1. A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows Alexander Vikhansky Department of Engineering, Queen Mary, University of London

2. A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows Lattice-Boltzmann method

3. A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows Boltzmann equation

4. A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows NS equations

5. A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows Plan of the presentation

6. A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows Plan of the presentation

7. A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows Knudsen number: Boltzmann equation

8. A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows Chapman-Enskog expansion

9. A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows Kinetic effects: 1. Knudsen slip (Kn), 2. Thermal slip (Kn). Knudsen layer (Kn 2 )

10. A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows Kinetic effects: 3. Thermal creep (Kn).

11. A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows Kinetic effects: 4. Thermal stress flow (Kn 2 ).

12. A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows Discrete ordinates equation

13. A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows Collision operator BGK model:

14. A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows Boundary conditions

15. A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows Boundary conditions: bounce-back rule

16. A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows Method of moments 1. Euler set: 2. Grad set: – 5 equations; – 13 equations; 3. Grad-26, Grad-45, Grad-71.

17. A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows Method of moments 1. Euler set: 2. Grad set: 3. Grad-26: 4. Grad-45, Grad-71: The error:

18. A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows Simulation of thermophoretic flows Velocity set:

19. A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows M. Young, E.P. Muntz, G. Shiflet and A. Green Knudsen compressor

20. A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows Knudsen compressor

21. A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows

24. Effect of the boundary conditions

25. A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows Semi-implicit lattice-Boltzmann method for non-Newtonian flows From the kinetic theory of gases: Constitutive equation:

26. A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows Newtonian liquid: Bingham liquid: Semi-implicit lattice-Boltzmann method for non-Newtonian flows General case:

27. A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows Semi-implicit lattice-Boltzmann method for non-Newtonian flows Velocity set (3D): Velocity set (2D): Equilibrium distribution: Post-collision distribution:

28. A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows Semi-implicit lattice-Boltzmann method for non-Newtonian flows Bingham liquidPower-law liquid

29. A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows Flow of a Bingham liquid in a constant cross-section channel

30. A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows Creep flow through mesh of cylinders

31. A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows Flow through mesh of cylinders

32. A. Vikhansky, Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows Continuous in time and space discrete ordinate equation is used as a link from the LB to Navier-Stokes and Boltzmann equations. This approach allows us to increase the accuracy of the method and leads to new boundary conditions. The method was applied to simulation of a very subtle kinetic effect, namely, thermophoretic flows with small Knudsen numbers. The new implicit collision rule for non-Newtonian rheology improves the stability of the calculations, but requires the solution of a (one-dimensional) non-linear algebraic equation at each point and at each time step. In the practically important case of Bingham liquid this equation can be solved analytically. CONCLUSIONS