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

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Lattice-Boltzmann method for non-Newtonian and
non-equilibrium flows
Alexander Vikhansky Department of
Engineering, Queen Mary, University of London
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Lattice-Boltzmann method
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Boltzmann equation
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NS equations
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Plan of the presentation
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Plan of the presentation
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Boltzmann equation
Knudsen number
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Chapman-Enskog expansion
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Kinetic effects
Knudsen layer (Kn2)
1. Knudsen slip (Kn), 2. Thermal slip (Kn).
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Kinetic effects
3. Thermal creep (Kn).
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Kinetic effects
4. Thermal stress flow (Kn2).
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Discrete ordinates equation
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Collision operator
BGK model
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Boundary conditions
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Boundary conditions bounce-back rule
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Method of moments
5 equations
1. Euler set
2. Grad set
13 equations
3. Grad-26, Grad-45, Grad-71.
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Method of moments
The error
1. Euler set
2. Grad set
3. Grad-26
4. Grad-45, Grad-71
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Simulation of thermophoretic flows
Velocity set
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Knudsen compressor
M. Young, E.P. Muntz, G. Shiflet and A. Green
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Knudsen compressor
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Effect of the boundary conditions
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Semi-implicit lattice-Boltzmann method for
non-Newtonian flows
From the kinetic theory of gases
Constitutive equation
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Semi-implicit lattice-Boltzmann method for
non-Newtonian flows
Newtonian liquid
Bingham liquid
General case
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Semi-implicit lattice-Boltzmann method for
non-Newtonian flows
Equilibrium distribution
Velocity set (3D)
Velocity set (2D)
Post-collision distribution
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Semi-implicit lattice-Boltzmann method for
non-Newtonian flows
Bingham liquid
Power-law liquid
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Flow of a Bingham liquid in a constant
cross-section channel
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Creep flow through mesh of cylinders
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Flow through mesh of cylinders
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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.