Bernoulli Law

Aggression ^[19]
Aggression
Dimension
Units
SI	meter
Notes
	Total pressure divided by specific gravity .

Total pressure
Total pressure
Dimension
Units
SI	J / m 3 = Pa
GHS	erg / cm 3
Notes
	Constantly along the streamline of the stationary flow of an incompressible fluid .

Figure from D. Bernoulli 's “Hydrodynamics”: due to the flow through the pipe, which compensates for the flow through the right hole O, the pressure in the pipe is less than in the vessel on the left.

The Bernoulli law ^[1] (also the Bernoulli equation ^[2] ^[3] , the Bernoulli theorem ^[4] ^[5], or the Bernoulli integral ^[2] ^[6] ^[7] ) establishes the relationship between the stationary fluid flow rate and its pressure . According to this law, if along the flow line the fluid pressure increases, then the flow velocity decreases, and vice versa. The quantitative expression of the law in the form of the Bernoulli integral is the result of the integration of the hydrodynamic equations of an ideal fluid ^[2] (that is, without viscosity and thermal conductivity ).

History

For the case of an incompressible fluid, a result equivalent to the modern Bernoulli equation was published in 1738 by Daniel Bernoulli ^{[K 1]} . In its modern form, the integral was published by Johann Bernoulli in 1743 ^[11] for the case of an incompressible fluid, and for some cases of compressible fluid flows by Euler in 1757 ^[12] .

Bernoulli integral in incompressible fluid

For the stationary flow of an incompressible fluid, the Bernoulli equation can be obtained as a consequence of the law of conservation of energy . Bernoulli's law states that the magnitude $\rho v^{2}/2+\rho gh+p$ ${\ displaystyle \ rho v ^ {2} / 2 + \ rho gh + p}$ $\rho v^{2}/2+\rho gh+p$ keeps a constant value along the streamline:

{\frac {\rho v^{2}}{2}}+\rho gh+p={\text{const}}.

{\ displaystyle {\ frac {\ rho v ^ {2}} {2}} + \ rho gh + p = {\ text {const}}.}

{\frac {\rho v^{2}}{2}}+\rho gh+p={\text{const}}.

Here

\rho

{\ displaystyle \ rho}

- fluid density ;

v

{\ displaystyle v}

- flow rate ;

h

{\ displaystyle h}

- height;

p

{\ displaystyle p}

- pressure ;

g

{\ displaystyle g}

- acceleration of gravity .

An elementary derivation of the Bernoulli equation from the law of conservation of energy

An elementary derivation of the Bernoulli equation from the energy conservation law is given, for example, in the textbook of D. V. Sivukhin ^[13] . The stationary motion of the fluid along the streamline shown in the figure is considered. Left to the volume of fluid originally enclosed between two sections $A_{1}$ ${\ displaystyle A_ {1}}$ and $A_{2}$ ${\ displaystyle A_ {2}}$ force is acting $F_{1}=p_{1}A_{1}$ ${\ displaystyle F_ {1} = p_ {1} A_ {1}}$ and on the right - in the opposite direction $F_{2}=-p_{2}A_{2}$ ${\ displaystyle F_ {2} = - p_ {2} A_ {2}}$ . Speed $v$ ${\ displaystyle v}$ and pressure $p$ ${\ displaystyle p}$ in sections 1 and 2, as well as their areas are indicated by the lower indices 1 and 2. For an infinitely short time $\Delta t$ ${\ displaystyle \ Delta t}$ the left boundary of this volume of fluid has shifted a small distance $s_{1}=v_{1}\Delta t$ ${\ displaystyle s_ {1} = v_ {1} \ Delta t}$ and the right one is at a distance $s_{2}=v_{2}\Delta t$ ${\ displaystyle s_ {2} = v_ {2} \ Delta t}$ . Work done by pressure is equal to:

W=F_{1}s_{1}+F_{2}s_{2}=\Delta t\left(v_{1}A_{1}p_{1}-v_{2}A_{2}p_{2}\right).

{\ displaystyle W = F_ {1} s_ {1} + F_ {2} s_ {2} = \ Delta t \ left (v_ {1} A_ {1} p_ {1} -v_ {2} A_ {2} p_ {2} \ right).}

