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![Page 1: Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013 Hans J. Herrmann Fluids in Curved Space Computational Physics IfB, ETH Zürich, Switzerland.](https://reader036.fdocuments.in/reader036/viewer/2022081602/551bf93d550346a34f8b4820/html5/thumbnails/1.jpg)
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Hans J. HerrmannHans J. Herrmann
Fluids in Fluids in Curved SpaceCurved Space
Fluids in Fluids in Curved SpaceCurved Space
Computational Physics IfB, ETH Zürich, Switzerland and Departamento de Física Univ. Fed. do Ceará, Fortaleza
Complex SystemsComplex SystemsFoundations and Applications Foundations and Applications
Rio de Janeiro Rio de Janeiro Oct. 29 - Nov. 1, 2013Oct. 29 - Nov. 1, 2013
Feliz cumpleaños
Constantino !
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Comparado con la amistad,un Nature no vale nada
Montreal, Rio de Janeiro, Paris, Boston, Cali, Erice, Mexico, Sao Paulo, Tel Aviv, Habana, Natal, Cancun, Jülich, Maceió,Budapest, Bariloche,Brasilia, Bangalore,Stuttgart, Fortaleza,Mar del Plata, Iguaçu, Catania, Zürich, Manaus, Larnaca,....
y como se llamaba este lugar ?
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
CollaboratorsCollaborators
Miller Mendoza Farhang Mohseni Sauro Succi
Bruce Boghosian Nuno Araújo Ilya Karlin
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Examples for Fluid DynamicsExamples for Fluid Dynamicsin Curved Spacesin Curved Spaces
graphene semiconductor
Möbius band
generalized curved spacesvessels
curved boundary conditions
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Measuring Distance in Curved Space
infinitesimal surface element :
gives the parametrization, and is the trajectory joining them.
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Working with Contravariant Components of Vectors
metric tensor
covariant contravariant
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Geodesics: Shortest PathGeodesics: Shortest Path
For a particle:
geodesic equation containsinertial forces:
geodesic equation:
1
2i im ik ml klkl l k m
g g gg
x x x
:Christoffel symbols
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
ijikR g R
Curved Spaces and Curved Spaces and Curvilinear CoordinatesCurvilinear Coordinates
Ricci scalar or curvature scalar
Ricci curvature tensorSpherical and cylindrical coordinates represent flat spaces. One can demonstrate:
Curved spaces, e.g. 2d surface of a sphere:
l ll m m lik il
ik ik lm il kml kR
x x
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Navier-Stokes Equations Navier-Stokes Equations on Manifoldson Manifolds
mass conservation
momentum conservation
mass density fluid velocity
pressure metric tensor
shear viscosity
determinant of metric tensor
covariant derivatives
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Using Boltzmann’s Equation Using Boltzmann’s Equation on Manifoldson Manifolds
Taking into account that particles move along geodesics:
in thermodynamic equilibrium
BGK:
add a forcing term
anisotropic Gaussian shape: Hermite polynomials expansion possible !!
This is also the case for the Boltzmann equation in curvilinear coordinates (polar, cylindrical and spherical coordinates).
: microscopic velocity
: macroscopic velocity
: normalized temperature
P. J. Love and D. Cianci, Phil. Trans., of the Royal Soc. A 369, 2362 (2011).
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Boundary ConditionsBoundary Conditions
contravariant coordinates transformation (removing poles).
real geometry
brute force approximation of a sphere
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Advantages of Lattice BoltzmannAdvantages of Lattice Boltzmann
- Lattice Boltzmann computations in curved spaces and/or curvilinear (cylindrical, polar, etc.) coordinates.
- Curved spaces in Cartesian grids due to the contravariant components.
- The instabilities due to non-inertial forces are automatically included.
- Low relativistic flow through intrinsically curved spaces, e.g. interstellar media.
- Metric tensor and Christoffel symbols can vary with time. Modeling of elastic pipes, vessels, and flow within deformable membranes.
- “Exact” representation of the geometry of complex boundaries by using contravariant coordinates.
