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Historical development of classical fluid dynamics
http://hdl.handle.net/10748/4129
http://hdl.handle.net/10748/4129757b43ee-c2a1-4185-a3f7-0f69666721ab
| 名前 / ファイル | ライセンス | アクション |
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10260-001.pdf (39.8 MB)
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| Item type | 学位論文 / Thesis or Dissertation(1) | |||||||||||
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| 公開日 | 2011-09-28 | |||||||||||
| タイトル | ||||||||||||
| タイトル | Historical development of classical fluid dynamics | |||||||||||
| 言語 | en | |||||||||||
| 言語 | ||||||||||||
| 言語 | eng | |||||||||||
| キーワード | ||||||||||||
| 主題Scheme | Other | |||||||||||
| 主題 | Exact differential | |||||||||||
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| 主題Scheme | Other | |||||||||||
| 主題 | complete differential | |||||||||||
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| 主題Scheme | Other | |||||||||||
| 主題 | fluid dynamics | |||||||||||
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| 主題Scheme | Other | |||||||||||
| 主題 | fluid mechanics | |||||||||||
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| 主題Scheme | Other | |||||||||||
| 主題 | microscopic allydescriptive equations | |||||||||||
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| 主題Scheme | Other | |||||||||||
| 主題 | hydrostatics | |||||||||||
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| 主題Scheme | Other | |||||||||||
| 主題 | hydrodynamics | |||||||||||
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| 主題Scheme | Other | |||||||||||
| 主題 | hydromechanics | |||||||||||
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| 主題Scheme | Other | |||||||||||
| 主題 | mathematical history | |||||||||||
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| 主題Scheme | Other | |||||||||||
| 主題 | The Navier-Stokes equations | |||||||||||
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| 主題Scheme | Other | |||||||||||
| 主題 | two-constant theory | |||||||||||
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| 主題Scheme | Other | |||||||||||
| 主題 | tensor function | |||||||||||
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| 主題Scheme | Other | |||||||||||
| 主題 | rapidly decreasing function | |||||||||||
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| 主題Scheme | Other | |||||||||||
| 主題 | The Boltzmann equations | |||||||||||
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| 主題Scheme | Other | |||||||||||
| 主題 | the transport equations | |||||||||||
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| 主題Scheme | Other | |||||||||||
| 主題 | gas theory | |||||||||||
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| 主題Scheme | Other | |||||||||||
| 主題 | weak solution | |||||||||||
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| 主題Scheme | Other | |||||||||||
| 主題 | strong solution | |||||||||||
| 資源タイプ | ||||||||||||
| 資源タイプ識別子 | http://purl.org/coar/resource_type/c_46ec | |||||||||||
| 資源タイプ | thesis | |||||||||||
| 著者 |
増田, 茂
× 増田, 茂
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| 抄録 | ||||||||||||
| 内容記述タイプ | Abstract | |||||||||||
| 内容記述 | Part 1. Exact differentials in fluid dynamics are important quantities in any mathematical analysis of continuous systems; for example, we may need to know if udx + vdy + wdz satisfies exact, or equivalently complete, differentiability in three dimensions. In the hands of d'Alembert, Euler, Lagrange, Laplace, Cauchy, Poisson and Stokes, these practitioners have succeeded in developing its theoretical consequences. From the geometric point of view, Gauss and Riemann had applied such constructs, while Helmholtz and W. Thomson applied these to the theory of vortices. Although Helmholtz's vorticity equation was strongly criticized by Bertrand, Saint-Venant sided with Helmholtz. Here, we would like to review from the historical viewpoint the study of exact differential in fluid mechanics. In §2, we present proofs of the eternal existence of unique exact differentials by L agrange, Cauchy and Stokes. From a separate development, the formulation of the two-constant theory in equilibrium/motion had been deduced by Navier, Poisson, Cauchy, Saint-Venant and Stokes. Today's Navier-Stokes equations were formulated and used in practice. An up-to-the present study is given in papers to follow. Part 2. The “two-constant” theory introduced first by Laplace in 1805 still forms the basis of current theory describing isotropic, linear elasticity. The Navier-Stokes equations in incompressible case ∂_tu - μΔu + u・∇u + ∇p=f, div u=0. as presented in final form by Stokes in 1845, were derived in the course of the development of the “twoconstant” theory. Following in historical order the various contributions of Navier, Cauchy, Poisson, Saint-Venant and Stokes over the intervening period, we trace the evolution of the equations, and note concordances