At the beginning of the time interval $\Delta t$ ${\ displaystyle \ Delta t}$ fluid volume enclosed between two surfaces $A_{1}$ ${\ displaystyle A_ {1}}$ and $A_{2}$ ${\ displaystyle A_ {2}}$ , consists of the left blue element and the middle blue part, at the end of this interval, the displaced volume consists of the middle blue part and the right blue element. Since the flow is stationary, the contribution of the blue fragment to the energy and mass of the discussed volume of liquid does not change, and the conservation of mass allows us to conclude that the mass of the left blue element is equal to the mass of the right blue element: $\Delta m=\Delta tv_{1}A_{1}\rho _{1}=\Delta tv_{2}A_{2}\rho _{2}.$ ${\ displaystyle \ Delta m = \ Delta tv_ {1} A_ {1} \ rho _ {1} = \ Delta tv_ {2} A_ {2} \ rho _ {2}.}$ . Therefore, the work of forces, the expression for which can be converted to: $\Delta W=\Delta m\left({\frac {p_{1}}{\rho _{1}}}-{\frac {p_{2}}{\rho _{2}}}\right),$ ${\ displaystyle \ Delta W = \ Delta m \ left ({\ frac {p_ {1}} {\ rho _ {1}}} - {\ frac {p_ {2}} {\ rho _ {2}}} \ right),}$ equal to the change in energy, equal, in turn, to the energy difference of the right blue element $\Delta E_{2}$ ${\ displaystyle \ Delta E_ {2}}$ and left blue element $\Delta E_{1}$ ${\ displaystyle \ Delta E_ {1}}$ .

For an incompressible fluid, it is possible, firstly, to put $\rho _{1}=\rho _{2}=\rho$ ${\ displaystyle \ rho _ {1} = \ rho _ {2} = \ rho}$ and secondly, in the expression for the energy of the liquid element, we restrict ourselves to the kinetic and potential energy: $\Delta E_{1}=\Delta m\left({\frac {v_{1}^{2}}{2}}+gh_{1}\right),$ ${\ displaystyle \ Delta E_ {1} = \ Delta m \ left ({\ frac {v_ {1} ^ {2}} {2}} + gh_ {1} \ right),}$ $\Delta E_{2}=\Delta m\left({\frac {v_{2}^{2}}{2}}+gh_{2}\right).$ ${\ displaystyle \ Delta E_ {2} = \ Delta m \ left ({\ frac {v_ {2} ^ {2}} {2}} + gh_ {2} \ right).}$ After this equality $\Delta W=\Delta E_{2}-\Delta E_{1}$ ${\ displaystyle \ Delta W = \ Delta E_ {2} - \ Delta E_ {1}}$ gives: $p_{1}+\rho gh_{1}+{\frac {\rho v_{1}^{2}}{2}}=p_{2}+\rho gh_{2}+{\frac {\rho v_{2}^{2}}{2}}$ ${\ displaystyle p_ {1} + \ rho gh_ {1} + {\ frac {\ rho v_ {1} ^ {2}} {2}} = p_ {2} + \ rho gh_ {2} + {\ frac {\ rho v_ {2} ^ {2}} {2}}}$ , or $p+\rho gh+{\frac {\rho v^{2}}{2}}={\rm {const}}$ ${\ displaystyle p + \ rho gh + {\ frac {\ rho v ^ {2}} {2}} = {\ rm {const}}}$ .

The constant on the right-hand side (may vary for different streamlines) is sometimes called total pressure ^[2] . The terms “weight pressure” may also be used. $\rho gh$ ${\ displaystyle \ rho gh}$ , "Static pressure" $p$ ${\ displaystyle p}$ and "dynamic pressure" $\rho v^{2}/2$ {\ displaystyle \ rho v ^ {2} / 2} . According to D. V. Sivukhin ^[13] , the irrationality of these concepts was noted by many physicists.

The dimension of all terms is a unit of energy per unit volume. The first and second terms in the Bernoulli integral have the meaning of the kinetic and potential energy per unit volume of the liquid. The third term in its origin is the work of pressure forces (see the above conclusion of the Bernoulli equation), but in hydraulics it can be called the “pressure energy” and part of the potential energy ^[14] ).

Derivation of the Torricelli Formula from Bernoulli's Law

Torricelli Formula Illustration

As applied to the flow of an ideal incompressible fluid through a small hole in the side wall or the bottom of a wide vessel, Bernoulli's law gives the equality of the total pressures on the free surface of the liquid and at the exit of the hole:

\rho gh+p_{0}={\frac {\rho v^{2}}{2}}+p_{0},

{\ displaystyle \ rho gh + p_ {0} = {\ frac {\ rho v ^ {2}} {2}} + p_ {0},}

Where

h

{\ displaystyle h}

- the height of the liquid column in the vessel, counted from the level of the hole,

v

{\ displaystyle v}

- fluid flow rate,

p_{0}

{\ displaystyle p_ {0}}

- atmospheric pressure .

From here: $v={\sqrt {2gh}}$ ${\ displaystyle v = {\ sqrt {2gh}}}$ . This is the Torricelli formula . It shows that when the fluid expires, it acquires the speed that a body would freely fall from a height $h$ ${\ displaystyle h}$ . Or, if the jet flowing from a small hole in a vessel is directed upward, at the upper point (neglecting losses), the jet will reach the level of the free surface in the vessel ^[15] .

