May 2008 IPAM 2008

Sediment Transport in Viscous Fluids

Andrea Bertozzi

UCLA Department of Mathematics

Collaborators:

Junjie Zhou, Benjamin Dupuy, and A. E. Hosoi MIT

Ben Cook, Natalie Grunewald, Matthew Mata, Thomas Ward, Oleg Alexandrov, Chi Wey, UCLA

Rachel Levy, Harvey Mudd College

Thanks to NSF and ONR

Thin film and fluid instabilities:a breadth of applications

Spin coating microchips

De-icing airplanes

Paint design

Lung surfactants

Nanoscale fluid coatingsGene-chip design

Shocks in particle laden thin films

J. Zhou, B. Dupuy, ALB, A. E. Hosoi, Phys. Rev. Lett. March 2005

Experiments show different settling regimes

Model is a system of conservation laws Two wave solution involves classical

shocks

Experimental Apparatus and Parameters

30cmX120cm acrylic sheet

Adjustable angle 0o-60o

Polydisperse glass beads (250-425 m)

PDMS 200-1000 cSt

Glycerol

Experimental Phase Diagram

clear fluid

well mixed fluid particle ridge

glycerol

PDMS

Model Derivation I-Particle Ridge Regime

Flux equations div +g = 0, div j = 0 = -pI + )(grad j + (grad j)T) stress tensor j = volume averaged flux, =effective density = effective viscosity p = pressure = particle concentration jp = vp , jf=(1-) vf , j=jp+jf

Model Derivation II-particle ridge regime

Particle velocity vR relative to fluid

w(h) wall effect

Richardson-Zaki correction m=5.1

Flow becomes solid-like at a critical particle concentration

2

2

2

)(1)(

)()()(

9

2

2Ah

ahhw

ghwfa

vf

fpR

2max )/1()(

)1()(

mf

= viscosity, a = particle size = particle concentration

Lubrication approximation

0)(

)())((

)(

)(

8

5))((

)(

)()(

)(

)())(( 32

433

x

xxxxx hhhhDhht

h

x

xxxxx hhhDhht

h

))(()(8

5))((

)()(

)(

)( 433

0)()(3

2

)(

)(

)(3

x

s hwhfVh

dimensionless variables as in clear fluid*

*D() = (3Ca)1/3cot(), Ca=fU/, - Bertozzi & Brenner Phys. Fluids 1997

)()1()( 2

2

ff

H

aV

f

fps

Dropping higher

order terms

Reduced model

0)()(3

2

)(

)()(

0)(

)())((

3

32

x

s

x

hwhfVht

h

ht

h

Remove higher order terms

System of conservation laws for u=h and v=h

0),(

0),(

x

x

vuGt

v

vuFt

u

Comparison between full and reduced models macroscopic dynamics well described by reduced model

reduced model

full model

Double shock solution

Riemann problem can have double shock solution

Four equations in four unknowns (s1,s2,ui,vi)

ri

rrir

ri

rrir

li

llii

li

llii

vv

vuGvuG

uu

vuFvuFs

vv

vuGvuG

uu

vuFvuFs

),(),(),(),(

),(),(),(),(

2

1

=15%

=30%

Singular behavior at contact line

Shock solutions for particle laden films

SIAM J. Appl. Math 2007, Cook, ALB, Hosoi

Improved model for volume averaged velocities

Richardson-Zacki settling model produces singular shocks for small precursor

Propose alternative settling model for high concentrations – no singular shocks, but still singular depedence on precursor

May 2008 IPAM 2008

Volume averaged model

May 2008 IPAM 2008

Fullmodel

Reducedmodel

Hugoniot locus for Riemann problem – Richardson-Zacki settling

May 2008 IPAM 2008

When b is small there are no connections from the h=1 state.

Singular shock formation

May 2008 IPAM 2008

Modified settling as an alternative

May 2008 IPAM 2008

R. Buscall et al JCIS 1982

Modified Hugoniot locus:

Double shock solutions exist for arbitrarily small precursor.

