Naudotojas:Laws of physics/Bernulio dėsnis

Skysčių dinamikoje Bernulio dėsnis teigia , kad padidėjus skysčio tekėjimo greičiui proporcingai nukrenta skysčio slėgis , kitais žodžiai nukrenta skysčio potencinė energija. Dėsnis pavadintas olandų fiziko , matematiko Danieliaus Bernulio garbei , kuris dėsnį apraše savo knygoje  Hydrodynamica 1738m.

Bernulio dėsnis, gali būti taikomas ivairių skysčių srauto matavime, todėl yra daugelis Bernulio lygčių; yra įvairių Bernulio lygčių matuoti skirtingus srautus. Paprasta forma Bernulio lygtys galioja incompressible srautus (pvz., dauguma skysčių  ir dujos juda ). Labiau išsivysčiusias formas gali būti taikomas compressible srautų aukštojo Macho skaičius (žr. į daiktavardžiai, Bernulio lygtis).

v - skysčio srauto greitis taške-supaprastinti,

g yra pagreičio dėl svorio,

z yra aukštis nuo tos vietos, virš atskaitos plokštumos, su teigiamu z-kryptimi nukreipta aukštyn – taigi, priešinga gravitacijos pagreitis,

p - slėgis pasirinktą tašką, ir

ρ - tankis, skystis ne visi taškai skystyje.