Other manifestations and applications of Bernoulli's law

Bernoulli's law explains the Venturi effect : in the narrow part of the pipe, the fluid flow rate is higher and the pressure is less than in the wide

The incompressible fluid approximation, and with it the Bernoulli law, is also valid for laminar gas flows, provided that the flow velocities are small compared to the speed of sound ^[16] .

Coordinate along the horizontal pipe $z$ ${\ displaystyle z}$ constant and the Bernoulli equation takes the form ${\frac {\rho v^{2}}{2}}+p={\text{const}}$ ${\ displaystyle {\ frac {\ rho v ^ {2}} {2}} + p = {\ text {const}}}$ . It follows that with a decrease in the flow cross section due to an increase in velocity, the pressure decreases. The effect of pressure reduction with increasing flow rate is the basis for the operation of a Venturi flow meter ^[17] and a jet pump ^[1] .

Bernoulli’s law explains why ships traveling in a parallel course can be attracted to each other (for example, such an incident occurred with the Olympic liner) ^[18] .

Hydraulic Application

The consistent application of Bernoulli's law led to the emergence of a technical hydromechanical discipline - hydraulics . For technical applications, the Bernoulli equation is often written in the form in which all terms are divided by “ specific gravity ” $\rho g$ ${\ displaystyle \ rho g}$ :

H=h+{\frac {p}{\rho g}}+{\frac {v^{2}}{2g}}={\text{const}},

{\ displaystyle H = h + {\ frac {p} {\ rho g}} + {\ frac {v ^ {2}} {2g}} = {\ text {const}},}

where the members of length dimension in this equation may have the following names:

H

{\ displaystyle H}

- hydraulic height ^[4] or pressure ^[19] ,

h

{\ displaystyle h}

- leveling height ^[4] ,

{\frac {p}{\rho g}}

{\ displaystyle {\ frac {p} {\ rho g}}}

- piezometric height ^[4] or (in total with leveling height) hydrostatic pressure ^[19] ,

{\frac {v^{2}}{2g}}

{\ displaystyle {\ frac {v ^ {2}} {2g}}}

- speed height ^[4] or pressure head ^[19] .

Bernoulli's law is valid only for ideal fluids in which there is no loss of viscous friction . To describe the flows of real liquids in technical hydromechanics (hydraulics), the Bernoulli integral is used with the addition of terms that approximately take into account various “ hydraulic pressure losses ” ^[19] .

Bernoulli integral in barotropic flows

The Bernoulli equation can also be derived from the equation of fluid motion ^{[K 2]} ^{[K 3]} . In this case, the flow is assumed to be stationary and barotropic . The latter means that the density of the liquid or gas is not necessarily constant (as in the previously assumed incompressible liquid), but is a function of pressure only: $\rho =\rho (p)$ ${\ displaystyle \ rho = \ rho (p)}$ that allows you to enter the pressure function ^[22] ${\mathcal {P}}=\int {\frac {\mathrm {d} p}{\rho (p)}}.$ ${\ displaystyle {\ mathcal {P}} = \ int {\ frac {\ mathrm {d} p} {\ rho (p)}}.}$ Under these assumptions, the quantity

{\frac {v^{2}}{2}}+gh+{\mathcal {P}}={\text{const}}

{\ displaystyle {\ frac {v ^ {2}} {2}} + gh + {\ mathcal {P}} = {\ text {const}}}

constant along any streamline and any vortex line . The relation is valid for the flow in any potential field , while $g h$ ${\ displaystyle gh}$ replaced by mass power potential $\varphi$ ${\ displaystyle \ varphi}$ .

Derivation of the Bernoulli integral for a barotropic flow

The Gromeka – Lamb equation ^[23] ^[24] (square brackets denote the vector product ) has the form:

{\frac {\partial {\vec {v}}}{\partial t}}+\operatorname {grad} \left({\frac {v^{2}}{2}}\right)+\left[\mathrm {rot} \,{\vec {v}},{\vec {v}}\right]=-{\frac {1}{\rho }}\operatorname {grad} p+{\vec {F}}

{\ displaystyle {\ frac {\ partial {\ vec {v}}} {\ partial t}} + \ operatorname {grad} \ left ({\ frac {v ^ {2}} {2}} \ right) + \ left [\ mathrm {rot} \, {\ vec {v}}, {\ vec {v}} \ right] = - {\ frac {1} {\ rho}} \ operatorname {grad} p + {\ vec {F}}}

By virtue of the assumptions made ${\frac {\partial {\vec {v}}}{\partial t}}=0,$ ${\ displaystyle {\ frac {\ partial {\ vec {v}}} {\ partial t}} = 0,}$ ${\frac {\operatorname {grad} p}{\rho }}=\operatorname {grad} {\cal {P}}$ ${\ displaystyle {\ frac {\ operatorname {grad} p} {\ rho}} = \ operatorname {grad} {\ cal {P}}}$ and ${\vec {F}}=-\operatorname {grad} \varphi$ ${\ displaystyle {\ vec {F}} = - \ operatorname {grad} \ varphi}$ (in the particular case of uniform gravity, its potential is $\varphi =g\,h$ ${\ displaystyle \ varphi = g \, h}$ ), so that the Gromeki-Lamb equation takes the form:

