic coarsenning from simulations.
Conjecture (Krug): conserved 1-d models exhibit either unlimited coarsenning or unlimited steepenning, not stable periodic solutions.
April 04, 2005
Tue 19 April 10am Small Seminar Room
J. Krug, Universitaet zu Koeln.
"Mass transfer models for sand ripples"
Wed 20 April
the session on Friction and Control advertised for 1pm Small Seminar Room will take place later in the programme
Thu 21 April 10am Program Seminar, KITP Auditorium
Prof. J. Langer, UCSB.
Plastic Deformation in Amorphous Solids: Effective Temperature and Shear Localization.
Thur 21 April: Programme dinner
California Pizza Kitchen (SB).
Please advise RCB of numbers ASAP for our reserevation.
I hope to book for 6.45. That fits with catching the express bus from UCSB North Hall at 6.15, for which meet KITP Courtyard/Common Room 6pm. I will also canvas whether we have enough cars, but the bus ride is straightforward.
Fri 22 April 10am Seminar, Small Seminar Room
Prof. J. Gollub, Haverford College and Penn
Experiments in granular flows