Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013 Hans J. Herrmann Fluids in...

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 !


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 ?


CollaboratorsCollaborators

Miller Mendoza Farhang Mohseni Sauro Succi

Bruce Boghosian Nuno Araújo Ilya Karlin


Examples for Fluid DynamicsExamples for Fluid Dynamicsin Curved Spacesin Curved Spaces

graphene semiconductor

Möbius band

generalized curved spacesvessels

curved boundary conditions


Measuring Distance in Curved Space

infinitesimal surface element :

gives the parametrization, and is the trajectory joining them.


Working with Contravariant Components of Vectors

metric tensor

covariant contravariant


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


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


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


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).


Boundary ConditionsBoundary Conditions

contravariant coordinates transformation (removing poles).

real geometry

brute force approximation of a sphere


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.


Some ApplicationsSome Applications

discretizing in 19 velocities on cubic lattice


Stretching: Poiseuille FlowStretching: Poiseuille Flow


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


Validation and New ResultsValidation and New Results

cylinders spheres tori


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

‘


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


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


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.


Relativistic Fluid DynamicsRelativistic Fluid Dynamics


Fluid Dynamics ExamplesFluid Dynamics Examples

quark-gluon plasma

Au Au

electronic flow in graphene

supernovae


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


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


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.


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)


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)


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:


Some ApplicationsSome Applications

discretizing in 19 velocities on cubic lattice


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)


Application to Supernova ExplosionsApplication to Supernova Explosions

pressure

particle density

temperature


Application to GrapheneApplication to Graphene

M. Müller, J. Schmalian, and L. Fritz, Phys. Rev. Lett. 103, 025301 (2009)


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)


Changing the Geometry Allows Changing the Geometry Allows Preturbulence at Re = 25 Preturbulence at Re = 25




Preturbulence in Graphene: Preturbulence in Graphene: Relativistic EffectsRelativistic Effects

Re = 3000 St: Strouhal number (adimensional vortex shedding frequency)

shift in the vortex shedding frequencies




Richtmyer-Meshkov InstabilityRichtmyer-Meshkov Instability

relativistic case non-relativistic case

F. Mohseni, M. Mendoza, H.J.H, preprint


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


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


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


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.


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.


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


• 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


Bom aniversario

Constantino !

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