In fluid dynamics, Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy.^{[1]}^{: Ch.3 }^{[2]}^{: 156–164, § 3.5 } The principle is named after the Swiss mathematician and physicist Daniel Bernoulli, who published it in his book Hydrodynamica in 1738.^{[3]} Although Bernoulli deduced that pressure decreases when the flow speed increases, it was Leonhard Euler in 1752 who derived Bernoulli's equation in its usual form.^{[4]}^{[5]} The principle is only applicable for isentropic flows: when the effects of irreversible processes (like turbulence) and nonadiabatic processes (e.g. thermal radiation) are small and can be neglected.
Bernoulli's principle can be applied to various types of fluid flow, resulting in various forms of Bernoulli's equation. The simple form of Bernoulli's equation is valid for incompressible flows (e.g. most liquid flows and gases moving at low Mach number). More advanced forms may be applied to compressible flows at higher Mach numbers.
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.^{[2]}^{: § 3.5 } Thus an increase in the speed of the fluid—implying an increase in its kinetic energy (dynamic pressure)—occurs with a simultaneous decrease in (the sum of) its potential energy (including the static pressure) 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.^{[6]}^{: Example 3.5 }
Bernoulli's principle can also be derived directly from Isaac Newton's second Law of Motion. 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.^{[a]}^{[b]}^{[c]}
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.^{[10]}
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 flows. 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:

(A) 
where:
The constant on the righthand side of the equation depends only on the streamline chosen, whereas v, z and p depend on the particular point on that streamline.
The following assumptions must be met for this Bernoulli equation to apply:^{[2]}^{: 265 }
For conservative force fields (not limited to the gravitational field), Bernoulli's equation can be generalized as:^{[2]}^{: 265 }
By multiplying with the fluid density ρ, equation (A) can be rewritten as:
The constant in the Bernoulli equation can be normalized. A common approach is in terms of total head or energy head H:
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.
In many applications of Bernoulli's equation, the change in the ρgz 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 ρgz term can be omitted. This allows the above equation to be presented in the following simplified form:
The simplified form of Bernoulli's equation can be summarized in the following memorable word equation:^{[1]}^{: § 3.5 }
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 brought to rest at some point, this point is called a stagnation point, and at this point the static pressure is equal to the stagnation pressure.
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".^{[1]}^{: 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, Bernoulli's principle importantly does not apply in the boundary layer or in fluid flow through long pipes.
The Bernoulli equation for unsteady potential flow is used in the theory of ocean surface waves and acoustics. For an irrotational flow, the flow velocity can be described as the gradient ∇φ of a velocity potential φ. In that case, and for a constant density ρ, the momentum equations of the Euler equations can be integrated to:^{[2]}^{: 383 }
which is a Bernoulli equation valid also for unsteady—or time dependent—flows. Here ∂φ/∂t denotes the partial derivative of the velocity potential φ with respect to time t, and v = ∇φ is the flow speed. The function f(t) depends only on time and not on position in the fluid. As a result, the Bernoulli equation at some moment t does not only apply along a certain streamline but in the whole fluid domain. This is also true for the special case of a steady irrotational flow, in which case f and ∂φ/∂t are constants so equation (A) can be applied in every point of the fluid domain.^{[2]}^{: 383 } Further f(t) can be made equal to zero by incorporating it into the velocity potential using the transformation:
Note that the relation of the potential to the flow velocity is unaffected by this transformation: ∇Φ = ∇φ.
The Bernoulli equation for unsteady potential flow also appears to play a central role in Luke's variational principle, a variational description of freesurface flows using the Lagrangian mechanics.