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Some ApplicationsSome Applications
discretizing in 19 velocities on cubic lattice
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Stretching: Poiseuille FlowStretching: Poiseuille Flow
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Solution in Curvilinear Coordinates Solution in Curvilinear Coordinates Taylor-Couette InstabilityTaylor-Couette Instability
r
z
t1 t2
square of velocity
M. Mendoza, S. Succi, H.J.H., Sci. Rep., in print, arXiv:1201.6581
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Validation and New ResultsValidation and New Results
cylinders spheres tori
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
The Campylotic MediumThe Campylotic Medium(Randomly Curved Space)(Randomly Curved Space)
0 1
01N
ij ij n
nr r rg a e
M. Mendoza, S. Succi, H.J.H., Sci. Rep., in print, arXiv:1201.6581
καμπυλος
N local curvatures (impurities) at
random positions
‘
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Curvature Disordered MediaCurvature Disordered Media
flux in absence of impurities
flux
number of impurities
range of curvature perturbation
Ricci or curvature tensor
Ricci or curvature scalar
l ll m m lik il
ik ik lm il kml kR
x x
ij
ikR g R
0
0 2
01
N NA
N N
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Curvature Disordered MediaCurvature Disordered Media
flux in absence of impurities
flux
number of impurities
range of curvature perturbation
Ricci or curvature tensor
Ricci or curvature scalar
l ll m m lik il
ik ik lm il kml kR
x x
ij
ikR g R
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Hydrodynamics in ManifoldsHydrodynamics in Manifolds(Summary)(Summary)
1. We have developed a Lattice Boltzmann Model (LBM) for general manifolds:
a) It allows to make computations in virtually any curvilinear coordinate system (polar, cylindrical, spherical, etc.) with LBM.
b) The LBM for manifolds can represent very complex geometries “exactly” in a cubic lattice due to the fact that it works in the contravariant coordinate system, and avoids a stair case approximation for curved boundary conditions.
c) Non-inertial forces are automatically included via the Christoffel symbols.
2. Flow through randomly curved spaces can present very unusual behavior.
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Relativistic Fluid DynamicsRelativistic Fluid Dynamics
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Fluid Dynamics ExamplesFluid Dynamics Examples
quark-gluon plasma
Au Au
electronic flow in graphene
supernovae
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Relativistic Navier-Stokes EquationsRelativistic Navier-Stokes Equations
energy conservation
number density fluid velocity
pressureenergy density
viscous tensor (still controversial)
correction term
Lorentz’s factor:
number of particles
energy-momentum conservation
21 1v v c
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Comparison Non- and Relativistic Comparison Non- and Relativistic Hydrodynamics Hydrodynamics
non-relativistic fluids: 5 equations
conservation of particle number
energy and momentum conservation
equation of state
relativistic fluids: 6 equations
conservation of mass
momentum conservation
equation of state
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Why Lattice Boltzmann?Why Lattice Boltzmann?
•The non-linear effects are intrinsically included in the Boltzmann equation.
•All the information about the system is contained in the particle distribution functions.
•It is a hyperbolic equation, in contrast to the relativistic Navier-Stokes equations, which are parabolic, and therefore could violate causality.
•It has already a natural speed limit (lattice speed), a property that it shares with relativity.
