*Week 2*

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## April 08, 2005

### Constitutive Relations and Granular Flow

Discussion led by Thomas Halsey, 10am 8 April 2005

1. Chute flow: Pouliquen flow rule:

u/\sqrt{gh}=F(h/h_{stop}(\theta)) = 0.14 h/h_{stop}

u=mass flux/h

h=flow depth

\theta=slope angle

steady flow for h>h_{stop}, but accelerating beyond a threshold of \theta.

Chute MUST have a rough base.

Behaviour close to threshold h_{stop} has been queried; the result may only be asymptotic in h/h_{stop}.

Coefficients not universal, but 0.14 seems characteristic of spherical particles; only weekly dependent on friction.

For sand, RHS \propto (h/h_{stop} – 1).

GDR Midi claim to obtain this from a local constitutive rheology which (they claim) addresses a much broader range of expts.

- Key new ingredient is h
_{stop}as a scaling length in the problem.

2. Plane shear. GDRM have some 2D numerical simulations for comparison.

Identify Ft/Fn=\mu_{eff} where F_ are driving normal and shear forces at free surface (per unit area).

mu_{eff} \propto I +const

where I= d s/\sqrt{p/\rho}

d=particle diameter

s=shear rate

p = Fn approx.

*Claim by TCH:*

**all this behaviour associated with coherent motion in flows**.

Timescale for loaded ball to collapse onto base wall

\tau = 2mv_{0}/F (1-e)/e 1-e=restitution c't.

Adapt this to relative motion of balls in a flowe:

\tau = \rho d^{2} s/p f(e, \phi)

\phi =volume fraction (varies little)

Local collapse -> velocity correlation out to lengthscale \ell_{E} (alluding to earlier ideas of eddies).

Length set by competition between accretion (modelled by diffusion) and shear, leading to

\ell_{E} = \sqrt{d^{2}/\tau 1/s}==\sqrt{p/\rho}/s g(e).

Notice this offers the interpretation that an earlier dimensionless number is given by I=d/ \ell .

Next question is nature of correlations. Early interpretation of vortex structures disputed.

For two particles in long term contact, relative motion constrained by consistency at contact:

v'-v"=1/2 (w'+w")x(r'-r")

[v= velocity, w=angular velocity, r=position]

Corresponding relative acceleration equation also useful.

GOAL: show how this physical picture leads both the Pouliquen flow rule and to shear phenomenology.

Can apply Edwards counting argument:

Reduction in d.o.f. Nf/N = 3/2(4-z) d=3, rough grains.

Forces determiined from three (non-slip) equations at each contact.

Integration forwards obstructed by:

– new contacts

– failure of contact by slip or negative normal load.

Shear Rheology: generic parameterisation:

\sigma = \mu s

\mu = \rho \ell_{\mu}^{2} s

debatably, \ell_{\mu}=A \ell_{E} (1+B d/\ell_{E} + …)

Sign of first finite size effect term matters in what follows.

Simple Bagnold argument has \ell_{\mu}=d and it clearly makes no sense to allow the theory to go below this!

Hence constitutive relation comes out in terms of:

\mu = ~~\rho s~~ A^{2} 1/f(e) p/~~\rho~~ 1/s^{2} (1+Bf(e) d s/\sqrt{p/\rho}+...)

Returning to shear flow, this essentially recovers GDRM.

The analysis also goes through for the Pouliquen (chute) flow rule. \ell_{\E} plays the role of h_{stop} and the two can be consistently identified.

*Daniel Lhuillier* made the point that stresses other than shear are not predicted, and [anomalously for a constitutive relation] remain in the theory.

*Robin Ball* would turn that point around: the theory is a prediction of shear [rate] given the stress. This interestingly matches how David Pine finds controlled stress the better way to probe rheology of jamming suspensions. **Is it profound that such systems should be analysed in terms of flow given the stress, rather than stress given the flow?**

### Programme Seminar 8 April 05

Robin Ball

*From Granular Statics towards Plasticity and Yield*

Link to full media on KITP website

## April 06, 2005

### Boundary induced instabilities in fluids, and analogous computations for granular materials.

Wed 6 April 1pm KITP SMall Seminar Room

Jean Carlson Group (plus John Doyle and Bassam Bamieh) Open Meeting

###### Beyond Stability Analysis: Robustness Analysis in Transition [for hydrodynamic flows]

###### Bassam Bamieh, UCSB

Focus on stabililty of response at infinite time

Rc = critical raeynolds no. at which instability appears

RT = Reynolds no at which transition to unsteady flow occurs

Experimentally these do not match well – suggesting intrinsically non-linear issues at satake.

