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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
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
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.
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
Comparison With Clear Film
h
x
x
particle-laden film
no particles(same viscosity)
clear fluid simulated by removing settling term
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:
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
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.