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Formulation/Statement of the problems

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.

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Formulation/Statement of the problems

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  1. Slip boundary conditions on thermal viscous incompressible flows Luisa Consiglieri Department of Mathematics and CMAF Formulation/Statement of the problems Existence results and open problems

  2. Governing equations INCOMPRESSIBILITY MOTION EQUATIONS ENERGY EQUATION internal energy: e fluid velocity: deviator stress tensor:

  3. No-slip boundary conditions

  4. Navier slip boundary condition (1823) The fluid cannot penetrate the solid wall and the inadequacy of the adherence condition

  5. Slip boundary conditions FRICTIONAL BOUNDARY CONDITION s=2, linear Navier law s=3, Chezy-Manning law s=1: [C. le Roux, 1999] … fluids with slip boundary conditions [Jager & Mikelic, 2001] On the roughness-induced effective boundary conditions …

  6. Energy boundary conditions EXAMPLE (convective-radiation coefficient) l=1: h= convective heat transfer coefficient l=4: h=Stefan-Boltzmann constant

  7. Constitutive law for the heat flux EXAMPLE heat capacity FOURIER LAW(q=2)

  8. Assumptions ENERGY-DEPENDENT PARAMETERS

  9. THEOREM for Navier-Stokes-Fourier flows Under the assumptions then there exists a weak solution to the coupled system

  10. Bingham fluid [Duvaut & Lions, 1972] Transfert de chaleur dans un fluide de Bingham... (constant plasticity threshold, without convective terms, and DIRICHLET condition for fluid motion)

  11. The asymptotic limit case of a high diffusity [Ladyzhenskaya, 1970] New equation for description of motion ... [J.F. Rodrigues and i, 2003 & 2005] On stationary flows ...

  12. And so many other models … Taking the asymptotic limit of a high diffusity when it follows

  13. Fluids with shear thinning behaviour p =3/2 p =2

  14. non-Newtonian fluids POWER LAWS (Ostwald & de Waele) p>2: dilatant fluid 1<p<2: pseudo-plastic fluid p=1: Bingham fluid p=2: Navier-Stokes fluid

  15. (p-q) ASSUMPTIONS

  16. (p-q) relations n=3 under Dirichlet boundary condition and without Joule effect [2006] Math. Mod. and Meth. in Apppl. Sci. 16 :12, 2013--2027. http://dx.doi.org/10.1142/S0218202506001790

  17. Theorem Under the above assumptions then there exists a weak solution to the coupled system [2008] J. Math. Anal. Appl. 340 :1 (2008), 183--196. http://dx.doi.org/10.1016/j.jmaa.2007.07.080

  18. Open problem under the assumptions then there exists a weak solution? i believe YES!

  19. Greater range of (p-q) exponents ? as in the Dirichlet boundary value problem:

  20. 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 [2008] Annali Mat. Pura Appl. http://dx.doi.org/10.1007/s10231-007-0060-3

  21. Present goal

  22. Acknowledgement: Université de Pau et des Pays de L’Adour

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