1. 16/05/2006 Cardiff School of Mathematics 1 Modelling Flows of Viscoelastic Fluids with Spectral Elements: a first approach Giancarlo Russo, supervised by Prof. Tim Phillips

2. 216/05/2006Cardiff School of Mathematics - The Project - The development of models to describe and solve the flows of viscoelastic fluids with free surfaces; - a parallel theoretical and numerical (spectral elements) analysis of the problems; - the die-swell problem; - the filament stretching problem. Work Plan – First Year - - Newtonian fluids; - Analysis of the general formulation of a free surface Stokes problem; - Weak two and three fields formulations: compatible approximation spaces, compatibility conditions and error estimates; - Development of a code to solve 1-D and 2-D steady and unsteady Stokes flow.

3. 316/05/2006Cardiff School of Mathematics Outline The problems Theoretical analysis of the weak formulation Results for the 2-fields problem Results for the 3-fields problem A stability estimate for the stress tensor A model to investigate the Extrudate Swell problem The 1-D and 2-D discretization processes : some results from the codes

4. 416/05/2006Cardiff School of Mathematics Formulations of the problem Definition of the stress tensor for a Newtonian fluidDefinition of the stress tensor for a Newtonian fluid Two fields formulationsTwo fields formulations Three fields formulationsThree fields formulations Two free surface boundary conditionsTwo free surface boundary conditions - Stress tensor - Velocity - Pressure - External Loads - Normal Vectors - Curvature Radii - Rate of Strain - Kinematic Viscosity Variables and Parameter (1) (2) (3) (4)

5. 516/05/2006Cardiff School of Mathematics The Weak Formulation We look forsuch that for all the following equations are satisfied : where 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)

6. 616/05/2006Cardiff School of Mathematics The spurious modes for the pressure: the compatibility condition The bilinear form in the 2-fields formulation : The weak problem for the velocity : Coupling with the pressure : the spurious modes space The compatibility (LBB) condition (see [1] and [2] ) :[1][2] (7) (8) (9) (10)

7. 716/05/2006Cardiff School of Mathematics Some results for the 2-fields formulation A compatibility result (see [3]) : A compatibility result (see [3]) :[3] The spaces and as approximating spaces for the the velocity and pressure fields respectively, are compatible in the sense of the LBB condition. Some approximation results (see [4]) : Some approximation results (see [4]) :[4] (11) (12) (13) (14) REMARK : in (14) P N has been used as approximating space for the pressure. It’s non-optimal.

8. 816/05/2006Cardiff School of Mathematics More results for the 3-fields formulation Compatible spaces for the 3-fields formulation (see [5]) : Compatible spaces for the 3-fields formulation (see [5]) :[5] The space together with the spaces are compatible spaces for the approximation of the stress, velocity and pressure fields respectively according to the LBB and the previous condition. and A velocity – stress compatibility condition (see [5]) : A velocity – stress compatibility condition (see [5]) :[5] Some approximation results (see [5]) : Some approximation results (see [5]) :[5] (15) (16) (17) (18) REMARK : in (18) P N-2 has been used as approximating space for the pressure. It’s non-optimal, But slightly better (a factor ½) than (14).

9. 916/05/2006Cardiff School of Mathematics A stability estimate for the stress tensor The compatibility condition (15) has been derived with an abstract approach, namely, using the Closed Range Theorem. In a similar fashion a stability estimate for the stress tensor is here derived. According to the closed range theorem (see [6] for details) we can swap velocity and stress in (15) without changing our constant β; the same condition still also holds when we approximate the problem in the discrete subspace, and it finally reads as follows :[6] (19) We can now replace the integral in (19) using the first equation of (5), namely, the weak momentum equation, then, exploiting the continuity of the bilinear form (7) and finally the stability estimate for the velocity in (11) we can derive the following estimate: (20) Note that in (20) C const and C stab are the continuity constant for the bilinear form in (7) and the stability constant for the velocity in (11), while β 1 =β/C, where C is the continuity constant for the orthogonal projector in

10. 1016/05/2006Cardiff School of Mathematics A model to investigate the Extrudate Swell problem I (see e.g. [7] and [8] for an analysis and some data) [7][8][7][8] Key points: Tracking the free surface Capturing the large stress gradient at the exit, analyzing the angle of swelling a ; Estimate the final diameter D final corresponding to a total stress relaxation. The Sturm-Liouville problem and the Jacobi polynomials

11. 1116/05/2006Cardiff School of Mathematics A model to investigate the Extrudate Swell problem II (see e.g. [7] and [8] for an analysis and some data) [7][8][7][8] Following an approach proposed in [9] for the flow past a sphere, we parametrize[9]by and setsuch that at each step k the weight function can follow the behaviour of the analytic singularity (while with the parametrization above we approach the geomtrical one). The main idea is to approximate the free surface problems with a sequence of rigid boundary problems, for which we have the values of the power of the singularity at the die; this values are different from fluid to fluid, but always (approximatively) in the range (-0.3, -0.7 ). Pattern of the algorithm : construct the new boundary from the previous step (the first comes from a fully developed Poiseuille flow, so it’s a stick-slip problem); find the value of the singular corner as the tangent to the boundary at the die; find the necessary eigenvalues to compute the power of the singularity (see e.g. [8] );[8] Construct the correspondent Jacobi polynomials for the corner element; Solve the problem with the updated boundary and Jacobi polynomial; Check the value of the normal velocity.

12. 1216/05/2006Cardiff School of Mathematics 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)

13. 1316/05/2006Cardiff School of Mathematics The 2-D discretization process I The 2-D spectral (tensorial) expansion : The 2-D spectral (tensorial) expansion :

14. 1416/05/2006Cardiff School of Mathematics The 2-D discretization process II The problem :The problem : The 2-D expansion :The 2-D expansion : The discrete problem:The discrete problem: Approximation of u 1 Approximation of u 2

15. 1516/05/2006Cardiff School of Mathematics Coming soon (hopefully…) Complete the codes including pressure gradient and stress field Adapt them to the Extrudate Swell problem (in the newtonian case) Generalize to the unsteady cases Try some preconditioners on the CG routine Start with non-newtonian fluids

16. 1616/05/2006Cardiff School of Mathematics References [1] BABUSKA I., The finite element method with Lagrangian multipliers, Numerical Mathematics, 1973, 20: 179-192. [2] BREZZI F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers, R.A.I.R.O., Anal. Numer., 1974, R2, 8, 129-151. [3] MADAY Y, PATERA A.T., RØNQUIST E.M., The PN ×PN−2 method for the approximation of the Stokes problem. Technical Report 92025, Laboratoire d‘Analyse Numrique, Universitet Pierre et Marie Curie, 1992. [4] BERNARDI C., MADAY Y. Approximations spectrales de problemes aux limites elliptiques, Springer Verlag France, Paris, 1992. [5] 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. [6] SHWAB C., p- and hp- Finite Element Methods, Theory and Applications in Solid and Fluid Mechanics, Oxford University Press, New York, 1998. [7] OWENS R.G., PHILLIPS T. N. Computational Rheology, Imperial College Press, 2002. [8] TANNER R.I., Engineering Rheology - 2nd Edition, Oxford University Press, 2000. [9] GERRITSMA M.I., PHILLIPS T.N., Spectral elements for axisymmetric Stokes problem, Journal of Computational Physics 2000, 164 : 81-103. [10] KARNIADAKIS G., SHERWIN S.J., Spectral / hp elements for computational fluid dynamics - 2nd Edition, Oxford University Press, 2005.