Particle-Laden Thin Film Flow and the Gulf of
Mexico Oil Spill
Oil washes up with the tide
across a beach at the
mouth of the Mississippi
River near Venice, LA.
What happens when
oil and sand mix?
Photo from USA Today
Image from USA today
•
Large oil plumes can be seen floating near the surface of the water
about 6-10 miles south of Pensacola Pass, near Pensacola, Fla.
Waters south of Venice, La and Grand Isle, La.
Crude oil washes ashore in Orange Beach, Ala. Large
amounts of the oil are arriving along the Alabama coast,
leaving deposits of the slick mess 4-6 inches thick at some
beach locations.
Photo from USA Today online
Thick oil from the Deepwater Horizon spill is found on a beach
in Gulfport, Miss.
Photo from USA Today
Morphological environments on contaminated
beach, July 1
Courtesy of
Ping Wang, USF
After Hurricane Alex
Deposition of tar balls and oil stain in association with
individual wave runup. High waves associated with the
hurricane transported the oil contamination over a
distance of up to 30 m landward of the active berm
crest, covering an active turtle nest. Photo taken July 1,
2010.
Courtesy of Ping Wang US
Types and cross-shore distribution of oil
contamination
June 30, 2010 during hurricane.
June 24 pre-hurricane
(a) Concentrated tar balls
and tar patties at the
maximum high-tide runup
and patchy tar balls
distribution between the
active berm crest and the
upper limit of wave runup.
(b) Oil stains and tar balls
distribute on the surface
between the active berm
crest and the maximum
high-tide runup. Note:
The zone of oil
contamination of (a) is
much wider than (b), due
to variations in wave
energy. Courtesy of Ping
Buried Oil
UCLA slurry flow research group
• Collaborator: Peko Hosoi (MIT)
• Former postdocs: Natalie Grunewald, Thomas
Ward
• Current postdoc: Nebo Murisic
• Visitors: Dirk Peschka, Benoit Pausader
• Former students: Ben Cook, Chi Wey, Rob
Glidden
• Current students: Matthew Mata, Paul Latterman
Basic research on sand/oil mixtures – MIT exp from
2003
well mixed fluid particle ridge
clear fluid
PDMS
glycerol
2003
particles
Very different
dynamics
Santa Monica Beach sand and PDMS
D
Dirk Peschka movies
Role of sand concentration
Santa Monica Beach Sand
and PDMS
Dirk Peschka
• Water-particulate mixtures have a long
history of research – in the context of
landscape erosion, sand bar formation, and
mud slides.
• In contrast, oil (or viscous fluids in general)
particulate mixtures are less well-studied at
least for geological problems.
• Gulf Oil Spill has exhibited some very
complex dynamics of oil washing up on
beaches and suggests a number of
problems for further study.
• This talk is mainly about research pre-Deep
Model derivation
•
Flux equations
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div P+r(f)g = 0, div j = 0
P = -pI + m(f)(grad j + (grad j)T) stress tensor
j = volume averaged flux,
r=effective density
m = effective viscosity
p = pressure
f = particle concentration
jp = fvp , jf=(1-f) vf , j=jp+jf
J. Zhou et al PRL
Original model
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Model Derivation II
Particle velocity vR
relative to fluid
2 ( r p  r f )a
vR 
f (f ) w(h) g
9
mf
2
w(h) 
w(h) wall effect
Richardson-Zaki
correction m=5.1
Flow becomes solidlike at a critical
particle concentration
2
ah
1  ( Ah )
2
f (f)  (1  f)
2
m
m(f)  (1  f / fmax )
m(f) = viscosity, a = particle size
f= particle concentration
2
Lubrication approximation
 ( r (f )h)  r (f ) 3

h hxxx
t
 m (f )
dimensionless variables as in
2
 r (f ) 3

5 rfluid*
(f ) 4
r (f ) 3 
clear
 D(  )
h ( r (f )h) 
h ( r (f )) 
h
 m (f )

x
8 m (f )
x


m (f )
 0
x
 f

 (fh)  f
5 f
3
3
4

h hxxx  D(  ) 
h ( r (f )h) x 
h ( r (f )) x  
t
8 m (f )
 m (f )
x
 m (f )
fr(f ) 3 2


h  Vsfhf (f )w(h)  0
3
 m(f )
x
r p  r f a2
Vs 
2
rf H
f (f )  (1  f ) f (f )

Dropping higher
order terms
*D() = (3Ca)1/3cot(), Ca=mfU/g,
- Bertozzi & Brenner Phys. Fluids 1997
Reduced model
Remove higher
order terms
 ( r (f )h)  r (f ) 2 3 

h  0
t
 m (f )
x

 (fh) fr (f ) 3 2

h  Vsfhf (f ) w(h)  0
t
3
 m (f )
x
u
System of conservation
 F(u,v)x  0
t
laws for u=r(f)h and v=fh
v
 G(u,v)x  0
t
Comparison between full and reduced models
macroscopic dynamics well described by reduced
model
full model
reduced
model
Double
shock
solution
• Riemann
problem can
have double
shock solution
•
f=15%
f=30%
Four equations
in four
unknowns
F (ui , vi )  F (ul , vl ) G(ui , vi )  G(ul , vl )
(s1,s2,ui,vi)
s1 

