Formulation/Statement of the problems
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Formulation/Statement of the problems Slip boundary conditions on thermal viscous incompressible flows Luisa Consiglieri Department of Mathematics and CMAF Existence results and open problems
description
Slip boundary conditions on thermal viscous incompressible flow s. Luisa Consiglieri Department of Mathematics and CMAF. Formulation/Statement of the problems. Existence results and open problems. Governing equations. INCOMPRESSIBILITY. MOTION EQUATIONS. - PowerPoint PPT Presentation
Transcript of Formulation/Statement of the problems
Formulation/Statement of the problems
Slip boundary conditionson thermal viscous incompressible flows
Luisa Consiglieri
Department of Mathematics and CMAF
Existence results and open problems
Governing equations
)3,2( nIRboundedOpen n
gDeee
)(:)](),([1
uau
fluid velocity:
deviator stress tensor:
internal energy: eniiu ,...,1)( u
jiij ,)( pressure
D
u)uu T
2/(
u inx
un
i i
i
10INCOMPRESSIBILITY
MOTION EQUATIONS
ENERGY EQUATION
fuu
)(1
No-slip boundary conditions
outin
domain
in
inflowout
outflow
0)( 00 measLipschitz
0: u On conditionDirichlet
Navier slip boundary condition (1823)
On
u On
:
0:0
conditionDirichlet
The fluid cannot penetrate the solid wall
,0: nuNu
vectornormal
outwardunitN
n
nT
:
and the inadequacy of the adherence condition
TT u ,0
Slip boundary conditions
On
u On
:
0:0
conditionDirichlet
TTT
TT
u ||
u ||
,0),(
0),(
e
e
conditionCoulombfrictionntwisePoi
FRICTIONAL BOUNDARY CONDITION
sN eu ||),(,0: TT u nu
[C. le Roux, 1999] … fluids with slip boundary conditions [Jager & Mikelic, 2001] On the roughness-induced effective boundary conditions …
s=2, linear Navier laws=3, Chezy-Manning laws=1:
Energy boundary conditions
On
On
:
0:0
econditionDirichlet
EXAMPLE (convective-radiation coefficient)
l=1: h= convective heat transfer coefficient
l=4: h= Stefan-Boltzmann constant
eeehe l 1||)},({),(
TT una ),()(),( eee
Constitutive law for the heat flux
heat capacity
FOURIER LAW (q=2)
EXAMPLE
2||)( qa
e
cp )(
tyconductiviheat
pce
q )()(,
Assumptions
ENERGY-DEPENDENT PARAMETERS
#
##
),(0,:
),(0:
eIRIR
eIRIR
:friction ryCarathéodo
, :diffusity ryCarathéodo
1),1|(||),(
||)(),(,:#
#
lee
eesigneIRIRl
l
|
:radiation-convective ryCarathéodo
IReesigne ,,0)()),(),((
THEOREM for Navier-Stokes-Fourier flows
Under the assumptions
then there exists a weak solution
to the coupled system
)(),( 12 Lg Lf
11
n
nr
)(),( a uDe
otherwise and if , ,1)1(
1
snppn
nps
## ),(0: eIRIR , :viscosity ryCarathéodo
0,1
01
0:)()(
0v,0,0:)(
onLWe
ononinlr
N
vvHvu
0|)(||)(|
)()()()( u if
u
uu D
D
DeDeI
)( plasticityofthreshold
[Duvaut & Lions, 1972] Transfert de chaleur dans un fluide de Bingham ... (constant plasticity threshold, without convective terms, and DIRICHLET condition for fluid motion)
Bingham fluid
Vvuvfuv|
uv
uvuuvu
TT
)(|||)(
|)(||)(|)(
)(:)()()(:
DD
DDD
inesourceheatandn
econditionNeumannsHomogeneou
g
0
The asymptotic limit case of a high diffusity
Taking
|)(|)(|)(|)(|| 2 uu DD
[Ladyzhenskaya, 1970] New equation for description of motion ...[J.F. Rodrigues and i, 2003 & 2005] On stationary flows ...
And so many other models …
Taking the asymptotic limit of a high diffusity when
it follows
u T
insourceheatand
een
econditionFourier
0g
||)(
||)(|)(|)(|)(|)(|| 2
Tuuu DD
Fluids with shear thinning behaviour
]2,1[|)|1(
)(1
0
pDD
eIp
u u
p =3/2
p =2
non-Newtonian fluids
uDDfeI II )()(
POWER LAWS (Ostwald & de Waele)
p>2: dilatant fluid1<p<2: pseudo-plastic fluid p=1: Bingham fluid p=2: Navier-Stokes fluid
pddF
ddF
)(
)(
2
21
|)](|)()(|),([ 21 u,|u DFeDFeItensorstressdeviator
fabledifferentiF obtain we ||
(p-q) ASSUMPTIONS
)(),( 1' Lgp Lf
conditions F-S-N satisfy ,,,
uuu u DDDDFe :|||)],(|),([ 2
)1()(: ##0 pp ddFdIRIRF :convex strictly
)1|(||)(|0)())()((
)(,||)(:1#
1#
q
qnn LIRIR
a aa
a :continuous a
(p-q) relations
,
npn
q
npn
n
nnp
pnq
21
2
3,
)1(
)12(
under Dirichlet boundary conditionand without Joule effect
[2006] Math. Mod. and Meth. in Apppl. Sci. 16 :12, 2013--2027. http://dx.doi.org/10.1142/S0218202506001790
n=3
Theorem
Under the above assumptionsthen there exists a weak solution
to the coupled system
npjiij
nnp
lr
Np
LL
onLWe
ononin
))((,:))((
0:)()(
0v,0,0:)(
''
0,1
0,1
:Y
vvWvu
p'
1
)1(1
n
qnr
[2008] J. Math. Anal. Appl. 340 :1 (2008), 183--196. http://dx.doi.org/10.1016/j.jmaa.2007.07.080
Open problem
under the assumptions
then there exists a weak solution? i believe YES!
)(),( uDe
:continuous
0)())()((
)1|(||)(|
)(,||)(
:
1#
1#
p
p
nnnn
L
MM
Greater range of (p-q) exponents ?
as in the Dirichlet boundary value problem:
)(),(
)1(2
3
1
2
)(,
)2(
2
2
3
0
1
CC
nnp
pnq
n
np
n
n
LW
nnp
pnqnp
n
n
lqp
div0,v functionstest
Vv functionstest
The non-stationary case
Existence result holds provided that
Strong monotone property for the motion and heat fluxes
for some (p-q) relationship
and convective exponent: l=1
2,||)())()((
2,||)())()((
q
pAAq
p
if aa
if
|)](|),([)(),( uu DFeDAe
[2008] Annali Mat. Pura Appl. http://dx.doi.org/10.1007/s10231-007-0060-3
Present goal
domains unbounded in fluids rotating
[...]
regularity
Acknowledgement:
Université de Pau et des Pays de L’Adour
