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Bernoulli's principle states that an increase in the speed of a fluid simultaneously takes place with a decrease in static pressure or a decrease in the fluid's potential energy. Although the principle was formulated by Daniel Bernoulli, it was Leonhard Euler who derived Bernoulli’s equation in its usual form in the year 1752.
NCERT Solutions of: Class 11 Physics Chapter 10 Mechanical Properties of Fluids
Key Terms: Bernoulli’s Principle, Bernoulli’s Equation, Principle of continuity, Application of Bernoulli’s principle, Bernoulli’s equation for Statis fluids, Conservation of energy.
What is Bernoulli’s Principle?
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Bernoulli’s principle, also known as Bernoulli’s equation states that:
| An increase in the speed of fluid takes place simultaneously with a decrease in pressure or a decrease in the fluid's potential energy. Bernoulli’s principle applies to liquids in a perfect state, so pressure and density are inversely proportional to each other, which means that low-velocity liquids exert more pressure than faster-moving liquids. |
Daniel Bernoulli, an 18th-century Swiss mathematician, and physicist, while conducting experiments about the conservation of energy, discovered this principle. His results were published in Hydrodynamica which considered the basic properties of fluid flow, pressure, density, and velocity. Bernoulli’s principle is the only principle that explains how heavier-than-air objects can fly.
Bernoulli’s Theorem Video Explanation
Bernoulli’s Principle: Explanation
- The liquid should be understood as not only a liquid but also a gas. This principle is at the core of many applications. Some very common examples are an airplane trying to stay high, or even the most common everyday items, such as a shower curtain that folds inward.
- When the width of the river changes, the same thing will happen to the river. The water speed decreases in a larger area, while the water speed increases in a narrow area.
- You would think that the pressure in the liquid is increasing; however, contrary to what has been said above, the pressure of the liquid in the narrow part of the river will decrease, and the pressure of the liquid in the widest part of the river will increase.
- The Swiss scientist Daniel Bernoulli discovered this concept while experimenting with liquids in pipes. In his experiments, he found that the velocity of the liquid increased, but the internal pressure of the liquid decreased. He called this concept Bernoulli's principle.
- This concept is definitely difficult to understand and complex. You may think that the water pressure in the smallest space increases, but in fact the water pressure in the smallest space increases, but the pressure in the water does not increase.
- As a result, the pressure in the liquid increases. Changes in pressure also change the velocity of the fluid. Now let us figure out this concept.
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Bernoulli’s Principle Formula
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Bernoulli’s principle formula shows the relationship between pressure, kinetic energy, and gravitational potential energy of a fluid in a container. The formula for Bernoulli’s principle is given as follows:
| p + 1/2pv2 + pgh = constant |
Where,
p = the pressure exerted by the fluid,
v = the velocity of the fluid,
ρ = the density of the fluid and
h = the height of the container.
Derivation of Bernoulli’s Equation
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The mechanism of fluid flow is a complicated process, but some important properties related to aerodynamic flow can be obtained by using the concept of energy conservation. Take any liquid moving in a pipe as an example. The cross-sectional area of different parts and different heights. See below.
Now consider the incompressible fluid flowing through this tube with a constant motion, but according to the concept of the continuity equation, the fluid velocity must be changed to produce an acceleration, but the force is very important. This is possible due to the liquid surrounding it, but the pressure must be different in different parts.
Bernoulli's equation is a general equation, which describes the pressure difference between two different points in the pipeline as related to the change of velocity or kinetic energy and the change of height or potential energy. This connection was proposed in 1738 by Bernoulli, a Swiss physicist, and mathematician.
Read More: Reynolds Number
General Expression of Bernoulli’s Equation
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- Let's look at the two different regions in the image above; we call the first region BC, and the second region DE. Now let us assume that the liquid was between B and D before. However, this liquid moves at tiny intervals (infinite) (Δt).
- If the fluid velocity at point B is equal to v1 and the fluid velocity at point D is equal to v2, then when fluid B initially moves in the direction of C, the distance is equal to v1Δt, but v1Δt is very small. We can assume that this is a constant. Cross-section through area BC.
- In the same way, within the same time interval Δt, the liquid that was previously at point D is now at point E. The distance traveled is v2Δt. The pressures P1 and P2 act on the two areas A1 and A2, thereby connecting the two parts. The complete circuit is shown in the figure below.
Findings of the Work Done
First we calculate the work done by the fluid in the BC zone (W1). Work done is:
W1 = P1A1 (v1Δt) = P1ΔV
If we also consider the continuity equation, the same volume of liquid will flow through BC and DE. Therefore, the work done by the liquid is on the right side of the pipe or in the DE area
W2 = P2A2 (v2Δt) = P2ΔV
Therefore, we can consider the work done on the liquid-P2ΔV. Therefore, the total work done to the fluid is
W1 – W2 = (P1 − P2) ΔV
All the work done helps to convert the potential gravitational and kinetic energy of the fluid. Now let us consider that the density of the liquid is ρ and the mass flowing through the pipe in the time interval Δt is Δm.
Hence, Δm = ρA1 v1Δt = ρΔV
Also, Check the: Units of Pressure
Limitations of the Applications of Bernoulli’s Principle
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- Due to internal friction, some energy is lost during fluid flow, because fluid has different layers, and each layer flows at a different speed, so each layer exerts a specific friction force.
- On the other hand, they are losing energy. A suitable term for this fluid characteristic is viscosity.
- Bernoulli’s equation is applicable only to incompressible fluids because it does not take into account the elastic energy of fluids.
- It is applicable only to streamline the flow of a fluid and not when the flow is turbulent.
- The principle does not take into consideration the angular momentum of the fluid. So it cannot be applied when the fluid flows along a curved path.
Read More: Bulk Modulus
Principle of Continuity
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The principle of continuity tells us that in this volume, things that flow into a certain volume at a given time should be accumulated, and things that flow out of that volume at that point in time should be subtracted. If the accumulation becomes negative, the matter in that volume will decrease. Bernoulli's principle is the result of the law of conservation of mass, which fully describes the behavior of liquid motion together with the second equation based on Newton's second law of motion and the third equation based on the conservation of motion-energy.
From the above situation, it can be concluded that the mass of liquid inside the container remains the same.
The rate of mass entering = Rate of mass leaving
The rate of mass entering = ρA1V1Δt—– (1)
The rate of mass entering = ρA2V2Δt—– (2)
Using the above equations,
ρA1V1=ρA2V2
This equation is known as the Principle of Continuity.
Read more: Hydraulic Machines
The Relation Between Conservation of Energy and Bernoulli’s Equation
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The Bernoulli equation can be expressed or defined as a statement about the law of conservation of energy applicable to flowing liquids. Qualitative behavior, commonly referred to as the Bernoulli effect, is the decrease in fluid pressure in an area where the flow rate increases. The pressure drop that narrows the flow path may seem contradictory, but when you think of pressure as energy density, the situation is less contradictory. The kinetic energy must be increased by the pressure energy in the high-speed flow through contraction.
Read More: Hooke’s Law
Bernoulli’s Equation at Constant Depth
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When the liquid is moving but its depth does not change, that is, h1 = h2. In this case, the Bernoulli equation has the form P1 + 1 / 2pv21 = P1 + 1 / 2pv22. The fact that the fluid flows at a constant depth is so important that this equation is often called Bernoulli's principle. This is the Bernoulli equation for a constant depth liquid (note that if we follow it along its path, this is correct for a small volume of liquid).
Bernoulli’s Equation for Static Fluids
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Consider the following equation, where the fluid is static, that is
| v1 = v2 = 0 |
In this case, the Bernoulli equation looks like this: P1+ pgh1 = P2 + ρgh2
We further simplify the equation by taking h2 = 0 (we can always choose a height equal to 0, as we often do in other gravity situations, we assume that all other heights are related to it). We get P2 = P1 + ρgh1
This equation shows that in a static liquid, the pressure increases with depth. If we go from point 1 to point 2 in the liquid, the depth increases by h1, so P2 is greater than P1 by pgh1. In the simplest case, P1 is equal to 0p at the top of the liquid, and we conclude that the family relationship is as follows:
P = ρ g h
PEg= m g h
The equation includes the fact that the pressure due to the weight of the liquid is ρ * g * h.
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Points to Remember
- Bernoulli’s principle applies to liquids in a perfect state for which pressure and density are inversely proportional to each other.
- This means that low-velocity liquids exert more pressure than faster-moving liquids.
- Liquid should be understood as not only a liquid but also gas.
- This principle is at the core of many applications. Some very common examples are an airplane trying to stay high, or even the most common everyday items, such as a shower curtain that folds inward.
- One limitation is that due to internal friction, some energy is lost during fluid flow, because fluid has different layers, and each layer flows at a different speed, so each layer exerts a specific friction force.
- The Bernoulli equation can be expressed or defined as a statement about the law of conservation of energy applicable to flowing liquids.
Previous Years’ Questions
- The coefficient of viscosity has following dimensions … [KCET 1994]
- The ratio of inertial force to viscous force of a fluid … [KEAM]
- When the temperature increases, the viscosity of … [JKCET 2006]
- If a liquid does not wet glass, its angle of contact is …
- An object of mass 26 kg floats in air and it is in equilibrium state … [UPSEE 2017]
- Viscosity decreases with increase in temperature is the reason for … [COMEDK UGET 2009]
- The viscous force acting on a rain drop of radius … [COMEDK UGET 2004]
- A square plate of 0.1 m side moves parallel to a second plate … [AP EAPCET]
- A rain drop of radius 0.3mm has a terminal velocity … [VITEEE 2019]
- Raindrops are falling from a certain height … [DUET 2009]
- Two non-mixing liquids of densities … [NEET 2016]
- Two small spherical metal balls, having equal masses, are made … [NEET 2019]
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- If Y, K and η are the values of Young's modulus … [JEE Mains 2021]
Sample Questions
Ques. Which of the following fluid flows does not conform to Bernoulli’s equation? (1 mark)
Unsteady
Rotational
Turbulent
All of the above
Ans. Option 4: This equation only applies to a continuous and steady flow.
