Fluids in
Curved Space
Feliz cumpleaños
Hans J. Herrmann
Constantino !
Computational Physics
IfB, ETH Zürich, Switzerland
and
Departamento de Física
Univ. Fed. do Ceará, Fortaleza
Complex Systems
Foundations and Applications
Rio de Janeiro
Oct. 29 - Nov. 1, 2013
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 ?
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Collaborators
Miller Mendoza
Farhang Mohseni
Sauro Succi
Bruce Boghosian
Nuno Araújo
Ilya Karlin
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Examples for Fluid Dynamics
in Curved Spaces
vessels
curved boundary conditions
generalized curved spaces
graphene semiconductor
Möbius band
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Measuring Distance in Curved Space
infinitesimal surface element :
ds 2 = gij dx i dx j
Ds = ò
W
dx i dx j
gij
dl
dl dl
l
gives the parametrization, and
is the trajectory joining them.
W
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Working with Contravariant
Components of Vectors
v = v ei = vi e
i
contravariant
covariant
v = v v ei × e j = v v gij
2
i
j i
j i
metric tensor
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Geodesics: Shortest Path
Christoffel symbols :
1 im  gik gml gkl 
i
kl  g  l  k  m 
2
x
x 
 x
geodesic equation:
d (Ds) = d ò
W
2 i
For a particle:
dx i dx j
gij
dl = 0
dl dl
k
d x
dx
i dx
= -G kl
2
dl
dl dl
l
geodesic equation contains
inertial forces:
dpi
= -G ikl p k pl + Fexti
dt
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Curved Spaces and
Curvilinear Coordinates
R  g Rik
ij


l
m
m l
Rik 

 ik lm  il  km
x
x
l
ik
l
l
il
k
Curved spaces, e.g. 2d surface of a sphere:
Ricci scalar or
curvature scalar
Ricci curvature tensor
R¹0
Spherical and cylindrical coordinates
represent flat spaces. One can demonstrate:
R=0
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Navier-Stokes Equations
on Manifolds
¶r
+ ( ru i );i = 0
¶t
mass conservation
shear
viscosity
fluid velocity
mass density
k
æ
ö
¶( ru k )
1
¶
¶
r
u
kj
k j
ij
+ ( P g + ru u ); j = m
gg
i ç
¶t
¶x j ÷ø
g ¶x è
momentum conservation
determinant of
metric tensor
pressure metric tensor
( ru )
i
;i
covariant derivatives
¶ ru
i
k
=
+
G
r
u
ik
¶x i
i
T ik ;k
¶T ik
= k + G imkT mk + G kmkT im
¶x
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Using Boltzmann’s Equation
on Manifolds
Taking into account that particles move along geodesics:
add a forcing term
BGK
:
This is also the case for the Boltzmann equation in curvilinear
coordinates (polar, cylindrical and spherical coordinates).
in thermodynamic equilibrium
P. J. Love and D. Cianci, Phil. Trans., of the Royal Soc. A 369, 2362 (2011).
: microscopic velocity
anisotropic Gaussian shape:
Hermite polynomials expansion possible !!
: macroscopic velocity
: normalized temperature
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Boundary Conditions
contravariant coordinates
transformation (removing poles).
real geometry
brute force approximation
of a sphere
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Advantages of Lattice Boltzmann
-
Lattice Boltzmann computations in curved spaces and/or curvilinear
(cylindrical, polar, etc.) coordinates.
-
Curved spaces
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.
in
Cartesian
grids
due
to
the
contravariant
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Some Applications
discretizing in 19 velocities on cubic lattice
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Stretching: Poiseuille Flow
æ 1 0 0 ö
gik = ç 0 1 0 ÷
ç
÷
0
0
1
è
ø
x
æ 2 0 0
gik = ç 0 1 0
ç
è 0 0 1
ö
÷
÷
ø
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Solution in Curvilinear Coordinates
Taylor-Couette Instability
M. Mendoza, S. Succi, H.J.H., Sci. Rep., in print, arXiv:1201.6581
square of velocity
z
r
æ 1 0
gik = ç 0 r 2
ç
è 0 0
0
0
1
ö
÷
÷
ø
t1
t2
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Validation and New Results
æ 1 0
gik = ç 0 r 2
ç
è 0 0
0
0
1
cylinders
ö
÷
÷
ø
æ 1
0
gik = ç 0 r 2 sin 2 q
ç
çè 0
0
0
0
r2
ö
÷
÷
÷ø
æ 1
0
ç
gik = ç 0 (R + r cosf )2
çè 0
0
spheres
0 ö
÷
0 ÷
r 2 ÷ø
tori
Vr
R
f
r
q
Vq
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
The Campylotic Medium
(Randomly Curved Space)
‘
καμπυλος
N local curvatures
(impurities) at
random positions

gij   ij 1  a0  n 1 e
N
 r  rn r0
M. Mendoza, S. Succi, H.J.H., Sci. Rep., in print, arXiv:1201.6581
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013