Bernoulli developed his principle from observations on liquids, and Bernoulli's equation is valid for ideal fluids: those that are incompressible, irrotational, inviscid, and subjected to conservative forces. It is sometimes valid for the flow of gases: provided that there is no transfer of kinetic or potential energy from the gas flow to the compression or expansion of the gas. If both the gas pressure and volume change simultaneously, then work will be done on or by the gas. In this case, Bernoulli's equation—in its incompressible flow form—cannot be assumed to be valid. However, if the gas process is entirely isobaric, or isochoric, then no work is done on or by the gas (so the simple energy balance is not upset). According to the gas law, an isobaric or isochoric process is ordinarily the only way to ensure constant density in a gas. Also the gas density will be proportional to the ratio of pressure and absolute temperature; however, this ratio will vary upon compression or expansion, no matter what nonzero quantity of heat is added or removed. The only exception is if the net heat transfer is zero, as in a complete thermodynamic cycle or in an individual isentropic (frictionless adiabatic) process, and even then this reversible process must be reversed, to restore the gas to the original pressure and specific volume, and thus density. Only then is the original, unmodified Bernoulli equation applicable. In this case the equation can be used if the flow speed of the gas is sufficiently below the speed of sound, such that the variation in density of the gas (due to this effect) along each streamline can be ignored. Adiabatic flow at less than Mach 0.3 is generally considered to be slow enough.^{[15]}
It is possible to use the fundamental principles of physics to develop similar equations applicable to compressible fluids. There are numerous equations, each tailored for a particular application, but all are analogous to Bernoulli's equation and all rely on nothing more than the fundamental principles of physics such as Newton's laws of motion or the first law of thermodynamics.
For a compressible fluid, with a barotropic equation of state, and under the action of conservative forces,^{[16]}
In engineering situations, elevations are generally small compared to the size of the Earth, and the time scales of fluid flow are small enough to consider the equation of state as adiabatic. In this case, the above equation for an ideal gas becomes:^{[1]}^{: § 3.11 }
In many applications of compressible flow, changes in elevation are negligible compared to the other terms, so the term gz can be omitted. A very useful form of the equation is then:
where:
The most general form of the equation, suitable for use in thermodynamics in case of (quasi) steady flow, is:^{[2]}^{: § 3.5 }^{[17]}^{: § 5 }^{[18]}^{: § 5.9 }
Here w is the enthalpy per unit mass (also known as specific enthalpy), which is also often written as h (not to be confused with "head" or "height").
Note that
The constant on the righthand side is often called the Bernoulli constant and denoted b. For steady inviscid adiabatic flow with no additional sources or sinks of energy, b is constant along any given streamline. More generally, when b may vary along streamlines, it still proves a useful parameter, related to the "head" of the fluid (see below).
When the change in Ψ can be ignored, a very useful form of this equation is:
When shock waves are present, in a reference frame in which the shock is stationary and the flow is steady, many of the parameters in the Bernoulli equation suffer abrupt changes in passing through the shock. The Bernoulli parameter remains unaffected. An exception to this rule is radiative shocks, which violate the assumptions leading to the Bernoulli equation, namely the lack of additional sinks or sources of energy.
For a compressible fluid, with a barotropic equation of state, the unsteady momentum conservation equation
With the irrotational assumption, namely, the flow velocity can be described as the gradient ∇φ of a velocity potential φ. The unsteady momentum conservation equation becomes
In this case, the above equation for isentropic flow becomes:
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.
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 crosssectional area A, the length of the parcel is dx, and the volume of the parcel A dx. If mass density is ρ, the mass of the parcel is density multiplied by its volume m = ρA dx. The change in pressure over distance dx is dp and flow velocity v = 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. If the pressure decreases along the length of the pipe, dp is negative but the force resulting in flow is positive along the x axis.
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 crosssectional position x(t).
With density ρ constant, the equation of motion can be written as
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.
Another way to derive Bernoulli's principle for an incompressible flow is by applying conservation of energy.^{[19]} In the form of the workenergy theorem, stating that^{[20]}
Therefore,
The system consists of the volume of fluid, initially between the crosssections A_{1} and A_{2}. In the time interval Δt fluid elements initially at the inflow crosssection A_{1} move over a distance s_{1} = v_{1} Δt, while at the outflow crosssection the fluid moves away from crosssection A_{2} over a distance s_{2} = v_{2} Δt. The displaced fluid volumes at the inflow and outflow are respectively A_{1}s_{1} and A_{2}s_{2}. The associated displaced fluid masses are – when ρ is the fluid's mass density – equal to density times volume, so ρA_{1}s_{1} and ρA_{2}s_{2}. By mass conservation, these two masses displaced in the time interval Δt have to be equal, and this displaced mass is denoted by Δm:
The work done by the forces consists of two parts:

(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:

(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 z_{elevation}.
