Abstract
This paper is concerned with the instability and stability of the trivial steady states of the incompressible Navier–Stokes equations with Navier-slip boundary conditions in a slab domain in dimension two. The main results show that the stability (or instability) of this constant equilibrium depends crucially on whether the boundaries dissipate energy and the strengthen of the viscosity and slip length. It is shown that in the case that when all the boundaries are dissipative, then nonlinear asymptotic stability holds true. Otherwise, there is a sharp critical viscosity, which distinguishes the linear and nonlinear stability from instability.
Similar content being viewed by others
Change history
23 July 2022
A Correction to this paper has been published: https://doi.org/10.1007/s00021-022-00700-8
References
Achdou, Y., Pironneau, O., Valentin, F.: Effective boundary conditions for laminar flow over periodic rough boundaries. J. Comput. Phys. 147, 187–218 (1998)
Amrouche, C., Rejaiba, A.: \(L^p\)-theory for Stokes and Navier–Stokes equations with Navier boundary condition. J. Differ. Eqs. 256, 1515–1547 (2014)
Amrouche, C., Seloula, N.H.: On the Stokes equations with the Navier-type boundary conditions. Differ. Eqs. Appl. 3(4), 581–607 (2011)
Antontsev, S., de Oliveira, H.: Navier-Stokes equations with absorption under slip boundary conditions: existence, uniqueness and extinction in time. RIMS Kôkyûroku Bessatsu B1, 21–41 (2007)
Bänsch, E.: Finite element discretization of the Navier–Stokes equations with free capillary surface. Numer. Math. 88, 203–235 (2001)
Beavers, G., Joseph, D.: Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30, 197–207 (1967)
Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. International Series of Monographs on Physics. Clarendon Press, Oxford (1961)
Chauhan, D.S., Shekhawat, K.S.: Heat transfer in Couette flow of a compressible Newtonian fluid in the presence of a naturally permeable boundary. J. Phys. D Appl. Phys. 26, 933–936 (1993)
Drazin, P., Reid, W.: Hydrodynamic Stability, 2nd edn. Cambridge University Press, Cambridge (2004)
Gie, G.M., Kelliher, J.P.: Boundary layer analysis of the Navier–Stokes equations with generalized Navier boundary conditions. J. Differ. Eqs. 253, 1862–1892 (2012)
Guo, Y., Hallstrom, C., Spirn, D.: Dynamics near unstable, interfacial fluids. Commun. Math. Phys. 270, 635–689 (2007)
Guo, Y., Han, Y.: Critical Rayleigh number in Rayleigh–Bénard convection. Q. Appl. Math. 68(1), 149–160 (2010)
Guo, Y., Tice, I.: Linear Rayleigh–Taylor instability for viscous, compressible fluid. SIMA J. Math. Anal. 42(4), 1688–1720 (2010)
Guo, Y., Tice, I.: Compressible, inviscid Rayleigh–Taylor instability. Indiana Univ. Math. J. 60(2), 677–712 (2011)
Guo, Y., Tice, I.: Stability of contact line in fluids: 2D STOKES flow (2017). arXiv:1603.03721v1
Haase, A.S., Wood, J.A., Lammertink, R.G.H., Snoeijer, J.H.: Why bumpy is better: the role of the dissipaption distribution in slip flow over a bubble mattress. Phys. Rev. Fluid 1, 054101 (2016)
Jäger, W., Mikelić, A.: On the Roughness-induced effective boundary conditions for an incompressible viscous flow. J. Differ. Eqs. 170, 96–122 (2001)
Jäger, W., Mikelić, A.: On the interface boundary condition of Beavers, Joseph, and Saffman. SIAM J. Appl. Math. 60, 1111–1127 (2000)
Jiang, F., Jiang, S., Ni, G.: Nonlinear instability for nonhomogeneous incompressible viscous fluids. Sci. China Math. 56(4), 665–686 (2013)
Jiang, F., Jiang, S.: On instability and stability of three-dimensional gravity driven viscous flows in a boundary domain. Adv. Math. 264, 831–863 (2014)
John, V.: Slip with friction and penetration with resistance boundary conditions for the Navier–Stokes equation-numerical test and aspect of the implementation. J. Comput. Appl. Math. 147, 287–300 (2002)
Kelliher, J.P.: Navier–Stokes equations with Navier boundary conditions for a bounded domain in plane. SIAM J. Math. Anal. 38(1), 210–232 (2006)
Ladyzhenskaya, O.A.: Mathematical Theory of Viscous Incompressible Flow. Gordon, New York (1969)
Li, H., Zhang, X.: Stability of plane Couette flow for the compressible Navier–Stokes equations with Navier-slip boundary (2016). Preprint
Magnaudet, J., Riverot, M., Fabre, J.: Accelerated flows past a rigid sphere or a spherical bubble. Part 1. Steady straining flow. J. Fluid Mech. 284, 97–135 (1995)
Navier, C.: Sur les lois de léquilibre et du mouvement des corps élastiques. Mem. Acad. R. Sci. Inst. France 6, 369 (1827)
Qian, T., Wang, X., Sheng, P.: Molecular scale contact line hydrodynamics of immiscible flows. Phys. Rev. E 68, 016306 (2003)
Serrin, J.: Mathematical principles of classical fluid mechanics. In: Truesdell, C. (ed.) Fluid Dynamics I. Encyclopedia of Physics, pp. 125–263. Springer, Berlin (1959)
Solonnikov, V., Ščadilov, V.: A certain boundary value problem for the stationary system of Navier–Stokes equations. Trudy Mat. Inst. Steklov. 125, 196–210 (1973); translation in Proc. Steklov Inst. Math., 125, 1973, 186-199
Temam, R.: Navier–Stokes Equations, Studies in Mathematics and its Applications 2. North-Holland, Amsterdam (1984)
da Veiga, H.B.: On the regularity of flows with Ladyzhenskaya shear-dependent viscosity and slip or nonslip boundary conditions. Commun. Pure Appl. Math. LVIII, 552–577 (2005)
Wang, Y., Tice, I.: The viscous surface-internal wave problem: nonlinear Rayleigh–Taylor instability. Commun Partial Differ. Eqs. 37, 1967–2028 (2012)
Wang, Yanjin, Tice, Ian, Kim, Chanwoo: The viscous surface-internal wave problem: global well-posedness and decay. Arch. Ration. Mech. Anal. 212, 1–92 (2014)
Wang, Y., Xin, Z.: Vanishing viscosity and surface tension limits of incompressible viscous surface waves. arXiv:1504.00152
Xiao, Y., Xin, Z.: On the vanishing viscosity limit for the 3D Navier–Stokes equations with a slip boundary condition. Commun. Pure Appl. Math. 60, 1027–1055 (2007)
Xiao, Y., Xin, Z.: On the inviscid limit of the 3D Navier–Stokes equations with generalized Navier-slip boundary conditions. Commun. Math. Stat. 1(3), 259–279 (2013)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by G. P. Galdi
Rights and permissions
About this article
Cite this article
Ding, S., Li, Q. & Xin, Z. Stability Analysis for the Incompressible Navier–Stokes Equations with Navier Boundary Conditions. J. Math. Fluid Mech. 20, 603–629 (2018). https://doi.org/10.1007/s00021-017-0337-2
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00021-017-0337-2