Bernoulli’s principle

Hello dear students we know that flow of liquid in the following types,

  1. Streamline flow
  2. Turbulent flow

The flow of liquid in which all the layers of liquid are flows with constant velocities and the velocity of that individual layers is constant at all the points also different layers of liquid appear like the parallel lines, known as stream and the flow is called as streamline flow or laminar flow.

In this article we are going to learn very important principle related to streamline flow. i.e. Bernoulli’s  principle.

Let’s discuss the concept of Bernoulli’s principle.……………!

Consider the liquid of density ‘ρ’ is flowing through the pipe of different cross-sectional areas kept on the plane horizontal surface as shown in fig. below,

Let,

A1 and A2 be the cross-sectional areas of pipe as shown in the fig.

v1 and v2 of liquid when flows through cross-sections of area A1 and A2.

h1  and h2   be the height of open ends of pipe from the surface.

If ρ is the density of liquid, then

∴ ρ = m/V

Then the velocity of flow from point A to B is given by,

Liquid exerts pressures  openings A1 and A2 due to forces F1 and F2 respectively.

Then be definition forces at the upper and lower ends are

According to equation of continuity the volume of liquid that flows through different cross-section in equal interval of time is always same,

∴ Volume of liquid, V = A1 x1 = A2 x2

The liquid travels distance ∆x1  and ∆x2   cross-sections of area A1 and A2 in equal interval of time ∆t, then the corresponding work done is given by,

The resultant work done during the flow of liquid is,

∴∆W = W1 – W2

∴∆W = p1V – p2V

∴∆W = (p1 – p2)V         ……………………………(A)

Now the potential energy of liquid at the upper and lower ends can be given as,

∴PE1 = ρV gh1             ∵ mass of liquid = ρV

∴ PE2 = ρV gh2

Since h1 > h2, Resultant potential energy between two cross-sections is ,

∴∆PE1 = ρV gh2 – ρV gh1

∴∆PE1 = (h2 -h1 )ρVg   ……………………………(B)

Let us find the kinetic energy of flow at different positions,

According to law of conservation of energy, the total work done is nothing but the mechanical energy used in the system, which is sum of potential energy and kinetic energy.

∴ Mechanical energy = Potential energy + kinetic energy

Hence we can state the Bernoulli’s principle as,

For streamline flow, the sum of pressure energy, potential energy per unit volume and kinetic energy per unit volume is same at all the points in the flow.

Go through the following applications of Bernoulli’s principle…..!

  1. Venturimeter: – The instrument used to find the speed of flow of liquid is based on Bernoulli’s principle.
  2. Bunsen’s Burner: – Nozzle of burner having small aperture is used to make sharp flow of fuel used in cylinder.
  3. Blood flow: – The blood flow in arteries and veins in human body obeys Bernoulli’s principle. The cause of heart can be explained with the help of Bernoulli’s principle.

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