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Dynamics and Patterns in Sheared Granular Fluid : Order Parameter Description and Bifurcation Scenario NDAMS Workshop @ YITP 1 st November 2011 Meheboob Alam and Priyanka Shukla Engineering Mechanics Unit Jawaharlal Nehru Centre for Advanced

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Meheboob Alam and Priyanka Shukla

Engineering Mechanics Unit Jawaharlal Nehru Centre for Advanced Scientific Research Bangalore, India

1st November 2011 NDAMS Workshop @ YITP

Dynamics and Patterns in Sheared Granular Fluid : Order Parameter Description and Bifurcation Scenario

1 Wednesday 2 November 11

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Outline of Talk

  • Shear-banding phenomena
  • Gradient Banding and Patterns in 2D-gPCF
  • Vorticity Banding in 3D-gPCF
  • Theory for Mode Interactions
  • Spatially Modulated Patterns (CGLE)
  • Summary
  • Possible Connection: Saturn’s Ring

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Gradient Banding in 2D-gPCF

Order-parameter description of shear-banding?

Shukla & Alam (2009, 2011) Saitoh &Hayakawa (2011)

x

Alam 2003

x

y

Tan & Goldhirsch 1997 3 Wednesday 2 November 11

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Balance Equations

Navier-Stokes Order Constitutive Model

Granular Hydrodynamic Equations

(Savage, Jenkins, Goldhirsch, ...)

Flux of pseudo-thermal energy

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Control paramters

Restitution Coeff. Volume fraction or mean density Couette Gap

Uniform Shear Solution

d : Particle diameter

  • Base Flow Assumption: Steady, Fully developed.
  • Boundary condition: No Slip, Zero heat flux.

Plane Couette Flow (gPCF)

Reference Length Reference velocity Reference Time

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Linear Stability

Perturbation If the disturbances are infinitesimal

‘Nonlinear terms’ of the disturbance

  • eqns. can be ‘neglected’.

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Can ‘Linear Stability Analysis’ able to predict ‘Shearbanding’ in Granular Couette flow as

  • bserved in Particle Simulations?

y

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Can ‘Linear Stability Analysis’ able to predict ‘Shearbanding’ in Granular Couette flow as

  • bserved in Particle Simulations?

Not for all flow regime

y

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Linear Theory

Tan & Goldhirsch 1997 Phys. Fluids, 9

We must look beyond Linear Stability

?

Particle Simulation

STABLE UNSTABLE

Shukla & Alam 2009, PRL, 103, 068001

Density segregated solutions are not possible in dilute limit Flow is ‘non-uniform’ in dilute limit Density Segregated solutions are possible in dilute limit Flow remains ‘uniform’ in dilute limit

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Dynamics close to critical situation is dominated by finitely many “critical” modes.

Nonlinear Stability Analysis: Center Manifold Reduction

(Carr 1981; Shukla & Alam, PRL 2009)

Z : complex amplitude of finite amplitude perturbation

A m p l i t u d e L i n e a r E i g e n v e c t

  • r

Critical Mode Non-Critical Mode Disturbance First Landau Coefficient Second Landau Coefficient

Taking the inner product of slow mode equation with adjoint eigenfunction

  • f the linear problem and separating the like-power terms in amplitude,

we get Landau equation

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Cont…

Other perturbation methods can be used: e.g. Amplitude expansion method and multiple scale analysis

Represent all non-critical modes Enslaved Equation

Distortion of mean flow Second harmonic Adjoint

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1st Landau Coefficient

Analytical expression of first Landau coefficient

Analytical solution exists.

Linear Problem Second Harmonic Distortion to mean flow Distortion to fundamental

Analytically solvable

Shukla & Alam (JFM 2011a)

We have also developed a spectral based numerical code to calculate Landau coefficients.

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Numerical Method: comparison with analytical solution

Spectral collocation method, SVD for inhomogeneous eqns. & Gauss-Chebyshev quadrature for integrals.

This validates spectral-based numerical code.

Shukla & Alam JFM (2011a)

Real part of first Landau coefficient Distorted density eigenfunction

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Equilibrium Amplitude and Bifurcation

Cubic Solution

Real amplitude eqn. Phase eqn.

