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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.

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Presentation on theme: "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."— Presentation transcript:

1 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Hans J. Herrmann Fluids in Curved Space Computational Physics IfB, ETH Zürich, Switzerland and Departamento de Física Univ. Fed. do Ceará, Fortaleza Complex Systems Foundations and Applications Rio de Janeiro Oct Nov. 1, 2013 Feliz cumpleaños Constantino !

2 Constantino’s birthday party, Rio de Janeiro, Oct 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 ?

3 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Collaborators Miller Mendoza Farhang Mohseni Sauro Succi Bruce Boghosian Nuno Araújo Ilya Karlin

4 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Examples for Fluid Dynamics in Curved Spaces graphene semiconductor Möbius band generalized curved spaces vessels curved boundary conditions

5 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Measuring Distance in Curved Space infinitesimal surface element : gives the parametrization, and is the trajectory joining them.

6 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Working with Contravariant Components of Vectors metric tensor covariant contravariant

7 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Geodesics: Shortest Path For a particle: geodesic equation contains inertial forces: geodesic equation :

8 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Curved Spaces and Curvilinear Coordinates Ricci scalar or curvature scalar Ricci curvature tensor Spherical and cylindrical coordinates represent flat spaces. One can demonstrate: Curved spaces, e.g. 2d surface of a sphere :

9 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Navier-Stokes Equations on Manifolds mass conservation momentum conservation mass density fluid velocity pressure metric tensor shear viscosity determinant of metric tensor covariant derivatives

10 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Using Boltzmann’s Equation on 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).

11 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Boundary Conditions contravariant coordinates transformation (removing poles). real geometry brute force approximation of a sphere

12 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Advantages 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.

13 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Some Applications discretizing in 19 velocities on cubic lattice

14 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Stretching: Poiseuille Flow

15 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Solution in Curvilinear Coordinates Taylor-Couette Instability r z t1t1 t2t2 square of velocity M. Mendoza, S. Succi, H.J.H., Sci. Rep., in print, arXiv:

16 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Validation and New Results cylindersspherestori

17 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 The Campylotic Medium (Randomly Curved Space) M. Mendoza, S. Succi, H.J.H., Sci. Rep., in print, arXiv: καμπυλος N local curvatures (impurities) at random positions ‘

18 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Curvature Disordered Media flux in absence of impurities flux number of impurities range of curvature perturbation Ricci or curvature tensor Ricci or curvature scalar

19 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Curvature Disordered Media flux in absence of impurities flux number of impurities range of curvature perturbation Ricci or curvature tensor Ricci or curvature scalar

20 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Hydrodynamics in Manifolds (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.

21 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Relativistic Fluid Dynamics

22 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Fluid Dynamics Examples quark-gluon plasma Au electronic flow in graphene supernovae

23 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Relativistic Navier-Stokes Equations energy conservation number density fluid velocity pressure energy density viscous tensor (still controversial) correction term Lorentz’s factor: number of particles energy-momentum conservation

24 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Comparison Non- and Relativistic 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

25 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 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.

26 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Relativistic Lattice Boltzmann Method Marle model 4-dimensional system C. Marle, C. R. Acad. Sc. Paris 260, 6539 (1965)

27 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Relativistic Lattice Boltzmann Method macroscopic variables: equilibrium distribution function: M. Mendoza. B. Boghosian, S. Succi, H.J.H., Phys. Rev. Lett. 105, (2010); Phys.Rev.D 82, (2010)

28 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Relativistic Equilibrium Distribution 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, (2013) Maxwell - Jüttner distribution : λ = 0 expansion in orthogonal polynomials :

29 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Some Applications discretizing in 19 velocities on cubic lattice

30 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Validation with Quark-Gluon Plasma BAMPS: Boltzmann Approach of Multi- Parton Scattering. ratio between time consumptions : 1 : 20 : ( single CPU ) for RLB : vSHASTA : BAMPS M. Mendoza. B. Boghosian, S. Succi, H.J.H., Phys. Rev. Lett. 105, (2010) D. Hupp. M. Mendoza, S. Succi, H.J.H., Phys. Rev. D 84, (2011)

31 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, Application to Supernova Explosions pressure particle density temperature

