Direction: A syringe with a frictionless plunger contains water and has at its end a 100 mm long needle of 1 mm diameter. The internal diameter of the syringe is 10 mm. Water density is 1000 kg/m3. The plunger is pushed in at 10 mm/s and the water comes out as a jet. 
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Neglect losses in the cylinder and assume fully developed laminar viscous flow throughout the needle: the Darcy friction factor is 64/Re. where Re is the Reynolds number. Given that the viscosity of water is 1.0 × 10-3 kg/ms, the force F in newtons required on the plunger is
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- 0.13
- 0.16
- 0.3
- 4.4
- 0.13
Correct Option: C
Given, v = velocity of water = 10 × 10–3 kg/sm
| Now , Re = | ||
| v |
... since = v2 = 1
Re = 1000
Now, Darcy’s friction factor,
| 4f = | = | = 0.064 | ||
| Re | 1000 |
| Head loss in needle = ht = | ||
| 2gD |
| = | = 0.3265 | |
| 2 × 9.8 × 0.001 |
Applying Bernoulli’s equation at points 1 and 2, we have
| + | + z1 = | + | + z2 + h1 | |||||
| ρg | 2g | ρg | 2g |
Since z1 = z2 and P2 = 0
| ∴ | = | + hl | ||
| ρg | 2g |
| P1 = | (v2² - v1²) + ρghl | |
| 2 |
| P1 = | [ (1)² - (0.01)²] + 1000 × 9.8 × 0.3265 | |
| 2 |
= 499.95 + 3199.7 = 3699.65
Now force required on plunger = P1 × A1
| = 3699.65 × | × (0.01)2 = 0.3 N | |
| 4 |
