## Fluid Mechanics - Dec 2015

### Mechanical Engineering (Semester 4)

TOTAL MARKS: 100

TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.

(2) Attempt any **four** from the remaining questions.

(3) Assume data wherever required.

(4) Figures to the right indicate full marks.
**1(a)** Derive the continuity equation in Cartesian co-ordinates.(7 marks)
**1(b)** Define or explain following terms

1. Newton's law of viscosity

4. Dimensional homogeneity

2. Capillarity

5. Water hammer

3. Hydraulic radius

6. Summit

7. Derived units(7 marks)
**2(a)** Derive the expression for time required to emptying a tank through an orifice at its bottom.(7 marks)

### Solved any one question from Q.2(b) & Q.2(c)

**2(b)** A block of wood has a horizontal cross section 500 mm × 500 mm and height h. it floats vertically in water. If the specific gravity of wood is 0.6, find the maximum height of block so that it can remain in stable equilibrium.(7 marks)
**2(c)** A barge in the shape of a rectangular block 8 m wide, 12.8 m long and 3 m deep floats in water with a draft of 1.8 m. the centre of gravity of the barge is 0.3 m above the water surface. State whether the barge is in stable equilibrium.
Calculate the righting moment when the barge heels by 10^{°}(7 marks)

### Solved any one question from Q.3 & Q.4

**3(a)** Derive an expression for loss of head due to friction in pipe flow.(7 marks)
**3(b)** A pipe of diameter 225 m is attached to a 150 mm diameter pipe in a straight line by means of a flange. Water flows at the rate of 0.05 m^{3} /s. the pressure loss at transition as indicated by differential gauge length on a water-mercury manometer connected between two pipes equals 35 mm. calculate the loss of head due to contraction.(7 marks)
**4(a)** Derive the relation among different energies using Bernoulli's theorem.(7 marks)
**4(b)** A supersonic aircraft flies at an altitude of 1.8 km where temperature is 4° C. determine the speed of the aircraft if its sound is heard 4 seconds after its passage over the head of an observer. Take R = 287 J/kg K and γ = 1.4.(7 marks)

### Solved any one question from Q.5 & Q.6

**5(a)** Define boundary layer thickness and also explain following terms.
1. Displacement thickness (δ * )

2. Momentum thickness (θ)

3. Energy thickness (δ e )(9 marks)
**5(b)** A 150 mm diameter vertical cylinder rotates concentrically inside cylinder of diameter 151 mm. both the cylinders are 250 mm high. The space between the
cylinders is filled with a liquid whose viscosity is unknown. If a torque of 12 Nm is required to rotate the inner cylinder at 100 rpm, determine the viscosity of the liquid.(5 marks)
**6(a)** Enlist the various Mechanical gauges for pressure measurement and describe their working with suitable diagram.(9 marks)
**6(b)** Velocity for a two dimensional flow field is given by V = (3+2xy+4t^{2})i+ (xy^{2}+3t)j

Find the velocity and acceleration at appoint (1, 2) after 2 sec.(5 marks)

### Solved any one question from Q.7 & Q.8

**7(a)** State the Karman-Prandtl equation for the velocity distribution near hydro
dynamically smooth boundaries.(7 marks)
**7(b)** Using the method of dimensional analysis obtain an expression for the discharge Q over a rectangular weir. The discharge depends on the head H over the weir, acceleration due to gravity g, length of the weir crest over the channel bottom Z and the kinematic viscosity v of the liquid.(7 marks)
**8(a)** Derive the Euler's equation for motion.(7 marks)
**8(b)** A crude oil of viscosity 0.9 poise and relative density 0.9 is flowing through a horizontal circular pipe of diameter 120 mm and length 12 m. calculate the difference of pressure at the two ends of the pipe, if 785 N of the oil is collected in tank in 25 seconds(7 marks)