and differences between each contributor. In particular, from the historical perspective of these equations we look for evidence for the notion of tensor. Also in the formulation of equilibrium equations, we obtain the competing theories of the “twoconstant” theory in capillary action of Laplace and Gauss. After Stokes' linear equations, the equations of gas theories were deduced by Maxwell in 1865, Kirchhoff in 1868 and Boltzmann in 1872. They contributed to formulate the fluid equations and to fix the NS equations, when Prandtl stated the today's formulation in using the nomenclature as the “socalled NS equations” in 1934, in which Prandtl included the three terms of nonlinear and two linear terms with the ratio of two coefficients as 3 : 1, which arose Poisson in 1831, Saint-Venant in 1843, and Stokes in 1845. Prandtl says, “The following differential equation, known as the equation of Navier-Stokes, is the fundamental equation of hydrodynamics,”Dw/dt = g - 1/pgrad p + 1/3ν grad div Δw+νΔw, where, Dw/dt = ∂w/∂t + w・∇w, ν = μ/p, w = (u, v, w), g = (X, Y, Z) In the appendices, we show the process of formulation citing their main papers of Navier, Cauchy, Poisson, Laplace and Gauss with our commentary. In addition to, from the viewpoint of mathematics, several important topics such as integral theory in §E.17 and §E.23 which is Gauss' selling point. We show his unique RDF and reduction of integral from sextuple to quadruple, in the sections §E.2, §E.16 and §E.17. In and after §E.18, we show his calculus of variations in the capillarity against the RDF and calculation of the capillarity by Laplace. Finally, for the question to be solved by variational equation introduced in §E.18 and §E.19, we sketch his method deduced from the previous work of theory in curved surface [15], to the capillary problems including the height of fluid and the tangent angle made between the fluid surface and the wall in §E.28 and §E.29. Part 3. The microscopically-description of hydromechanics equations are followed by the description of equations of gas theory by Maxwell, Kirchhoff and Boltzmann. Above all, in 1872, Boltzmann formulated the Boltzmann equations, expressed by the following today's formulation : ∂_tf + v・∇_xf = Q(f, 9), t > 0, x, v ∈ R^n(n ≥ 3), x = (x, y, z), v =(ξ, η, ζ) , (1) Q(f,g)(t, x, v) = ∫_<R^3>∫_<s^2> B(v— v_*, σ) {g(v´_*)f(v´) - g(v_*)f(v)}dσdv_*, g(v´_*) = g(t, x, v´_*), etc. (2) These equations are able to be reduced for the general form of the hydrodynamic equations, after the formulations by Maxwell and Kirchhoff, and from which the Euler equations and the Navier-Stokes equations are reduced as the special case. After Stokes' linear equations, the equations of gas theories were deduced by Maxwell in 1865, Kirchhoff in 1868 and Boltzmann in 1872, They contributed to formulate the fluid equations and to fix the Navier-Stokes equations, when Prandtl stated the today's formulation in using the nomenclature as the “so-called Navier-Stokes equations” in 1934, in which Prandtl included the three terms of nonlinear and two linear terms with the ratio of two coefficients as 3 : 1, which arose from Poisson in 1831, Saint-Venant in 1843, and Stokes in 1845. Part 4. After the NS equations were fixed or so, many researchers of hydrodynamics studied the mathematical analyses, in particular, the functional analysis on the solutions of the NS equations. From the viewpoint of the mathematics, the full-scale studies have been begun to the weak solutions by Leray [12, 13, 14] in 1933/34 and by Hopf [4] in 1950/51. And soon after that, A.A.Kiselev [5, 7] in 1954/55, and Kiselev and Ladyzhenskaya [8] in 1957 and Ladyzhenskaya [11] in 1959 constructed the generalized solutions / the strong solutions. Prodi [23] and J.L.Lions [15] discussed the uniqueness of the solution of the NS equations in the three dimensions. We sketch these historical facts at the beginning of the study on the solutions of the NS equations. Finally, we show two sort of translations into English on the solutions of the NS equations , viz. ・ from Hopf's German paper [4] only on the existence of a weak solution like Leray ・ from Ladyzhenskaya's Russian paper [11] of a generalized / strong solution like Kiselev in the first time We think that both are notable and full-scale studies not only of the NS equations or of the mathematical history, but also of the pure mathematics like functional analysis. | |||||||||||
| 内容記述 | ||||||||||||
| 内容記述タイプ | Other | |||||||||||
| 内容記述 | 首都大学東京, 2011-03-25, 博士（理学） | |||||||||||
| 書誌情報 |
発行日 2011-03-25 |
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| 著者版フラグ | ||||||||||||
| 出版タイプ | VoR | |||||||||||
| 出版タイプResource | http://purl.org/coar/version/c_970fb48d4fbd8a85 | |||||||||||
| その他のタイトル | ||||||||||||
| その他のタイトル | 流体数理の古典理論 | |||||||||||
| 言語 | ja | |||||||||||
| 国立国会図書館分類 | ||||||||||||
| 主題Scheme | NDLC | |||||||||||
| 主題 | UT51 | |||||||||||
