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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 Alexander Vikhansky Department of Engineering, Queen Mary, University of London
Boltzmann equation Knudsen number:
Kinetic effects: Knudsen layer (Kn2) 1. Knudsen slip (Kn), 2. Thermal slip (Kn).
Kinetic effects: 3. Thermal creep (Kn).
Kinetic effects: 4. Thermal stress flow (Kn2).
Collision operator BGK model:
Method of moments – 5 equations; 1. Euler set: 2. Grad set: – 13 equations; 3. Grad-26, Grad-45, Grad-71.
Method of moments The error: 1. Euler set: 2. Grad set: 3. Grad-26: 4. Grad-45, Grad-71:
Simulation of thermophoretic flows Velocity set:
Knudsen compressor M. Young, E.P. Muntz, G. Shiflet and A. Green
Semi-implicit lattice-Boltzmann method for non-Newtonian flows From the kinetic theory of gases: Constitutive equation:
Semi-implicit lattice-Boltzmann method for non-Newtonian flows Newtonian liquid: Bingham liquid: General case:
Semi-implicit lattice-Boltzmann method for non-Newtonian flows Equilibrium distribution: Velocity set (3D): Velocity set (2D): Post-collision distribution:
Semi-implicit lattice-Boltzmann method for non-Newtonian flows Bingham liquid Power-law liquid
Flow of a Bingham liquid in a constant cross-section channel
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.