Fluid Mechanics and Hydraulic Machinery Miscellaneous
 Consider steady, incompressible and irrotational flow through a reducer in a horizontal pipe where the diameter is reduced from 20 cm to 10 cm. The pressure in the 20 cm pipe just upstream of the reducer is 150 kPa. The fluid has a vapour pressure of 50 kPa and a specific weight of 5 kN / m^{3} . Neglecting frictional effects, the maximum discharge (in m^{3} / s) that can pass through the reducer without causing cavitation is

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Considering potential head difference = 0,
i.e z_{1} = z_{2}
Apply Bernoulli’s theoremp + v^{2} = C ρg 2g p_{1} + v_{1}² = p_{2} + v_{2}² w_{1} 2g w_{2} 2g
But w_{1} = w_{2} = w = 5 (incompressible flow)∴ 150 + v_{1}² = 50 + v_{2}² 5 2g 5 2g or v_{2}²  v_{1}² = 150  50 2g 5 or v_{2}²  v_{1}² = 20 m ....(1) 2g Also, discharge, Q = π d_{1}² v_{1} = π d_{2}² v_{2} 4 4 or v_{1} = d_{2} ² = 10 ² v_{2} d_{1} 20 or v_{1} = v_{2} ......(2) 4
From equations (1) and (2)v_{2}²  v_{2}² 16 = 20 2g or 15v_{2}² = 20 32g
or v_{2} = √(20 × 32g) / 15 = 20.45 m /s∴ Discharge , Q = π d_{2}^{2} v_{2} 4 = π (0.1)^{2} × 20.45 = 0.16 m^{3} /sec 4
Correct Option: B
Considering potential head difference = 0,
i.e z_{1} = z_{2}
Apply Bernoulli’s theoremp + v^{2} = C ρg 2g p_{1} + v_{1}² = p_{2} + v_{2}² w_{1} 2g w_{2} 2g
But w_{1} = w_{2} = w = 5 (incompressible flow)∴ 150 + v_{1}² = 50 + v_{2}² 5 2g 5 2g or v_{2}²  v_{1}² = 150  50 2g 5 or v_{2}²  v_{1}² = 20 m ....(1) 2g Also, discharge, Q = π d_{1}² v_{1} = π d_{2}² v_{2} 4 4 or v_{1} = d_{2} ² = 10 ² v_{2} d_{1} 20 or v_{1} = v_{2} ......(2) 4
From equations (1) and (2)v_{2}²  v_{2}² 16 = 20 2g or 15v_{2}² = 20 32g
or v_{2} = √(20 × 32g) / 15 = 20.45 m /s∴ Discharge , Q = π d_{2}^{2} v_{2} 4 = π (0.1)^{2} × 20.45 = 0.16 m^{3} /sec 4
 The following data about the flow of liquid was observed in a continuous Chemical process plant:
Mean flow rate of the liquid is

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∴ Mean flow rate = ∑fx = 652.8 = 8.16 ∑f 80 Correct Option: C
∴ Mean flow rate = ∑fx = 652.8 = 8.16 ∑f 80
 Navier Stoke's equation represents the conservation of

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Momentum
Correct Option: D
Momentum
 Bernoulli's equation can be applied between any two points on a stream line for a rotational flow field. State:

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TRUE
Correct Option: A
TRUE
Direction: The gap between a moving circular plate and a stationary surface is being continuously reduced, as the circular plate comes down at a uniform speed V towards the stationary bottom surface, as shown in the figure. In the process, the fluid contained between the two plates flows out radially. The fluid is assumed to be incompressible and inviscid.
 The radial component of the fluid acceleration at r = R is

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Radial acceleration
a_{r} = V_{r} × ∂V_{r} + ∂V_{r} ∂r ∂t a_{r} = V.r × ∂ V.r + ∂ V.r 2h ∂r 2h ∂t 2h ∂h = V ∂t ∴ a_{r} = V.r × V.r + V.r × 1 ∂h 2h 2h 2 h^{2} ∂t ∴ a_{r} = V^{2}r + 2V^{2}r 4h^{2} 4h^{2} a_{r} = 3V^{2}r 4h^{2}
Correct Option: A
Radial acceleration
a_{r} = V_{r} × ∂V_{r} + ∂V_{r} ∂r ∂t a_{r} = V.r × ∂ V.r + ∂ V.r 2h ∂r 2h ∂t 2h ∂h = V ∂t ∴ a_{r} = V.r × V.r + V.r × 1 ∂h 2h 2h 2 h^{2} ∂t ∴ a_{r} = V^{2}r + 2V^{2}r 4h^{2} 4h^{2} a_{r} = 3V^{2}r 4h^{2}