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