State and explain Bernoulli’s theorem. Mention any one application of Bernoulli’s theorem.
Answers
Answer:
In 1738, the Swiss scientist Daniel Bernoulli developed a relationship for the flow of fluid through a pipe of varying cross section. He proposed a theorem for the streamline flow of a liquid based on the law of conservation of energy.
Bernoulli’s theorem
According to Bernoulli’s theorem, the sum of pressure energy, kinetic energy, and potential energy per unit mass of an incompressible, non-viscous fluid in a streamlined flow remains a constant. Mathematically,
This is known as Bernoulli’s equation.
Proof:
Let us consider a flow of liquid through a pipe AB as shown in Figure 7.33. Let V be the volume of the liquid when it enters A in a time t which is equal to the volume of the liquid leaving B in the same time. Let aA, vA and PA be the area of cross section of the tube, velocity of the liquid and pressure exerted by the liquid at A respectively.
Let the force exerted by the liquid at A is
FA = PAaA
Distance travelled by the liquid in time t is
d = vA t
Therefore, the work done is
W = FAd = PAaAvA t
But aAvAt = aAd =V, volume of the liquid entering at A.
Thus, the work done is the pressure energy (at A),
W = FAd = PAV
Since m is the mass of the liquid entering at A in a given time, therefore, pressure energy of the liquid at A is
Potential energy of the liquid at A,
PEA = mg hA,
Due to the flow of liquid, the kinetic energy of the liquid at A,
Therefore, the total energy due to the flow of liquid at A, EA = EPA + KEA + PEA
Similarly, let aB, vB, and PB be the area of cross section of the tube, velocity of the liquid, and pressure exerted by the liquid at B. Calculating the total energy at EB, we get
From the law of conservation of energy,
EA = EB
Thus, the above equation can be written as
The above equation is the consequence of the conservation of energy which is true until there is no loss of energy due to friction. But in practice, some energy is lost due to friction. This arises due to the fact that in a fluid flow, the layers flowing with different velocities exert frictional forces on each other. This loss of energy is generally converted into heat energy. Therefore, Bernoulli’s relation is strictly valid for fluids with zero viscosity or non-viscous liquids. Notice that when the liquid flows through a horizontal pipe, then h = 0
Applications of Bernoulli’s Theorem
Blowing off roofs during wind storm
In olden days, the roofs of the huts or houses were designed with a slope. One important scientific reason is that as per the Bernoulli’s principle, it will be safeguarded except roof during storm or cyclone.
During cyclonic condition, the roof is blown off without damaging the other parts of the house. In accordance with the Bernoulli’s principle, the high wind blowing over the roof creates a low-pressure P1. The pressure under the roof P2 is greater. Therefore, this pressure difference (P2–P1) creates an up thrust and the roof is blown off.
Answer:
The Bernoulli equation simply states that total energy per unit mass of flowing fluid at any point in the subsurface is the sum of the kinetic, potential, and fluid-pressure energies and is equal to a constant value.
Explanation:
According to Bernoulli's theorem, the sum of the energies possessed by a flowing ideal liquid at a point is constant if the liquid is incompressible, non-viscous, and flows in streamline.
Potential energy + Kinetic energy + Pressure energy = Constant. P+21pv2+pgh=Constant.
Application of Bernoulli’s theorem:
The sum of pressure energy, kinetic energy, and potential energy per unit mass of an incompressible, non-viscous fluid in a streamlined flow remains constant, according to Bernoulli's theorem. This is known as Bernoulli's equation in mathematics.
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