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

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Presentation transcript:

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

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

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

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

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

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

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

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

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 )

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

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

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

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

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

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

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.

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:

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

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

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

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

Effect of the boundary conditions

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:

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:

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:

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

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

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

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

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