1. 13/04/2007 UK Numerical Analysis Day, Oxford Computing Laboratory 1 Spectral Elements Method for free surface and viscoelastic flows Giancarlo Russo, supervised by Prof. Tim Phillips

2. 213/04/2007 UK Numerical Analysis Day, Oxford Computing Laboratory Outline Theoretical Issue: Theoretical Issue: existence and uniqueness of a solution for a steadystate die swell problem From the free to the fixed boundary Setting up the problems with new variables and new operator Necessary and sufficient conditions for a solution to exist Uniqueness Numerical Issue: Numerical Issue: Matrix-Logarithm approach and free surface’s tracking The log-conformation representation Channel flow test Free surface tracking for die swell and filament stretching.

3. 313/04/2007 UK Numerical Analysis Day, Oxford Computing Laboratory The Problem in in on on Domain change and set of admissible functions for the free surface

4. 413/04/2007 UK Numerical Analysis Day, Oxford Computing Laboratory New Variables New Differential Operators New Equations New Operators The modified problem Compatibility conditions for existence and uniqueness of a solution

5. 513/04/2007 UK Numerical Analysis Day, Oxford Computing Laboratory Key steps for conditions (1) and (2) to be satisfied The proof of condition (1) in the case of the usual divergence lies on the application that the operator is an isomorphism, and the whole point is the application of the divergence theorem. To prove (2), which involves the gradient, we have to apply the Poincare’ inequality instead. What we need then for these conditions to hold for our operators defined in (3) and (4) is for G, as defined in (5), to be bounded. Using the hypotesis on the domain and the free surface, we can bound the following quantities:

6. 613/04/2007 UK Numerical Analysis Day, Oxford Computing Laboratory So far we have proved that, picked a free surface map, the corresponding problem (with that surface) has got a unique solution; now we have to prove that such a map always exists, which means that the Cauchy problem (C1)-(C2) has a unique solution. Applying the Schauder’s fixed point theorem we have to prove that the following operator Existence and uniqueness of a solution for the free boundary problem This operator has to be: 1) CONTINOUS 2)COMPACT 3)A CONTRACTION 1) CONTINUITY: consider the following sequence, supposing it converges strongly in the let’s say. There will be a corresponding sequence u N satisfying the intial (weak) problem, and which will have the following properties: From the definition of the operator E, and taking the limit of the sequence of weak problems, we can deduce that namely E is continuous.

7. 713/04/2007 UK Numerical Analysis Day, Oxford Computing Laboratory 2)COMPACTNESS : the operator E is the composition of a continuous function and a compact embedding, therefore is compact. More precisely: Existence and uniqueness of a solution for the free boundary problem II 3)CONTRACTION : We have to prove that the operator E is a contraction, namely: If we write and combine with (OpE) we obtain Finally, expanding the continuity equation and since G is invertible, we write We remark that the y-component of the velocity field eventually vanishes when we approach the total relaxation stress configuration.

8. 813/04/2007 UK Numerical Analysis Day, Oxford Computing Laboratory Setting up the spectral approximation: the weak formulation for the Oldroyd-B model We look forsuch that for all the following equations are satisfied : where h includes the UCD terms (which are approximated with a 1st order OIFS /Euler) and b, c, d, and l are defined as follows : Remark:it simply means the velocity fields has to be chosen according to the boundary conditions, which in the free surface case are the ones given in (4). (5) (6)

9. 913/04/2007 UK Numerical Analysis Day, Oxford Computing Laboratory 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)

10. 1013/04/2007 UK Numerical Analysis Day, Oxford Computing Laboratory The 2-D discretization process I The 2-D spectral (tensorial) expansion : The 2-D spectral (tensorial) expansion :

11. 1113/04/2007 UK Numerical Analysis Day, Oxford Computing Laboratory 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: 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 it looks like the oscillations due to the use of polynomials to approximate exponential behaviours are deeply reduced.

12. 1213/04/2007 UK Numerical Analysis Day, Oxford Computing Laboratory First results from the log-conformation channel flow: Re =1, We =5, Parabolic Inflow/Outflow, 2 Elements, N=6

13. 1313/04/2007 UK Numerical Analysis Day, Oxford Computing Laboratory In the figure we remark the approach we are going to follow to track the free surfaces in the die swell and filament stretching problems: this is a completely “wet” approach, it means the values of the fields in blue nodes, the ones on the free surfaces, are extrapolated from the values we have in the interior nodes (the black ones) at each time step. After a certain number of timesteps we then redistribute the nodes to avoid big gaps between the free surface nodes an the neighbours. This approach has been proposed by Webster & al. in a finite differences context. Free surface problems: a complete “wet” approach

14. 1413/04/2007 UK Numerical Analysis Day, Oxford Computing Laboratory Future Work Analyze the unsteady free surface die swell problem Testing the log-conformation method for higher We and different geometries Implementing the free surface “wet” approach in a SEM framework for the die swell and the filament stretching Eventually join the latter with the log-conformation method for the constitutive equation

15. 1513/04/2007 UK Numerical Analysis Day, Oxford Computing Laboratory [1] ISTRATESCU V.I., Fixed Point Theory, An Introduction, Redidel Publishing Company, 1981. [2] CABOUSSAT A. Analysis and numerical simulation of free surface flows, Ph.D. thesis, ´Ecole Polytechnique Federale de Lausanne, Lausanne, 2003. [3] GERRITSMA M.I., PHILLIPS T.N., Compatible spectral approximation, for the velocity-pressure-stress formulation of the Stokes problem, SIAM Journal of Scientific Computing, 1999, 20 (4) : 1530-1550. [4] FATTAL R.,KUPFERMAN R. Constitutive laws for the matrix logarithm of the conformation tensor, Journal of Non-Newtonian Fluid Mechanics,2004, 123: 281- 285. [5] 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, [6] 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. [7] VAN OS R. Spectral Element Methods for predicting the flow of polymer solutions and melts, Ph.D. thesis, The University of Wales, Aberystwyth, 2004. [8] WEBSTER M., MATALLAH H., BANAAI M.J., SUJATHA K.S., Computational predictions for viscoelastic filament stretching flows: ALE methods and free- surface techniques (CM and VOF), J. Non-Newtonian Fluid Mechanics, 137 (2006): 81-102. References