A free falling mass from an elevation z > 0 (in a vacuum) will reach a speed
The hydrostatic pressure p is defined as

(Eqn. 2b) 
If Eqn. 1 is multiplied by the density of the fluid, an equation with three pressure terms is obtained:

(Eqn. 3) 
Note that the pressure of the system is constant in this form of the Bernoulli equation. If the static pressure of the system (the third term) increases, and if the pressure due to elevation (the middle term) is constant, then the dynamic pressure (the first term) must have decreased. In other words, if the speed of a fluid decreases and it is not due to an elevation difference, 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.
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 A_{1} is equal to the amount of mass passing outwards through the boundary defined by the area A_{2}:
An equivalent expression can be written in terms of fluid enthalpy (h):
In modern everyday life there are many observations that can be successfully explained by application of Bernoulli's principle, even though no real fluid is entirely inviscid,^{[22]} and a small viscosity often has a large effect on the flow.
One of the most common erroneous explanations of aerodynamic lift asserts that the air must traverse the upper and lower surfaces of a wing in the same amount of time, implying that since the upper surface presents a longer path the air must be moving faster over the top of the wing than the bottom. Bernoulli's principle is then cited to conclude that the pressure must be lower on top of the wing than the bottom.^{[26]}^{[27]}
However, there is no physical principle that requires the air to traverse the upper and lower surfaces in the same amount of time. In fact, theory predicts and experiments confirm that the air traverses the top surface in a shorter time than it traverses the bottom surface, and this explanation based on equal transit time is false.^{[28]}^{[29]}^{[30]} While this explanation is false, it is not the Bernoulli principle that is false, because this principle is well established; Bernoulli's equation is used correctly in common mathematical treatments of aerodynamic lift.^{[31]}^{[32]}
There are several common classroom demonstrations that are sometimes incorrectly explained using Bernoulli's principle.^{[33]} One involves holding a piece of paper horizontally so that it droops downward and then blowing over the top of it. As the demonstrator blows over the paper, the paper rises. It is then asserted that this is because "faster moving air has lower pressure".^{[34]}^{[35]}^{[36]}
One problem with this explanation can be seen by blowing along the bottom of the paper: if the deflection was caused by faster moving air, then the paper should deflect downward; but the paper deflects upward regardless of whether the faster moving air is on the top or the bottom.^{[37]} Another problem is that when the air leaves the demonstrator's mouth it has the same pressure as the surrounding air;^{[38]} the air does not have lower pressure just because it is moving; in the demonstration, the static pressure of the air leaving the demonstrator's mouth is equal to the pressure of the surrounding air.^{[39]}^{[40]} A third problem is that it is false to make a connection between the flow on the two sides of the paper using Bernoulli's equation since the air above and below are different flow fields and Bernoulli's principle only applies within a flow field.^{[41]}^{[42]}^{[43]}^{[44]}
As the wording of the principle can change its implications, stating the principle correctly is important.^{[45]} What Bernoulli's principle actually says is that within a flow of constant energy, when fluid flows through a region of lower pressure it speeds up and vice versa.^{[46]} Thus, Bernoulli's principle concerns itself with changes in speed and changes in pressure within a flow field. It cannot be used to compare different flow fields.
A correct explanation of why the paper rises would observe that the plume follows the curve of the paper and that a curved streamline will develop a pressure gradient perpendicular to the direction of flow, with the lower pressure on the inside of the curve.^{[47]}^{[48]}^{[49]}^{[50]} Bernoulli's principle predicts that the decrease in pressure is associated with an increase in speed; in other words, as the air passes over the paper, it speeds up and moves faster than it was moving when it left the demonstrator's mouth. But this is not apparent from the demonstration.^{[51]}^{[52]}^{[53]}
Other common classroom demonstrations, such as blowing between two suspended spheres, inflating a large bag, or suspending a ball in an airstream are sometimes explained in a similarly misleading manner by saying "faster moving air has lower pressure".^{[54]}^{[55]}^{[56]}^{[57]}^{[58]}^{[59]}^{[60]}^{[61]}
Streamlines are closer together above the wing than they are below so that Bernoulli's principle predicts the observed upward dynamic lift.
One of the most widely circulated, but incorrect, explanations can be labeled the “Longer Path” theory, or the “Equal Transit Time” theory.
