Asymptotic behavior of the incompressible Navier-Stokes fluid with degree of freedom in porous medium

Abstract

The authors study the asymptotic behavior of the incompressible Navier-Stokes fluid with degree of freedom in the porous medium in ℝ ⁿ with n = 2 or 3. They derive the Darcy law as ε, the character size of the hole, tends to zero. Moreover, the authors obtain the expression of the degree of freedom from the homogenized model.

Keywords

Homogenization / Navier-Stokes fluid / Darcy law / Degree of freedom

References

[1] Sanchez-Palencia E.. Nonhomogeneous media and vibration theory, Lecture Notes in Physics, 1979, Amsterdam: Springer-Verlag, North Holland

[2] Keller J. B.. Darcy’s law for flow in porous media and the two-space method, Nonlinear Partial Differential Equations in Engineering and Applied Science. Proc. Conf., Univ. Rhode Island, Kingston. RI, 1979, 1980 429-443

[3] Bensoussan A., Lions J. L., Papanicolaou G.. Asymptotic analysis for periodic structures, Studies in Mathematics and Its Applications, 1978

[4] Tartar L.. Incompressible fluid flow in a porous medium convergence of the homogenization process, Appendix to Lecture Notes in Physics, 1980

[5] Allaire G.. Homogenization of the Stokes flow in a connected porous medium. Asymptot. Anal., 1989, 2: 203-222

[6] Allaire G.. Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes I-II. Arch. Rat. Mech. Anal., 1991, 113: 209-259

[7] Allaire G.. Homogenization of the Navier-Stokes equations with a slip boundary condition. Comm. Pure. Appl. Math., 1991, 44: 605-641

[8] Mikelic A.. Homogenization of nonstationary Navier-Stokes equations in a domain with a grained boundary. Ann. Mat. Pura et. appl., 1991, 158(4): 167-179

[9] Mikelic A., Aganovic I.. Homogenization of stationary of miscible fluids in a domain boundary. SIAM J. Math. Anal., 1988, 19: 287-294

[10] Masmoudi N.. Homogenization of the compressible Navier-Stokes equations in a porous medium. ESIAM Control Optim. Calc. Var., 2002, 8: 885-906

[11] Zhao H., Zheng-An Y.. Homogenization of the time discretized compressible Navier-Stokes system. Nonlinear Analysis, 2012, 75: 2486-2498

[12] Brinkman H. C.. A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res., 1947, A1: 27-34

[13] Lions P. L.. Mathematical Topics in Fluid Mechanics, 1996, New York: The Clarendon Press Oxford University Press

[14] Lipton R., Avellanda M.. Darcy’s law for slow viscous flow past a stationary array of bubbles. Proc. Roy. Soc. Edinbergh Sect. A, 1990, 114(1–2): 71-79

[15] Hornung U.. Homogenization and porous media, Interdisciplinary Applied Mathematics Series, 1997, New York: Springer-Verlag

[16] Temam R.. Navier-Stokes Equations, 1979, Amsterdam: North Holland

[17] Nečas J.. Sur les normes équivalentes dans W_p ^k(Ω) et sur la coercivité des formes formellement positives, Séminaire Equations aux Dérivées partielles, 1966 102-128