Transcript otws3 7274
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 May 2008 IPAM 2008 Thin film and fluid instabilities: a breadth of applications Spin coating microchips De-icing airplanes " Nanoscale fluid coatings "Gene-chip design " Paint design Lung surfactants 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 Experimental Apparatus and Parameters 30cmX120cm acrylic sheet Adjustable angle 0o-60o Polydisperse glass beads (250425 mm) PDMS 200-1000 cSt Glycerol Experimental Phase Diagram well mixed fluid particle ridge clear fluid PDMS glycerol 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 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 2 ( r r ) a 2 p f vR f (f ) w(h) g 9 mf w(h) ah 2 1 ( Ah 2 ) 2 f (f ) (1 f ) m m (f ) (1 f / fmax ) m(f) = viscosity, a = particle size f = particle concentration 2 Lubrication approximation dimensionless variables as in clear fluid* r (f ) 3 r (f ) 2 3 ( r (f )h) r (f ) 3 5 r (f ) 4 h hxxx D( ) h ( r (f )h) x h ( r (f )) x h 0 t 8 m (f ) m (f ) m (f ) x m (f ) f 3 (fh) f 3 5 f 4 h hxxx D( ) h ( r (f )h) x h ( r (f )) x t 8 m (f ) m (f ) x m (f ) f fr (f ) 3 2 h Vsfhf (f ) w(h) 0 3 m (f ) m (f ) x r p r f a2 Vs rf H2 f (f ) (1 f ) f (f ) *D() = (3Ca)1/3cot(), Ca=mfU/g, - Bertozzi & Brenner Phys. Fluids 1997 Dropping higher order terms Reduced model Remove higher order terms ( r (f )h) r (f ) 2 3 h 0 t m (f ) x (fh) fr (f ) 3 2 h Vsfhf (f ) w(h) 0 t 3 m (f ) x System of conservation laws for u=r(f)h and v=fh u F (u , v)x 0 t v G (u , v)x 0 t Comparison between full and reduced models macroscopic dynamics well described by reduced model full model reduced model Double shock solution f=15% Riemann problem can have double shock solution f=30% F (ui , vi ) F (ul , vl ) G (ui , vi ) G (ul , vl ) Four equations s1 in four ui ul vi vl unknowns (s1,s2,ui,vi) F (ur , vi ) F (ur , vr ) G (ur , vi ) G (ur , vr ) s2 ui u r Singular behavior at contact line vi vr 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 Volume averaged model Full model Reduced model May 2008 IPAM 2008 Hugoniot locus for Riemann problem – Richardson-Zacki settling When b is small there are no connections from the h=1 state. May 2008 IPAM 2008 Singular shock formation May 2008 IPAM 2008 Modified settling as an alternative R. Buscall et al JCIS 1982 Modified Hugoniot locus: Double shock solutions exist for arbitrarily small precursor. May 2008 IPAM 2008 Two Dimensional Instability of Particle-Laden Thin Films Benjamin Cook, Oleg Alexandrov, and Andrea Bertozzi Submitted to Eur. Phys. J. 2007 UCLA Mathematics Department 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 Lubrication model for particle-rich ridge as described in ZDBH 2005 Unstratified film: concentration f assumed independent of depth volume-averaged velocity effective mixture viscosity relative velocity Stokes settling velocity “wall effect” hindered settling 2x2 conservation laws: Double Shock Solutions from Cook, Bertozzi, and Hosoi, SIAM J. Appl. Math., submitted. numerical (Lax-Friedrichs) 1-shock 2-shock fR=fL fL=0.3 b=0.01 Effect of Precursor values of h and f at ridge f - modified settling f - original settling h - original settling h - modified settling fmax 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 Incorporating Particle Diffusion diffusivity: particle radius a dimensionless diffusion coefficient: equations become: Leighton and Acrivos, J. Fluid Mech. 1987 shear rate Time-Dependent Base State 4th-order equations 1st-order equations h x f x Comparison With Clear Film particle-laden film no particles (same viscosity) h x f clear fluid simulated by removing settling term x Linear Stability Analysis Introduce perturbation: derive evolution equations: extract growth rate: Evolution of Perturbation t=4000 h g x after t=4500 perturbation is largest at trailing shock t Perturbation Growth Rates maximum growth rate is reduced, and occurs at longer wavelength no particles particles 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 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 no variation in x direction no settling in x direction settling in z direction balanced by shearinduced diffusion figure from SAZ 1990 Properties of SAZ 1990 Model velocities are weighted averages: diffusive flux: diffusive flux balancing gravity implies df/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 Migration Model shear-induced flux: * gravity flux: non-dimensionalize: balance equations: * Phillips, Armstrong, Brown, Graham, and Abbott, Phys. Fluids A, 1992 Depth Profiles (velocities relative to homogeneous mixture) Velocity Ratio 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 Critical Concentration Phase Diagram 45 30 15 f 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 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.