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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 Bernoulli theorem ： ウィキペディア英語版 Bernoulli's principle In fluid dynamics, Bernoulli's principle states that for an inviscid flow of a non-conducting fluid, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy.〔Clancy, L.J., ''Aerodynamics'', Chapter 3.〕〔Batchelor, G.K. (1967), Section 3.5, pp. 156–64.〕 The principle is named after Daniel Bernoulli who published it in his book ''Hydrodynamica'' in 1738.〔(【引用サイトリンク】 title=Hydrodynamica )〕 Bernoulli's principle can be applied to various types of fluid flow, resulting in what is loosely denoted as Bernoulli's equation. In fact, there are different forms of the Bernoulli equation for different types of flow. The simple form of Bernoulli's principle is valid for incompressible flows (e.g. most liquid flows and gases moving at low Mach number). More advanced forms may in some cases be applied to compressible flows at higher Mach numbers (see the derivations of the Bernoulli equation). Bernoulli's principle can be derived from the principle of conservation of energy. This states that, in a steady flow, the sum of all forms of energy in a fluid along a streamline is the same at all points on that streamline. This requires that the sum of kinetic energy, potential energy and internal energy remains constant.〔 Thus an increase in the speed of the fluid – implying an increase in both its dynamic pressure and kinetic energy – occurs with a simultaneous decrease in (the sum of) its static pressure, potential energy and internal energy. If the fluid is flowing out of a reservoir, the sum of all forms of energy is the same on all streamlines because in a reservoir the energy per unit volume (the sum of pressure and gravitational potential ''ρ g h'') is the same everywhere.〔Streeter, V.L., ''Fluid Mechanics'', Example 3.5, McGraw–Hill Inc. (1966), New York.〕 Bernoulli's principle can also be derived directly from Newton's 2nd law. If a small volume of fluid is flowing horizontally from a region of high pressure to a region of low pressure, then there is more pressure behind than in front. This gives a net force on the volume, accelerating it along the streamline.〔"If the particle is in a region of varying pressure (a non-vanishing pressure gradient in the x-direction) and if the particle has a finite size l, then the front of the particle will be ‘seeing’ a different pressure from the rear. More precisely, if the pressure drops in the x-direction (dp/dx < 0) the pressure at the rear is higher than at the front and the particle experiences a (positive) net force. According to Newton’s second law, this force causes an acceleration and the particle’s velocity increases as it moves along the streamline... Bernoulli's equation describes this mathematically (see the complete derivation in the appendix)."〕〔"Acceleration of air is caused by pressure gradients. Air is accelerated in direction of the velocity if the pressure goes down. Thus the decrease of pressure is the cause of a higher velocity." 〕〔" The idea is that as the parcel moves along, following a streamline, as it moves into an area of higher pressure there will be higher pressure ahead (higher than the pressure behind) and this will exert a force on the parcel, slowing it down. Conversely if the parcel is moving into a region of lower pressure, there will be an higher pressure behind it (higher than the pressure ahead), speeding it up. As always, any unbalanced force will cause a change in momentum (and velocity), as required by Newton’s laws of motion." ''See How It Flies'' John S. Denker http://www.av8n.com/how/htm/airfoils.html〕 Fluid particles are subject only to pressure and their own weight. If a fluid is flowing horizontally and along a section of a streamline, where the speed increases it can only be because the fluid on that section has moved from a region of higher pressure to a region of lower pressure; and if its speed decreases, it can only be because it has moved from a region of lower pressure to a region of higher pressure. Consequently, within a fluid flowing horizontally, the highest speed occurs where the pressure is lowest, and the lowest speed occurs where the pressure is highest.〔Resnick, R. and Halliday, D. (1960), section 18-4, ''Physics'', John Wiley & Sons, Inc.〕 == Incompressible flow equation == In most flows of liquids, and of gases at low Mach number, the density of a fluid parcel can be considered to be constant, regardless of pressure variations in the flow. Therefore, the fluid can be considered to be incompressible and these flows are called incompressible flow. Bernoulli performed his experiments on liquids, so his equation in its original form is valid only for incompressible flow. A common form of Bernoulli's equation, valid at any arbitrary point along a streamline, is: where: :$v\backslash ,$ is the fluid flow speed at a point on a streamline, :$g\backslash ,$ is the acceleration due to gravity, :$z\backslash ,$ is the elevation of the point above a reference plane, with the positive ''z''-direction pointing upward – so in the direction opposite to the gravitational acceleration, :$p\backslash ,$ is the pressure at the chosen point, and :$\backslash rho\backslash ,$ is the density of the fluid at all points in the fluid. For conservative force fields, Bernoulli's equation can be generalized as:〔Batchelor, G.K. (1967), §5.1, p. 265.