Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion An Experimental and Theoretical Investigation of Gravity-Driven Single and Bidisperse Flows Matt Hin 1 , Kaiwen Huang 2 , Shreyas Kumar 1 , Gilberto Urdaneta 2 Aliki Mavromoustaki 2 , Sungyon Lee 2 , Andrea L. Bertozzi 2 1 Harvey Mudd College and 2 UCLA August 7, 2013
Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion Table of Contents Motivation 1 Experimentation 2 Bidisperse Bifurcation 3 Prefactor Characterization 4 Particle Fronts 5 Theory 6 Particle Concentrations 7 Conclusion and Future Work 8
Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion Motivation (a) Oil Spills (b) Mudslides Figure: Why our research matters.
Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion Experimental Apparatus
Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion Flow Progression; Settled
Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion Flow Progression; Ridged
Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion Regime Diagram
Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion Prefactor Characterization Ward et al. (2009) and Murisic et al. (2012) establish a x ∝ t 1 / 3 relation. Hence, x = ct 1 / 3 where c is the prefactor. A larger c indicates that the flow is faster.
Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion Extracting the Prefactor
Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion Prefactor vs Angle
Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion Theoretical Predictions for the Prefactor Huppert developed a theory for a fluid flowing down an incline � 1 / 3 � 9 a 2 g sin α t 1 / 3 x = 4 ν Murisic et al. generalized this to slurry flows � 1 / 3 ρ ( φ )9 a 2 g sin α � t 1 / 3 x = 4 µ ( φ )
Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion Prefactor vs Angle - Comparison to Theory
Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion Prefactor vs Lambda
Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion Prefactor vs Lambda
Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion Visible Fronts
Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion Settled Regime
Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion Ridged Regime
Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion Particle Equations Particle Transport Equation φ t + u · ∇ φ + ∇ · J = 0
Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion Particle Equations Particle Transport Equation φ t + u · ∇ φ + ∇ · J = 0 where J = J grav
Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion Particle Equations Particle Transport Equation φ t + u · ∇ φ + ∇ · J = 0 where J = J grav + J coll
Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion Particle Equations Particle Transport Equation φ t + u · ∇ φ + ∇ · J = 0 where J = J grav + J coll + J visc
Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion Particle Fluxes J grav = d 2 φ ( ρ P − ρ L ) f ( φ ) g 18 µ L J coll = − K c d 2 ( φ ∇ ˙ γ + φ ˙ γ ∇ φ ) , (1) 4 J visc = − K v d 2 4 µ ( φ ) φ 2 ˙ γµ φ ∇ φ. where, f ( φ ) - hindrance settling function, d - particle diameter, ρ P - particle density, ρ L - liquid density, K c , K v - empirical constants, γ - shear rate (here, ˙ ˙ γ ∼ ∂ u /∂ z )
Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion Assumptions for the Continuum Model
Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion Assumptions for the Continuum Model Time-scales Normal ( z ) direction: fast ‘diffusion’ dynamics → rapid averaging of φ in z
Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion Assumptions for the Continuum Model Time-scales Normal ( z ) direction: fast ‘diffusion’ dynamics → rapid averaging of φ in z Axial ( x ) direction: slow flow dynamics down the incline
Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion Assumptions for the Continuum Model Time-scales Normal ( z ) direction: fast ‘diffusion’ dynamics → rapid averaging of φ in z Axial ( x ) direction: slow flow dynamics down the incline Limits η = d H
Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion Assumptions for the Continuum Model Time-scales Normal ( z ) direction: fast ‘diffusion’ dynamics → rapid averaging of φ in z Axial ( x ) direction: slow flow dynamics down the incline Limits η = d H As d → H , η → 1 → continuum hypothesis breaks down
Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion Assumptions for the Continuum Model Time-scales Normal ( z ) direction: fast ‘diffusion’ dynamics → rapid averaging of φ in z Axial ( x ) direction: slow flow dynamics down the incline Limits η = d H As d → H , η → 1 → continuum hypothesis breaks down As d → 0, η → 0 → Brownian diffusion becomes important
Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion Assumptions for the Continuum Model Time-scales Normal ( z ) direction: fast ‘diffusion’ dynamics → rapid averaging of φ in z Axial ( x ) direction: slow flow dynamics down the incline Limits η = d H As d → H , η → 1 → continuum hypothesis breaks down As d → 0, η → 0 → Brownian diffusion becomes important H L ≪ η 2 ≪ 1
Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion z-direction ODE Leading Order ODE for z-direction σ z + (1 + ρ s φ ) = 0 (2) � 2 φ � K v − K c �� + φσ z +2 ρ s cot α σφ z 1 + (1 − φ ) = 0 (3) ( φ − φ m ) 9 K c K c
Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion Suspension Flow in x Flow velocity ODE solutions give φ ( z ) and σ ( z ) Since σ = µ ( φ ) u z , then u ( z ) can be found using the no-slip BC Coupled system of PDEs h t + F x = 0 ( φ 0 h ) t + G x = 0 (4) � 1 � 1 where F = h 3 z , G = h 3 ˜ ˜ u d ˜ φ ˜ u d ˜ z and ˜ z = z / h 0 0
Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion Monodisperse Bifurcation
Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion Monodisperse Bifurcation
Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion Bidisperse Flow in x h t + F x = 0 ( h φ 1 ) t + G x = 0 (5) ( h φ 2 ) t + J x = 0 (6) � 1 � 1 � 1 where where F = h 3 z , G = h 3 ˜ z , J = h 3 ˜ ˜ u d ˜ φ 1 ˜ u d ˜ φ 2 ˜ u d ˜ z , 0 0 0 and ˜ z = z / h
Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion Bidisperse Bifurcation
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