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Modelling Flows of Viscoelastic Fluids with Spectral Elements: a first approach

Modelling Flows of Viscoelastic Fluids with Spectral Elements: a first approach. Giancarlo Russo, supervised by Prof. Tim Phillips.

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Modelling Flows of Viscoelastic Fluids with Spectral Elements: a first approach

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

  2. 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. Cardiff School of Mathematics

  3. 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 Cardiff School of Mathematics

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

  5. The Weak Formulation We look for such that for all the following equations are satisfied : (5) where b, c, d, and l are defined as follows : (6) 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) . Cardiff School of Mathematics

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

  7. Some results for the 2-fields formulation • A compatibility result (see [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]) : (11) (12) (13) (14) REMARK : in (14) PN has been used as approximating space for the pressure. It’s non-optimal. Cardiff School of Mathematics

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

  9. 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 : (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) Cconst andCstab are the continuity constant for the bilinear form in (7) and the stability constant for the velocity in (11) , while β1 =β/C, where C is the continuity constant for the orthogonal projector in Cardiff School of Mathematics

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

  11. A model to investigate the Extrudate Swell problem II(see e.g. [7] and [8] for an analysis and some data) 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 ). Following an approach proposed in [9] for the flow past a sphere, we parametrize by and set such 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). • 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] ); • 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. Cardiff School of Mathematics

  12. 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: Cardiff School of Mathematics

  13. The 2-D discretization process I • The 2-D spectral (tensorial) expansion : Cardiff School of Mathematics

  14. The 2-D discretization process II • The problem : Approximation of u1 • The 2-D expansion : • The discrete problem: Approximation of u2 Cardiff School of Mathematics

  15. 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 Cardiff School of Mathematics

  16. 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. Cardiff School of Mathematics

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