ui  ul
vi  vl
F (ur , vi )  F (ur , vr ) G(ur , vi )  G(ur , vr )
s2 

ui  ur
vi  vr
Singular behavior at contact line
UCLA
Riemann Connections
Richardson-Zaki Settling model
• Cook, ALB, Hosoi, SIAP, 2008
Riemann Connections
Richardson-Zaki Settling model- singular shocks
• Cook, ALB, Hosoi, SIAP, 2008
Riemann Connections
Buscall et al settling model (goes to zero at maximum
packing fraction).
• Cook, ALB, Hosoi, SIAP, 2008
Constant Volume Model
Clear flow – analysis by Huppert, Nature 1980s, rarefaction-shock
similarity solution – no free parameters
What happens to this solution when you add particles?
Natalie Grunewald, Rachel Levy, Matthew Mata, Thomas Ward, and
Andrea L. Bertozzi, Self-similarity in particle laden flows at constant
volume, J. Eng. Math., 2009.
-departure from similarity solution in model – but modest change
Thomas Ward, Chi Wey, Robert Glidden, A. E. Hosoi, and A. L. Bertozzi,
Experimental study of gravitation effects in the flow of a particle-laden
thin film on an inclined plane, Physics of Fluids, 21, 083305, 2009.
-use experiment as a way to measure effective viscosity of mixture
from Huppert solution.
-experiments show that effective viscosity at high particle
concentration is greatly affected by particle settling.
2D Evolution
Equations
The evolution equations for the mixture as a whole for the particles are
given by
ht  div(hvav )  0,
(fh)t  div(fh(vav  (1  f )vrel )  f pt  0
where the volume-averaged velocity of the two phases is given by
 h3
 r (f ) 3
h3
5 h4
2
vav 
  D( ) 
( r (f )h) 
( r (f )) 
h xˆ
m (f )
8 m (f )
 m (f )
 m (f )
the settling velocity of the particles relative to the liquid is
vrel  Vs f (f ) w(h) xˆ
and the particle flux term for shear-induced diffusion is
f pt
3 2
h 2 r (f )
1/ 3 ˆ
  a (3Ca ) D(f )
f
2
m (f )
Equations from Cook, Alexandrov, and Bertozzi (2009),
ADI scheme simulation by Matthew Mata (2010).
Film Thickness 0.1
Film Thickness,
Precursor=0.01
Particle Concentration,
Precursor=0.1
Particle Concentration,
Precursor=0.01
Experiments
- Motivation: older experiments (Hosoi MIT, 04)
- Our experiments: carried out at Applied Math
Lab, UCLA, summer/fall 2009
- Main goals: i) obtain detailed data on evolution
of the contact line region (regimes);
ii) study influence of incl. angle &
particle concentration on contact line dynamics;
iii) also influence of liquid viscosity
& particle size;
iv) study details of fingering instab.
- β ∈ [10deg,55deg] (incl. angle)
- ϕ ∈ [0.25,0.55] (particle volume fraction)
- μ = 100, 1000 cSt (PDMS viscosity)
- a ∈ [150,850μm] (particle size)
Undergraduate REU: Joyce Ho, Paul Latterman,
Stephen Lee, Kanhui Lin, Vincent Hu, mentor:Nebo
Experiments (cont.)
small β, small ϕ
large β, large ϕ
intermediate values of β & ϕ
Particle Volume Fraction Model
- Main question: Will particle settle out of the flow or remain in the suspension?
- Simple model: equilibrium balance of particle settling against shear induced migration normal
to substrate (Ben Cook PRE 2008, tested against old data from MIT). New experiments varying
bead size and viscosity of oil.
- Particle volume fraction model:
- The flux: i) Peclet number large ⇒ ignore Brownian motion
ii) settling function based on Stokes setting velocity (Richardson-Zaki hinderance &
solid wall effect)
iii) shear-induced migration (collisions & viscosity contributions) as in Phillips etal.
iv) consider a flat film (away from contact line), and an equilibrium situation: a
balance in contributions to flux J ⇒ no LHS
Particle Volume Fraction Model (cont.)
- Concentrate on z-direction (cross-section of the film) & after some manipulation (Cook)
- Result: system of two BVPs (for concentration and shear rate)
- May be solved for particle volume fraction, shear stress, σ/rate (particle velocity) given
inclination angle, height of liquid column, and hight-averaged particle volume fraction in the
column (R-K & shooting)
Particle Volume Fraction Model (cont.)
Settled: β = 15 deg, ϕ = 0.25
Particle Volume Fraction Model (cont.)
Ridged: β = 45 deg, ϕ = 0.475
Particle Volume Fraction Model (cont.)
- Consider a solution of the system where there is no variation of ϕ in z-direction (ϕ` = 0;
well mixed case):
High viscosity
Small beads-143umMedium beads-337um
Large beads-625um
urisic et al submitted to Physica D special Childress Issue
Particle Volume Fraction Model (cont.)
Small beads
Medium beads
Low viscosity oil
Conclusions and Future Work
- We carried out detailed experimental study of particle-laden thin film flow down an
incline
- Three distinct regimes of flow near contact line: presence of particles suppresses fingering
instability
- Simple theoretical model agrees very well with the experimental data (at least in
predicting the range of values for which transition in regimes occurs)
- Still to do: analysis of the motion of the front; more detailed study of the fingering
instability
- We also need to model the full system: liquid + particles
Additional research problems
motivated
by
oilspill
• Current theory is for dry sand mixed with oil.
Experiments suggest even trace amounts of water
in the mixture cause instabilities in flow on incline.
How to model?
• Current theory is for separation of mixtures. More
relevant to spill is clear oil deposited on top of
sand and ensuing dynamics. This can be done in
the lab to some degree.
• Layering of oil and sand in berm must be
understood, in particular to locate hidden oil on
beaches.