Historical development of classical fluid dynamics

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 ...