Supercritical Bifurcation Subcritical Bifurcation Pitchfork (stationary) bifurcation Hopf (oscillatory) bifurcation

Cubic Landau Eqn

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Phase Diagram

Shearbanding in dilute flows

This agrees with MD simulations of Tan & Goldhirsch 1997

Constitutive equations are function of radial distribution function (RDF) Nonlinear Stability theory and MD simulations both support gradient banding in 2D-GPCF (PRL 2009)

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Cont… (JFM 2011a)

Carnahan-Starling RDF

Change of constitutive relations lead to three degenerate points

Stable Solutions Unstable Solutions Subcritical -> supercritical Supercritical-> subcritical Subcritical -> supercritical

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Paradigm of Pitchfork Bifurcations

Supercritical Subcritical Supercritical Subcritical Bifurcation from infinity

JFM, 2011a

Khain2007 Tan & Goldhirsch1997

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Conclusions

  • Order-parameter equation i.e. Landau equation describes shear-banding

transition in a sheared granular fluid.

  • Landau coefficients suggest that there is a “sub-critical” (bifurcation from infinity)

finite amplitude instability for “dilute’’ flows even though the dilute flow is stable according to linear theory.

  • This result agrees with previous MD-simulation of gPCF.
  • gPCF serves as a paradigm of pitchfork bifurcations.
  • Analytical solutions have been obtained.
  • An spectral based numerical code has been validated.

References: Shukla & Alam (2011a), J. Fluid Mech., vol 666, 204-253 Shukla & Alam (2009) Phys. Rev. Lett., vol 103 , 068001.

  • Problem is analytically solvable.

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``Gradient-banding’’ and Saturn’s Ring?

References: Schmitt & Tscharnuter (1995, 1999) Icarus Salo, Schmidt & Spahn (2001) Icarus, Schmidt & Salo (2003) Phys. Rev. Lett.

  • Self gravity, corriolis and tidal forces?...

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Patterns in 2D-gPCF

Flow is unstable due to stationary and traveling waves, leading to particle clustering along the flow and gradient directions (Alam 2006)

Shukla & Alam , JFM (2011b) vol. 672, 147-195

x y

Modulation in ‘y’-direction Modulation in ‘x’-direction

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Particle Simulations of Granular PCF (Conway and Glasser 2006)

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Amplitude Expansion Method

(Stuart, Watson 1960, Reynolds and Potter 1967, Shukla & Alam, JFM 2011a )

Solvability Condition

: Real amplitude

Landau coefficient

Equivalent to “center manifold reduction’’ Assumption

For

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Linear Theory

Standing wave instability

2nd peak

Traveling wave instability

Phase velocity

Long-wave instability

1st peak

Wavenumber

Growth rate Phase velocity

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Long-Wave Instabilities

TW Density Patterns SW Density Patterns

Growth Rate

Real and Imag. Part

  • f first LC

Amplitude

Supercritical pitchfork/Hopf bifurcation

Linear Non-linear Non-linear Linear

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Stationary Instability

SW density patterns Supercritical pitchfork bifurcation

Structural features are different from long-wave stationary instability

Amplitude Real of first LC

Linear Non-linear

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Travelling Instabilities

Supercritical Hopf bifurcation

Nonlinear patterns are slightly affected by nonlinear corrections

Linear Non-linear

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Dominant Stationary Instabilities

Density patterns are structurally similar at all densities

Resonance Non-linear Non-linear Non-linear

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Dominant Traveling Instabilities

Stable

Supercritical Hopf Bifurcation Subcritical Hopf Bifurcation

Resonance

Unstable

Non-linear Non-linear

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Evidence for Resonance

Evidence

Jump in first Landau coefficient

Subcritical region

Criterion for mean flow resonance Criterion for 1:2 resonance

Interaction of linear mode with a shear banding mode Distortion of mean flow Eqn. Second Harmonic Eqn.

Origin Subcritical region

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Evidence for Resonance

Multiple resonance in subcritical region Single mode analysis is not valid at the resonance point

Coupled Landau Equations

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Conclusions

  • The origin of nonlinear states at long-wave lengths is tied to the

corresponding subcritical / supercritical nonlinear gradient-banding solutions (discussed in 1st Part of talk).

  • For the dominant stationary instability nonlinear solutions appear via

supercritical bifurcation.

  • Structure of patterns of supercritical stationary solutions look similar

at any value of density and Couette gap.

  • For the dominant traveling instability, there are supercritical and

subcritical Hopf bifurcations at small and large densities.

  • Uncovered mean flow resonance at quadratic order.

References: Shukla & Alam (2011b), J. Fluid Mech., vol. 672, p. 147-195. Shukla & Alam (2011a), J. Fluid Mech., vol 666, p. 204-253. Shukla & Alam (2009) Phys. Rev. Lett., vol 103 , 068001.

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Vorticity Banding in 3D-gPCF

Shukla & Alam (2011c) (Submitted) Streamwise Gradient Vorticity

Pure Spanwise gPCF

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Linear Vorticity Banding

Gradient-banding modes stationary modes at all density. Vorticity-banding modes stationary at dilute limit & traveling in moderate-to-dense limit.

