Bernoulli's Theorem and its Applications | Physics Class 11 - NEET PDF Download

If you hang two balloons vertically with separate threads and blow out air with your mouth in between the gap of two balloons, to your surprise the balloons will come closer to each other rather than going far off. This is an illustration of Bernoulli's Equation, which we are going to study in detail in this document.

When the person blows air in between the balloons, the pressure drops and they come closer.

What is Bernoulli's Principle?

Bernoulli's principle states that: 

The sum of pressure energy, kinetic energy and potential energy per unit volume of an incompressible, non-viscous fluid in a streamlined irrotational flow remains constant along a streamline.

Mathematically, it can be expressed as, 

Applications of Bernoulli’s Principle and Equation

Bernoulli's Principle, which describes the behavior of fluids, has a range of practical applications in everyday life and various fields. Here are some notable applications: 

1. Airplane Lift: Bernoulli's Principle is crucial in aviation for understanding how airplane wings generate lift. The shape of the wings causes the air above them to move faster, creating lower pressure and providing the lift needed for flight.
2. Venturi Tubes and Flowmeters: In fluid dynamics, instruments like venturi tubes and flowmeters use Bernoulli's Principle. These devices, designed with constricted sections, exploit changes in fluid speed to measure flow rates accurately, commonly used in plumbing and industrial applications.
3. Wind Turbines: The principle is applied to study the airflow around wind turbine blades. By relating air velocity to pressure differences, engineers can optimize the design and efficiency of wind turbines for power generation.
4. Sailing: Bernoulli's Principle helps explain how sails on a boat work. As wind flows across the curved surface of a sail, it generates lower pressure on one side, propelling the boat forward.
5. Blood Flow in the Human Body: The principle is relevant to understanding blood flow. As blood moves through blood vessels with varying diameters, changes in speed affect pressure, influencing circulation.
6. Aerospace Design: Engineers use Bernoulli's Principle in the design of aircraft and spacecraft, considering how air flows over surfaces to optimize performance and fuel efficiency.
7. Spray Nozzles: Devices like spray nozzles use Bernoulli's Principle to create a fine mist. As a fluid passes through a constriction, its speed increases, leading to a decrease in pressure that results in atomization.
8. Weather Phenomena: Bernoulli's Principle is involved in the study of weather patterns and phenomena, including the formation of clouds and the dynamics of air masses.

Proof of Bernoulli's Principle 

Assumptions 

The fluid is incompressible, non-viscous, non-rotational, and streamlined flow. 

Bernoulli's Theorem

Mass of the fluid entering from side S
dm₁ = p A₁ dx₁ = p dV₁
The work done in this displacement dx₁ at point S is
Wp₁ = F₁dx₁ = P₁A₁dx₁
Wp₁ = P₁dV₁    {∵ A₁dx₁ = dV₁}
At the same time, the amount of fluid that moves out of the tube at point T is dm₂ = pdV₂

According to the equation of continuity  

The work done in the displacement of dm₂ mass at point T

Wp₂ = P₂dV₂

Now apply the work-energy theorem.

h = Gravitational head
 = volume head

Principle of Continuity

According to the principle of continuity:

If the fluid is in streamlined flow and is incompressible then we can say that the mass of fluid passing through different cross sections is equal.

The rate of mass of liquid entering = Rate of mass of liquid leaving.

The rate of mass entering = ρA₁V₁Δt

The rate of mass entering = ρA₂V₂Δt

Using the above equations, ρA₁V₁=ρA₂V₂

This equation is known as the Principle of Continuity.

Bernoulli’s Equation at Constant Depth

When the fluid moves but its depth is constant—that is, ℎ₁=ℎ₂.

Under that condition, Bernoulli’s equation becomes:

Bernoulli’s Equation for Static Fluids

When the fluid is static, then v₁ = v₂ = 0, then Bernoulli’s equation is given as:

Application of Bernoullis principle 

1. Magnus effect: 

When a spinning ball is thrown, it deviates from its usual path in flight. This effect is called the Magnus effect and plays an important role in tennis, cricket soccer, etc., as by applying appropriate spin the moving ball can be made to curve in any desired direction.

If a ball is moving from left to right and also spinning about a horizontal axis perpendicular to the direction of motion as shown in the figure, then relative to the ball air will be moving from right to left.

(A)            (B)             (C)The resultant velocity of air above the ball will be (V rw) while below it (V -rw) (shown in figure). So in accordance with Bernoulli's principle pressure above the ball will be less than below it. Due to this difference of pressure, an upward force will act on the ball and hence the ball will deviate from its usual path OA₀ and will hit the ground at A₁ following the path OA₁ (figure shown) i.e., if a ball is thrown with backspin, the pitch will curve less sharply prolonging the flight.

Similarly, if the spin is clockwise, i.e., the ball is thrown with topspin, the force due to pressure difference will act in the direction of gravity and so the pitch will curve more sharply shortening the flight.

Furthermore, if the ball is spinning about a vertical axis, the curving will be sideways as shown in the figure. producing the so-called out swing or in the swing.

Action of Atomiser: The action of the aspirator, carburetor, paint gun, scent spray, or insect sprayer is based on Bernoulli's principle. In all these by means of motion of a piston P in a cylinder C high-speed air is passed over a tube T dipped in liquid L to be sprayed. High-speed air creates low pressure over the tube due to which liquid (paint, scent, insecticide, or petrol) rises in it and is then blown off in very small droplets with expelled air.