\operatorname {grad} \left({\frac {v^{2}}{2}}+\varphi +{\cal {P}}\right)+\left[\mathrm {rot} \,{\vec {v}},{\vec {v}}\right]=0

{\ displaystyle \ operatorname {grad} \ left ({\ frac {v ^ {2}} {2}} + \ varphi + {\ cal {P}} \ right) + \ left [\ mathrm {rot} \, {\ vec {v}}, {\ vec {v}} \ right] = 0}

The scalar product of this equation by the unit vector ${\vec {l}}={\frac {\vec {v}}{v}},$ ${\ displaystyle {\ vec {l}} = {\ frac {\ vec {v}} {v}},}$ tangent to the streamline, gives:

{\frac {\partial }{\partial l}}\left({\frac {v^{2}}{2}}+\varphi +{\cal {P}}\right)=0

{\ displaystyle {\ frac {\ partial} {\ partial l}} \ left ({\ frac {v ^ {2}} {2}} + \ varphi + {\ cal {P}} \ right) = 0}

since the product of the gradient by the unit vector gives the derivative in the direction ${\frac {\partial }{\partial l}}$ ${\ displaystyle {\ frac {\ partial} {\ partial l}}}$ , and the vector product is perpendicular to the direction of speed. Consequently, along the streamline ${\frac {v^{2}}{2}}+\varphi +{\cal {P}}=\mathrm {const} .$ ${\ displaystyle {\ frac {v ^ {2}} {2}} + \ varphi + {\ cal {P}} = \ mathrm {const}.}$ This relation is also valid for a vortex line, the tangent vector to which at each point is directed along $\mathrm {rot} \,{\vec {v}}.$ ${\ displaystyle \ mathrm {rot} \, {\ vec {v}}.}$

For vortex-free barotropic flows, the velocity of which can be expressed as a gradient of the velocity potential ${\vec {v}}=\operatorname {grad} \psi$ ${\ displaystyle {\ vec {v}} = \ operatorname {grad} \ psi}$ , Bernoulli integral in the form ${\frac {\partial \psi }{\partial t}}+{\frac {\left(\operatorname {grad} \psi \right)^{2}}{2}}+gh+{\cal {P}}=\mathrm {const}$ ${\ displaystyle {\ frac {\ partial \ psi} {\ partial t}} + {\ frac {\ left (\ operatorname {grad} \ psi \ right) ^ {2}} {2}} + gh + {\ cal {P}} = \ mathrm {const}}$ ^{[K 4] is} also preserved in non-stationary flows, and the constant on the right-hand side has the same value for the entire flow ^[25] .

Formula Saint-Venant - Wanzel

If the adiabatic law is satisfied during a perfect gas ^[26]

p={\frac {p_{0}}{\rho _{0}}}\rho ^{\gamma },\qquad \rho ={\frac {\rho _{0}}{p_{0}^{1/\gamma }}}p^{1/\gamma },\qquad {\cal {P}}=-{\frac {\gamma }{\gamma -1}}{\frac {p_{0}}{\rho _{0}}}\left[1-\left({\frac {p}{p_{0}}}\right)^{(\gamma -1)/\gamma }\right],

{\ displaystyle p = {\ frac {p_ {0}} {\ rho _ {0}}} \ rho ^ {\ gamma}, \ qquad \ rho = {\ frac {\ rho _ {0}} {p_ { 0} ^ {1 / \ gamma}}} p ^ {1 / \ gamma}, \ qquad {\ cal {P}} = - {\ frac {\ gamma} {\ gamma -1}} {\ frac {p_ {0}} {\ rho _ {0}}} \ left [1- \ left ({\ frac {p} {p_ {0}}} \ right) ^ {(\ gamma -1) / \ gamma} \ right],}

then the Bernoulli equation is expressed as follows ^[27] (the contribution from gravity can usually be neglected):

{\frac {v^{2}}{2}}-{\frac {\gamma }{\gamma -1}}{\frac {p_{0}}{\rho _{0}}}\left[1-\left({\frac {p}{p_{0}}}\right)^{(\gamma -1)/\gamma }\right]=\mathrm {const}

{\ displaystyle {\ frac {v ^ {2}} {2}} - {\ frac {\ gamma} {\ gamma -1}} {\ frac {p_ {0}} {\ rho _ {0}}} \ left [1- \ left ({\ frac {p} {p_ {0}}} \ right) ^ {(\ gamma -1) / \ gamma} \ right] = \ mathrm {const}}

along a streamline or vortex line. Here

\gamma ={\frac {C_{p}}{C_{V}}}

{\ displaystyle \ gamma = {\ frac {C_ {p}} {C_ {V}}}}

Is the adiabatic index of the gas, expressed in terms of heat capacity at constant pressure and at a constant volume,

p,\ \rho

{\ displaystyle p, \ \ rho}

- pressure and gas density,

p_{0},\ \rho _{0}

{\ displaystyle p_ {0}, \ \ rho _ {0}}

- conditionally selected constants (the same for the entire flow) of pressure and density.

