Wavelet-vaguelette decomposition and its application to rheometry

Transcription

1 Wavelet-vaguelette decomposition and its application to rheometry Christophe Ancey École Polytechnique Fédérale de Lausanne WavE 2006 conference, July Wavelet-vaguelette decomposition and its application to rheometry p. 1/21

2 Contents Basics of rheometry: definition and objectives of rheometry The Couette inverse problem: mathematical formulation, usual methods Wavelet-vaguelette decomposition Continuous and discrete formulation Application to rheometry and comparison with Tikhonov regularization Wavelet-vaguelette decomposition and its application to rheometry p. 2/21

3 The basics of fluid rheometry For most fluids, there is a relation between stress and rate of strain. For one-dimensional flows, this relation takes the form τ = f( γ), τ where γ is the shear rate ( γ = du/dy in a Cartesian frame). For instance, for a Newtonian fluid: τ = µ γ Flow curve: relation τ = f( γ) γ Rheometry: the experimental science investigating the flow properties Wavelet-vaguelette decomposition and its application to rheometry p. 3/21

4 The basics of fluid rheometry (2) Experimentally, we use rheometers to create one-dimensional (rotational) flows. A rheometer combines an electric engine and very accurate measurement system, imposing and measuring the torque M and rotational velocity ω. Usually, we cannot directly measure τ and γ and therefore, we need to find relations between the quantities of interest (τ, γ) and the measured values (M, ω). Bohlin CVOR rheometer (cap removed to show the engine) Wavelet-vaguelette decomposition and its application to rheometry p. 4/21

5 The basics of fluid rheometry (3) Parallel plate rheometer Couette rheometer (inner cylinder replaced with a vane) Wavelet-vaguelette decomposition and its application to rheometry p. 5/21

6 Couette viscometric flows How to relate the rheological properties ( γ, τ) to the bulk measurements (M, ω)? Solving the governing equations in a cylindrical coordinate system (r, θ, z) ϱ du dt = ϱg + ( p1 + σ) The orthoradial projection is r r τ + 2τ θ p = 0, from which we deduce τ = c/r 2, with c a constant. The torque on a cylinder of radius r and height h is then M(r) = h 0 (2πr)rτ dz. In this coordinate system, the shear rate is defined as γ = rd(u(r)/r)/dr. Wavelet-vaguelette decomposition and its application to rheometry p. 6/21

7 The Couette problem ω = R2 R 1 γ(r) r dr Ω τ = M 2 π R 2 1 h ω rotational velocity M measured/imposed torque R1 inner radius R2 outer radius γ shear rate Ω Usually ω 1 = 0 and ω 2 = ω with: β = (R 1 /R 2 ) 2 ω(τ) = K γ = 1 2 τ βτ γ(s) ds (1) S τ = α 1 M is the shear stress, α 1 = 1/(2πR 2 1) Wavelet-vaguelette decomposition and its application to rheometry p. 7/21

8 A longstanding problem Mooney (1931): approximation when β 1 (common for most geometries) ω(τ) = 1 2 τ βτ γ(s) S ds = 1 β 2 γ(τ) + O ( (1 β) 2) Krieger & Maron (1952), Krieger & Eldrod (1953), Yang & Krieger (1978): asymptotic expansion into a β series Tanner & Williams (1970): iterative procedure (Landweber s procedure) Galerkin s approach: Macsporran (1989) with spline functions Specific procedures when data are contaminated with noise (Borgia & Spera, 1990) or for viscoplastic materials (Nguyen & Boger, 1992) Wavelet-vaguelette decomposition and its application to rheometry p. 8/21

9 Least-square approach with Tikhonov regularization: Yeow et al. (2000), i.e. finding γ such that: This involves: discretizing the integral term ω K γ 2 + λ γ 2 is minimum adding a term that penalizes non-smooth solutions tuning the Lagrangian multiplier to find the best compromise between agreement and smoothness iterating the procedure when there is a yield stress Yeow, Y. L., W. C. Ko, and P. P. P. Tang, Solving the inverse problem of Couette viscometry by Tikhonov regularization, Journal of Rheology, 44, , Wavelet-vaguelette decomposition and its application to rheometry p. 9/21

10 Shortcomings of the current approaches lack of versatility: recipes rather than robust methods iterative procedure (slow convergence, accuracy, etc.) noise amplification or influence how to cope with complex materials when smoothing the rheological response? How to recover the shear rate from data (without noise amplication) that may represent irregular (non-smooth) rheological behaviours? Wavelet-vaguelette decomposition (WVD)... D.L. Donoho, Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition, Applied and Computational Harmonic Analysis, 2 (1995) Wavelet-vaguelette decomposition and its application to rheometry p. 10/21

11 The idea finding an appropriate family of functions Ψ k that can provide a sparse representation of γ Daubechies wavelet D8 (mother wavelet) expressing (projecting) γ in terms of Ψ k : γ k a k Ψ k Wavelet-vaguelette decomposition and its application to rheometry p. 11/21

