EPSRC Portfolio Partnership in Complex Fluids and Complex Flows Computational Rheology The Numerical Prediction of Complex Flows of Complex Fluids Gigantic number of microstructural degrees of freedom Broad range of time scales: s  10 3 s Broad range of length scales: m  1m Quantum mechanics – ab initio methods Atomistic modelling – non-equilibrium molecular dynamics Kinetic theory – stochastic simulation or Brownian dynamics Continuum mechanics – grid-based methods The polymer molecules are represented by a backbone segment connecting two identical pom- poms. This coarse-grained molecular model is: strain-hardening in extension strain-softening in shear The differential approximation to the original model predicted: zero second normal stress difference; orientation is unbounded at high strain rates; discontinuity in the derivative of the extensional viscosity. Verbeeten et al. (2001) introduced modifications to the differential approximation of McLeish and Larson. Conservation of mass and momentum Constitutive equation for the XPP model Here s is the orientation tensor, λ is the backbone stretch, λ 0b is the relaxation time for the orientation of the backbone, λ os is the relaxation for the stretch, and α is the anisotropy parameter. Non-dimensional governing equations for the XPP model: Non-dimensional parameters for XPP model We = 0.1 We = 5 We = 1 We =10 We = 0.1 We = 5 We = 1 We =10 Weak formulation of the governing equations. Transfinite mapping of physical elements onto the parent element. Decoupled velocity/pressure and stress computations. Mass and momentum equations solved using the PCG method with efficient preconditioners. Constitutive equation is solved using a BiCGSTAB method with the stress mass matrix as preconditioner. Local upwinding factors (LUST) are used in the SUPG discretization of the constitutive equation for enhanced stability. Flow past a Cylinder Transfinite Mapping We=1 and ε=0.33 We=3 and ε=0.6 Modelling HierarchyMajor Challenges Dynamics of Polymer Melts The pom-pom model of McLeish and Larson (1998) Extended Pom-Pom (XPP) Model Governing Equations Dimensionless Equations FV control volume and MDC for FE/FVFE with 4 fv sub-cells for FE/FV T3T3 T2T2 T1T1 T6T6 T5T5 T4T4 l fe triangular element fv triangular sub-cells fe vertex nodes (p, u,  ) fe midside nodes (u,  ) fv vertex nodes (  ) Finite Volume Grid for SLFV i, j + 2 i, j - 2 i, j + 1 i + 2, ji - 2, ji, ji - 1, ji + 1, j i, j - 1 U V P,  xx,  yy,  xy FE/FV Spatial Discretisation and Median Dual Cell SLFV Spatial Discretisation 4:1 Planar Contraction - Streamlines  = 1/9,  = 1/3,  = 0.15, q = 2. Re = 0 4:1 Planar Contraction - Streamlines  = 1/9,  = 1/3,  = 0.15, q = 2. Re = 1 Comparison of the Predictions of the Salient Corner Vortex Intensity and Cell Size for the FE/FV and SLFV Schemes  = 1/9,  = 1/3,  = 0.15, q = 2. Salient corner vortex intensitySalient corner vortex cell size Spectral Element Method Contours of Stretch Two beads connected by a spring. The equation of motion of each bead contains contributions from the tension force in the spring, the viscous drag force, and the force due to Brownian motion. Q The dimensionless form of the Fokker-Planck equation for homogeneous flows is  RJRJ RBRB e x Flow between Eccentrically Rotating Cylinders Dependence of Stretch on Cylinder as a Function of We Dumbbell Models for Polymer Solutions The equivalence between the Fokker-Planck equation and a stochastic differential equations is used. The method of BCFs was devised by Hulsen et al (1997) to overcome the problem of tracking particle trajectories Based on the evolution of a number of continuous configuration fields Dumbbell connectors with the same initial configuration and subject to same random forces throughout the domain are combined to form a configuration field The evolution of an ensemble of configuration fields provides the polymer dynamics Method of Brownian Configuration Fields Oldroyd B Hookean For the Hookean dumbbell model Contour Maps of (  p ) xy for the Eccentrically Rotating Cylinder Problem PRIFYSGOL CYMRU ABERTAWE UNIVERSITY OF WALES SWANSEA