The airfoil of the airplane wing, according to the textbook explanation that is more or less standard in the United States, has a special shape with more curvature on top than on the bottom; consequently, the air must travel farther over the top surface than over the bottom surface. Because the air must make the trip over the top and bottom surfaces in the same elapsed time ..., the velocity over the top surface will be greater than over the bottom. According to Bernoulli's theorem, this velocity difference produces a pressure difference which is lift.^{[permanent dead link]}
...it is often asked why fluid particles should meet up again at the trailing edge. Or, in other words, why should two particles on either side of the wing take the same time to travel from S to T? There is no obvious explanation and reallife observations prove that this is wrong.
It is then assumed that these two elements must meet up at the trailing edge, and because the running distance over the top surface of the airfoil is longer than that over the bottom surface, the element over the top surface must move faster. This is simply not true. Experimental results and computational fluid dynamic calculations clearly show that a fluid element moving over the top surface of an airfoil leaves the trailing edge long before its companion element moving over the bottom surface arrives at the trailing edge.
There is nothing wrong with the Bernoulli principle, or with the statement that the air goes faster over the top of the wing. But, as the above discussion suggests, our understanding is not complete with this explanation. The problem is that we are missing a vital piece when we apply Bernoulli’s principle. We can calculate the pressures around the wing if we know the speed of the air over and under the wing, but how do we determine the speed?
This occurs because of Bernoulli’s principle — fastmoving air has lower pressure than nonmoving air.
Fastermoving fluid, lower pressure. ... When the demonstrator holds the paper in front of his mouth and blows across the top, he is creating an area of fastermoving air.
Bernoulli's Principle states that faster moving air has lower pressure... You can demonstrate Bernoulli's Principle by blowing over a piece of paper held horizontally across your lips.
If the lift in figure A were caused by "Bernoulli's principle," then the paper in figure B should droop further when air is blown beneath it. However, as shown, it raises when the upward pressure gradient in downwardcurving flow adds to atmospheric pressure at the paper lower surface.
In fact, the pressure in the air blown out of the lungs is equal to that of the surrounding air...
...air does not have a reduced lateral pressure (or static pressure...) simply because it is caused to move, the static pressure of free air does not decrease as the speed of the air increases, it misunderstanding Bernoulli's principle to suggest that this is what it tells us, and the behavior of the curved paper is explained by other reasoning than Bernoulli's principle.
Make a strip of writing paper about 5 cm × 25 cm. Hold it in front of your lips so that it hangs out and down making a convex upward surface. When you blow across the top of the paper, it rises. Many books attribute this to the lowering of the air pressure on top solely to the Bernoulli effect. Now use your fingers to form the paper into a curve that it is slightly concave upward along its whole length and again blow along the top of this strip. The paper now bends downward...an oftencited experiment, which is usually taken as demonstrating the common explanation of lift, does not do so...
Blowing over a piece of paper does not demonstrate Bernoulli’s equation. While it is true that a curved paper lifts when flow is applied on one side, this is not because air is moving at different speeds on the two sides... It is false to make a connection between the flow on the two sides of the paper using Bernoulli’s equation.
An explanation based on Bernoulli’s principle is not applicable to this situation, because this principle has nothing to say about the interaction of air masses having different speeds... Also, while Bernoulli’s principle allows us to compare fluid speeds and pressures along a single streamline and... along two different streamlines that originate under identical fluid conditions, using Bernoulli’s principle to compare the air above and below the curved paper in Figure 1 is nonsensical; in this case, there aren’t any streamlines at all below the paper!
The wellknown demonstration of the phenomenon of lift by means of lifting a page cantilevered in one’s hand by blowing horizontally along it is probably more a demonstration of the forces inherent in the Coanda effect than a demonstration of Bernoulli’s law; for, here, an air jet issues from the mouth and attaches to a curved (and, in this case pliable) surface. The upper edge is a complicated vortexladen mixing layer and the distant flow is quiescent, so that Bernoulli’s law is hardly applicable.
Millions of children in science classes are being asked to blow over curved pieces of paper and observe that the paper 'lifts'... They are then asked to believe that Bernoulli's theorem is responsible... Unfortunately, the 'dynamic lift' involved...is not properly explained by Bernoulli's theorem.
Bernoulli’s principle is very easy to understand provided the principle is correctly stated. However, we must be careful, because seeminglysmall changes in the wording can lead to completely wrong conclusions.
