An Experimental and Theoretical Investigation of Gravity-Driven - - PowerPoint PPT Presentation

▶
an experimental and theoretical investigation of gravity
SMART_READER_LITE
LIVE PREVIEW

An Experimental and Theoretical Investigation of Gravity-Driven - - PowerPoint PPT Presentation

Mar 01, 2023 12 likes •457 views

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 ,

slide-1
SLIDE 1

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 Hin1, Kaiwen Huang2, Shreyas Kumar1, Gilberto Urdaneta2

Aliki Mavromoustaki2, Sungyon Lee2, Andrea L. Bertozzi2

1Harvey Mudd College and 2UCLA

August 7, 2013

slide-2
SLIDE 2

Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion

Table of Contents

1

Motivation

2

Experimentation

3

Bidisperse Bifurcation

4

Prefactor Characterization

5

Particle Fronts

6

Theory

7

Particle Concentrations

8

Conclusion and Future Work

slide-3
SLIDE 3

Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion

Motivation

(a) Oil Spills (b) Mudslides Figure: Why our research matters.

slide-4
SLIDE 4

Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion

Experimental Apparatus

slide-5
SLIDE 5

Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion

Flow Progression; Settled

slide-6
SLIDE 6

Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion

Flow Progression; Ridged

slide-7
SLIDE 7

Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion

Regime Diagram

slide-8
SLIDE 8

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 ∝ t1/3 relation. Hence, x = ct1/3 where c is the prefactor. A larger c indicates that the flow is faster.

slide-9
SLIDE 9

Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion

Extracting the Prefactor

slide-10
SLIDE 10

Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion

Prefactor vs Angle

slide-11
SLIDE 11

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 x = 9a2g sin α 4ν 1/3 t1/3 Murisic et al. generalized this to slurry flows x =

  • ρ(φ)9a2g sin α

4µ(φ) 1/3 t1/3

slide-12
SLIDE 12

Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion

Prefactor vs Angle - Comparison to Theory

slide-13
SLIDE 13

Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion

Prefactor vs Lambda

slide-14
SLIDE 14

Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion

Prefactor vs Lambda

slide-15
SLIDE 15

Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion

Visible Fronts

slide-16
SLIDE 16

Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion

Settled Regime

slide-17
SLIDE 17

Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion

Ridged Regime

slide-18
SLIDE 18

Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion

Particle Equations

Particle Transport Equation φt + u · ∇φ + ∇ · J = 0

slide-19
SLIDE 19

Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion

Particle Equations

Particle Transport Equation φt + u · ∇φ + ∇ · J = 0 where J = Jgrav

slide-20
SLIDE 20

Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion

Particle Equations

Particle Transport Equation φt + u · ∇φ + ∇ · J = 0 where J = Jgrav + Jcoll

slide-21
SLIDE 21

Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion

Particle Equations

Particle Transport Equation φt + u · ∇φ + ∇ · J = 0 where J = Jgrav + Jcoll + Jvisc

slide-22
SLIDE 22

Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion

Particle Fluxes

Jgrav = d2φ(ρP − ρL) 18µL f (φ)g Jcoll = −Kcd2 4 (φ∇˙ γ + φ˙ γ∇φ) , (1) Jvisc = − Kvd2 4µ(φ)φ2 ˙ γµφ∇φ.

where, f (φ) - hindrance settling function, d - particle diameter, ρP - particle density, ρL - liquid density, Kc, Kv - empirical constants, ˙ γ - shear rate (here, ˙ γ ∼ ∂u/∂z)

slide-23
SLIDE 23

Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion

Assumptions for the Continuum Model

slide-24
SLIDE 24

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

slide-25
SLIDE 25

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

slide-26
SLIDE 26

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

slide-27
SLIDE 27

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

slide-28
SLIDE 28

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

slide-29
SLIDE 29

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 ≪ η2 ≪ 1

slide-30
SLIDE 30

Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion

z-direction ODE

Leading Order ODE for z-direction σz + (1 + ρsφ) = 0 (2) σφz

  • 1 +

2φ (φ − φm) Kv − Kc Kc

  • +φσz+2ρs cot α

9Kc (1−φ) = 0 (3)

slide-31
SLIDE 31

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)uz, then u(z) can be found using the no-slip BC Coupled system of PDEs ht + Fx = 0 (φ0h)t + Gx = 0 (4)

where F = h3 1 ˜ u d˜ z, G = h3 1 ˜ φ ˜ u d˜ z and ˜ z = z/h

slide-32
SLIDE 32

Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion

Monodisperse Bifurcation

slide-33
SLIDE 33

Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion

Monodisperse Bifurcation

slide-34
SLIDE 34

Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion

Bidisperse Flow in x

ht + Fx = 0 (hφ1)t + Gx = 0 (5) (hφ2)t + Jx = 0 (6) where where F = h3

1 ˜ u d˜ z, G = h3 1 ˜ φ1 ˜ u d˜ z, J = h3 1 ˜ φ2 ˜ u d˜ z, and ˜ z = z/h

slide-35
SLIDE 35

Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion

Bidisperse Bifurcation

slide-36
SLIDE 36

Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion

Bidisperse Numerical Solution - Height

(a) Initial Condition (b) Solution for Fluid Height Figure: Numerical Solutions

slide-37
SLIDE 37

Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion

Fluorescence Imaging

(Loading Video)

slide-38
SLIDE 38

Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion

Results

slide-39
SLIDE 39

Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion

Correlating Intensity and Concentration

slide-40
SLIDE 40

Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion

Conclusion

Explored the parameter space for the bidisperse case Compared front positions against theoretical statements Imaged the second particle front of the heavier species Confirmed particles tend towards the downstream front Observed two distinct particle behaviors in fluorescent experiments

slide-41
SLIDE 41

Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion

Future Work

Continue to characterize prefactors for fluid and particle fronts for different regimes and dispersity Determine viable single-valued correlation curve Improve fluorescent experimental setup

slide-42
SLIDE 42

Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion

Acknowledgements Special thanks to Li Wang Jeffrey Wong

slide-43
SLIDE 43

Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion

Thank you for surviving. Any questions?

slide-44
SLIDE 44

Motivation Experimentation Bidisperse Bifurcation Prefactor Characterization Particle Fronts Theory Particle Concentrations Conclusion

Thank you for surviving. Any questions?

It’s Miller Time!