Boundary layers in incompressible Navier-Stokes equations with Navier boundary conditions for the vanishing viscosity limit
Boundary layers in incompressible Navier-Stokes equations with Navier boundary conditions for the vanishing viscosity limit
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不可压缩纳维-斯托克斯方程中的边界层，具有消失粘度极限的纳维边界条件
DOI：
10.4310/cms.2010.v8.n4.a10
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发表时间：
2010-12
期刊：
影响因子：
--
通讯作者：
Xiao-Ping Wang, Ya-Guang Wang, Zhouping Xin
中科院分区：
文献类型：
--
作者：
Xiao-Ping Wang, Ya-Guang Wang, Zhouping Xin
抽象的。本文研究了带摩擦边界条件的不可压Navier-Stokes方程的粘性消失极限。为了简化解的粘性展开式，我们只考虑Navier边界条件中滑移长度α是粘性函数的幂的情况，α= π。首先，通过多尺度分析，我们正式推导出γ= 1 2是决定边界层行为的关键。当γ > 1 2时，边界层出现在解的零阶项中，满足非线性Prandtl方程与无滑移情况下相同的边值问题;当γ= 1 2时，边界层也出现在解的零阶项中，满足非线性Prandtl方程，但切向速度剖面具有Robin边界条件。当γ < 1 2时，边界层以O（ψ）阶解出现，满足线性化Prandtl方程的边值问题。其次，当滑移长度大于垂向粘性的平方根时，利用能量方法严格证明了各向异性粘性不可压Navier-Stokes方程粘性消失极限的渐近性态。即使边界层出现在解的低阶项中，并满足线性问题，在粘性消失极限下，流动的涡度也是无界的。
Abstract. In this paper, we study the vanishing viscosity limit for the incompressible NavierStokes equations with the Navier friction boundary condition. To simplify the expansion of solutions in terms of the viscosity, we shall only consider the case that the slip length α in the Navier boundary condition is a power of the viscosity ǫ, α= ǫ . First, by multi-scale analysis we formally deduce that γ= 1 2 is critical in determining the boundary layer behavior. When γ > 1 2 , the boundary layer appears in the zero-th order terms of the expansion of solutions, and satisfies the same boundary value problem for the nonlinear Prandtl equations as in the non-slip case, when γ= 1 2 , the boundary layer also appears in the zero-th order terms of solutions, and satisfies the nonlinear Prandtl equations but with a Robin boundary condition for the tangential velocity profile, and when γ < 1 2 , the boundary layer appears in the order O(ǫ) terms of solutions, and satisfies a boundary value problem for the linearized Prandtl equations. Secondly, we justify rigorously the asymptotic behavior of the vanishing viscosity limit for the incompressible Navier-Stokes equations with anisotropic viscosities by using the energy method, when the slip length is larger than the square root of the vertical viscosity. Even though the boundary layer appears in the lower order terms of solutions and satisfies a linear problem, the vorticity of flow is unbounded in the vanishing viscosity limit.
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DOI：
10.1201/9780203749364
发表时间：
1999-05
期刊：
--
影响因子：
--
作者：
O. Oleinik;V. Samokhin
通讯作者：
O. Oleinik;V. Samokhin
DOI：
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发表时间：
2005
期刊：
SIAM J. Math. Anal.
影响因子：
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作者：
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通讯作者：
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影响因子：
3
作者：
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DOI：
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发表时间：
1961
期刊：
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影响因子：
--
作者：
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通讯作者：
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影响因子：
1.7
作者：
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