Structure does not agreee well between (in)stability analysis and experiments in transition.

More elborate theories consider a cascade of successive stability analyses.

**Possible Fixes**

– Investigate some norm of the transient behaviour post perturbation in linear approximation.

– Pseudo-spectral analysis (related)

– add exogenous inputs

These do not give Rc, but they do inidcate the 'right' flow structures.

Large linear growth of time integrated measures

Stability Theory does *not* explain/include:

-effects of free stream turbulence

-uncertain body forces

-wall roughness

-uncertainty in NS eqns: base flow, neglected terms

various 'hacks' address these effects.

Lyaponov stability – perturb init conditions

Dynamical/structural stability – perturb eqns

Exogenous inputs

– key thing is to move beyond just perturbing the initial conditions: structural stability is crucial.

###### Eric Dunham

###### A simple friction example of stabiity analysis

Standard block-slide rmodel with friction arbitrarily parameterised.

\tau decerasing with v and healing with time, all fit by Dietrerich-Rima law.

Stabiilty condition on spring stiffness.

Transient oscillations seen in energy norm.

Can extend analysis to problems like propagating rupture.

###### Lisa Manning

###### Channel flow over a flexible wall

?Drag reduction strategies: ribbed wall, compliant wall, activley driven wall.

Old expts suggest compliant walls can reduce drag.

Older theory assumed travelling wave solutions, and ignored wall shear stress.

Present model: spring-backed flexible membrane.

Assume streamwise-uniform solutions.

Transformations move coupling from inhomogenous boundary conditions into homogenous bc's with applied bulk terms.

###### Jim Langer

###### Glassy Materials

Expts now probe six orders of magnitude in strain rate on metallic glasses. Theoretical modelling now spans these.

Pushed harder, these systems fail under uniaxial compressive loading to give a 45deg. shear band.

Questions:

– criteria for this?

– relation to hydrodynamics and/or granular materials

Thickness of shearbands << thermal diffusion estimates, but new ideas suggest thinking instead in terms of 'effective' temperature related to disorder, having orders of magnitude slower diffusion.

## April 04, 2005

### Thomas halsey: KITP Director's Seminar 4 April 20005

#### The world in a grain of sand: a perspective on Granular Physics

link to full media in KITP archive

*A world in a grain of sand ...* WIlliam Blake

Key Granular Physics ideas:

- Separate particles, short range interactions.
- Energy scale of packing (e.g. mgd) >> kT athermal limit,

far out of eq'm. - Particles stiff: modulus E>>stress (TCH said 'rigid'),

but compliance may be a singular perturnbation.

– all a reasonable match to the beach!

Typical granular packing is quite random: is there some sense in which ideas of eq'm stat mech can be used?

Clear case of this is dilute granular gases, where kinetic theory can be used;

Also granular stat mech ideas for solids.

There is a shortage of dimensionless groups in granular phsyics and dynamics in particular. Hence an absence of intermediate lengthscales. Research response is granular fluid and solid mechanics, in which granular scal is explicit.

Physical issues to beware of

- Established microscopic theory known as Contact Mechanics. Many fundamental issues and computations are open here.

Ex. 1 *History dependence of contact.*

Load contact normally by N and tangentially by T.w, and T<=\mu N Coulomb limit. All trajectories in T,N plane must stay below the limit. Which material points are in contact (and hence the shear compliance of the contact) varies with the loading history. Mindlin, Deveziewicz 1950's.

Ex 2. *Rolling Friction*

Generally rolling leads to dissipation because Coulomb criterion violated at trailing contact. Rolling a balloon over a table gives sound from the stick-slip entailed. Still not all worked out for sphere on sphere.