Bernoulli equation for incompressible fluids

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 between two sections along a streamline, ignoring viscosity, compressibility, and thermal effects. Derivation through 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 x axis be directed down the axis of the pipe. Define a parcel of fluid moving through a pipe with cross-sectional area A, the length of the parcel is dxdx, and the volume of the parcel A dxA dx. If mass density is ρ, the mass of the parcel is density multiplied by its volume m = ρA dxm = ρA dx. The change in pressure over distance dxdx is dpdp and flow velocity v = dx/dtv = dx/dt. Apply Newton's second law of motion (force = mass × acceleration) and recognizing that the effective force on the parcel of fluid is −A dp−A dp. If the pressure decreases along the length of the pipe, dpdp is negative but the force resulting in flow is positive along the x axis. ${\displaystyle {\begin{aligned}m{\frac {\mathrm {d} v}{\mathrm {d} t}}&=F\\\rho A\mathrm {d} x{\frac {\mathrm {d} v}{\mathrm {d} t}}&=-A\mathrm {d} p\\\rho {\frac {\mathrm {d} v}{\mathrm {d} t}}&=-{\frac {\mathrm {d} p}{\mathrm {d} x}}\end{aligned}}}$ In steady flow the velocity field is constant with respect to time, v = v(x) = v(x(t)), so v itself is not directly a function of time t. It is only when the parcel moves through x that the cross sectional area changes: v depends on t only through the cross-sectional position x(t). ${\displaystyle {\begin{aligned}{\frac {\mathrm {d} v}{\mathrm {d} t}}&={\frac {\mathrm {d} v}{\mathrm {d} x}}{\frac {\mathrm {d} x}{\mathrm {d} t}}\\&={\frac {\mathrm {d} v}{\mathrm {d} x}}v\\&={\frac {\mathrm {d} }{\mathrm {d} x}}\left({\frac {v^{2}}{2}}\right).\end{aligned}}}$ With density ρ constant, the equation of motion can be written as ${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\left(\rho {\frac {v^{2}}{2}}+p\right)=0}$ by integrating with respect to x ${\displaystyle {\frac {v^{2}}{2}}+{\frac {p}{\rho }}=C}$ where C 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. Derivation by using conservation of energy Another way to derive Bernoulli's principle for an incompressible flow is by applying conservation of energy.[1] In the form of the work-energy theorem, stating that[2] the change in the kinetic energy EkinEkin of the system equals the net work W done on the system; ${\displaystyle W=\Delta E_{\text{kin}}.}$ Therefore, the work done by the forces in the fluid equals increase in kinetic energy. The system consists of the volume of fluid, initially between the cross-sections A1A1 and A2A2. In the time interval ΔtΔt fluid elements initially at the inflow cross-section A1A1 move over a distance s1 = v1 Δts1 = v1 Δt, while at the outflow cross-section the fluid moves away from cross-section A2A2 over a distance s2 = v2 Δts2 = v2 Δt. The displaced fluid volumes at the inflow and outflow are respectively A1s1A1s1 and A2s2A2s2. The associated displaced fluid masses are – when ρ is the fluid's mass density – equal to density times volume, so ρA1s1ρA1s1 and ρA2s2ρA2s2. By mass conservation, these two masses displaced in the time interval ΔtΔt have to be equal, and this displaced mass is denoted by ΔmΔm: ${\displaystyle {\begin{aligned}\rho A_{1}s_{1}&=\rho A_{1}v_{1}\Delta t=\Delta m,\\\rho A_{2}s_{2}&=\rho A_{2}v_{2}\Delta t=\Delta m.\end{aligned}}}$ The work done by the forces consists of two parts: The work done by the pressure acting on the areas A1A1 and A2A2 ${\displaystyle W_{\text{pressure}}=F_{1,{\text{pressure}}}s_{1}-F_{2,{\text{pressure}}}s_{2}=p_{1}A_{1}s_{1}-p_{2}A_{2}s_{2}=\Delta m{\frac {p_{1}}{\rho }}-\Delta m{\frac {p_{2}}{\rho }}.}$ The work done by gravity: the gravitational potential energy in the volume A1s1 is lost, and at the outflow in the volume A2s2 is gained. So, the change in gravitational potential energy ΔEpot,gravity in the time interval Δt is ${\displaystyle \Delta E_{\text{pot,gravity}}=\Delta m\,gz_{2}-\Delta m\,gz_{1}.}$ Now, the work by the force of gravity is opposite to the change in potential energy, Wgravity = −ΔEpot,gravity: while the force of gravity is in the negative z-direction, the work—gravity force times change in elevation—will be negative for a positive elevation change Δz = z2 − z1, while the corresponding potential energy change is positive.[3] So: ${\displaystyle W_{\text{gravity}}=-\Delta E_{\text{pot,gravity}}=\Delta m\,gz_{1}-\Delta m\,gz_{2}.}$ And therefore the total work done in this time interval Δt is ${\displaystyle W=W_{\text{pressure}}+W_{\text{gravity}}.}$ The increase in kinetic energy is ${\displaystyle \Delta E_{\text{kin}}={\tfrac {1}{2}}\Delta m\,v_{2}^{2}-{\tfrac {1}{2}}\Delta m\,v_{1}^{2}.}$ Putting these together, the work-kinetic energy theorem W = ΔEkin gives: ${\displaystyle \Delta m{\frac {p_{1}}{\rho }}-\Delta m{\frac {p_{2}}{\rho }}+\Delta m\,gz_{1}-\Delta m\,gz_{2}={\tfrac {1}{2}}\Delta m\,v_{2}^{2}-{\tfrac {1}{2}}\Delta m\,v_{1}^{2}}$ or ${\displaystyle {\tfrac {1}{2}}\Delta m\,v_{1}^{2}+\Delta m\,gz_{1}+\Delta m{\frac {p_{1}}{\rho }}={\tfrac {1}{2}}\Delta m\,v_{2}^{2}+\Delta m\,gz_{2}+\Delta m{\frac {p_{2}}{\rho }}.}$ After dividing by the mass Δm = ρA1v1 Δt = ρA2v2 Δt the result is: ${\displaystyle {\tfrac {1}{2}}v_{1}^{2}+gz_{1}+{\frac {p_{1}}{\rho }}={\tfrac {1}{2}}v_{2}^{2}+gz_{2}+{\frac {p_{2}}{\rho }}}$ or, as stated in the first paragraph: ${\displaystyle {\frac {v^{2}}{2}}+gz+{\frac {p}{\rho }}=C}$   (Eqn. 1), Which is also Equation (A) Further division by g produces the following equation. Note that each term can be described in the length dimension (such as meters). This is the head equation derived from Bernoulli's principle: ${\displaystyle {\frac {v^{2}}{2g}}+z+{\frac {p}{\rho g}}=C}$   (Eqn. 2a) The middle term, z, represents the potential energy of the fluid due to its elevation with respect to a reference plane. Now, z is called the elevation head and given the designation zelevation. A free falling mass from an elevation z > 0 (in a vacuum) will reach a speed ${\displaystyle v={\sqrt {{2g}{z}}},}$ when arriving at elevation z = 0. Or when we rearrange it as a head: ${\displaystyle h_{v}={\frac {v^{2}}{2g}}}$ The term v2/2g is called the velocity head, expressed as a length measurement. It represents the internal energy of the fluid due to its motion. The hydrostatic pressure p is defined as ${\displaystyle p=p_{0}-\rho gz,}$ with p0 some reference pressure, or when we rearrange it as a head: ${\displaystyle \psi ={\frac {p}{\rho g}}.}$ The term p/ρg is also called the pressure head, expressed as a length measurement. It represents the internal energy of the fluid due to the pressure exerted on the container. When we combine the head due to the flow speed and the head due to static pressure with the elevation above a reference plane, we obtain a simple relationship useful for incompressible fluids using the velocity head, elevation head, and pressure head. ${\displaystyle h_{v}+z_{\text{elevation}}+\psi =C}$   (Eqn. 2b) If we were to multiply Eqn. 1 by the density of the fluid, we would get an equation with three pressure terms: ${\displaystyle {\frac {\rho v^{2}}{2}}+\rho gz+p=C}$   (Eqn. 3) We note that the pressure of the system is constant in this form of the Bernoulli equation. If the static pressure of the system (the far right term) increases, and if the pressure due to elevation (the middle term) is constant, then we know that the dynamic pressure (the left term) must have decreased. In other words, if the speed of a fluid decreases and it is not due to an elevation difference, we know it must be due to an increase in the static pressure that is resisting the flow. All three equations are merely simplified versions of an energy balance on a system.