Curvature Disordered Media
0    A
N N0
1   N N0 
2
F 0 :flux in absence of impurities
F : flux
N : number of impurities
r0 : range of curvature perturbation
l :V 1/3 / N
Rik : Ricci or curvature tensor
R : Ricci or curvature scalar
R  g ij Rik
lik ill
m
Rik  l  k  ikl lm
 ilmlkm
x
x
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Curvature Disordered Media
F 0 :flux in absence of impurities
F : flux
N : number of impurities
r0 : range of curvature perturbation
l :V 1/3 / N
Rik : Ricci or curvature tensor
R : Ricci or curvature scalar
R  g ij Rik
lik ill
m
Rik  l  k  ikl lm
 ilmlkm
x
x
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - 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.
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Relativistic Fluid Dynamics
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Fluid Dynamics Examples
electronic flow in graphene
quark-gluon plasma
Au
Au
supernovae
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Relativistic Navier-Stokes Equations
number of particles
¶ng
+ Ñ ×(ng u) = 0
¶t
¶ éë(e + P)g 2 ùû
¶t
+ Ñ × éë(e + P)g 2u ùû =
¶P
¶t
energy conservation
viscous tensor
number density
fluid velocity
(still controversial)
energy-momentum conservation
i
i
ij
¶u
¶u
¶P
¶P
¶Õ
(e + P)g 2
+ (e + P)g 2u j j = -u i
- i+
¶t
¶x
¶t ¶x
¶x j
pressure
Lorentz’s factor:
energy
density   v   1
1 v c
2
correction term
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Comparison Non- and Relativistic
Hydrodynamics
non-relativistic fluids: 5 equations
conservation of mass
momentum conservation
equation of state
relativistic fluids: 6 equations
conservation of particle number
energy and momentum conservation
equation of state
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - 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.
clattice » c
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Relativistic Lattice Boltzmann Method
Marle model

 ( p f ) 
C. Marle, C. R. Acad. Sc. Paris 260, 6539 (1965)
m
M
( f eq  f )
4-dimensional system
  v c  0.6 ,   1  
1
fi ( x   x, t   t )  f i ( x, t )  ( f i eq ( x, t )  f i ( x, t ))

1
gi ( x   x, t   t )  gi ( x, t )  ( gieq ( x, t )  g i ( x, t ))


2 1 2
 1.4
 N   0
T   0
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Relativistic Lattice Boltzmann Method
P HYSICAL
REVIEW
L ETTERS
Member Subscription Copy
Library or Other Institutional Use Pr ohibited Until 2015
macroscopic variables:
Articles published week ending
2 JULY 2010
Published by the
American Physical Society
Volume 105, Number 1
equilibrium distribution function:
M. Mendoza. B. Boghosian, S. Succi, H.J.H., Phys. Rev. Lett. 105, 014502 (2010); Phys.Rev.D 82, 105008 (2010)
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Relativistic Equilibrium
Distribution in Velocity Space
Maxwell - Jüttner
distribution: λ = 0
mc 2
x=
kB T
d=1
f eq ( x , v , t ) 
A d  2 (v)
1  v  U

exp 
 (v) (U )   
 T

expansion in orthogonal polynomials:

f eq ( x , v , t )  A exp   (v)  1

  (v) (U )  3  (U )  (v) 



 v U  (U ) (v)
2

4


  vxU x v yU y  vxU x vzU z  v yU y vzU z   (U ) (v )
2
2
4 2 (v) 2 (U ) 
2
2
2
2
 4
v
U

v
U

v
U












x
x
y
y
z
z
  6 2  15 
 
 1  2

4  2 2
2
 (U ) 
 (v ) 

(
v
)
U

U


2
 

 
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)
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Some Applications
discretizing in 19 velocities on cubic lattice
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Validation with Quark-Gluon Plasma
ratio between time consumptions:
1 : 20 : 80000 (single CPU)
for RLB : vSHASTA : BAMPS
BAMPS: Boltzmann Approach of MultiParton Scattering.
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)
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Application to Supernova Explosions
pressure
particle density
temperature
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Application to Graphene
M. Müller, J. Schmalian, and L. Fritz, Phys. Rev. Lett. 103, 025301 (2009)
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Results with RLB for Graphene
Re = 100
L = 5 mm
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)
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Changing the Geometry Allows
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)
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Preturbulence in Graphene:
Relativistic Effects
shift in the vortex shedding frequencies
Re = 3000
M. Mendoza, H.J.H. S. Succi, Phys. Rev. Lett. 106, 156601 (2011)
M. Mendoza, H.J.H., Succi, Sci. Rep. 3, 1052 (2013)
St: Strouhal number
(adimensional vortex
shedding frequency)
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Richtmyer-Meshkov Instability
relativistic case
non-relativistic case
F. Mohseni, M. Mendoza, H.J.H, preprint
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q-Statistics and
Relativistic Fluid Dynamics
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
quark-gluon plasma
Au
Au
long range entanglement of
quarks and gluons
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
q-Relativistic Hydrodynamics
similarities with standard
differences with standard
relativistic hydrodynamics
relativistic hydrodynamics
conservation of particle number
energy and momentum conservation
equation of state
h ® h (q)
shear viscosity
x ® x (q)
bulk viscosity
k ® k (q)
thermal
conductivity
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Future work:
Relativistic Lattice q-Boltzmann
m
p
U
q
m
é
¶ m ê p ( f ) ùú = m é f eq
ë
û
t M êë
( )
q
-( f ) ù
úû
q
weight function
( )
f eq
q
¥
= w( p,q)å an Lqn
n=0
¥
3
d
p
q q
ò-¥ w( p,q)Ln Lm p0 = d mn
orthonormal polynomial expansion
1. q-Lattice Boltzmann Methods
2. expanding the non-equilibrium
distribution around the equilibrium
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
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.
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - 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.
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
References
•
•
•
•
•
•
•
•
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
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
References
•
•
•
•
•
•
•
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 KohnSham 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
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013
Bom aniversario
Constantino !
Constantino’s birthday party, Rio de Janeiro, Oct. 29 - Nov. 1, 2013