Bernoulli's principle
label: Bernoulliren printzipio @eu
label: Théorème de Bernoulli @fr
label: Bernoulli-Gleichung @de
label: Wet van Bernoulli @nl
label: Закон Бернулли @ru
label: مبدأ برنولي @ar
label: Νόμος του Μπερνούλι @el
label: Bernoullis ekvation @sv
label: Prionsabal Bernoulli @ga
label: Закон Бернуллі @uk
label: 베르누이 방정식 @ko
label: Principi de Bernoulli @ca
label: Equazione di Bernoulli @it
label: ベルヌーイの定理 @ja
label: Równanie Bernoulliego @pl
label: Prinsip Bernoulli @in
label: Principio de Bernoulli @es
label: Princípio de Bernoulli @pt
label: 伯努利定律 @zh
label: Bernoulliho rovnice @cs
subject: Category:Fluid dynamics
subject: Category:1738 in science
subject: Category:Eponymous laws of physics
subject: Category:Equations of fluid dynamics
proof: The derivation for compressible fluids is similar. Again, the derivation depends upon conservation of mass, and conservation of energy. Conservation of mass implies that in the above figure, in the interval of time , the amount of mass passing through the boundary defined by the area is equal to the amount of mass passing outwards through the boundary defined by the area : Conservation of energy is applied in a similar manner: It is assumed that the change in energy of the volume of the streamtube bounded by and is due entirely to energy entering or leaving through one or the other of these two boundaries. Clearly, in a more complicated situation such as a fluid flow coupled with radiation, such conditions are not met. Nevertheless, assuming this to be the case and assuming the flow is steady so that the net change in the energy is zero, where and are the energy entering through and leaving through , respectively. The energy entering through is the sum of the kinetic energy entering, the energy entering in the form of potential gravitational energy of the fluid, the fluid thermodynamic internal energy per unit of mass entering, and the energy entering in the form of mechanical work: where is a force potential due to the Earth's gravity, is acceleration due to gravity, and is elevation above a reference plane. A similar expression for may easily be constructed. So now setting : which can be rewritten as: Now, using the previously-obtained result from conservation of mass, this may be simplified to obtain which is the Bernoulli equation for compressible flow. An equivalent expression can be written in terms of fluid enthalpy :
proof: The Bernoulli equation for incompressible fluids can be derived by either integrating Newton's second law of motion or by applying the law of conservation of energy, ignoring viscosity, compressibility, and thermal effects. ; Derivation by integrating Newton's second law of motion The simplest derivation is to first ignore gravity and consider constrictions and expansions in pipes that are otherwise straight, as seen in Venturi effect. Let the axis be directed down the axis of the pipe. Define a parcel of fluid moving through a pipe with cross-sectional area , the length of the parcel is , and the volume of the parcel . If mass density is , the mass of the parcel is density multiplied by its volume . The change in pressure over distance is and flow velocity . Apply Newton's second law of motion and recognizing that the effective force on the parcel of fluid is . If the pressure decreases along the length of the pipe, is negative but the force resulting in flow is positive along the axis. In steady flow the velocity field is constant with respect to time, , so itself is not directly a function of time . It is only when the parcel moves through that the cross sectional area changes: depends on only through the cross-sectional position . With density constant, the equation of motion can be written as by integrating with respect to where is a constant, sometimes referred to as the Bernoulli constant. It is not a universal constant, but rather a constant of a particular fluid system. The deduction is: where the speed is large, pressure is low and vice versa. In the above derivation, no external work–energy principle is invoked. Rather, Bernoulli's principle was derived by a simple manipulation of Newton's second law. thumb|center|600px|A streamtube of fluid moving to the right. Indicated are pressure, elevation, flow speed, distance (), and cross-sectional area. Note that in this figure elevation is denoted as , contrary to the text where it is given by . ; Derivation by…
text: Static pressure + Dynamic pressure = Total pressure.
description: natuurkundige wetmatigheid @nl
description: משוואה בהידרודינמיקה ובאווירונאוטיקה @iw
description: comportamiento de una partícula fluida moviéndose a lo largo de líneas de corriente @es
description: এর গতিবিদ্যা সমন্ধের নীতি @bn
description: fysiikan laki, joka liittyy fluidin (nesteen tai kaasun) virtauksiin @fi
description: اصل مربوط به دینامیک سیالات @fa
description: fizikalni zakon, ki povezuje tlak in hitrost v tekoči tekočini @sl
description: Equazione descrivente le caratteristiche del moto di un fluido inviscido @it
description: द्रव गतिशीलतेशी संबंधित तत्त्व @mr
description: Grundgleichung der Strömungsmechanik @de
description: 瑞士流體物理學家丹尼爾·伯努利於1738年出版他的理論 @zh
description: jurnal percobaan bernoulli @in
description: principi que descriu el comportament d'un líquid que es mou al llarg d'un corrent d'aigua @ca
description: физический закон, связывающий давление и скорость в текущей жидкости @ru
description: równanie hydrodynamiki @pl
description: théorème en dynamique des fluides @fr
description: principle relating to fluid dynamics
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