A complete statement of Bernoulli's Theorem is as follows: 'In a flow where no energy is being added or taken away, the sum of its various energies is a constant: consequently where the velocity increases the pressure decreases and vice versa.'
...if a streamline is curved, there must be a pressure gradient across the streamline, with the pressure increasing in the direction away from the centre of curvature.
The curved paper turns the stream of air downward, and this action produces the lift reaction that lifts the paper.
The curved surface of the tongue creates unequal air pressure and a lifting action. ... Lift is caused by air moving over a curved surface.
Viscosity causes the breath to follow the curved surface, Newton's first law says there a force on the air and Newton’s third law says there is an equal and opposite force on the paper. Momentum transfer lifts the strip. The reduction in pressure acting on the top surface of the piece of paper causes the paper to rise.
'Demonstrations' of Bernoulli's principle are often given as demonstrations of the physics of lift. They are truly demonstrations of lift, but certainly not of Bernoulli's principle.
As an example, take the misleading experiment most often used to "demonstrate" Bernoulli's principle. Hold a piece of paper so that it curves over your finger, then blow across the top. The paper will rise. However most people do not realize that the paper would not rise if it were flat, even though you are blowing air across the top of it at a furious rate. Bernoulli's principle does not apply directly in this case. This is because the air on the two sides of the paper did not start out from the same source. The air on the bottom is ambient air from the room, but the air on the top came from your mouth where you actually increased its speed without decreasing its pressure by forcing it out of your mouth. As a result the air on both sides of the flat paper actually has the same pressure, even though the air on the top is moving faster. The reason that a curved piece of paper does rise is that the air from your mouth speeds up even more as it follows the curve of the paper, which in turn lowers the pressure according to Bernoulli.
Some people blow over a sheet of paper to demonstrate that the accelerated air over the sheet results in a lower pressure. They are wrong with their explanation. The sheet of paper goes up because it deflects the air, by the Coanda effect, and that deflection is the cause of the force lifting the sheet. To prove they are wrong I use the following experiment: If the sheet of paper is pre bend the other way by first rolling it, and if you blow over it than, it goes down. This is because the air is deflected the other way. Airspeed is still higher above the sheet, so that is not causing the lower pressure.
The Bernoulli effect is commonly—and incorrectly—invoked to explain: :why two suspended balloons or table tennis balls move toward each other when you blow air between them; :why paper rises when you blow air over it; :why a pitched baseball curves; :why a spoon is drawn toward a stream of water; :why a ball remains suspended in an air jet. Here’s the news: None of these phenomena is the result of the Bernoulli effect.
Finally, let’s go back to the initial example of a ball levitating in a jet of air. The naive explanation for the stability of the ball in the air stream, 'because pressure in the jet is lower than pressure in the surrounding atmosphere,' is clearly incorrect. The static pressure in the free air jet is the same as the pressure in the surrounding atmosphere...
Asymmetrical flow (not Bernoulli's theorem) also explains lift on the pingpong ball or beach ball that floats so mysteriously in the tilted vacuum cleaner exhaust...
Bernoulli’s theorem is often obscured by demonstrations involving nonBernoulli forces. For example, a ball may be supported on an upward jet of air or water, because any fluid (the air and water) has viscosity, which retards the slippage of one part of the fluid moving past another part of the fluid.
In a demonstration sometimes wrongly described as showing lift due to pressure reduction in moving air or pressure reduction due to flow path restriction, a ball or balloon is suspended by a jet of air.
A second example is the confinement of a pingpong ball in the vertical exhaust from a hair dryer. We are told that this is a demonstration of Bernoulli's principle. But, we now know that the exhaust does not have a lower value of ps. Again, it is momentum transfer that keeps the ball in the airflow. When the ball gets near the edge of the exhaust there is an asymmetric flow around the ball, which pushes it away from the edge of the flow. The same is true when one blows between two pingpong balls hanging on strings.
This demonstration is often incorrectly explained using the Bernoulli principle. According to the INCORRECT explanation, the air flow is faster in the region between the sheets, thus creating a lower pressure compared with the quiet air on the outside of the sheets.
Although the Bernoulli effect is often used to explain this demonstration, and one manufacturer sells the material for this demonstration as 'Bernoulli bags,' it cannot be explained by the Bernoulli effect, but rather by the process of entrainment.