Working of Aeroplane: This is also based on Bernouilli's principle. The wings of the aeroplane have tapering as shown in the figure. Due to this specific shape of wings when the airplane runs, air passes at higher speed over it as compared to its lower surface. This difference of air speeds above and below the wings, in accordance with Bernoulli's principle, creates a pressure difference, due to which an upward force called ' dynamic lift' ( = pressure difference × area of the wing) acts on the plane. If this force becomes greater than the weight of the plane, the plane will rise up.

Example 1: If pressure and velocity at point A are P₁ and V₁ respectively & at point B is P₂, V₂ is the figure as shown. Comment on P₁ and P₂.  

Solution. From the equation of continuity A₁V₁ = A₂V₂

here A₁ > A₂

⇒ V₁ < V₂ ....(1)

from Bernoulli's equation. We can write

2. Torricielli's Law of Efflux (Velocity of efflux)

The cross-sectional area of the hole at A is greater than B If water comes in the tank with velocity v_A and goes outside with velocity

on applying Bernoulli theorem at A and B
 

Range (R) 

Let us find the range R on the ground.

Considering the vertical motion of the liquid.

(H -h) =  or

Now, considering the horizontal motion,

R = vt or  or

From the expression of R, the following conclusions can be drawn,

(i) R_h = R_{H -h} 

as  and

This can be shown in Figure.

(ii) R is maximum at H/2 and R_max = H.

Proof : R² = 4 (Hh – h²)
For R to be maximum.  

or H - 2h = 0 or h = H/2

That is, R is maximum at h=H/2

and  

Example 2: A cylindrical dark 1 m in radius rests on a platform 5 m high. Initially, the tank is filled with water up to a height of 5 m. A plug whose area is 10^–4 m² is removed from an orifice on the side of the tank at the bottom Calculate 

(a) the initial speed with which the water flows from the orifice 

(b) the initial speed with which the water strikes the ground and 

(c) time taken to empty the tank to half its original volume 

(d) Does the time to empty the tank depend upon the height of the stand?

Solution. The situation is shown in the figure:

 
(a) As the speed of flow is given by

(b) As the initial vertical velocity of water is zero, so is its vertical velocity when it hits the ground

So the initial speed with which water strikes the ground.

(c) When the height of the water level above the hole is y, the velocity of flow will be  and so the rate of flow

Which on integration improper limits gives

 = 9.2 × 103s – ~ 2.5 h

(d) No, as an expression of t is independent of the height of the stand.

3. Venturimeter

The figure shows a venturi meter used to measure flow speed in a pipe of non-uniform cross-section. We apply Bernoulli's equation to the wide (point 1) and narrow (point 2) parts of the pipe, with h₁ = h₂

From the continuity equation

Substituting and rearranging, we get

Because A₁ is greater than A₂, v₂ is greater than v₁ and hence the pressure P₂ is less than P₁. A net force to the right acceleration the fluid as it enters the narrow part of the tube (called the throat) and a net force to the left slows as it leaves. The pressure difference is also equal to ρgh, where h is the difference in liquid level in the two tubes. Substituting in Eq. (1), we get

4. Pivot Tube

It is a device used to measure the flow velocity of fluid. It is a U-shaped tube that can be inserted in a tube or in the fluid-flowing space as shown in the figure shown. In the U tube a liquid that is immiscible with the fluid is filled up to a level C and the short opening M is placed in the fluid flowing space against the flow so that few of the fluid particles enter into the tube and exert a pressure on the liquid in limb A of U tube. Due to this the liquid level changes as shown in the figure show.

At end B fluid is freely flowing, which exerts approximately negligible pressure on this liquid. The pressure difference at ends A and B can be given by measuring the liquid level difference h as 

It is a gas, then 

P_A - P_B = ρhg

It is the liquid of density, then 

P_A - P_B = h(ρ - ρ_g)g

Now if we apply Bernoulli's equation at the ends of A and B we'll have

Now by using equations, we can evaluate the velocity v, with which the fluid is flowing.

Note: A pivot tube is also used to measure the velocity of airplanes with respect to the wind. It can be mounted at the top surface of the plain and hence the velocity of wind can be measured with respect to the plane.

5. Siphon

It is a pipe used to drain liquid at a lower height but the pipe initially rises and then comes down 

let velocity of outflow be v and the pipe is of uniform cross-section A. Applying Bernoulli's Equation between P (top of the tank) and R (Opening of pipe) we get 

⇒

here velocity is considered zero at P since the area of the tank is very large compared to the area of the pipe

Naturally for siphon to work 

h₁ > 0

Now as the area of the pipe is constant by the equation of continuity

as A_v = constant the velocity of flow inside the siphon is also constant between Q and R

(P + pgh + 1/2 pv²)_Q = (P + pgh + 1/2 pv²)R ⇒ P_Q + pgh₂ = P₀ - pgh₁ (v is same)

⇒ P_Q = P₀ - Pg (h₁ + h₂) as P_Q = 0    ⇒ P₀ > pg (h₁ + h₂) means (h₁ + h₂) should not be more than P/pg for siphon to work.

Summary

Bernoulli’s equation states that the sum on each side of the following equation is constant, or the same at any two points in an incompressible frictionless fluid: