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.
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 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 cross-sectional position x(t).
With density ρ constant, the equation of motion can be written as
by integrating with respect to x
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.
Another way to derive Bernoulli's principle for an incompressible flow is by applying conservation of energy.^{ [19] } In the form of the work-energy theorem, stating that^{ [20] }
Therefore,
The system consists of the volume of fluid, initially between the cross-sections A_{1} and A_{2}. In the time interval Δt fluid elements initially at the inflow cross-section A_{1} move over a distance s_{1} = v_{1} Δt, while at the outflow cross-section the fluid moves away from cross-section 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:
And therefore the total work done in this time interval Δt is
The increase in kinetic energy is
Putting these together, the work-kinetic energy theorem W = ΔE_{kin} gives:^{ [19] }
or
After dividing by the mass Δm = ρA_{1}v_{1} Δt = ρA_{2}v_{2} Δt the result is:^{ [19] }
or, as stated in the first paragraph:
| (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
when arriving at elevation z = 0. Or when rearranged as head:
The term v^{2}/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
with p_{0} some reference pressure, or when rearranged as head:
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. The head due to the flow speed and the head due to static pressure combined with the elevation above a reference plane, a simple relationship useful for incompressible fluids using the velocity head, elevation head, and pressure head is obtained.
| (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}:
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 A_{1} and A_{2} 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 ΔE_{1} and ΔE_{2} are the energy entering through A_{1} and leaving through A_{2}, respectively. The energy entering through A_{1} 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 (ε_{1}) entering, and the energy entering in the form of mechanical p dV work:
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 ΔE_{2} may easily be constructed. So now setting 0 = ΔE_{1} − ΔE_{2}:
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 (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] }
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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 real-life 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 — fast-moving air has lower pressure than non-moving air.
Faster-moving 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 faster-moving 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 downward-curving 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 often-cited 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 well-known 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 vortex-laden 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 seemingly-small 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 ping-pong ball or beach ball that floats so mysteriously in the tilted vacuum cleaner exhaust...
Bernoulli's theorem is often obscured by demonstrations involving non-Bernoulli 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 ping-pong 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 ping-pong 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.