Sediment Transport in Viscous Fluids

1. 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 IPAM 2008

2. Thin film and fluid instabilities:a breadth of applications • Spin coating microchips • De-icing airplanes • Paint design • Lung surfactants • Nanoscale fluid coatings • Gene-chip design

3. 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

4. Experimental Apparatus and Parameters • 30cmX120cm acrylic sheet • Adjustable angle 0o-60o • Polydisperse glass beads (250-425 mm) • PDMS 200-1000 cSt • Glycerol

5. Experimental Phase Diagram particle ridge well mixed fluid clear fluid PDMS glycerol

6. Model Derivation I-Particle Ridge Regime • Flux equations • 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

7. 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 m(f) = viscosity, a = particle size f = particle concentration

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

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

10. Comparison between full and reduced models macroscopic dynamics well described by reduced model full model reduced model

11. f=15% f=30% Double shock solution • Riemann problem can have double shock solution • Four equations in four unknowns (s1,s2,ui,vi) Singular behavior at contact line

12. 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 IPAM 2008

13. Volume averaged model Full model Reduced model IPAM 2008

14. Hugoniot locus for Riemann problem – Richardson-Zacki settling When b is small there are no connections from the h=1 state. IPAM 2008

15. Singular shock formation IPAM 2008

16. Modified settling as an alternative R. Buscall et al JCIS 1982 Modified Hugoniot locus: Double shock solutions exist for arbitrarily small precursor. IPAM 2008

17. Two Dimensional Instability of Particle-Laden Thin Films Benjamin Cook, Oleg Alexandrov, and Andrea Bertozzi Submitted to Eur. Phys. J. 2007 UCLA Mathematics Department

18. Background - Fingering Instability 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. image from Huppert 1982

19. Stokes settling velocity “wall effect” hindered settling Lubrication model for particle-rich ridge as described in ZDBH 2005 Unstratified film: concentration  assumed independent of depth volume-averaged velocity relative velocity effective mixture viscosity 2x2 conservation laws:

20. Double Shock Solutions from Cook, Bertozzi, and Hosoi, SIAM J. Appl. Math., submitted. numerical (Lax-Friedrichs) 1-shock 2-shock R=L L=0.3 b=0.01

21. Effect of Precursor values of h and  at ridge max  - modified settling  - original settling h - original settling h - modified settling

22. 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

23. Incorporating Particle Diffusion diffusivity: Leighton and Acrivos, J. Fluid Mech. 1987 shear rate particle radius a dimensionless diffusion coefficient: equations become:

24. Time-Dependent Base State 4th-order equations 1st-order equations h x  x

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

26. Linear Stability Analysis Introduce perturbation: derive evolution equations: extract growth rate:

27. Evolution of Perturbation t=4000 h g x after t=4500 perturbation is largest at trailing shock t

28. Perturbation Growth Rates maximum growth rate is reduced, and occurs at longer wavelength no particles particles

29. 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

30. 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

31. no variation in x direction no settling in x direction settling in z direction balanced by shear-induced diffusion figure from SAZ 1990

32. 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

33. Migration Model * shear-induced flux: gravity flux: non-dimensionalize: balance equations: * Phillips, Armstrong, Brown, Graham, and Abbott, Phys. Fluids A, 1992

34. Depth Profiles (velocities relative to homogeneous mixture)

35. Velocity Ratio

36. 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

37. Critical Concentration

38. 45 30 15 Phase Diagram  f

39. 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

40. 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.