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In physics, the Navier–Stokes equations , named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of viscous fluid substances. These balance equations arise from applying Newton's second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term—hence describing ''viscous flow''. The main difference between them and the simpler Euler equations for inviscid flow is that Navier–Stokes equations also in the Froude limit (no external field) are not conservation equations, but rather a dissipative system, in the sense that they cannot be put into the quasilinear homogeneous form: : Navier–Stokes equations are useful because they describe the physics of many things of scientific and engineering interest. They may be used to model the weather, ocean currents, water flow in a pipe and air flow around a wing. The Navier–Stokes equations in their full and simplified forms help with the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of pollution, and many other things. Coupled with Maxwell's equations they can be used to model and study magnetohydrodynamics. The Navier–Stokes equations are also of great interest in a purely mathematical sense. Somewhat surprisingly, given their wide range of practical uses, it has not yet been proven that in three dimensions solutions always exist (existence), or that if they do exist, then they do not contain any singularity (they are smooth). These are called the Navier–Stokes existence and smoothness problems. The Clay Mathematics Institute has called this one of the seven most important open problems in mathematics and has offered a US$1,000,000 prize for a solution or a counter-example.〔 〕 ==Flow velocity== The solution of the Navier–Stokes equations is a flow velocity. It is a field, since it is defined at every point in a region of space and an interval of time. Once the velocity field is calculated, other quantities of interest, such as pressure or temperature, may be found. This is different from what one normally sees in classical mechanics, where solutions are typically trajectories of position of a particle or deflection of a continuum. Studying velocity instead of position makes more sense for a fluid; however for visualization purposes one can compute various trajectories. 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「Navier–Stokes equations」の詳細全文を読む スポンサード リンク
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