Historical development of classical fluid dynamics ; 流体数理の古典理論

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 ...