I N N F M Computational Rheology Isaac Newton Institute Dynamics of Complex Fluids -10 Years on Institute of non-Newtonian Fluid Mechanics EPSRC Portfolio Partnership Juan P. Aguayo Hamid Tamaddon Mike Webster Schlumberger, UNAM-(Mexico), INNFM Mike Webster

I N N F M Computational Rheology – Some Outstanding Challenges  To achieve highly elastic, high strain-rate/deformation rate solutions (polymer melts & polymer solutions)  To quantitatively predict pressure-drop, as well as flow field structures (vortices, stress distributions) To accurately represent transient flow evolution in complex flows To quantitatively predict multiple-scale response (multi- mode) To achieve compressible viscoelastic representations

I N N F M TRANSIENT & STEADY Contraction Flows EPTT Oldroyd Axisymmetric Planar Axisymmetric

Fluid viscosity = 1.75Pa.s – 8:1 contraction, exit length 7.4mm Pressure-drop vs flow-rate in contractions Newtonian syrup Boger fluid Fluid viscosity = 1.75Pa.s – 20:1 contraction, exit length 40mm axisymmetricplanar

Pressure drop (epd) vs. We, 4:1:4 axisymmetric Szabo et al. with FENE-CR J. Non-Newt. Fluid Mech. 72:73-86, 1997 epd We

Schematic diagram for a) 4:1:4 contraction/expansion, b) 4:1 contraction Szabo et al. J. Non-Newt. Fluid Mech. 72:73-86, 1997 Rothstein and McKinley J. Non-Newt. Fluid Mech. 86:61-88, 1999 J. Non-Newt. Fluid Mech. 98:33-63, 2001 Wapperom and Keunings J. Non-Newt. Fluid Mech. 97: , 2001 : Total pressure drop Excess pressure drop (epd - P )

Pressure-drop (epd) vs. We, Oldroyd-B, a, c) axisymmetric, b, d) planar c) 4:1 a) d) b) A x i s y m m e t ri c PlanarPlanar 4:1:4 a) b)

Oldroyd-B,  =0.9 Pressure profile around constriction zone, 4:1:4 axisymmetric and planar case

Oldroyd-B,  =0.9 N1 p 3D view – 4:1:4 contraction/expansion A x i s y m m e t ri c PlanarPlanar

(P - P Newt ) and stress profiles along wall, 4:1 and 4:1:4 axisymmetric case Oldroyd-B,  =0.9 4:14:1:4

(P - P Newt ) and stress profiles along wall – 4:1:4 planar and axisymmetric case Oldroyd-B,  =0.9 PlanarAxisymmetric

epd We Pressure-drop (epd) vs. We, 4:1:4 axisymmetric, alternative models We

(P - P Newt ) profiles along wall – 4:1:4 axisymmetric, increasing  upturn epd monotonic decrease epd upturn & enhanced epd

Alternative differential pressure-drop measure Since & by calibration 

Rate of dissipation & pressure-drop, 4:1:4 0 rate of dissipation definition Seeking { P – 1} > 0,

epd Pressure-drop (epd) vs. a) We, b) upstream sampling distance, 4:1:4 axisymmetric We

4:1:4 a xisymmetric vortex cell size, Oldroyd-B,  change upturn & enhanced epd mono-dec epd  =0.99  =1/9

a) Oldroyd-B extensional viscosity,  b) Shear and extensional viscosity,  c) Shear and extensional viscosity,  Rheological properties: Oldroyd-B, LPTT, EPTT, SXPP

NEW BOGER fluid modelling & Pressure Drop Axisymmetric contraction Planar contraction

Centreline pressure gradient 4:1:4 axisymmetric, Oldroyd-B  =0.99  =0.9  =1/9