Using the obtained formula, the speed of the gas flowing out of the high-pressure vessel through a small hole is found. Conveniently, the pressure and density of the gas in the vessel, the gas velocity in which is equal to zero, is taken as $p_{0},\ \rho _{0},$ ${\ displaystyle p_ {0}, \ \ rho _ {0},}$ then the flow rate is expressed through external pressure $p$ ${\ displaystyle p}$ according to the formula of Saint-Venant - Vancel ^[28] :

v^{2}={\frac {2\gamma }{\gamma -1}}{\frac {p_{0}}{\rho _{0}}}\left[1-\left({\frac {p}{p_{0}}}\right)^{(\gamma -1)/\gamma }\right].

{\ displaystyle v ^ {2} = {\ frac {2 \ gamma} {\ gamma -1}} {\ frac {p_ {0}} {\ rho _ {0}}} left [1- \ left ( {\ frac {p} {p_ {0}}} \ right) ^ {(\ gamma -1) / \ gamma} \ right].}

Thermodynamics of Bernoulli's Law

From thermodynamics it follows that along the streamline of any stationary flow of an ideal fluid

{\frac {v^{2}}{2}}+w+\varphi ={\text{const}},\quad s={\text{const}},

{\ displaystyle {\ frac {v ^ {2}} {2}} + w + \ varphi = {\ text {const}}, \ quad s = {\ text {const}},}

Where $w$ ${\ displaystyle w}$ - enthalpy of unit mass , $\varphi$ ${\ displaystyle \ varphi}$ - gravitational potential (equal $g z$ ${\ displaystyle gz}$ for uniform gravity) $s$ ${\ displaystyle s}$ Is the entropy of a unit mass.

Derivation of Bernoulli's law from the Euler equation and thermodynamic relations

1. Euler equation for stationary ( $\partial {\vec {v}}/\partial t=0$ ${\ displaystyle \ partial {\ vec {v}} / \ partial t = 0}$ ) the motion of an ideal fluid in the field of gravity ^[29] has the form

({\vec {v}}\cdot \nabla ){\vec {v}}=-{\frac {1}{\rho }}\nabla p+{\vec {g}},

{\ displaystyle ({\ vec {v}} \ cdot \ nabla) {\ vec {v}} = - {\ frac {1} {\ rho}} \ nabla p + {\ vec {g}},}

where the acceleration of gravity can be expressed through the gravitational potential ${\vec {g}}=-\nabla \varphi$ ${\ displaystyle {\ vec {g}} = - \ nabla \ varphi}$ (for a uniform field $\varphi =gh$ ${\ displaystyle \ varphi = gh}$ ), the dot between the vectors in parentheses means their scalar product .

2. The scalar product of this equation by the unit vector ${\vec {l}}={\frac {\vec {v}}{v}},$ ${\ displaystyle {\ vec {l}} = {\ frac {\ vec {v}} {v}},}$ tangent to the streamline gives

{\frac {\partial }{\partial l}}\left({\frac {v^{2}}{2}}+\varphi \right)=-{\frac {1}{\rho }}{\frac {\partial p}{\partial l}},

{\ displaystyle {\ frac {\ partial} {\ partial l}} \ left ({\ frac {v ^ {2}} {2}} + \ varphi \ right) = - {\ frac {1} {\ rho }} {\ frac {\ partial p} {\ partial l}},}

since the product of the gradient by the unit vector gives the derivative in the direction ${\frac {\partial }{\partial l}}.$ ${\ displaystyle {\ frac {\ partial} {\ partial l}}.}$

3. Thermodynamic differential relation

\mathrm {d} w={\frac {1}{\rho }}\mathrm {d} p+T\mathrm {d} s,

{\ displaystyle \ mathrm {d} w = {\ frac {1} {\ rho}} \ mathrm {d} p + T \ mathrm {d} s,}

Where $w$ ${\ displaystyle w}$ - enthalpies of unit mass , $T$ ${\ displaystyle T}$ - temperature and $s$ ${\ displaystyle s}$ - entropy of a unit mass, gives

{\frac {\partial w}{\partial l}}={\frac {1}{\rho }}{\frac {\partial p}{\partial l}}+T{\frac {\partial s}{\partial l}},\quad

{\ displaystyle {\ frac {\ partial w} {\ partial l}} = {\ frac {1} {\ rho}} {\ frac {\ partial p} {\ partial l}} + T {\ frac {\ partial s} {\ partial l}}, \ quad}

so that

{\frac {\partial }{\partial l}}\left({\frac {v^{2}}{2}}+w+\varphi \right)=T{\frac {\partial s}{\partial l}}.

