1. A Study of Flow in a Corrugated Channel Xin Kai Li Institute of Simulation Sciences De Montfort University Leicester UK

2. Outline Introduction Mathematical model Spectral element method for a corrugated channel flow Numerical results Conclusion with further work

3. Mathematical Model Momentum equation Mass equation Constitutive Equation

4. Viscosity model Cross model

5. Spectral Element method Gauss-Lobatto-Legendre Polynomials

6. Basis functions (Gauss-Lobatto-Legendre polynomials) for the velocity and the extra-stress tensor Basis function for the pressure and the deformation tensor

7. Spectral element configuration (k=4, M=N=5) Gauss-Lobatto-Legendre collocation points for the velocity and stress tensor Interior Gauss-Lobatto-Legendre collocation points for the pressure p ( u,  )

8. Numerical algebraic equations A ---- Helmholtz operator B ---- Gradient operator C ---- Stress tensor matrix

9. Numerical example Flow geometry

10. Velocity profiles

11. Shear rate and viscosity

12. Velocity vector profiles

13. Velocity along the x-axis