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A liquid is flowing in a horizontal uniform capillary tube under a constant pressure difference P. The value of pressure for which the rate of flow of the liquid is doubled when the radius and length both are doubled is
- a)P
- b)3P/4
- c)P/2
- d)P/4
Correct answer is option 'D'. Can you explain this answer?
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A liquid is flowing in a horizontal uniform capillary tube under a con...
In fluid dynamics, the flow rate of a liquid through a capillary tube can be described using Poiseuille's law. According to this law, the flow rate \( Q \) is given by the formula:
Q = (π * r^4 * ΔP) / (8 * η * L)
where:
- \( Q \) = Flow rate
- \( r \) = Radius of the tube
- \( ΔP \) = Pressure difference
- \( η \) = Viscosity of the liquid
- \( L \) = Length of the tube
Now, let's analyze the problem step by step.
1. Initial Conditions
- Let the initial radius be \( r \) and length be \( L \).
- The initial flow rate \( Q_1 = (π * r^4 * P) / (8 * η * L) \).
2. New Conditions
- The radius is doubled (\( 2r \)) and the length is also doubled (\( 2L \)).
- The new flow rate \( Q_2 \) becomes:
Q_2 = (π * (2r)^4 * P') / (8 * η * (2L))
- Simplifying \( Q_2 \):
Q_2 = (π * 16r^4 * P') / (16 * η * L) = (π * r^4 * P') / (η * L)
3. Condition for Doubling the Flow Rate
To find the pressure \( P' \) that doubles the flow rate, set \( Q_2 = 2Q_1 \):
(π * r^4 * P') / (η * L) = 2 * [(π * r^4 * P) / (8 * η * L)]
Cancelling out common terms and simplifying gives:
P' = P/4
Conclusion
Thus, to double the flow rate when both the radius and length of the capillary tube are doubled, the pressure difference required is:
P' = P/4
Therefore, the correct answer is option **D: P/4**.
Q = (π * r^4 * ΔP) / (8 * η * L)
where:
- \( Q \) = Flow rate
- \( r \) = Radius of the tube
- \( ΔP \) = Pressure difference
- \( η \) = Viscosity of the liquid
- \( L \) = Length of the tube
Now, let's analyze the problem step by step.
1. Initial Conditions
- Let the initial radius be \( r \) and length be \( L \).
- The initial flow rate \( Q_1 = (π * r^4 * P) / (8 * η * L) \).
2. New Conditions
- The radius is doubled (\( 2r \)) and the length is also doubled (\( 2L \)).
- The new flow rate \( Q_2 \) becomes:
Q_2 = (π * (2r)^4 * P') / (8 * η * (2L))
- Simplifying \( Q_2 \):
Q_2 = (π * 16r^4 * P') / (16 * η * L) = (π * r^4 * P') / (η * L)
3. Condition for Doubling the Flow Rate
To find the pressure \( P' \) that doubles the flow rate, set \( Q_2 = 2Q_1 \):
(π * r^4 * P') / (η * L) = 2 * [(π * r^4 * P) / (8 * η * L)]
Cancelling out common terms and simplifying gives:
P' = P/4
Conclusion
Thus, to double the flow rate when both the radius and length of the capillary tube are doubled, the pressure difference required is:
P' = P/4
Therefore, the correct answer is option **D: P/4**.
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A liquid is flowing in a horizontal uniform capillary tube under a constant pressure difference P. The value of pressure for which the rate of flow of the liquid is doubled when the radius and length both are doubled isa)Pb)3P/4c)P/2d)P/4Correct answer is option 'D'. Can you explain this answer?
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A liquid is flowing in a horizontal uniform capillary tube under a constant pressure difference P. The value of pressure for which the rate of flow of the liquid is doubled when the radius and length both are doubled isa)Pb)3P/4c)P/2d)P/4Correct answer is option 'D'. Can you explain this answer? for NEET 2024 is part of NEET preparation. The Question and answers have been prepared according to the NEET exam syllabus. Information about A liquid is flowing in a horizontal uniform capillary tube under a constant pressure difference P. The value of pressure for which the rate of flow of the liquid is doubled when the radius and length both are doubled isa)Pb)3P/4c)P/2d)P/4Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for NEET 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A liquid is flowing in a horizontal uniform capillary tube under a constant pressure difference P. The value of pressure for which the rate of flow of the liquid is doubled when the radius and length both are doubled isa)Pb)3P/4c)P/2d)P/4Correct answer is option 'D'. Can you explain this answer?.
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