32 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Application to Graphene M. Müller, J. Schmalian, and L. Fritz, Phys. Rev. Lett. 103, (2009)

33 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Results with RLB for Graphene Re = 100 L = 5 u typ = 10 5 m/s M. Mendoza, H.J.H. S. Succi, Phys. Rev. Lett. 106, (2011) M. Mendoza, H.J.H., Succi, Sci. Rep. 3, 1052 (2013)

34 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Changing the Geometry Allows Preturbulence at Re = 25 M. Mendoza, H.J.H. S. Succi, Phys. Rev. Lett. 106, (2011) M. Mendoza, H.J.H., Succi, Sci. Rep. 3, 1052 (2013)

35 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Preturbulence in Graphene: Relativistic 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, (2011) M. Mendoza, H.J.H., Succi, Sci. Rep. 3, 1052 (2013)

36 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Richtmyer-Meshkov Instability relativistic case non-relativistic case F. Mohseni, M. Mendoza, H.J.H, preprint

37 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 q-Statistics and Relativistic Fluid Dynamics quark-gluon plasma 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, (2008) T.S. Biró and E. Molnár, Eur. Phys. J. A (2012) 48: 172

38 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 q-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

39 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Future work: Relativistic Lattice q-Boltzmann orthonormal polynomial expansion weight function 1. q-Lattice Boltzmann Methods 2. expanding the non-equilibrium distribution around the equilibrium

40 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Relativistic Hydrodynamics (Summary) Relativistic Hydrodynamics (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.

41 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Future 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.

42 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013References M. MENDOZA, B. BOGHOSIAN, H.J. HERRMANN, S. SUCCI, Fast Lattice Boltzmann solver for relativistic hydrodynamics, Phys. Rev. Lett. 105, (2010), arXiv: M. MENDOZA, B. BOGHOSIAN, H.J. HERRMANN, S. SUCCI, Derivation of the Lattice Boltzmann model for relativistic hydrodynamics, Phys. Rev.D 82, (2010), arXiv: v1 M. MENDOZA, H.J. HERRMANN, S. SUCCI, Preturbulent Regimes in Graphene Flows, Phys.Rev.Lett 106, (2011) M. MENDOZA, H.J. HERRMANN, S. SUCCI, Hydrodynamic approach to the conductivity in graphene, Sci. Rep. 3, 1052 (2013), arXiv: D. HUPP, M. MENDOZA, S. SUCCI, H.J. HERRMANN, Relativistic Lattice Boltzmann method for quark-gluon plasma simulations, Phys. Rev. D 84, (2011), arXiv: M. MENDOZA, S. SUCCI, H.J. HERRMANN, Flow through randomly curved manifolds, accepted for Sci. Rep. arXiv: S. PALPACELLI, M. MENDOZA, H.J. HERRMANN, S. SUCCI, Klein tunneling in the presence of random impurities, IJMPC 23, (2012) arXiv: M. MENDOZA, N.A.M. ARAÚJO, S. SUCCI, H.J. HERRMANN, Transition in the equilibrium distribution function of relativistic particles, Sci. Rep. 2, (2012), arXiv:

43 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 M. MENDOZA, I. KARLIN, S. SUCCI, H.J. HERRMANN, Ultrarelativistic transport coefficients in two dimensions, JSTAT P02036 (2013), arXiv: M. MENDOZA, I. KARLIN, S. SUCCI, H.J. HERRMANN, Relativistic Lattice Boltzmann Model with Improved Dissipation, Phys. Rev. D 87, (2013), arXiv: F. MOHSENI, M. MENDOZA, S. SUCCI, H.J. HERRMANN, Lattice Boltzmann model for ultra-relativistic flows, Phys. Rev. D 87, (2013), arXiv: D. ÖTTINGER, M. MENDOZA, H.J. HERRMANN, Gaussian quadrature and lattice discretization of the Fermi-Dirac distribution for graphene, Phys. Rev. E 88, (2013), arXiv: 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, (2013), arXiv: F. MOHSENI, M. MENDOZA, S. SUCCI, H.J. HERRMANN, Cooling of the quark-gluon plasma due to the Richtmyer-Meshkov instability, preprintReferences

44 Constantino’s birthday party, Rio de Janeiro, Oct Nov. 1, 2013 Bom aniversario Constantino !


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