{\ displaystyle {\ frac {\ partial} {\ partial l}} \ left ({\ frac {v ^ {2}} {2}} + w + \ varphi \ right) = T {\ frac {\ partial s} {\ partial l}}.}

In a stationary flow of an ideal fluid, all particles moving along a given streamline have the same entropy ^[30] ( $\partial s/\partial l=0$ ${\ displaystyle \ partial s / \ partial l = 0}$ ), therefore, along the streamline:

s={\text{const}},\quad {\frac {v^{2}}{2}}+w+\varphi ={\text{const}}.

{\ displaystyle s = {\ text {const}}, \ quad {\ frac {v ^ {2}} {2}} + w + \ varphi = {\ text {const}}.}

The Bernoulli integral is used in engineering calculations, including for environments that are very far in their properties from an ideal gas, for example, for water vapor used as a coolant in steam turbines. In this case, the so-called Mollier diagrams can be used, representing the specific enthalpy (along the ordinate ) as a function of specific entropy (along the abscissa ) and, for example, pressure (or temperature) in the form of a family of isobars ( isotherms ). In this case, the sequence of states along the streamline lies on some vertical line ( $s={\text{const}}$ ${\ displaystyle s = {\ text {const}}}$ ) The length of the segment of this line, cut off by two isobars, corresponding to the initial and final coolant pressure, is equal to half the change in the velocity square ^[31] .

Generalizations of the Bernoulli integral

The Bernoulli integral is also preserved when the flow passes through the front of the shock wave, in the reference frame in which the shock wave is at rest ^[32] . However, with such a transition, the entropy of the medium does not remain constant (increases); therefore, the Bernoulli relation is only one of the three Hugoniot relations , along with the laws of conservation of mass and momentum, relating the state of the medium behind the front with the state of the medium in front of the front and with the speed of the shock wave.

Generalizations of the Bernoulli integral are known for some classes of viscous fluid flows (for example, for plane-parallel flows ^[33] ), in magnetic hydrodynamics ^[34] , and hydrohydrodynamics ^[35] . In relativistic hydrodynamics, when the flow velocities become comparable to the speed of light $c$ ${\ displaystyle c}$ , the integral is formulated in terms of relativistically invariant ^[36] specific enthalpy and specific entropy ^[37] .

Comments

↑ In the record of D. Bernoulli, the internal pressure in the liquid did not explicitly appear ^[8] ^[9] ^[10] .
↑ “... [Derivation of the Bernoulli theorem from the energy equation] impoverishes the content of the Bernoulli theorem ... The Bernoulli integral, generally speaking, does not depend on the energy equation, although it really coincides with it for the isentropic and adiabatic motion of a perfect gas” ^[20] .
↑ “Two ... ways to get the Bernoulli equation are not equivalent. With the energy conclusion, there is no need for the assumption of isentropic flow. When integrating the equation of motion, Bernoulli integrals are obtained not only along streamlines, but also along vortex lines ” ^[21] .
↑ In Russian-language literature, the Bernoulli integral for potential flows of an incompressible or barotropic fluid is known as the Cauchy – Lagrange integral ^[25]

