Further Results on Steady-State Flow of a Navier-Stokes Liquid Around a Rigid Body. Existence of the Wake.(Kyoto Conference on the Navier-Stokes Equations and their Applications)
Further Results on Steady-State Flow of a Navier-Stokes Liquid Around a Rigid Body. Existence of the Wake.(Kyoto Conference on the Navier-Stokes Equations and their Applications)
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关于刚体周围纳维-斯托克斯液体稳态流动的进一步结果。
DOI：
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发表时间：
2007
期刊：
影响因子：
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通讯作者：
Ana L. Silverstre
中科院分区：
文献类型：
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作者：
G. Galdi;Ana L. Silverstre
A rigid body \mathcal{R} is moving in a Navier‐Stokes liquid \mathcal{L} that fills the whole space. We assume that all data with respect to a frame, \mathcal{F} , attached to \mathcal{R} , namely, the body force acting on \mathcal{L} , the boundary conditions on \mathcal{R} as well as the translational velocity, U , and the angular velocity, $\Omega$ , of \mathcal{R} are independent of time. We assume $\Omega$\neq 0 (the case $\Omega$=0 being already known) and take, without loss of generality, $\Omega$ parallel to the base vector e_{1} in \mathcal{F} . We show that, if the magnitude of these data is not too large, there exists at least one steady motion of \mathcal{L} in \mathcal{F} , such that the velocity field and its gradient decay like (1+|x|)^{-1}(1+2{\rm Re} s(x))^{-1} and (1+|x|)^{-\frac{3}{2}}(1+2{\rm Re} s(x))^{-\frac{3}{2}}, respectively, where {\rm Re} is the Reynolds number and s(x) :=|x|+x_{1} is representative of the wake behind the body. This motion is unique in the (larger) class of motions having velocity field decaying like |x|^{-1} . Since {\rm Re} is proportional to |U\cdot e_{1}| , the above formulas show that the \mathcal{L} exhibits a wake behind \mathcal{R} if and only if U is not orthogonal to $\Omega$.
A rigid body \mathcal{R} is moving in a Navier‐Stokes liquid \mathcal{L} that fills the whole space. We assume that all data with respect to a frame, \mathcal{F} , attached to \mathcal{R} , namely, the body force acting on \mathcal{L} , the boundary conditions on \mathcal{R} as well as the translational velocity, U , and the angular velocity, $\Omega$ , of \mathcal{R} are independent of time. We assume $\Omega$\neq 0 (the case $\Omega$=0 being already known) and take, without loss of generality, $\Omega$ parallel to the base vector e_{1} in \mathcal{F} . We show that, if the magnitude of these data is not too large, there exists at least one steady motion of \mathcal{L} in \mathcal{F} , such that the velocity field and its gradient decay like (1+|x|)^{-1}(1+2{\rm Re} s(x))^{-1} and (1+|x|)^{-\frac{3}{2}}(1+2{\rm Re} s(x))^{-\frac{3}{2}}, respectively, where {\rm Re} is the Reynolds number and s(x) :=|x|+x_{1} is representative of the wake behind the body. This motion is unique in the (larger) class of motions having velocity field decaying like |x|^{-1} . Since {\rm Re} is proportional to |U\cdot e_{1}| , the above formulas show that the \mathcal{L} exhibits a wake behind \mathcal{R} if and only if U is not orthogonal to $\Omega$.