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08-10/08/2007 EPFDC 2007, University of Birmingham 1 Flow of polymer solutions using the matrix logarithm approach in the spectral elements framework Giancarlo Russo, Prof. Tim Phillips Cardiff School of Mathematics

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208-10/08/2007EPFDC 2007, University of Birmingham Outline 1)The model and the spectral framework 2) The log-conformation representation 3) Planar channel flow simulations 4) Flow past a cylinder simulation 5) Some remarks about the code (discretization in time, upwinding, etc) 6) Things still to be sorted (hopefully SOON) and future work 7) References

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308-10/08/2007EPFDC 2007, University of Birmingham Setting up the spectral approximation: the weak formulation for the Oldroyd-B model Differential formulationWeak Formulation f includes the UCD terms and b, c, d, and l are defined as follows :

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408-10/08/2007EPFDC 2007, University of Birmingham 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 : 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: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: (5)

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508-10/08/2007EPFDC 2007, University of Birmingham The 2-D discretization process The 2-D spectral (tensorial) expansion : The 2-D spectral (tensorial) expansion :

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608-10/08/2007EPFDC 2007, University of Birmingham To model the dynamics of polymer solutions the Oldroyd-B model is often used as constitutive equation: To model the dynamics of polymer solutions the Oldroyd-B model is often used as constitutive equation: The Oldroyd-B model and the log-conformation representation A new equivalent constitutive equation is proposed by Fattal and Kupferman (see [2], [3]) : A new equivalent constitutive equation is proposed by Fattal and Kupferman (see [2], [3]) : where the relative quantities are defined as follows: where the relative quantities are defined as follows: 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. 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.

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708-10/08/2007EPFDC 2007, University of Birmingham 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)

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808-10/08/2007EPFDC 2007, University of Birmingham Results from the log-conformation channel flow: Re =1, beta=0.167, We =5, Parabolic Inflow/Outflow, 2 Elements, N=6

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908-10/08/2007EPFDC 2007, University of Birmingham 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)

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1008-10/08/2007EPFDC 2007, University of Birmingham 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)

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1108-10/08/2007EPFDC 2007, University of Birmingham 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)

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1208-10/08/2007EPFDC 2007, University of Birmingham 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. -OIFS 1 (Operator Integration Factor Splitting, 1 st order) is used to discretize the material derivative of velocity; Euler ( 1 st 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); U

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1308-10/08/2007EPFDC 2007, University of Birmingham 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 stretching problems

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1408-10/08/2007EPFDC 2007, University of Birmingham [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. References

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