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Slip Boundary Conditions in Thermal Viscous Incompressible Flows: Existence and Challenges

This study explores the formulation and implications of slip boundary conditions in thermal viscous incompressible flows. It discusses the governing equations, existence results, and open problems associated with the Navier slip conditions and their inadequacies. The analysis includes energy equations and constitutive laws for heat transfer, particularly in Bingham fluids and shear-thinning fluids. The paper aims to elucidate the weak solutions of coupled systems under various assumptions, providing insight into the complexities of boundary value problems in fluid dynamics.

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Slip Boundary Conditions in Thermal Viscous Incompressible Flows: Existence and Challenges

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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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