Notes

↑ ¹ ² Landsberg G.S. Bernoulli Law, 1985 .
↑ ¹ ² ³ ⁴ Vishnevetsky S. L. Bernoulli equation, 1988 .
↑ Titens O., Prandtl L. Hydro and aeromechanics, 1933 .
↑ ¹ ² ³ ⁴ ⁵ Loytsyansky L.G. Mechanics of liquid and gas, 2003 , §24. Bernoulli's theorem.
↑ Milne-Thomson L.M. Theoretical Hydrodynamics, 1964 .
↑ Sedov L.I. Continuum Mechanics, 1970 .
↑ Black G.G. Gas Dynamics, 1988 .
↑ Truesdell K. Essays on the History of Mechanics, 2002 .
↑ Mikhailov G.K. , 1999 , p. 17.
↑ Darrigol O. A history of hydrodynamics, 2005 , p. 9.
↑ Truesdell K. Essays on the History of Mechanics, 2002 , p. 255, 257.
↑ Euler L. Continuation des recherches, 1755 (1757) , p. 331.
↑ ¹ ² Sivukhin D.V. Mechanics, 1989 , §94. Stationary motion of an ideal fluid. Bernoulli equation.
↑ Chugaev R.R. Hydraulics. - L .: Energy , 1975 .-- 600 p.
↑ Sivukhin D.V. Mechanics, 1989 , §95. Examples for the application of the Bernoulli equation. Formula Torricelli.
↑ Sivukhin D.V. Mechanics, 1989 , §94, formula (94.6).
↑ Molokanov Yu. K. Processes and apparatuses for oil and gas processing . - M .: Chemistry, 1980 .-- S. 60. - 408 p.
↑ J.I. Perelman . Why are ships attracted? (Russian) . Date of treatment December 27, 2018.
↑ ¹ ² ³ ⁴ ⁵ Aggression, 1992 .
↑ Batchelor J. Introduction to fluid dynamics, 1973 , Note by G. Yu. Stepanova, p. 208.
↑ Goldstein R.V., Gorodtsov V.A. Mechanics of continuous media, 2000 , p. 104.
↑ Loitsyansky L.G. Fluid and gas mechanics, 2003 , §23, equation (9).
↑ Loitsyansky L.G. Fluid and gas mechanics, 2003 , §23, equation (7).
↑ Sedov L.I. Continuum Mechanics, 1970 , Chapter VIII. §2, equation (2.1).
↑ ¹ ² L. Loitsyansky. Mechanics of liquid and gas, 2003 , §42. Lagrange integral - Cauchy.
↑ Loitsyansky L.G. Fluid and gas mechanics, 2003 , §24, equation (29).
↑ L. Loitsyansky, Mechanics of Fluid and Gas, 2003 , §24, equation (30).
↑ Loitsyansky L.G. Fluid and gas mechanics, 2003 , §24, equation (31).
↑ Landau L.D., Lifshits E.M. Hydrodynamics, 2001 , Equation (2.4).
↑ Sedov L.I. Continuum Mechanics, 1970 , Chapter VII. §2. Pressure function.
↑ Paul R.V. , Mechanics, Acoustics, and the Doctrine of Heat, 2013 , p. 446.
↑ Landau L.D., Lifshits E.M. Hydrodynamics, 2001 , §85.
↑ Golubkin V.N., Sizykh G. B. On some general properties of plane-parallel viscous fluid flows // Bulletin of the USSR Academy of Sciences, series Fluid and gas mechanics: journal. - 1987. - No. 3 . - S. 176–178 . - DOI : 10.1007 / BF01051932 .
↑ Kulikovsky A.G. , Lyubimov G.A. Magnetic hydrodynamics. - M .: Fizmatlit , 1962 .-- S. 54. - 248 p.
↑ Rosenzweig R. Ferrohydrodynamics / Per. from English under the editorship of V.V. Gogosova. - M .: Mir , 1989 .-- S. 136. - 359 p. - ISBN 5-03-000997-3 .
↑ Zubarev D.N. , Relativistic thermodynamics, 1994 .
↑ Landau L.D., Lifshits E.M. Hydrodynamics, 2001 , Equation (134.11).

Literature

Batchelor J. Introduction to fluid dynamics / Per. from English under the editorship of G. Yu. Stepanova . - M .: Mir , 1973.- 760 p.
Vishnevetsky S. L. Bernoulli equation // Physical Encyclopedia / Ch. ed. A.M. Prokhorov . - M .: Soviet Encyclopedia , 1988. - T. 1: Aaronov-Bohm effect - Long lines. - S. 187. - 704 p.
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Links

Russian translation of the treatise of Daniel Bernoulli, in which Bernoulli's integral (law) first appears

[11] In the record of D. Bernoulli, the internal pressure in the liquid did not explicitly appear ^[8] ^[9] ^[10] .

[22] “... [Derivation of the Bernoulli theorem from the energy equation] impoverishes the content of the Bernoulli theorem ... The Bernoulli integral, generally speaking, does not depend on the energy equation, although it really coincides with it for the isentropic and adiabatic motion of a perfect gas” ^[20] .

[24] “Two ... ways to get the Bernoulli equation are not equivalent. With the energy conclusion, there is no need for the assumption of isentropic flow. When integrating the equation of motion, Bernoulli integrals are obtained not only along streamlines, but also along vortex lines ” ^[21] .

[29] In Russian-language literature, the Bernoulli integral for potential flows of an incompressible or barotropic fluid is known as the Cauchy – Lagrange integral ^[25]

[_954a3c17139ae86c-1] ¹ ² Landsberg G.S. Bernoulli Law, 1985 .

[_01ded71894cb92a8-2] ¹ ² ³ ⁴ Vishnevetsky S. L. Bernoulli equation, 1988 .

[_fd6c80b90c81e13e-3] Titens O., Prandtl L. Hydro and aeromechanics, 1933 .

[_3d55b95a28138758-4] ¹ ² ³ ⁴ ⁵ Loytsyansky L.G. Mechanics of liquid and gas, 2003 , §24. Bernoulli's theorem.

[_e630e4cd17ee5ed2-5] Milne-Thomson L.M. Theoretical Hydrodynamics, 1964 .

[_abf993350bde6824-6] Sedov L.I. Continuum Mechanics, 1970 .

