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Consider the following statements with reference to steady laminar flow
1. shear stress is zero at the centre.
2. discharge varies directly with viscosity of the fluid flowing.
3. velocity is maximum at the centre.
4. hydraulic gradient varies directly with the velocity.
1. shear stress is zero at the centre.
2. discharge varies directly with viscosity of the fluid flowing.
3. velocity is maximum at the centre.
4. hydraulic gradient varies directly with the velocity.
Which of these statements is/are correct?
- a)1,2 and 4
- b)1,3 and 4
- c)Both 1 and 3
- d)Both 3 and 4
Correct answer is option 'B'. Can you explain this answer?
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Explanation:
Statement 1: Shear stress is zero at the centre.
In steady laminar flow, the velocity profile is parabolic. As per the no-slip condition, the fluid in contact with the boundary (i.e., the pipe wall) has zero velocity. Therefore, the velocity of the fluid increases from zero at the wall to a maximum at the center. The shear stress, which is the force per unit area acting tangentially to the fluid's motion, is directly proportional to the velocity gradient. At the center of the pipe, the velocity gradient is zero, resulting in zero shear stress. Hence, statement 1 is correct.
Statement 2: Discharge varies directly with the viscosity of the fluid flowing.
The discharge (Q) is defined as the volume of fluid passing through a given cross-section per unit time. It is given by the equation Q = Av, where A is the cross-sectional area and v is the average velocity of the fluid. Viscosity is the measure of a fluid's resistance to flow. It is not directly related to the discharge but affects the velocity profile and shear stress distribution in the flow. Therefore, statement 2 is incorrect.
Statement 3: Velocity is maximum at the center.
As mentioned earlier, in steady laminar flow, the velocity profile is parabolic. The velocity increases from zero at the wall to a maximum at the center of the pipe. Hence, statement 3 is correct.
Statement 4: Hydraulic gradient varies directly with the velocity.
The hydraulic gradient is the change in hydraulic head per unit length of flow. It represents the energy loss or gain of the fluid as it flows through a pipe. The hydraulic gradient is directly proportional to the velocity head, which is the kinetic energy per unit weight of the fluid. Therefore, as the velocity increases, the hydraulic gradient also increases. Hence, statement 4 is correct.
Conclusion:
Based on the above analysis, statements 1, 3, and 4 are correct. Therefore, the correct option is B) 1, 3, and 4.
Statement 1: Shear stress is zero at the centre.
In steady laminar flow, the velocity profile is parabolic. As per the no-slip condition, the fluid in contact with the boundary (i.e., the pipe wall) has zero velocity. Therefore, the velocity of the fluid increases from zero at the wall to a maximum at the center. The shear stress, which is the force per unit area acting tangentially to the fluid's motion, is directly proportional to the velocity gradient. At the center of the pipe, the velocity gradient is zero, resulting in zero shear stress. Hence, statement 1 is correct.
Statement 2: Discharge varies directly with the viscosity of the fluid flowing.
The discharge (Q) is defined as the volume of fluid passing through a given cross-section per unit time. It is given by the equation Q = Av, where A is the cross-sectional area and v is the average velocity of the fluid. Viscosity is the measure of a fluid's resistance to flow. It is not directly related to the discharge but affects the velocity profile and shear stress distribution in the flow. Therefore, statement 2 is incorrect.
Statement 3: Velocity is maximum at the center.
As mentioned earlier, in steady laminar flow, the velocity profile is parabolic. The velocity increases from zero at the wall to a maximum at the center of the pipe. Hence, statement 3 is correct.
Statement 4: Hydraulic gradient varies directly with the velocity.
The hydraulic gradient is the change in hydraulic head per unit length of flow. It represents the energy loss or gain of the fluid as it flows through a pipe. The hydraulic gradient is directly proportional to the velocity head, which is the kinetic energy per unit weight of the fluid. Therefore, as the velocity increases, the hydraulic gradient also increases. Hence, statement 4 is correct.
Conclusion:
Based on the above analysis, statements 1, 3, and 4 are correct. Therefore, the correct option is B) 1, 3, and 4.
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Consider the following statements with reference to steady laminar flow1. shear stress is zero at the centre.2. discharge varies directly with viscosity of the fluid flowing.3. velocity is maximum at the centre.4. hydraulic gradient varies directly with the velocity.Which of these statements is/are correct?a)1,2 and 4b)1,3 and 4c)Both 1 and 3d)Both 3 and 4Correct answer is option 'B'. Can you explain this answer?
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