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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Slender-body theory ： ウィキペディア英語版 Slender-body theory In fluid dynamics and electrostatics, slender-body theory is a methodology that can be used to take advantage of the slenderness of a body to obtain an approximation to a field surrounding it and/or the net effect of the field on the body. Principal applications are to Stokes flow — at very low Reynolds numbers — and in electrostatics. == Theory for Stokes flow == Consider slender body of length $\backslash ell$ and typical diameter $2a$ with $\backslash ell\; \backslash gg\; a$, surrounded by fluid of viscosity $\backslash mu$ whose motion is governed by the Stokes equations. Note that the Stokes' paradox implies that the limit of infinite aspect ratio $\backslash ell/a\; \backslash rightarrow\; \backslash infty$ is singular, as no Stokes flow can exist around an infinite cylinder. Slender-body theory allows us to derive an approximate relationship between the velocity of the body at each point along its length and the force per unit length experienced by the body at that point. Let the axis of the body be described by $\backslash boldsymbol(s,t)$, where $s$ is an arc-length coordinate, and $t$ is time. By virtue of the slenderness of the body, the force exerted on the fluid at the surface of the body may be approximated by a distribution of Stokeslets along the axis with force density $\backslash boldsymbol(s)$ per unit length. $\backslash boldsymbol$ is assumed to vary only over lengths much greater than $a$, and the fluid velocity at the surface adjacent to $\backslash boldsymbol(s,t)$ is well-approximated by $\backslash partial\backslash boldsymbol/\backslash partial\; t$. The fluid velocity $\backslash boldsymbol(\backslash boldsymbol)$ at a general point $\backslash boldsymbol$ due to such a distribution can be written in terms of an integral of the Oseen tensor (named after Carl Wilhelm Oseen), which acts as a Greens function for a single Stokeslet. We have : $\backslash boldsymbol(\backslash boldsymbol)\; =\; \backslash int\_0^\backslash ell\; \backslash frac\; \backslash cdot\; \backslash left(\; \backslash frac\; -\; \backslash boldsymbol|\}\; +\; \backslash frac)(\backslash boldsymbol\; -\; \backslash boldsymbol)\}|^3\}\; \backslash right)\; \backslash ,\; \backslash mathrms$ where $\backslash mathbf$ is the identity tensor. Asymptotic analysis can then be used to show that the leading-order contribution to the integral for a point $\backslash boldsymbol$ on the surface of the body adjacent to position $s\_0$ comes from the force distribution at $|s-\; s\_0|\; =\; O(a)$. Since $a\; \backslash ll\; \backslash ell$, we approximate $\backslash boldsymbol(s)\; \backslash approx\; \backslash boldsymbol(s\_0)$. We then obtain : $\backslash frac\; \backslash sim\; \backslash frac\; \backslash boldsymbol(s)\; \backslash cdot\; \backslash Bigl(\; \backslash mathbf\; +\; \backslash boldsymbol\text{'}\backslash boldsymbol\text{'}\; \backslash Bigr)$ where $\backslash boldsymbol\text{'}\; =\; \backslash partial\; \backslash boldsymbol/\backslash partial\; s$. The expression may be inverted to give the force density in terms of the motion of the body: : $\backslash boldsymbol(s)\; \backslash sim\; \backslash frac\; \backslash frac\; \backslash cdot\; \backslash Bigl(\; \backslash mathbf\; -\; \backslash textstyle\backslash frac\; \backslash boldsymbol\text{'}\backslash boldsymbol\text{'}\; \backslash Bigr)$ Two canonical results that follow immediately are for the drag force $F$ on a rigid cylinder (length $\backslash ell$, radius $a$) moving a velocity $u$ either parallel to its axis or perpendicular to it. The parallel case gives : $F\; \backslash sim\; \backslash frac$ while the perpendicular case gives : $F\; \backslash sim\; \backslash frac$ with only a factor of two difference. Note that the dominant length scale in the above expressions is the longer length $\backslash ell$; the shorter length has only a weak effect through the logarithm of the aspect ratio. In slender-body theory results, there are $O(1)$ corrections to the logarithm, so even for relatively large values of $\backslash ell/a$ the error terms will not be that small. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Slender-body theory」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.