Water is flowing through a horizontal pipe in streamline flow at the narrowest part of the pipe:a)Both pressure and the velocity remains constantb)velocity is maximum and pressure is minimumc)both the pressure and velocity are maximumd)both the pressure and velocity are minimumCorrect answer is option 'B'. Can you explain this answer?

Question

Answer

In streamline flow, the product of cross section area and velocity remains constant (equation of continuity). So in the narrowest part of the pipe velocity is maximum.
And from Bernoulli's theorem, we know that the sum of potential energy, kinetic energy and pressure energy remains constant. Since pipe is horizontal potential energy is equal at all the points. So the narrowest part of pipe pressure (pressure energy) will be minimum because velocity (kinetic energy) is maximum in the narrowest part.

Answer

Equation of continuity says a1v1. =a2v2
area less(narrow) velocity is more
p1+1/2 roh v1^2=p2+1/2 ro v2^2
suppose v2>v1 due to A1>A2
if v2>v1 so 1/2ro v2^2 also > 1/2rov1^2
so p2 should be less for balance in Bernoulli equi

Answer

Streamline flow
In streamline flow, the fluid particles move along well-defined paths called streamlines. These streamlines do not cross each other, and the velocity of the fluid at any point is tangent to the streamline passing through that point.

Pressure and velocity in streamline flow
In streamline flow, the Bernoulli's equation is applicable. According to Bernoulli's equation, the sum of pressure energy, kinetic energy, and potential energy per unit volume of a fluid remains constant along a streamline.

The narrowest part of the pipe
When a fluid flows through a pipe, it experiences changes in velocity and pressure due to variations in the cross-sectional area of the pipe. At the narrowest part of the pipe, the cross-sectional area is smallest, which leads to an increase in velocity and a decrease in pressure.

Explanation of option 'B'
According to option 'B', the velocity is maximum and the pressure is minimum at the narrowest part of the pipe. This can be explained as follows:

- Velocity is maximum: At the narrowest part of the pipe, the cross-sectional area is smallest. According to the principle of continuity, which states that the mass flow rate of a fluid is constant, the velocity of the fluid must increase when the cross-sectional area decreases. Therefore, the velocity of the fluid is maximum at the narrowest part of the pipe.

- Pressure is minimum: According to Bernoulli's equation, as the velocity of a fluid increases, the pressure decreases. This is because the kinetic energy of the fluid increases at the expense of its pressure energy. Therefore, at the narrowest part of the pipe where the velocity is maximum, the pressure is minimum.

Conclusion
In streamline flow at the narrowest part of a horizontal pipe, the velocity is maximum and the pressure is minimum. This can be explained by the principle of continuity and Bernoulli's equation.

Answer

C

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