Bernoulli equation for compressible fluids

The derivation for compressible fluids is similar. Again, the derivation depends upon (1) conservation of mass, and (2) conservation of energy. Conservation of mass implies that in the above figure, in the interval of time Δt, the amount of mass passing through the boundary defined by the area A1 is equal to the amount of mass passing outwards through the boundary defined by the area A2: ${\displaystyle 0=\Delta M_{1}-\Delta M_{2}=\rho _{1}A_{1}v_{1}\,\Delta t-\rho _{2}A_{2}v_{2}\,\Delta t}$. 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 A1 and A2 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, ${\displaystyle 0=\Delta E_{1}-\Delta E_{2}}$ where ΔE1 and ΔE2 are the energy entering through A1 and leaving through A2, respectively. The energy entering through A1 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 (ε1) entering, and the energy entering in the form of mechanical p dV work: ${\displaystyle \Delta E_{1}=\left({\tfrac {1}{2}}\rho _{1}v_{1}^{2}+\Psi _{1}\rho _{1}+\epsilon _{1}\rho _{1}+p_{1}\right)A_{1}v_{1}\,\Delta t}$ where Ψ = gz is a force potential due to the Earth's gravity, g is acceleration due to gravity, and z is elevation above a reference plane. A similar expression for ΔE2 may easily be constructed. So now setting 0 = ΔE1 − ΔE2: ${\displaystyle 0=\left({\tfrac {1}{2}}\rho _{1}v_{1}^{2}+\Psi _{1}\rho _{1}+\epsilon _{1}\rho _{1}+p_{1}\right)A_{1}v_{1}\,\Delta t-\left({\tfrac {1}{2}}\rho _{2}v_{2}^{2}+\Psi _{2}\rho _{2}+\epsilon _{2}\rho _{2}+p_{2}\right)A_{2}v_{2}\,\Delta t}$ which can be rewritten as: ${\displaystyle 0=\left({\tfrac {1}{2}}v_{1}^{2}+\Psi _{1}+\epsilon _{1}+{\frac {p_{1}}{\rho _{1}}}\right)\rho _{1}A_{1}v_{1}\,\Delta t-\left({\tfrac {1}{2}}v_{2}^{2}+\Psi _{2}+\epsilon _{2}+{\frac {p_{2}}{\rho _{2}}}\right)\rho _{2}A_{2}v_{2}\,\Delta t}$ Now, using the previously-obtained result from conservation of mass, this may be simplified to obtain ${\displaystyle {\tfrac {1}{2}}v^{2}+\Psi +\epsilon +{\frac {p}{\rho }}={\text{constant}}\equiv b}$ which is the Bernoulli equation for compressible flow. An equivalent expression can be written in terms of fluid enthalpy (h): ${\displaystyle {\tfrac {1}{2}}v^{2}+\Psi +h={\text{constant}}\equiv b}$

• Terminologijos skysčių dinamika
• Navier–Stokso lygtis – srautas per klampus skystis
• Oilerio lygtis – už srautų inviscid skystis
• Hidraulika – taikomos skysčių mechanika skysčių
• Torricelli teisė – ypatingas atvejis, Bernulio principas
• Daniel Bernulio
• Coandă poveikis