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- W3204602423 abstract "Bounded solutions of the stationary axially symmetric Navier-Stokes equations in an infinite pipe, equipped with Navier-slip boundary condition, are considered in this paper. Here smooth means the velocity is continuous up to second-order derivatives, and bounded means the velocity itself and its gradient field are bounded. It is shown that such solutions with zero flux at one cross section, must be swirling solutions: $u=(-Cx_2,Cx_1,0)$. A slight modification of the proof will show that for an alternative slip boundary condition, solutions will be identically zero. Meanwhile, if the horizontal swirl component of the axially symmetric solution, $u_theta$, is independent of the vertical variable $z$, it is proven that such solutions must be helical solutions: $u=(-C_1x_2,C_1x_1,C_2)$. In this case, boundedness assumptions on solutions can be relaxed extensively to the following growing conditions: With respect to the distance to the origin, the vertical component of the velocity, $u_z$, is sublinearly growing, the horizontal radial component of the velocity, $u_r$, is exponentially growing, and the swirl component of the vorticity, $omega_theta$, is polynomially growing at any order. Also, by constructing a counterexample, we show that the growing assumption on $u_r$ is optimal." @default.
- W3204602423 created "2021-10-11" @default.
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- W3204602423 date "2021-10-06" @default.
- W3204602423 modified "2023-09-27" @default.
- W3204602423 title "Characterization of bounded smooth solutions to the axially symmetric Navier-Stokes equations in an infinite pipe with Navier-slip boundary" @default.
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