无限长收缩通道中的普朗特边界层
Prandtl Boundary Layers in An Infinitely Long Convergent Channel
AI总结：
本文证明了在曲率递减的无限长收缩喷管中，小黏性Navier-Stokes流可由Euler流与Prandtl流叠加一致逼近，并揭示了质量通量对顶点及无穷远处奇异渐近行为的决定作用。
摘要：
本文研究二维稳态Navier-Stokes方程在无限长收缩通道中大Reynolds数极限与解的渐近行为。研究表明，对于一般的收缩型无限长喷管，当其边界曲线满足曲率递减条件且给定任意有限的负质量通量时，Prandtl黏性边界层理论成立：即存在满足无滑移边界条件的小黏性Navier-Stokes流，该流动可由Euler流与Prandtl流的叠加一致逼近。此外，Navier-Stokes方程的解在喷管顶点附近及无穷远处的奇异渐近行为由给定的质量通量决定，这一点对Prandtl近似解的构造也至关重要，因为顶点处可能存在奇异性且喷管具有非紧性。我们分析中的一个关键要素是：收缩喷管边界曲线上的曲率递减条件保证了极限无黏流动具有顺压梯度，并在Prandtl展开和稳定性分析中都发挥了关键作用。
英文摘要：
This paper concerns the large Reynold number limits and asymptotic behaviors of solutions to the 2D steady Navier-Stokes equations in an infinitely long convergent channel. It is shown that for a general convergent infinitely long nozzle whose boundary curves satisfy curvature-decreasing and any given finite negative mass flux, the Prandtl's viscous boundary layer theory holds in the sense that there exists a Navier-Stokes flow with no-slip boundary condition for small viscosity, which is approximated uniformly by the superposition of an Euler flow and a Prandtl flow. Moreover, the singular asymptotic behaviors of the solution to the Navier-Stokes equations near the vertex of the nozzle and at infinity are determined by the given mass flux, which is also important for the constructions of the Prandtl approximation solution due to the possible singularities at the vertex and non-compactness of the nozzle. One of the key ingredients in our analysis is that the curvature-decreasing condition on boundary curves of the convergent nozzle ensures that the limiting inviscid flow is pressure favourable and plays crucial roles in both the Prandtl expansion and the stability analysis.
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