On the regularity of shear thickening viscous fluids

Abstract

The aim of this note is to improve the regularity results obtained by H. Beirão da Veiga in 2008 for a class of p-fluid flows in a cubic domain. The key idea is exploiting the better regularity of solutions in the tangential directions with respect to the normal one, by appealing to anisotropic Sobolev embeddings.

Keywords

Non-Newtonian fluids / Shear dependent viscosity / Boundary regularity

References

[1] Acerbi E., Fusco N.. Partial regularity under anisotropic (p, q) growth conditions. J. Diff. Eqs., 1994, 107(1): 46-67

[2] Astarita G., Marucci G.. Principles of Non-Newtonian Fluid Mechanics, 1974, London: McGraw-Hill

[3] Beirão da Veiga H.. On the regularity of flows with Ladyzhenskaya shear dependent viscosity and slip or non-slip boundary conditions. Comm. Pure Appl. Math., 2005, 58: 552-577

[4] Beirão da Veiga, H., Navier-Stokes equations with shear thickening viscosity, regularity up to the boundary, J. Math. Fluid Mech. DOI: 10.1007/s00021-008-0257-2

[5] Beirão da Veiga, H., Navier-Stokes equations with shear thinning viscosity, regularity up to the boundary, J. Math. Fluid Mech. DOI: 10.1007/s00021-008-0258-1

[6] Beirão da Veiga H.. On the Ladyzhenskaya-Smagorinsky turbulence model of the Navier-Stokes equations in smooth domains, the regularity problem. J. Eur. Math. Soc., 2009, 11: 127-167

[7] Beirão da Veiga H.. Concerning the Ladyzhenskaya-Smagorinsky turbulence model of the Navier-Stokes equations. CR Acad. Sci. Paris, 2007, 345(1): 249-252

[8] Beirão da Veiga H.. On the global regularity of shear-thinning flows in smooth domains. J. Math. Anal. Appl., 2009, 349: 335-360

[9] Berselli, L. C., On the W^2,q-regularity of incompressible fluids with shear-dependent viscosities: The shear thinning case, J. Math. Fluid Mech. DOI: 10.1007/s00021-008-0254-5

[10] Crispo F.. Shear thinning viscous fluids in cylindrical domains. Regularity up to the boundary. J. Math. Flui Mech., 2008, 10: 311-325

[11] Crispo F.. Global regularity of a class of p-fluid flows in cylinders. J. Math. Anal. Appl., 2008, 341: 559-574

[12] Fuchs M., Seregin G.. Variational Methods for Problems from Plasticity Theory and for Generalized Newtonian Fluids, Lecture Notes in Mathematics, 1749, 2000, Berlin: Springer-Verlag

[13] Galdi G. P. Galdi G. P., Robertson A. M., Rannacher R. Mathematical problems in classical and non-Newtonian fluid mechanics, Hemodynamical Flows: Modeling, Analysis and Simulation (Oberwolfach Seminars), 2007, Basel: Birkhaeuser-Verlag

[14] Lions J. L.. Quelques Méthodes de Résolution des Problèmes aux Limites Non-linéaires, 1969, Paris: Dunod and Gauthier-Villars

[15] Málek J., Nečas J., Rokyta M., Růžička M.. Weak and measure-valued solutions to evolutionary PDEs, 1996, London: Chapman and Hall

[16] Málek J., Nečas J., Růžička M.. On the Non-Newtonian Incompressible Fluids. Math. Models Methods Appl. Sci., 1993, 3: 35-63

[17] Nečas J.. équations aux Dérivées Partielles, 1965, Montréal: Presses de l’Université de Montréal

[18] Parés C.. Existence, uniqueness and regularity of solution of the equations of a turbulence model for incompressible fluids. Appl. Anal., 1992, 43(3–4): 245-296

[19] Rajagopal, K. R., Mechanics of non-newtonian fluids, Recent Developments in Theoretical FluidMechanics, G. P. Galdi and J. Nečas (eds.), Pitman Research Notes in Mathematics Series, 291, Longman, 1993, 129–162.

[20] Troisi M.. Teoremi di inclusione per spazi di Sobolev non isotropi. Ricerche Mat., 1969, 18: 3-24