ParticleLaden Thin Film Flow and the Gulf of

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 PDMS D Dirk Peschka movies

Role of sand concentration Santa Monica Beach Sand 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 • • div P+r(f)g = 0, div j = 0 P = -p. I + 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 2005 Original model

• • Model Derivation II Particle velocity v. R relative to fluid w(h) wall effect Richardson-Zaki correction m=5. 1 Flow becomes solidlike at a critical particle concentration m(f) = viscosity, a = particle size f = particle concentration

Lubrication approximation dimensionless variables as in clear fluid* Dropping higher order terms *D(b) = (3 Ca)1/3 cot(b), Ca=mf. U/g, - Bertozzi & Brenner Phys. Fluids 1997

Reduced model Remove higher order terms System of conservation laws for u=r(f)h and v=fh

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 (s 1, s 2, ui, vi) 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 1980 s, 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.

2 D Evolution Equations The evolution equations for the mixture as a whole for the particles are given by where the volume-averaged velocity of the two phases is given by the settling velocity of the particles relative to the liquid is and the particle flux term for shear-induced diffusion is 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 ∈ [10 deg, 55 deg] (incl. angle) - ϕ ∈ [0. 25, 0. 55] (particle volume fraction) - μ = 100, 1000 c. St (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-143 um. Medium beads-337 um Large beads-625 um 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 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.