- Dynamical friction not simple; better commonly modelled through material viscoelasticity.
- Surface Roughness another important complication
- Interstitial Fluids. Thes can be single phase, or two phase leading to capillary bridging interactions.
- Non-sphericity. Quartz particles rather prismatic in shape – harder to model. Recent work by Chaiken and Torquado on the M&M problem – packing vs aspect ratios. Claim that ellipsoids can pack more efficiently than spheres in some limit. [M&M's are remarkably perfect ellipsoids.]
- Electrostatic, van der Waals and other long-ranged interactions lead into the Powder regime. Industrially Powder and Granular are regarded as distinct regimes.
- Particle surface state plays a strong role in many granular properties, making universality harder to find.

Our task: *to build robust microscopically well-founded phenomenologies that can be easily calibrated against experiment*.

Engineers good at this, but Physicists are the specialists at moving from scale to scale.

### "Liquid Physics" – free surface flows.

Simplest case is **chute flow**.

Phase diagram (depth of layer, angle) searates three types of behaviour:

(a) no flux (up to angle of repose line)

(b) steady state flow (beyond critical angle line, where flow initiates)

(c) accelerating flow – dissipation fails to match gravitational work released

Pouliquen Flow Rule (PFR) in steady state regime

Froude no u/\sqrt{gh} = F(h/h_stop(angle) ~= \beta h/h_stop – \gammma

Sand: \beta, \gamma ~=0.7

Glass beads: \beta _{=0.14, \gamma}=0

but this difference between angular and round particles remains to be explained.

The PFR does have a growing theoretical literature.

TCH: correlated motion -> prantdl mixing length. This length then controls bothe the flow threshold and the scaling fo fully developed flow.

Simple analysis -> \ell ~= \sqrt{p/\rho} 1/s where s=shear rate.

Group de Recherche Midi claim in a lenght recent paper that a variety of flows (not all open surface) understood in terms of control by the parameter s d/sqrt{p/\rho}= d/\ell where d=particle diameter. The paper may be vulnerable to experimental critiique.

### Granular Packings and their statisitics

Packing P of particles in mechanical equilibrium: what contraints can one place on mean coordinaton number?

Nf = 1/2 z N d number of forces from the contacts [assuming friction acting]

Nd = N(d + 1/2 d(d-1)) degrees of freedom from the particles

=> z>= d+1

Numerical evidence is that packing does not typically lead to isostatic packing for particles with friction.

0.55 Random loose packing < \phi <0.64 Random close packing

Nowak et al tapping expt: cylindrical particle bed tapped with fiexd amplitude \gamma=a/g. Observed irreversible and reversible regimes.

Irreversible rise of \phi with \gamma until it joins a reversible curve which has slightly negative slope.

Edwards hypothesis to explain this.

a= jammed state. Pa=exp(-Va/X) / normalisation

X=compactivity

Remarkable but proving hard to verify. Essentially says that basins of attraction of differnet jammed state depend (relevantly) only on the volume of that jammed state.

Open agenda: construct simpler test cases in which the hypothesis can be tested more controllably.

Other upcoming issues:

Granular gasses.

Analogy between granular jamming and glass (and spin glass) transitiosn.

Suspensions: from colloidal systems to geophysical flows.

## April 01, 2005

### scheduled events 4–8 April

Mon 4 April KITP Seminar 12.15 KITP Auditorium (1403)

Thomas Halsey

*The world in a grain of sand: a perspective on Granular Physics.*

Tues 5 April 10am Small Seminar Room

Introduction of new participants and open discussion.

Wed 6 April 1pm KITP SMall Seminar Room

Jean Carlson Group (plus John Doyle and Bassam Bamieh) Open Meeting

*Boundary induced instabilities in fluids, and analogous computations for granular materials.*

Thurs 7 April 10am G. Program Seminar KITP Auditorium

Robin Ball

*From Granular Statics towards Plasticity and Yield.*

Thurs 7 April 7pm Programme Dinner

La Superica, 622 N Milpas, downtown SB.

Fri 8 April 10am Small Seminar Room

Jerry Gollub, Thomas Halsey …

*Discussion on Constitutive Relations in granular shear and flow.*