1. May 2008 IPAM 2008 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

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 0 o -60 o Polydisperse glass beads (250- 425  m) PDMS 200-1000 cSt Glycerol

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

6. Model Derivation I-Particle Ridge Regime Flux equations div  +  g = 0, div j = 0  = -pI +  )(grad j + (grad j) T ) stress tensor j = volume averaged flux,  =effective density  = effective viscosity p = pressure  = particle concentration j p =  v p, j f =(1-  ) v f, j=j p +j f

7. Model Derivation II-particle ridge regime Particle velocity v R relative to fluid w(h) wall effect Richardson-Zaki correction m=5.1 Flow becomes solid-like at a critical particle concentration  = viscosity, a = particle size  = particle concentration

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

9. Reduced model Remove higher order terms System of conservation laws for u=  h and v=  h

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

11. Double shock solution Riemann problem can have double shock solution Four equations in four unknowns (s 1,s 2,u i,v i )  =15%  =30% 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 May 2008 IPAM 2008

13. Volume averaged model May 2008 IPAM 2008 Full model Reduced model

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

15. Singular shock formation May 2008 IPAM 2008

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

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 image from Huppert 1982 instability caused by h 2 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.

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

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

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

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

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

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

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

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

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

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

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 a 2, stratified flow is independent of a In settling model  appears only in time scale, while  is crucial in stratified model

37. Critical Concentration

38. Phase Diagram 15 30 45  

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.