〕 :$+\backslash Psi+=\backslash text$ where ''Ψ'' is the force potential at the point considered on the streamline. ''E.g.'' for the Earth's gravity ''Ψ'' = ''gz''. The following two assumptions must be met for this Bernoulli equation to apply:〔 * the flow must be incompressible – even though pressure varies, the density must remain constant along a streamline; * friction by viscous forces has to be negligible. By multiplying with the fluid density $\backslash rho$, equation () can be rewritten as: :$$ \tfrac12\, \rho\, v^2\, +\, \rho\, g\, z\, +\, p\, =\, \text\, or: :$$ q\, +\, \rho\, g\, h\, =\, p_0\, +\, \rho\, g\, z\, =\, \text\, where: :$q\backslash ,\; =\backslash ,\; \backslash tfrac12\backslash ,\; \backslash rho\backslash ,\; v^2$ is dynamic pressure, :$h\backslash ,\; =\backslash ,\; z\backslash ,\; +\backslash ,\; \backslash frac$ is the piezometric head or hydraulic head (the sum of the elevation ''z'' and the pressure head)〔, 410 pages. See pp. 43–44.〕〔, 650 pages. See p. 22.〕 and :$p\_0\backslash ,\; =\backslash ,\; p\backslash ,\; +\backslash ,\; q\backslash ,$ is the total pressure (the sum of the static pressure ''p'' and dynamic pressure ''q''). The constant in the Bernoulli equation can be normalised. A common approach is in terms of total head or energy head ''H'': :$H\backslash ,\; =\backslash ,\; z\backslash ,\; +\backslash ,\; \backslash frac\backslash ,\; +\backslash ,\; \backslash frac\backslash ,\; =\backslash ,\; h\backslash ,\; +\backslash ,\; \backslash frac,$ The above equations suggest there is a flow speed at which pressure is zero, and at even higher speeds the pressure is negative. Most often, gases and liquids are not capable of negative absolute pressure, or even zero pressure, so clearly Bernoulli's equation ceases to be valid before zero pressure is reached. In liquids – when the pressure becomes too low – cavitation occurs. The above equations use a linear relationship between flow speed squared and pressure. At higher flow speeds in gases, or for sound waves in liquid, the changes in mass density become significant so that the assumption of constant density is invalid. === Simplified form === In many applications of Bernoulli's equation, the change in the ''ρ'' ''g'' ''z'' term along the streamline is so small compared with the other terms that it can be ignored. For example, in the case of aircraft in flight, the change in height ''z'' along a streamline is so small the ''ρ'' ''g'' ''z'' term can be omitted. This allows the above equation to be presented in the following simplified form: :$p\; +\; q\; =\; p\_0\backslash ,$ where ''p''0 is called 'total pressure', and ''q'' is 'dynamic pressure'.〔(【引用サイトリンク】 publisher = NASA Glenn Research Center )〕 Many authors refer to the pressure ''p'' as static pressure to distinguish it from total pressure ''p''0 and dynamic pressure ''q''. In ''Aerodynamics'', L.J. Clancy writes: "To distinguish it from the total and dynamic pressures, the actual pressure of the fluid, which is associated not with its motion but with its state, is often referred to as the static pressure, but where the term pressure alone is used it refers to this static pressure."〔Clancy, L.J., ''Aerodynamics'', Section 3.5.〕 The simplified form of Bernoulli's equation can be summarized in the following memorable word equation: :''static pressure + dynamic pressure = total pressure''〔 Every point in a steadily flowing fluid, regardless of the fluid speed at that point, has its own unique static pressure ''p'' and dynamic pressure ''q''. Their sum ''p'' + ''q'' is defined to be the total pressure ''p''0. The significance of Bernoulli's principle can now be summarized as ''total pressure is constant along a streamline.'' If the fluid flow is irrotational, the total pressure on every streamline is the same and Bernoulli's principle can be summarized as ''total pressure is constant everywhere in the fluid flow.''〔Clancy, L.J. ''Aerodynamics'', Equation 3.12〕 It is reasonable to assume that irrotational flow exists in any situation where a large body of fluid is flowing past a solid body. Examples are aircraft in flight, and ships moving in open bodies of water. However, it is important to remember that Bernoulli's principle does not apply in the boundary layer or in fluid flow through long pipes. If the fluid flow at some point along a stream line is brought to rest, this point is called a stagnation point, and at this point the total pressure is equal to the stagnation pressure. 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「Bernoulli's principle」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.