[_0b3e4b49e0683b34-7] Black G.G. Gas Dynamics, 1988 .

[_3389cf4d8f3f818c-8] Truesdell K. Essays on the History of Mechanics, 2002 .

[_b4ab867d093d70a4-9] Mikhailov G.K. , 1999 , p. 17.

[_8449e15dd1fbc246-10] Darrigol O. A history of hydrodynamics, 2005 , p. 9.

[_0fc8a1189324a1ea-12] Truesdell K. Essays on the History of Mechanics, 2002 , p. 255, 257.

[_f1ea317e3b172481-13] Euler L. Continuation des recherches, 1755 (1757) , p. 331.

[_071fd9b3358a863c-14] ¹ ² Sivukhin D.V. Mechanics, 1989 , §94. Stationary motion of an ideal fluid. Bernoulli equation.

[15] Chugaev R.R. Hydraulics. - L .: Energy , 1975 .-- 600 p.

[_93eca86edff7da3e-16] Sivukhin D.V. Mechanics, 1989 , §95. Examples for the application of the Bernoulli equation. Formula Torricelli.

[_a4bd9b6bc66809c1-17] Sivukhin D.V. Mechanics, 1989 , §94, formula (94.6).

[18] Molokanov Yu. K. Processes and apparatuses for oil and gas processing . - M .: Chemistry, 1980 .-- S. 60. - 408 p.

[19] J.I. Perelman . Why are ships attracted? (Russian) . Date of treatment December 27, 2018.

[_213b2f74fa25edeb-20] ¹ ² ³ ⁴ ⁵ Aggression, 1992 .

[_37296d5370307687-21] Batchelor J. Introduction to fluid dynamics, 1973 , Note by G. Yu. Stepanova, p. 208.

[_d32fc1e389c3afb7-23] Goldstein R.V., Gorodtsov V.A. Mechanics of continuous media, 2000 , p. 104.

[_8723d75582005649-25] Loitsyansky L.G. Fluid and gas mechanics, 2003 , §23, equation (9).

[_4ba63702278e7983-26] Loitsyansky L.G. Fluid and gas mechanics, 2003 , §23, equation (7).

[_0f186e073e04fba3-27] Sedov L.I. Continuum Mechanics, 1970 , Chapter VIII. §2, equation (2.1).

[_f151943804fc3cee-28] ¹ ² L. Loitsyansky. Mechanics of liquid and gas, 2003 , §42. Lagrange integral - Cauchy.

[_cf5a5592736d08ea-30] Loitsyansky L.G. Fluid and gas mechanics, 2003 , §24, equation (29).

[_3f049151955f5990-31] L. Loitsyansky, Mechanics of Fluid and Gas, 2003 , §24, equation (30).

[_aea8db49478652dd-32] Loitsyansky L.G. Fluid and gas mechanics, 2003 , §24, equation (31).

[_801ea2ecec1aaddc-33] Landau L.D., Lifshits E.M. Hydrodynamics, 2001 , Equation (2.4).

[_84eba299c7efe4c6-34] Sedov L.I. Continuum Mechanics, 1970 , Chapter VII. §2. Pressure function.

[_7a5f5775019e3af2-35] Paul R.V. , Mechanics, Acoustics, and the Doctrine of Heat, 2013 , p. 446.

[_d665517686fc6f7d-36] Landau L.D., Lifshits E.M. Hydrodynamics, 2001 , §85.

[37] Golubkin V.N., Sizykh G. B. On some general properties of plane-parallel viscous fluid flows // Bulletin of the USSR Academy of Sciences, series Fluid and gas mechanics: journal. - 1987. - No. 3 . - S. 176–178 . - DOI : 10.1007 / BF01051932 .

[38] Kulikovsky A.G. , Lyubimov G.A. Magnetic hydrodynamics. - M .: Fizmatlit , 1962 .-- S. 54. - 248 p.

[39] Rosenzweig R. Ferrohydrodynamics / Per. from English under the editorship of V.V. Gogosova. - M .: Mir , 1989 .-- S. 136. - 359 p. - ISBN 5-03-000997-3 .

[_bde3458ac0eb0b93-40] Zubarev D.N. , Relativistic thermodynamics, 1994 .

[_a23c2bb89cf16ae6-41] Landau L.D., Lifshits E.M. Hydrodynamics, 2001 , Equation (134.11).

Total pressure
Dimension	$L^{-1}MT^{-2}$ ${\ displaystyle L ^ {- 1} MT ^ {- 2}}$ $L^{-1}MT^{-2}$
Units
SI	J / m ³ = Pa
GHS	erg / cm ³
Notes
Constantly along the streamline of the stationary flow of an incompressible fluid .

Aggression ^[19]
Dimension	$L$ ${\ displaystyle L}$
Units
SI	meter
Notes
Total pressure divided by specific gravity .