12 The idea introducing the family of functions v k, image of Ψ k through K: v k = KΨ k the problem now is as follows ( ) ω = K γ K a k Ψ k = k k a k v k so we are interested in expressing ω in terms of v k, but this is not an orthogonal base! Wavelet-vaguelette decomposition and its application to rheometry p. 12/21

13 The idea introducing a dual basis u k such that u i, v j = δ ij (bi-orthogonality) implying that Ψ k = K u k (K : adjoint operator) and a k = γ, Ψ k = K γ, u k = ω, u k the coefficients a k are given by the inner products ω, u k. A few questions: How to select Ψ k? Reply: wavelet with finite support (e.g., Daubechies such as D8 wavelets) and the vaguelette u k? u k (x) = i=0 ( ) 1 τ β i (τψ k(τ)) β i Wavelet-vaguelette decomposition and its application to rheometry p. 13/21

14 Alternative technique: discrete formulation We introduce the discrete operator K n that maps L 2 (R) to R n and its adjoint operator K n K n : f(x) y : y i = (Kf)(x i ) = xi f(z) βx i z dz, 1 i n, K n : y = (y i ) 1 i n h(z) = n i=1 y i z H[(x i z)(z βx i )], where H denotes the Heavyside function. The shear rate is retrieved from measurements by inverting: ω i = (K n γ)(τ i ). The image of a basis function Ψ k denoted by e k = K n Ψ k. The dual vector is ẽ k = M 1 n e k, where M n is Gram matrix of the operator K n Kn (usually well-conditioned). γ(τ) = n i=1 [w, ẽ i ]Ψ i (τ). (2) w = ω 1, ω 2,, ω n Wavelet-vaguelette decomposition and its application to rheometry p. 14/21

15 T p = 2 4 wavelets used (D8). Casson model (dashed line); WVD solution (solid line). 1 T p = 2 5 = 32 wavelets used. Casson model (dashed line); WVD solution (solid line) T u p = 2 7 = 128 wavelets used. Casson model (dashed line); WVD solution (solid line) x Comparison between the dual function u 4,8 and its discrete counterpart ẽ 25. Wavelet-vaguelette decomposition and its application to rheometry p. 15/21

16 Applications Viscoplastic material (carbopol). Paar Physica MC1+ rheometer with a Couette geometry (R 1 = 1.25 cm and β = 0.26) and a Bohlin CVOR200 rheometer (R 1 = cm and β = 0.06). Velocity-profile measurements in a similar geometry (R 1 = 4 cm and β = 0.44) using Magnetic Resonance Imaging (MRI) techniques. Granular suspensions. Haake Rotovisco MV5 rheometer with a vane (R 1 = 3 cm and β = 0.09). The suspension was made up of glass beads (diameter 0.8 mm) in a 98.5% water/glycerol solution at the solid concentration of 60%. J.C. Baudez, S. Rodts, X. Chateau and P. Coussot, New technique for reconstructing instantaneous profiles from viscometric tests: application to pasty materials, Journal of Rheology, 48 (2004) C. Ancey, Role of lubricated contacts in concentrated polydisperse suspensions, Journal of Rheology, 45 (2001) Wavelet-vaguelette decomposition and its application to rheometry p. 16/21

17 Raw data Τ (Pa) (rad/s) Polymeric gel (carbopol) [Courtesy of P. Coussot] Τ (Pa) Granular suspension (rad/s) Interpolation curves obtained by the optimized Gasser-Müller kernel approach (see Hart J., Nonparametric Smoothing, Springer, N.-Y., 1999) Wavelet-vaguelette decomposition and its application to rheometry p. 17/21

18 Comparison for the polymeric gel Τ (Pa) Γ. (1/s) Granular suspension Wavelet-vaguelette decomposition and its application to rheometry p. 18/21

19 Comparison for the granular suspension Τ (Pa) Γ. (1/s) Granular suspension Wavelet-vaguelette decomposition and its application to rheometry p. 19/21

20 Summary, conclusion, and perspectives WVD decomposition makes it possible to recover the shear rate from the rotational velocity γ(τ) = k g k Ψ k (τ), where Ψ k is a wavelet basis and g k = ω(τ), u k. No additional assumption needed. Sparse representation and high accuracy and adaptivity (without tuning parameters). Controlled convergence (convergence rate known in advance). Better control of errors induced by noise. Wavelet-vaguelette decomposition and its application to rheometry p. 20/21

21 but some drawbacks Difficult to implement. Preconditioning the signal (noise removing) is essential. Time-consuming procedure: typically a few minutes on a PC against a few seconds for the Tikhonov regularization method! C. Ancey, Journal of Rheology 49 (2005) Mathematica package available from Wavelet-vaguelette decomposition and its application to rheometry p. 21/21