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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. Lattice-Boltzmann method. Boltzmann equation. NS equations. Plan of the presentation. Plan of the presentation. Boltzmann equation.

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Lattice-Boltzmann method for non-Newtonian and non-equilibrium flows

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

  2. Lattice-Boltzmann method

  3. Boltzmann equation

  4. NS equations

  5. Plan of the presentation

  6. Plan of the presentation

  7. Boltzmann equation Knudsen number:

  8. Chapman-Enskog expansion

  9. Kinetic effects: Knudsen layer (Kn2) 1. Knudsen slip (Kn), 2. Thermal slip (Kn).

  10. Kinetic effects: 3. Thermal creep (Kn).

  11. Kinetic effects: 4. Thermal stress flow (Kn2).

  12. Discrete ordinates equation

  13. Collision operator BGK model:

  14. Boundary conditions

  15. Boundary conditions: bounce-back rule

  16. Method of moments – 5 equations; 1. Euler set: 2. Grad set: – 13 equations; 3. Grad-26, Grad-45, Grad-71.

  17. Method of moments The error: 1. Euler set: 2. Grad set: 3. Grad-26: 4. Grad-45, Grad-71:

  18. Simulation of thermophoretic flows Velocity set:

  19. Knudsen compressor M. Young, E.P. Muntz, G. Shiflet and A. Green

  20. Knudsen compressor

  21. Effect of the boundary conditions

  22. Semi-implicit lattice-Boltzmann method for non-Newtonian flows From the kinetic theory of gases: Constitutive equation:

  23. Semi-implicit lattice-Boltzmann method for non-Newtonian flows Newtonian liquid: Bingham liquid: General case:

  24. Semi-implicit lattice-Boltzmann method for non-Newtonian flows Equilibrium distribution: Velocity set (3D): Velocity set (2D): Post-collision distribution:

  25. Semi-implicit lattice-Boltzmann method for non-Newtonian flows Bingham liquid Power-law liquid

  26. Flow of a Bingham liquid in a constant cross-section channel

  27. Creep flow through mesh of cylinders

  28. Flow through mesh of cylinders

  29. CONCLUSIONS • 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.

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