RHEOLOGICAL MEASUREMENTS OF LIQUID-SOLID FLOWS WITH INERTIA AND PARTICLE SETTLING Esperanza Linares, Melany L. Hunt, Roberto Zenit 1 Mechanical Engineering Caltech 1 1 UNAM, Mexico

RHEOLOGICAL MEASUREMENTS OF LIQUID-SOLID FLOWS WITH INERTIA AND PARTICLE SETTLING Esperanza Linares, Melany L. Hunt, Roberto Zenit 1 Mechanical Engineering Caltech 2 1 UNAM, Mexico

Rheology: suspension to a granular flow 3 Dilute non-colloidal suspensions Viscous interactions T =   ; P    - corrected viscosity,  =  f(  ) with  as the solid fraction Granular Flows Collisional interactions T, P   p d 2  2  p particle density d particle diameter SHEAR RATE,  Shear and Normal Stresses, T and P We are interested in this transition.

Prior studies liquid+solids Particle diameters > 0.5 mm (except Acrivos.,) Bagnold (1954) Savage & McKeown (1983) Hanes & Inman (1985) Acrivos, Fan & Mauri (1994) Prasad & Kytomaa (1995) Dijksman, et al. (2010) Boyer, Guazzelli & Pouliquen (2011) 4 PARAMETERS: Particle Reynolds number, Re d = f  d 2 / Solid Fraction,  Density ratio,  p / f Gap Reynolds number, Re gap = f r o (r o -r i )/

5 Acrivos, Fan, Mauri 1994 Boyer, Guazzelli & Pouliquen 2011 Bagnold 1954, “macroviscous” Prasad & Kytomaa 1995 Koos, Linares, Hunt, Brennen 2012 Hanes & Inman 1985 Re=ρ f d 2  / Dijksman, et al 2010 Experiments

6 Our rheometer 37 cm 19 cm3 cm outer rotating cylinder fluid and particles in annular gap inner stationary cylinder Torque measured in central section Guard cylinders Koos, Linares, Hunt, Brennen, Physics Fluids, 2012 Optical probes

Our pure fluid tests 7

8 Flows with ρ p ≈ ρ f and smooth walls Torque 3.3-mm polystryrene cylindrical particles St = ρ p d 2 γ / (9μ) = 0.3 = 0.4 Koos, et al, PF, 2012

9 Results for flows with ρ p ≈ ρ f and smooth walls Non-dimensional Torque 3.3-mm polystryrene cylindrical particles St = ρ p d 2 γ / (9μ) = 0.3 = 0.4 Koos, et al, PF, 2012

10  l is random loose pack solid fraction Nylon spheres 6.4 mm diameter:  l =0.57 SAN spheres 3.2 mm diameter  l =0.61 Polystyrene cylinders 3.3 mm:  l =0.55 Results: different particles with ρ p ≈ ρ f and smooth walls; Bagnold macro-viscous data Koos, et al, PF, 2012

11 Results for smooth & rough walls Correlation for low-Re suspensions studies, such as Acrivos, et al & Boyer, et al Rough Smooth Settling particles Difference is slip

12 Acrivos, Fan, Mauri 1994 Boyer, Guazzelli & Pouliquen 2011 Re=ρ f d 2 / Neutrally-buoyant experiments Bagnold 1954, “macroviscous” Koos, Linares, Hunt, Brennen 2012 Current work; rough walls

13 Acrivos, Fan, Mauri 1994 Boyer, Guazzelli & Pouliquen 2011 Re=ρ f d 2 / Neutrally-buoyant experiments + simulations Bagnold 1954, “macroviscous” Koos, Linares, Hunt, Brennen 2012 Current work; rough walls Simulations: Kulkarni & Morris 2008 Yeo & Maxey 2013 Picano, et al 2013

Effective viscosity Particle contribution to stress = sum stress over all particles + stress due to particle acceleration + Reynolds stress 14 Simulations Kulkarni & Morris 2008 ; Yeo & Maxey 2013; Picano, et al 2013 Eiler:  eff /=[1+1.25*/(1-/ m )] 2 Re d = d 2 /

Neutrally-buoyant; rough walls 15 3 mm Polystyrene; alcohol- glycerine;  p = f

16 Neutrally-buoyant; rough walls Picano, et al, 2013 Polystyrene; glycerol + alcohol

Neutrally-buoyant; rough walls 17 3 mm Polystyrene; water- glycerol

18 Neutrally buoyant Water glycerine Alcohol glycerine

Transition to turbulence; pure fluid 19 laminar turbulent rough smooth Alcohol glycerine Water glycerine

Turbulent 20 G=T/(H 2 ); G=A Re  =1 laminar; =1.75 turbulent Re>4000 (Ravelet, et al 2010) =1.68; pure fluid

Torque normalized by torque turbulent 21

With particle settling Replace inner cylinder clear cylinder to visualize expansion as the flow is sheared from outer cylinder 22

Summary Re d >1, inertial contribution to the effective viscosity that depends on Reynolds number Re gap >Re cr, turbulence & particle contribution and perhaps  eff =  turb f(  ) at lowest solid fraction Particle settling – complicates the flow Collisions: St>20 Single particle collisions in a liquid 23

Questions? 24