Density

Pitchfork bifurcation Supercritical Hopf bifurcation Stable Unstable Analytically solvable

Dispersion relation

Pure spanwise GPCF

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Nonlinear Stability

Linear Problem Second Harmonic Distortion to mean flow Distortion to fundamental Analytical expression for first Landau coefficient

Analytically solvable

Analytical solution exists at any order in amplitude.

Shukla & Alam (2011) (Submitted)

Adjoint Eigenfunction

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Nonlinear Vorticity Banding

Supercritical Pithfork Bifurcation Subcritical Pitchfork Bifurcation

Density

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Vorticity Banding in Dilute 3D Granular Flow

(Conway and Glasser. Phys. Fluids, 2004)

Length Width Depth

Particle density iso-surfaces for Particle

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Conclusions

Higher order nonlinear terms are important to get correct bifurcation scenario.

Supercritical Region Subcritical Region

Analytical solution exists at any order.

Pitchfork Bifurcation Hopf Bifurcation

Density

Subcritical and supercritical Subcritical

Vorticity Banding Gradient Banding

Pitchfork Bifurcation

Density

Shukla & Alam (2011 Submitted)

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Theory for Mode Interaction

(via Coupled Landau Equations) Gradient Banding Vorticity Banding In dilute-regime both gradient and vorticity banding modes exist

Coupled Landau Equations for non-resonating modes

Case1

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Single mode analysis fails

Coupled Landau Equations for resonating modes Case 2

Wavenumber Growth rate

Condition for 1:n resonance.

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Theory for Mode Interaction

Center-manifold reduction (Carr 1981)

Amplitude of 1st mode Amplitude of 2nd mode

Two dimensional Center Manifold EVP EVP

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Mode Interaction and Coupled Landau Eqn.

Coupled Landau Equation Non-resonating modes Coupled Landau Equation 1:2 resonance Coupled Landau Equation “mean flow” resonance Shukla & Alam (2011) (Preprint) Numerical results awaited

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Conclusions

  • Coupled Landau equations have been derived for

both cases: resonating mode interaction and non-resonating mode interaction.

  • Analytical solutions for the coefficients of coupled

Landau equations have been derived for the gradient- banding problem (first problem of the talk).

  • Detailed numerical results awaited.

Shukla and Alam (preprint 2011)

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Theory for Spatially Modulated Patterns

Complex Ginzburg Landau Equation (CGLE) Ordinary differential equation Partial differential equation

Landau Equation Complex Ginzburg Landau Equation

Holds for spatially periodic patterns Holds for spatially modulated patterns

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Under which condition CGLE arises?

Neutral Stability Curve

For all modes are decaying : Homogeneous state is stable, at a critical wave number gains neutral stability, there is a narrow band of wavenumbers around the critical value where the growth rate is slightly positive. width of the unstable wavenumbers:

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Theory (Multiple scale analysis)

Growth rate is of order Stewartson & Stuart (1971)

The timescale at which nonlinear interaction affects the evolution of fundamental mode is of order 1/(growth rate) Slow time scale Slow length scale Group velocity

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Conclusions

  • Complex Ginzburg Landau equation has been derived that describes

spatio-temporal patterns in a ``bounded’’ sheared granular fluid.

Patterns in Vibrated Bed

Recent work of Saitoh and Hayakawa (Granular Matter 2011) on CGLE in ``unbounded’’ shear flow.

  • Numerical results awaited...

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Summary

  • Landau-type order parameter theory for the gradient banding in gPCF has been

developed using center manifold reduction. Ref: PRL, vol. 103, 068001, (2009)

  • Analytical solution for the shearbanding instability, comparison with numerics &

bifurcation scenario have been unveiled. Ref: JFM, vol. 666, 204-253, (2011a)

  • The order parameter theory for 2D-gPCF has been developed. Nonlinear patterns

and bifurcations have been studied. Ref: JFM, vol. 672, 147-195 (2011b)

  • Nonlinear analysis for the gradient and vorticity banding in 3D-GPCF has been

carried out. Submitted (2011c)

  • Coupled Landau equations for resonating and non-resonating cases have been derived.

Preprint

  • Complex Ginzburg Landau equation has been derived for bounded shear flow.

Preprint

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Revisit nonlinear theory of Saturn’s Ring

  • Non-isothermal model with spin, stress

anisotropy...

  • Self-gravity, Corriolis and Tidal forces ...??
  • Spatially modulated waves (Joe’s talk)...
  • Wave interactions (Jurgen’s comment)...
  • Secondary instability, ....

THANK YOU

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