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Relativistic Lattice Boltzmann MethodRelativistic Lattice Boltzmann Method
( ) ( )eq
M
mp f f f
Marle model
1( , ) ( , ) ( ( , ) ( , ))eqi i i if x x t t f x t f x t f x t
4-dimensional system
1( , ) ( , ) ( ( , ) ( , ))eqi i i ig x x t t g x t g x t g x t
1 220.6 , 1 1.4v c
0T
0N
C. Marle, C. R. Acad. Sc. Paris 260, 6539 (1965)
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Relativistic Lattice Boltzmann Method Relativistic Lattice Boltzmann Method
macroscopic variables:
equilibrium distribution function:
M. Mendoza. B. Boghosian, S. Succi, H.J.H., Phys. Rev. Lett. 105, 014502 (2010); Phys.Rev.D 82, 105008 (2010)
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Relativistic Equilibrium Relativistic Equilibrium Distribution in Velocity SpaceDistribution in Velocity Space
d=1
M. Mendoza, N. Araújo, S. Succi, H.J.H. Scientific Reports 2, 611 (2012)F. Mohseni, M. Mendoza, S. Succi, H.J.H., Phys. Rev. D 87, 083003 (2013)
Maxwell - Jüttner distribution: λ = 0
2 ( )( , , )
1exp ( ) ( )
deq A vf x v t
v Uv U
T
2 22 22 2 2 2
4 2
2 22
2
( , , ) exp ( ) 1
( ) ( ) 3 ( ) ( )( ) ( )
2 4( ) ( )
4 ( ) ( )
6 151 4 2
( ) ( ) ( )
eq
x x y y x x z z y y z z
x x y y z z
f x v t A v
v U U vv U U v
v U v U v U v U v U v U U vv U
v U v U v U
U v v U U
expansion in orthogonal polynomials:
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Some ApplicationsSome Applications
discretizing in 19 velocities on cubic lattice
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Validation with Quark-Gluon PlasmaValidation with Quark-Gluon Plasma
BAMPS: Boltzmann Approach of Multi-
Parton Scattering.
ratio between time consumptions: 1 : 20 : 80000 (single CPU)for RLB : vSHASTA : BAMPS
M. Mendoza. B. Boghosian, S. Succi, H.J.H., Phys. Rev. Lett. 105, 014502 (2010)
D. Hupp. M. Mendoza, S. Succi, H.J.H., Phys. Rev. D 84, 125015 (2011)
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 201331
Application to Supernova ExplosionsApplication to Supernova Explosions
pressure
particle density
temperature
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Application to GrapheneApplication to Graphene
M. Müller, J. Schmalian, and L. Fritz, Phys. Rev. Lett. 103, 025301 (2009)
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Results with RLB for GrapheneResults with RLB for Graphene
Re = 100L = 5 utyp = 105 m/s
M. Mendoza, H.J.H. S. Succi, Phys. Rev. Lett. 106, 156601 (2011)
M. Mendoza, H.J.H., Succi, Sci. Rep. 3, 1052 (2013)
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Changing the Geometry Allows Changing the Geometry Allows Preturbulence at Re = 25 Preturbulence at Re = 25
M. Mendoza, H.J.H. S. Succi, Phys. Rev. Lett. 106, 156601 (2011)
M. Mendoza, H.J.H., Succi, Sci. Rep. 3, 1052 (2013)
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Preturbulence in Graphene: Preturbulence in Graphene: Relativistic EffectsRelativistic Effects
Re = 3000 St: Strouhal number (adimensional vortex shedding frequency)
shift in the vortex shedding frequencies
M. Mendoza, H.J.H. S. Succi, Phys. Rev. Lett. 106, 156601 (2011)
M. Mendoza, H.J.H., Succi, Sci. Rep. 3, 1052 (2013)
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Richtmyer-Meshkov InstabilityRichtmyer-Meshkov Instability
relativistic case non-relativistic case
F. Mohseni, M. Mendoza, H.J.H, preprint
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
qq-Statistics and -Statistics and Relativistic Fluid DynamicsRelativistic Fluid Dynamics
quark-gluon plasma
Au Au
long range entanglement of quarks and gluons
C. Tsallis, Eur. Phys. J. A 40, 257 (2009)T. Osada, G. Wilk, Phys. Rev. C 77, 044903 (2008)
T.S. Biró and E. Molnár, Eur. Phys. J. A (2012) 48: 172
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
qq-Relativistic Hydrodynamics-Relativistic Hydrodynamics
conservation of particle number
energy and momentum conservation
equation of state
shear viscosity
bulk viscosity
thermal conductivity
similarities with standard
relativistic hydrodynamics
differences with standard
relativistic hydrodynamics
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Future work: Future work: Relativistic Lattice Relativistic Lattice qq-Boltzmann-Boltzmann
orthonormal polynomial expansion
weight function
1. q-Lattice Boltzmann Methods2. expanding the non-equilibrium distribution around the equilibrium
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Relativistic HydrodynamicsRelativistic Hydrodynamics(Summary)(Summary)
1. The relativistic lattice Boltzmann model has applications in quark-gluon plasma, supernova explosions, and electronic gas in graphene.
a) The RLB is four orders of magnitude faster than other relativistic kinetic models.
b) Complex geometries in relativistic systems can be treated easy.
