Flow of polymer solutions using the matrix logarithm approach in the spectral elements framework

1. Flow of polymer solutions using the matrix logarithm approach in the spectral elements framework Giancarlo Russo, Prof. Tim Phillips Cardiff School of Mathematics EPFDC 2007, University of Birmingham

2. Outline 1)The model and the spectral framework 2) The log-conformation representation 3) Planar channel flow simulations 4) Flow past a cylinder simulation5) Some remarks about the code (discretization in time, upwinding, etc) 6) Things still to be sorted (hopefully SOON) and future work 7) References EPFDC 2007, University of Birmingham

3. Setting up the spectral approximation: the weak formulation for the Oldroyd-B model EPFDC 2007, University of Birmingham

4. The 1-D discretization process(note: all the results are obtained for N=5 and an error tolerance of 10°-05 in the CG routine) • The spectral (Lagrange) basis : • Approximating the solution: replacing velocity, pressure and stress by the following expansions and the integral by a Gaussian quadrature on the Gauss-Lobatto-Legendre nodes, namely the roots of L’(x), (5) becomes a linear system: EPFDC 2007, University of Birmingham

5. The 2-D discretization process • The 2-D spectral (tensorial) expansion : EPFDC 2007, University of Birmingham

6. The Oldroyd-B model and the log-conformation representation To model the dynamics of polymer solutions the Oldroyd-B model is often used as constitutive equation: where the relative quantities are defined as follows: A new equivalent constitutive equation is proposed by Fattal and Kupferman (see [2], [3]) : The main aim of this new approach is the chance of modelling flows with much higher Weissenberg number, because advecting psi instead of tau (or sigma) reduces the discrepancy in balancing deformation through advection. EPFDC 2007, University of Birmingham

7. The example of a 1-D toy problem Comparing the one dimensional problems on the left we can have an idea of how the use of logarithmic transformation weakens the stability constraint. • a(x) plays the role of u (convection) • b(x) is grad u instead (exponential growth) EPFDC 2007, University of Birmingham

8. Results from the log-conformation channel flow: Re =1, beta=0.167, We =5, Parabolic Inflow/Outflow, 2 Elements, N=6 EPFDC 2007, University of Birmingham

9. Matrix-Logarithm Approach(left, usual OLD-B right) : Flow past a cylinder(1:1): Re =0. 1, We =5, Parabolic Inflow/Outflow, 20 Elem, N=6, beta=0.15, deltaT=10d-2, eps=10d-9 (10d-8 tau, p); OLDROYD-B,(I, velocity) EPFDC 2007, University of Birmingham

10. Matrix-Logarithm Approach (left): Flow past a cylinder(1:1): Re =0. 1, We =5, Parabolic Inflow/Outflow, 20 Elem, N=6, beta=0.15, deltaT=10d-2, eps=10d-9 (10d-8 tau, p); OLDROYD-B,(I, velocity) EPFDC 2007, University of Birmingham

11. Matrix-Logarithm Approach: Flow past a cylinder(1:1): Re =0. 1, We =5, Parabolic Inflow/Outflow, 20 Elem, N=6, beta=0.15, deltaT=10d-2, eps=10d-9 (10d-8 tau, p); OLDROYD-B,(II, pressure and pressure gradient) EPFDC 2007, University of Birmingham

12. Remarks about the code In order to use (1) to eliminate the velocity, we have to premultiply by the inverse of H , and then by D; so we’ll finally obtain the following equation to recover the pressure. The matrix acting on the pressure now is known as the UZAWA (U) operator. U • OIFS 1 (Operator Integration Factor Splitting, 1st order) is used to discretize the material derivative of velocity; Euler ( 1st order ) for the stress in the const. eq.;- LUST ( Local Upwinding Spectral Technique) is used ( see [1] ); - To invert H, a Schur complement method is used to reduce the size of the problem, then a direct LU factorization is performed.- To invert U, since is symmetric, Preconditioned Conjugate Gradient methods is used; - The constitutive equation is solved via BiConjugate Gradient Stabilized (non-symm); EPFDC 2007, University of Birmingham

13. Future Work - Fix the stress profiles and then calculate the correct drag ; - Testing the log-conformation method for higher We and different geometries;- Find a general expression of the constitutive equation to apply the matrix logarithm method to a broader class of constitutive models (i.e. XPP and PTT ) in order to simulate polymer melts flows;- Eventually join the free surface “wet” approach with the log-conformation method in a SEM framework to investigate the extrudate swell and the filament stretchingproblems EPFDC 2007, University of Birmingham

14. References [1] OWENS R.G., CHAUVIERE C., PHILLIPS T.N., A locally upwinded spectral technique for viscoelastic flows, Journal of Non-Newtonian Fluid Mechanics, 108:49-71, 2002. [2] FATTAL R.,KUPFERMAN R. Time-dependent simulation of viscoelastic flows at high Weissenberg number using the log-conformation representation , Journal of Non-Newtonian Fluid Mechanics,2005, 126: 23-37. [3] HULSEN M.A.,FATTAL R.,KUPFERMAN R. Flow of viscoelastic fluids past a cylinder at high Weissenberg number: stabilized simulations using matrix logarithms, Journal of Non-Newtonian Fluid Mechanics,2005, 127: 27-39. [4] VAN OS R. Spectral Element Methods for predicting the flow of polymer solutions and melts, Ph.D. thesis, The University of Wales, Aberystwyth, 2004. EPFDC 2007, University of Birmingham