Two Dimensional Instability of Particle-Laden Thin Films

Benjamin Cook, Oleg Alexandrov, and Andrea Bertozzi

Submitted to Eur. Phys. J. 2007

UCLA Mathematics Department

Background - Fingering Instability

image from Huppert 1982

instability caused by h2 velocity

stabilized by surface tension at short wavelengths

observed by H. Huppert, Nature 1982.

references:

Troian, Safran, Herbolzhiemer, and Joanny, Europhys. Lett., 1989.Jerrett and de Bruyn, Phys. Fluids 1992.Spaid and Homsy, Phys. Fluids 1995.Bertozzi and Brenner, Phys. Fluids 1997.Kondic and Diez, Phys. Fluids 2001.

Unstratified film: concentration assumed independent of depth

effective mixture viscosity

Stokes settling velocity

hindered settling“wall effect”

relative velocityvolume-averaged velocity

2x2 conservation laws:

Lubrication model for particle-rich ridge

as described in ZDBH 2005

Double Shock Solutions

b=0.01

L=0.3

1-shock 2-shock

R=L

numerical (Lax-Friedrichs)

from Cook, Bertozzi, and Hosoi, SIAM J. Appl. Math., submitted.

Effect of Precursor

- original settling

h - original settlingh - modified settling

- modified settling

maxvalues of h and at ridge

Fourth Order Equations

add surface tension:

velocities are:

modified capillary number:

relative velocity is still unregularized - this leads to instability in the numerical solution

a likely regularizing effect is shear-induced diffusion

Incorporating Particle Diffusion

equations become:

dimensionless diffusion coefficient:

particle radius a

diffusivity: Leighton and Acrivos, J. Fluid Mech. 1987

shear rate

Time-Dependent Base State

h

x

x

4th-order equations

1st-order equations

Comparison With Clear Film

h

x

x

particle-laden film

no particles(same viscosity)

clear fluid simulated by removing settling term

Linear Stability Analysis

Introduce perturbation:

derive evolution equations:

extract growth rate:

Evolution of Perturbation

x

h

g

t=4000

t

after t=4500 perturbation is largest at trailing shock

Perturbation Growth Rates

particles

no particles

maximum growth rate is reduced,and occurs at longer wavelength

Conclusion

Lubrication model predicts the same qualitative effects of settling on the contact-line instability: longer wavelengths and more stable

Unclear if the predicted effects are of sufficient magnitude to explain experimental observations

Model for a Stratified Film due to Ben Cook (preprint 07)

Necessary to explain phase diagram

May change relative velocity (top layers move faster)

Stratified films have been observed for neutrally buoyant particles: B. D. Timberlake and J. F. Morris, J. Fluid Mech. 2005

no variation in x direction

no settling in x direction

settling in z direction balanced by shear-induced diffusion

figure from SAZ 1990

Properties of SAZ 1990 Model

velocities are weighted averages:

diffusive flux:

diffusive flux balancing gravity implies d/dz < 0

therefore particles move slower than fluid

possibly appropriate for normal settling regime: particles left behind

with non-diffusive migration, particles may move faster

Migration Model

* Phillips, Armstrong, Brown, Graham, and Abbott, Phys. Fluids A, 1992

shear-induced flux: *

gravity flux:

balance equations:

non-dimensionalize:

Depth Profiles

(velocities relative to homogeneous mixture)

Velocity Ratio

How to distinguish between settling and stratified flow?

Settling rate is proportional to a2, stratified flow is independent of a

In settling model appears only in time scale, while is crucial in stratified model

Critical Concentration

Phase Diagram

15

30

45

Conclusions The migration/diffusion model predicts

both faster and slower particles, depending on average concentration

Velocity differences due to stratification may be more significant than settling

This model is consistent with the phase diagram of ZDBH 2005

Conclusions

Double shock solution agrees extremely well with both reduced model and full model dynamics.

Explains emergence of particle-rich ridge Provides a theory for the front speed Similar to double shocks in thermocapillary-gravity flow These new shocks are classical, NOT undercompressive Result from different settling rates (2X2 system) Singular behavior at contact line seen even in reduced model

(no surface tension) – different from other driven film problems. Fingering (2D) problem can be analyzed but only qualitatively

explained by this theory Shear induced migration seems to play a role at lower angles

and particle concentrations.