Ques. The water flows through a pipe with a diameter of 1 m at a speed of 2 m3/s. If the pressure is 80 kPa at this time, what is the pressure of the water after the pipe is reduced to a diameter of 0.5 m? (1 mark)
76.2kPa
31.6kPa
93.5kPa
100.7kPa
61.2kPa
Ans. 76.2kPa
Ques. Suppose a huge 50 m high reservoir filled with water is open to the atmosphere and hit by a bullet. The bullet passes through the side of the reservoir to let the water flow away. The pit is 2m above the ground. When the hole is small. Compared with the size of the water tank, how fast does the water flow out of the water tank?(1 mark)
21.69 m/s
940.8 m/s
30.67 m/s
112.8 m/s
Ans. 30.67 m/s
Ques. The venturi is a T-shaped tube with a vertical tube in the water. The high-speed air flows through the horizontal pipe. As a result, the water rises from the vertical pipe as shown in the figure. For a horizontal pipe with v = 7 m / s, how high is the water in the vertical pipe? (1 mark)
h=4.52cm
h=3.22cm
h=1.23cm
None of the above
Ans. h=3.22cm
Ques. Bernoulli principle is applicable to which of the following fluid conditions? (1 mark)
Compressible, steady flow
Incompressible flow with internal friction
Incompressible, steady flow with no internal friction
Incompressible, steady flow
Ans. Option 3. Incompressible, steady flow with no internal friction
Ques. Explain why:
To keep a piece of paper horizontal, we should blow over it and not under it.
When we try to close a water tap with our fingers, fast jets of water gush through the openings between our fingers. (3 mark)
Ans. a) The velocity of air increases when we blow over the piece of paper which is why the pressure on it decreases in accordance with Bernoulli's theorem. While, the pressure below remains the same (atmospheric pressure). Therefore, the paper remains horizontal.
b) If we try to close a water tap with our fingers, by doing so the area of the outlet of the water jet is reduced, thus velocity of water increases according to the equation of continuity, av = constant.
Ques. The size of a needle of a syringe controls flow rate better than the thumb pressure exerted by a doctor at the time of administering an injection. Explain why? (3 mark)
Ans. Bernoulli's theorem is expressed for a constant height as P + ½ ρ v2 = constant. Here, in the equation, the pressure P takes place with a single power while the velocity occurs with a square power. This is why the velocity has more effect compared to the pressure and it is due to this reason that the needle of the syringe controls the flow rate better than the thumb pressure exerted by the doctor.
Ques. Why does a fluid coming out of a small hole in a vessel results in a backward thrust on the vessel? (2 mark)
Ans. A fluid coming out of a small hole in a vessel results in a backward thrust on the vessel because of the principle of conservation of momentum. Whereas the flowing fluid carries forward momentum, the vessel gets a backward momentum.
Ques. State the limitations of Bernoulli’s equation. (5 marks)
Ans.
- Bernoulli’s equation ideally applies to fluids with zero viscosity or non-viscous fluids. In the case of viscous fluids, we need to take into account the work done against viscous drag.
- Bernoulli’s equation has been derived on the assumption that there is no loss of energy due to friction. But in practice, when fluids flow, some of their kinetic energy gets converted into heat due to work done against the internal forces of friction or viscous forces.
- Bernoulli’s equation is applicable only to incompressible fluids because it does not take into account the elastic energy of fluids.
- Bernoulli’s equation is applicable only to streamline the flow of a fluid and not when the flow is turbulent.
- Bernoulli’s equation does not take into consideration the angular momentum of the fluid. So it cannot be applied when the fluid flows along a curved path.
Ques. Consider a liquid of density 1200 kg m-3 flowing steadily in a tube of the varying cross-section. The cross-section at point A is 1.0 cm2 and that at B is 20 mm2, points A and B are in the same horizontal plane. The speed of liquid at A is 10 cm s-1. Calculate the difference in pressure at A and B. (3 marks)
Ans. From the equation of continuity, the speed v2 at B is given by,
A1v1 = A2v2
or, (0.1 cm2) (10cm s-1 ) = (20 mm2)v2
or, v2= 1.0 cm220 mm2 x 10 cm s-1.
By Bernoulli equation
P1 + 1/2 ρ vâ2 + ρ gh1 = P2 + 1/2 ρ vâ2 + ρ gh2
Here h1 = h2 . Thus,
P1 – P2 = 1/2 ρ vâ2 - 1/2 ρ vâ2
= 144 Pa.
Ques. What are the 3 heads in Bernoulli’s principle? (3 marks)
Ans. The pressure head defines the flow energy of a column of fluid whose weight is equivalent to the pressure of the fluid. The sum of the elevation head, velocity head, and pressure head of a fluid is known as the total head. Therefore, Bernoulli's equation states that the total head of the fluid is constant.
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