2. Turbulent phenomena can produce noticeable electrical current fluctuations due to contact points and/or other kind of impurities in graphene samples.
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Future ChallengesFuture Challenges
1. Entropic formulations of the LB methods.
2. General relativity and coupling with Einstein equations.
3. Fluid structures interactions.
4. Flow through deformable pipes, e.g. vessels.
5. Deriving the orthogonal basis of q-polynomials with the weight being the relativistic equilibrium distribution at rest.
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
ReferencesReferences
• M. MENDOZA, B. BOGHOSIAN, H.J. HERRMANN, S. SUCCI, Fast Lattice Boltzmann solver for relativistic hydrodynamics, Phys. Rev. Lett. 105, 014502 (2010), arXiv:0912.2913
• M. MENDOZA, B. BOGHOSIAN, H.J. HERRMANN, S. SUCCI, Derivation of the Lattice Boltzmann model for relativistic hydrodynamics, Phys. Rev.D 82, 105008 (2010), arXiv:1009.0129v1
• M. MENDOZA, H.J. HERRMANN, S. SUCCI, Preturbulent Regimes in Graphene Flows, Phys.Rev.Lett 106, 156601 (2011)
• M. MENDOZA, H.J. HERRMANN, S. SUCCI, Hydrodynamic approach to the conductivity in graphene, Sci. Rep. 3, 1052 (2013), arXiv:1301.3428
• D. HUPP, M. MENDOZA, S. SUCCI, H.J. HERRMANN, Relativistic Lattice Boltzmann method for quark-gluon plasma simulations, Phys. Rev. D 84, 125015 (2011), arXiv:1109.0640
• M. MENDOZA, S. SUCCI, H.J. HERRMANN, Flow through randomly curved manifolds, accepted for Sci. Rep. arXiv:1201.6581
• S. PALPACELLI, M. MENDOZA, H.J. HERRMANN, S. SUCCI, Klein tunneling in the presence of random impurities, IJMPC 23, 1250080 (2012) arXiv:1202.6217
• M. MENDOZA, N.A.M. ARAÚJO, S. SUCCI, H.J. HERRMANN, Transition in the equilibrium distribution function of relativistic particles, Sci. Rep. 2, 00611 (2012), arXiv:1204.1889
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
• M. MENDOZA, I. KARLIN, S. SUCCI, H.J. HERRMANN, Ultrarelativistic transport coefficients in two dimensions, JSTAT P02036 (2013), arXiv:1301.3420
• M. MENDOZA, I. KARLIN, S. SUCCI, H.J. HERRMANN, Relativistic Lattice Boltzmann Model with Improved Dissipation, Phys. Rev. D 87, 065027 (2013), arXiv:1301.3423
• F. MOHSENI, M. MENDOZA, S. SUCCI, H.J. HERRMANN, Lattice Boltzmann model for ultra-relativistic flows, Phys. Rev. D 87, 083003 (2013), arXiv:1302.1125
• D. ÖTTINGER, M. MENDOZA, H.J. HERRMANN, Gaussian quadrature and lattice discretization of the Fermi-Dirac distribution for graphene, Phys. Rev. E 88, 013302 (2013), arXiv:1305.0373
• M. MENDOZA, S. SUCCI, H.J. HERRMANN, Kinetic formulation of the Kohn-Sham equations for ab initio electronic structure calculations, preprint
• F. FILLION-GOURDEAU, H.J. HERRMANN, M. MENDOZA, S. PALPACELLI, S. SUCCI, Formal analogy between the Dirac equation in its Majorana form and the discrete-velocity version of the Boltzmann kinetic equation, Phys. Rev. Lett. 111, 160602 (2013), arXiv:1310.0686
• F. MOHSENI, M. MENDOZA, S. SUCCI, H.J. HERRMANN, Cooling of the quark-gluon plasma due to the Richtmyer-Meshkov instability, preprint
ReferencesReferences
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Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Bom aniversario
Constantino !